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Thermal residual stresses in bonded composite repairs on cracked metal structures Albat, Andreas Michael 1998

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THERMAL RESIDUAL STRESSES IN BONDED COMPOSITE REPAIRS ON CRACKED METAL STRUCTURES by  ANDREAS MICHAEL ALBAT Dipl.-Ing., Technische Universitat Braunschweig, 1993  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R O F PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the, required standard  T H E UNIVERSITY O F BRITISH C O L U M B I A September 1998 © Andreas Michael Albat, 1998  In  presenting  degree  this  at the  thesis  in  University pf British Columbia,  freely available for reference copying  of  department publication  this or of  partial fulfilment  and study.  this  his  or  her  representatives.  Mechanical  Engineering  The University of British Columbia Vancouver, Canada  Date  DE-6  (2/88)  20.09.1998  requirements  I agree  that the  may be It  thesis for financial gain shall not  permission.  Department of  the  1 further agree  thesis for scholarly purposes by  of  is  that  for  an  advanced  Library shall make it  permission for extensive  granted by the understood  that  be allowed without  head  of  my  copying  or  my written  Abstract  ii  Abstract The objective of this research is to determine the thermal residual stresses and strains in bonded composite repairs on cracked metal structures. This work is an essential contribution to a fatigue damage initiation model for bonded composite repair, where knowledge of the initial stress/strain state after an elevated temperature cure is important. Furthermore, this work is an elementary part for the development of a generic certification approach to bonded composite repairs. Accounting properly for thermal residual stresses in test specimens and in real applications will assist in determining the true feasibility of a bonded composite repair.  The objective of this work was realized in four stages of research. In the first stage, seven A M R L sandwich type composite bonded repair specimens were manufactured, of which one was instrumented by placing 44 strain gauges at eight planar locations and within five different interfaces. Residual strains at ambient temperature (including both thermal residual strains and other process induced strains) were measured during the manufacturing process.  In the second stage, the stress free temperature for the repaired specimen was  experimentally determined and the thermal residual strains measured as a function of operating temperature. In the third stage, a theoretical analysis was carried out to estimate the thermal residual stress and strain distributions in various bonded repairs. This analysis also addressed the effect of symmetrical disbonds around the crack. Finally, a finite element analysis was carried out to assess the limitations of the theoretical analysis as well as to provide a more detailed insight into the complex thermal residual stress and strain state of the A M R L sandwich type specimen.  During this work it was found that high thermal residual strains (reaching 15% of the yield strain) are present in the bonded repair specimen at ambient temperature. Previous  Abstract  iii  analysis schemes predicted results nearly 60% higher. The thermal residual strain versus temperature measurement showed that only very small changes in thermal residual strains occurred above 90°C leading to a defined effective stress free temperature of 85.8°C for the employed adhesive F M 73M. By utilizing an effective stress free temperature, a linear-elastic approach was used to model thermal residual stresses and strains in composite bonded repairs. Major achievements in the theoretical analysis include a linear-elastic closed form solution for tapered joints and reinforcements without the need for a numerical solution scheme, a stress field prediction ahead of the crack tip for the metal substrate of a bonded repair based on a concise complete solution of the classical fracture mechanics problem of a center crack in an infinite plate and, an extended Rose model for the prediction of the stress intensity factor of a bonded repair with symmetrical disbonds showing the severity of thermal residual stresses especially for partially disbonded composite repairs to cracked metal specimens. The key to precise predictions of thermal residual stresses in bonded composite repairs is the knowledge of the adhesive behaviour at elevated temperatures under thermal residual stress loading. A generic type specimen is presented which allows to investigate the relevant adhesive behaviour.  Table of Contents  iv  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  viii  List of Figures List of Symbols Acknowledgements  ix xviii xxi  1  Introduction  1  2  Literature Review  6  2.1  Introduction  6  2.2  Bonded Repairs Versus Riveted Repairs  8  2.3  Analysis of Bonded Repairs  12  2.4  Test Specimens for Bonded Repairs  17  2.5  Experimental Testing of Bonded Repairs  20  2.6  Standards and Certification Issues for Bonded Repairs  25  2.7  Summary and Preview  27  3  A M R L Specimen M a n u f a c t u r i n g Procedure  29  3.1  Introduction  29  3.2  A M R L Specimen Specifications  31  3.3  Constant Load Amplitude Pre-cracking Procedure  32  3.3.1  Experimental Set-up  37  3.3.2  Crack Growth Rates  40  3.4  Aluminum Surface Preparation Procedure  44  3.4.1  Clad Removal  44  3.4.2  Grit Blasting  46  3.4.3  Preparation and Application of Silane Coupling Agent  47  3.4.4  Preparation and Application of BR127 Corrosion Inhibiting Primer  49  3.4.5  Boeing Wedge Test  52  Table of Contents 3.5  3.6  3.7  4  5  v  Boron Patch Fabrication Procedure  53  3.5.1  Cutting of Boron 5521/4 Prepreg and FM73M Adhesive  54  3.5.2  Lay-up of Boron 5521/4 Prepreg and FM73M Adhesive  55  3.5.3  Vacuum Bagging of Boron 5521/4 Prepreg and F M 73M Adhesive .  56  3.5.4  Cocure Cycle of Boron 5521/4 Prepreg and F M 73M Adhesive . . .  58  3.5.5  Ultrasonic Inspection of the Cocured Patches  60  A M R L Specimen Assembly Procedure  62  3.6.1  Fixture Design  62  3.6.2  Preparation of the Honeycomb  63  3.6.3  Assembling the A M R L Specimen  64  3.6.4  Ultrasonic Inspection of the Boron Patch Repair  72  Instrumented A M R L Specimen Construction  73  3.7.1  Instrumented Boron Patch Fabrication Procedure  75  3.7.2  Instrumented A M R L Specimen Assembly Procedure  78  Residual and T h e r m a l Residual Strain Measurements  81  4.1  Introduction  81  4.2  Residual Strain Measurement Procedure  82  4.2.1  Approach and Experimental Set-up  82  4.2.2  Data Analysis  83  4.2.3  Results  85  4.3  Stress Free Temperature Measurement Procedure  87  4.4  Thermal Residual Strain Measurement Procedure  88  4.4.1  Approach  88  4.4.2  Experimental Setup  90  4.4.3  Data Analysis  92  4.4.4  Results  102  4.5  Discussion and Estimation of the Process Induced Strains  109  4.6  Effective Stress Free Temperature  Ill  Theoretical Analysis of T h e r m a l Residual Stresses in B o n d e d Repair Specimens  116  5.1  116  Introduction  Table of Contents 5.2  vi  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements  5.3  5.4  5.2.1  General Approach for Double Symmetric Joints and Reinforcements  120  5.2.2  Double-sided Reinforcements with Uniform Thickness  123  5.2.3  Double-doubler Joints with Uniform Thickness  125  5.2.4  Interface Conditions between Steps for Tapered Joints and Reinforcements  126  5.2.5  Tapered Double-sided Reinforcements  130  5.2.6  Tapered Double-doubler Joints  132  5.2.7  Correction for Shear-lag in the Adherends  134  5.2.8  Results  137  Application of the Rose Model for Thermal Residual Stress Loading . . . .  143  5.3.1  Stage I: Stress at the Prospective Crack Location  144  5.3.2  Stage II: Upper Bounds for K  147  T  Estimation of the Stress Field around the Crack Tip  154  5.4.1  Theoretical Derivation  157  5.4.2  Results  163  5.5  Comparison between Closed Form Theoretical and Experimental Results  5.6  Generalization of the Rose Model for Partially Disbonded Patches  174  5.6.1  Double-sided Uniform Thickness Repair with Disbonds  175  5.6.2  Double-sided Tapered Repair with Disbonds  186  5.7 6  118  Summary and Discussion  .  166  198  F i n i t e Element Analysis of the A M R L Specimen  201  6.1  Introduction  201  6.2  Verification Models  204  6.2.1  Constant Thickness Double-sided Reinforcement  205  6.2.2  Tapered Reinforcement with a Reversed Stacking Sequence  208  6.2.3  Crack Mesh Verification  218  6.2.4  Patched Crack Model  223  6.2.5  Half Circular Patch Model  233  A M R L Sandwich Type Specimen Model  235  6.3  Table of Contents 7  vii  Discussion and Conclusion  252  7.1  Introduction  252  7.2  A M R L Specimen Manufacturing Procedure  253  7.3  Residual and Thermal Residual Strain Measurements  254  7.4  Theoretical Analysis of Thermal Residual Stresses in Bonded Repair Specimens  257  7.5  Finite Element Analysis of the A M R L Specimen  259  7.6  Implementation in a Design Procedure  261  7.7  Recommendations for Future Work  263  Bibliography  266  Appendix A  A M R L - S p e c i m e n Drawings  275  Appendix B  Pre-cracking Documentation  291  B.l  Pre-crack Data  292  B.2  Pre-crack Diagrams  306  B.3  Pre-crack Pictures  313  Appendix C Cl  Ultrasonic C-Scans  Ultrasonic C-Scans of the Precured Patches  Appendix D  M a t e r i a l Properties  317 318 323  D.l  Mechanical Properties of Boron/epoxy 5521/4 Laminates  324  D.2  Mechanical Properties of Aluminum 2024-T3  326  D.3  Mechanical Properties of F M 73M  328  Appendix E  M e a s u r e d T h e r m a l and T h e r m a l Residual Strains  329  List of Tables  viii  List of Tables Table 3.1  Pre-cracking load sequence  36  Table 3.2  M D T load control input  37  Table 3.3  A M R L specimen component designation  42  Table 3.4  Schedule for surface preparation procedure  51  Table 4.1  Residual Strains  86  Table 5.1  Comparison of the thermal residual strains at 21°C between experimental results, the one-dimensional analysis and the fracture mechanics analysis  Table 5.2  171  Comparison of the thermal residual strains at — 56.5°C between experimental results, the one-dimensional analysis and the fracture mechanics analysis  Table 6.1  172  Comparison of the thermal residual strains at 21°C between finite element results, experimental results, the one-dimensional analysis and the fracture mechanics analysis  246  Table B . l  Pre-crack data for face-sheet 1  292  Table B.2  Pre-crack data for face-sheet 2  293  Table B.3  Pre-crack data for face-sheet 3  294  Table B.4  Pre-crack data for face-sheet 4  295  Table B.5  Pre-crack data for face-sheet 5  296  Table B.6  Pre-crack data for face-sheet 6  297  Table B.7  Pre-crack data for face-sheet 7  298  Table B.8  Pre-crack data for face-sheet 8  299  Table B.9  Pre-crack data for face-sheet 9  300  Table B.10 Pre-crack data for face-sheet 10  301  Table B . l l Pre-crack data for face-sheet 11  302  Table B.l2 Pre-crack data for face-sheet 12  303  Table B.13 Pre-crack data for face-sheet 13  304  Table B.14 Pre-crack data for face-sheet 14  305  List of Figures  ix  List of Figures Figure 1.1  Small scale test specimen for composite repairs  3  Figure 2.1  Sketch of the Lockheed Electra lower wing skin  9  Figure 2.2  'Fingered' inner doubler for long joint fatigue life and external visual inspection  11  Figure 2.3  Minimum patch thickness design scheme  15  Figure 3.1  A M R L specimen  32  Figure 3.2  M T S 810  Figure 3.3  Indirect potential drop method  39  Figure 3.4  Crack propagation rates for all face-sheets  41  Figure 3.5  Pre-crack deviation (Face-sheet 3)  43  Figure 3.6  Effect of pre-cracking overload (Face-sheet 1)  43  Figure 3.7  Clad removal  45  Figure 3.8  Grit blasting set-up  46  Figure 3.9  Chemical bonding mechanism of the Silane treatment  47  38  Figure 3.10 Silane solution equipment  47  Figure 3.11 Silane coupling agent - brush method  48  Figure 3.12 Heat drying of the silane  49  Figure 3.13 Setup for BR127 priming  50  Figure 3.14 Boeing wedge test specimens  52  Figure 3.15 Temperature and humidity chamber  52  Figure 3.16 Cutting of the boron 5521/4 prepreg  55  Figure 3.17 Lay-up of the boron prepreg  56  Figure 3.18 Vacuum bagging procedure  57  Figure 3.19 Recommended boron 5521/4 cure cycle  58  Figure 3.20 Autoclave  59  Figure 3.21 Cured patches  60  Figure 3.22 C-scan of a cocured boron/epoxy patch  61  Figure 3.23 Setup for honeycomb cutting  64  Figure 3.24 A M R L specimen assembly (Face-sheet)  64  Figure 3.25 A M R L specimen assembly (Face-sheet mounting)  65  Figure 3.26 A M R L specimen assembly (Honeycomb mounting)  65  Figure 3.27 A M R L specimen assembly (2. Face-sheet mounting)  66  List of Figures  x  Figure 3.28 A M R L specimen assembly (Aluminum sheets)  67  Figure 3.29 A M R L specimen assembly ( F M 73M film adhesive)  67  Figure 3.30 A M R L specimen assembly (High temperature tape barrier)  68  Figure 3.31 A M R L specimen assembly (Patch fixation)  69  Figure 3.32 A M R L specimen assembly (Vacuum bag)  69  Figure 3.33 Cure cycle for the A M R L specimen  70  Figure 3.34 Placement of the strain gauges in the patch  77  Figure 3.35 X-ray of the instrumented patch  78  Figure 4.1  Initial patch strain measurement  82  Figure 4.2  Stress free temperature measurement setup  87  Figure 4.3  Titanium silicate bar with installed gauges  88  Figure 4.4  Thermal expansion characteristics of titanium silicate  89  Figure 4.5  A M R L specimen wrapped in breather cloth and aluminum foil  90  Figure 4.6  Regular and high heating/cooling rate cycles  91  Figure 4.7  Indicated strains (Gauge 3) for different temperature cycles  92  Figure 4.8  Origin and stress relaxation corrected indicated strains (Gauge 3) . . .  93  Figure 4.9  Superposition of all corrected indicated strains (Gauge 3)  94  Figure 4.10 Indicated strains for elevated temperature range (Gauge 38)  95  Figure 4.11 Indicated strains (Gauge 24)  96  Figure 4.12 Indicated thermal strains of titanium silicate (Gauge 45)  96  Figure 4.13 Indicated strains for low temperature range (Gauge 38)  97  Figure 4.14 Lead wire resistance change with temperature  99  Figure 4.15 Gauge locations  103  Figure 4.16 Thermal residual and total residual strains at ambient temperature due to an elevated temperature cure  104  Figure 4.17 Thermal residual strains at — 56.5°C due to elevated temperature cure .  105  Figure 4.18 Thermal residual strain versus temperature  106  Figure 4.19 Thermal residual strain versus temperature for strain gauges mounted in or on the boron/epoxy patch at locations B, C , D and H  108  Figure 4.20 Effective stress free temperature  112  Figure 4.21 Effective stress free temperature test specimen  113  Figure 4.22 Stress free temperature test specimen  114  Figure 5.1  Reinforcement (a) and double-doubler joint (b) with uniform substrate and patch thickness  120  List of Figures  xi  Figure 5.2  Force equilibrium in a bonded joint  121  Figure 5.3  Tapered reinforcement  130  Figure 5.4  Tapered double-doubler joint  132  Figure 5.5  Distribution of the shear stress through the thickness of a double-sided joint or reinforcement  Figure 5.6  Adhesive shear stress in double-sided reinforcements with uniform thickness and tapered edges  Figure 5.7  140  Normal stress in the substrate of double-sided reinforcements with uniform thickness and tapered edges  Figure 5.9  139  Normal stress in the patch of double-sided reinforcements with uniform thickness and tapered edges  Figure 5.8  135  140  Adhesive shear stress in double-doubler joints with uniform thickness and tapered edges  141  Figure 5.10 Normal stress in the patch of double-doubler joints with uniform thickness and tapered edges  142  Figure 5.11 Normal stress in the substrate of double-doubler joints with uniform thickness and tapered edges  142  Figure 5.12 Illustration of the superposition principle  149  Figure 5.13 Estimation and upper bounds for the stress intensity factor  153  Figure 5.14 Biaxial loaded center crack in an infinite plate  155  Figure 5.15 Approximate crack tip stress field solution  157  Figure 5.16 Stress parallel to a crack (—4.22mm < x < 4.22mm, y = 0mm) in an infinite plate with uniaxial loading of 40.43 M P a perpendicular to the crack  164  Figure 5.17 Stress perpendicular to a crack (—4.22mm < x < 4.22mm, y = 0mm) in an infinite plate with uniaxial loading of 40.43 M P a perpendicular to the crack  165  Figure 5.18 Shear stress around a crack (—4.22mm < x < 4.22mm, y = 0mm) in an infinite plate with uniaxial loading of 40.43 M P a perpendicular to the crack  166  Figure 5.19 Shear stress versus shear strain for the adhesive layer of a reinforcement during the cooling cycle  168  Figure 5.20 Illustration of the superposition principle for a composite repair with disbonds Figure 5.21 Free body diagram for a reinforcement with a disbond  174 176  List of Figures  xii  Figure 5.22 Stress intensity factor dependence on disbond length for a bonded repair under thermal residual stress loading ( A T = — 64.8°C)  183  Figure 5.23 Stress intensity factor dependence on disbond length for a bonded repair under remote stress loading (a^ = 40 MPa)  183  Figure 5.24 Stress intensity factor dependence on disbond length for a bonded repair under combined thermal residual stress and remote stress loading ( A T = -64.8°C, ^  = 40 MPa)  184  Figure 5.25 Patch efficiency for partially disbonded patches  185  Figure 5.26 Free body diagram for a tapered reinforcement with a disbond  187  Figure 5.27 Stress intensity factor dependencies on disbond length for a bonded tapered and uniform thickness patches under thermal residual stress loading ( A T = - 6 4 . 8 ° C )  195  Figure 5.28 Stress intensity factor dependencies on disbond length for bonded tapered and uniform thickness patches under remote stress loading  (CTQO  =  40 MPa)  196  Figure 5.29 Stress intensity factor dependencies on disbond length for bonded tapered and uniform thickness patches under combined thermal residual stress and remote stress loading ( A T = — 6 4 . 8 ° C ,  = 40 MPa)  . . . .  197  Figure 5.30 Patch efficiency for partially disbonded patches with tapered edges under remote stress loading (RS) and combined thermal residual stress and remote stress loading (TRS&RS) Figure 6.1  Adhesive shear stress for a uniform thickness reinforcement using element lengths between t and t /8 a  Figure 6.2  198  a  206  Adhesive shear stress for a uniform thickness reinforcement with a different number of elements through the thickness  207  Figure 6.3  Reverse and normal stacking sequence  208  Figure 6.4  Transverse shear stress at the adhesive mid-plane in the tapered region for different F E M meshes  Figure 6.5  Detailed view of the finite element mesh in the tapered section-reversed stacking  Figure 6.6  210  Thermal residual strains at the top surface of the patch in the tapered region for different F E M meshes  Figure 6.7  209  210  Thermal residual strains at the bottom surface of the patch in the tapered region for different F E M meshes  211  List of Figures Figure 6.8  xiii  Thermal residual strains at the top surface of the substrate in the tapered region for different F E M meshes  Figure 6.9  211  Thermal residual strains at the bottom surface of the substrate in the tapered region for different F E M meshes  212  Figure 6.10 Transverse shear stress at the adhesive mid-plane for reversed and normal stacking sequence  213  Figure 6.11 Thermal residual strains at the top surface of the patch for reversed and normal stacking sequence  214  Figure 6.12 Thermal residual strains at the bottom surface of the patch for reversed and normal stacking sequence  214  Figure 6.13 Thermal residual strain at the top surface of the substrate for reversed and normal stacking sequence  215  Figure 6.14 Thermal residual strain at the bottom surface of the substrate for reversed and normal stacking sequence  216  Figure 6.15 Peel stresses in the adhesive mid-plane for a reversed stacking sequence  216  Figure 6.16 Critical location for peel stresses in the adhesive mid-plane for the reversed stacking sequence  217  Figure 6.17 Stress intensity factors for different crack tip elements  220  Figure 6.18 Stress perpendicular to the crack (a ) for different crack tip elements .  222  Figure 6.19 Finite element mesh for the patched crack verification model  224  Figure 6.20 Stress in loading direction (a ) for the patched substrate along 9 = 0°.  226  Figure 6.21 Stress perpendicular to crack (a ) in the patched substrate for 9 = 90°  229  Figure 6.22 Stress parallel to the crack (a ) in the patched substrate for 9 = 0°  . .  230  Figure 6.23 Stress parallel to the crack (a ) in the patched substrate for 9 = 90° . .  230  Figure 6.24 Shear stress (r )  231  y  y  y  x  x  xy  in the patched substrate for 9 = 0°  Figure 6.25 Shear stress (T ) in the patched substrate for 9 = 90°  232  Figure 6.26 F E M mesh for circular patch verification model  233  Figure 6.27 Detailed view of tapered edge in the circular patch verification model .  234  xy  Figure 6.28 Shear stress (r ) yz  at R = 74.875 mm of a circular patch with a radius of  R = 75 mm for different angles 9  235  Figure 6.29 F E M mesh for A M R L sandwich type specimen  236  Figure 6.30 Thermal residual strains parallel to the fiber orientation (y-direction) for the A M R L sandwich type specimen in [/te] at the mid-plane of the substrate  238  List of Figures  xiv  Figure 6.31 Thermal residual strains parallel to the fiber orientation (y-direction) for the A M R L sandwich type specimen in [u-e] in the pach top-surface . . . Figure 6.32 Transverse shear stress r  yz  239  for the A M R L sandwich type specimen in  [MPa] in the mid-plane of the adhesive  241  Figure 6.33 Adhesive peel stress a for the A M R L sandwich type specimen in [MPa] z  at the adhesive/patch interface  242  Figure 6.34 Detailed view of the adhesive peel stress a for the A M R L sandwich type z  specimen in [MPa] for the ply drop-offs at R = 66 mm and R = 69 mm  243  Figure 6.35 Fractographic investigation patch/adhesive interface at the ply drop-off  244  Figure 7.1  262  Minimum patch thickness design scheme  Figure A . l A M R L specimen (Face-sheets)  276  Figure A.2 A M R L specimen (Spacer block)  277  Figure A.3 A M R L specimen (Honeycomb)  278  Figure A.4 A M R L specimen (Shims)  279  Figure A.5 A M R L specimen (Jig)  280  Figure A.6 A M R L specimen (Jig-bolts)  281  Figure A.7 Instrumented A M R L specimen (Aluminum)  282  Figure A.8 Instrumented A M R L specimen (Patch-adhesive interface)  283  Figure A.9 Instrumented A M R L specimen (1st boron/epoxy ply)  284  Figure A. 10 Instrumented A M R L specimen (Top boron/epoxy ply)  285  Figure A. 11 Instrumented A M R L specimen (Superimposed drawing)  286  Figure A.12 Instrumented A M R L specimen (Backside of face-sheet 7)  287  Figure A.13 Instrumented A M R L specimen (Gauge locations)  288  Figure A. 14 Instrumented A M R L specimen (Backside gauge locations)  289  Figure A.15 Instrumented A M R L specimen (Terminal locations)  290  Figure B . l  Crack propagation rates for face-sheet 1  306  Figure B.2  Crack propagation rates for face-sheet 2  306  Figure B.3  Crack propagation rates for face-sheet 3  307  Figure B.4  Crack propagation rates for face-sheet 4  307  Figure B.5  Crack propagation rates for face-sheet 5  308  Figure B.6  Crack propagation rates for face-sheet 6  308  Figure B.7  Crack propagation rates for face-sheet 7  309  Figure B.8  Crack propagation rates for face-sheet 8  309  List of Figures Figure B.9  xv  Crack propagation rates for face-sheet 9  310  Figure B.10 Crack propagation rates for face-sheet 10  310  Figure B . l l Crack propagation rates for face-sheet 11  311  Figure B.12 Crack propagation rates for face-sheet 12  311  Figure B.13 Crack propagation rates for face-sheet 13  312  Figure B.14 Crack propagation rates for face-sheet 14  312  Figure B.15 Face-sheet 1 pre-crack  313  Figure B.16 Face-sheet 2 pre-crack  313  Figure B.17 Face-sheet 3 pre-crack  313  Figure B.18 Face-sheet 4 pre-crack  313  Figure B.19 Face-sheet 5 pre-crack  314  Figure B.20 Face-sheet 6 pre-crack  314  Figure B.21 Face-sheet 7 pre-crack  314  Figure B.22 Face-sheet 8 pre-crack  314  Figure B.23 Face-sheet 9 pre-crack  315  Figure B.24 Face-sheet 10 pre-crack  315  Figure B.25 Face-sheet 11 pre-crack  315  Figure B.26 Face-sheet 12 pre-crack  315  Figure B.27 Face-sheet 13 pre-crack  316  Figure B.28 Face-sheet 14 pre-crack  316  Figure C l  C-scan of patch 1  318  Figure C.2  C-scan of patch 2  318  Figure C.3  C-scan of patch 3  318  Figure C.4  C-scan of patch 4  319  Figure C.5  C-scan of patch 5  319  Figure C.6  C-scan of patch 6  319  Figure C.7  C-scan of patch 7  320  Figure C.8  C-scan of patch 8  320  Figure C.9  C-scan of patch 9  320  Figure C I O C-scan of patch 10  321  Figure C . l l C-scan of patch 11  321  Figure C.12 C-scan of patch 12  321  Figure C.13 C-scan of patch 13  322  Figure C.14 C-scan of patch 14  :  322  List of Figures  xvi  Figure D . l Coefficient of thermal expansion for boron/epoxy 5521/4 parallel to the fiber direction based on different reference temperatures  324  Figure D.2 Coefficient of thermal expansion for boron/epoxy 5521/4 perpendicular to the fiber direction based on different reference temperatures  325  Figure D.3 Coefficient of thermal expansion for 2024-T3 aluminum parallel to the rolling direction based on different reference temperatures  326  Figure D.4 Coefficient of thermal expansion for 2024-T3 aluminum perpendicular to  Figure E . l  the rolling direction based on different reference temperatures  327  Gauge 1 - Thermal and thermal residual strains [Aluminum _L]  330  Figure E.2 Gauge 2 - Thermal and thermal residual strains [Aluminum ||]  330  Figure E.3 Gauge 3 - Thermal and thermal residual strains [Aluminum ||]  330  Figure E.4 Gauge 4 - Thermal and thermal residual strains [Aluminum _L]  331  Figure E.5 Gauge 5 - Thermal and thermal residual strains [Aluminum ||]  331  Figure E.6 Gauge 6 - Thermal and thermal residual strains [Aluminum ± ]  331  Figure E . 7 Gauge 7 - Thermal and thermal residual strains [Aluminum ||]  332  Figure E.8 Gauge 8 - Thermal and thermal residual strains [Aluminum J_]  332  Figure E.9 Gauge 9 - Thermal and thermal residual strains [Aluminum _L]  332  Figure E.10 Gauge 10 - Thermal and thermal residual strains [Aluminum ||]  . . . .  333  Figure E . l l Gauge 11 - Thermal and thermal residual strains [Aluminum _L] . . . .  333  Figure E.12 Gauge 12 - Thermal and thermal residual strains [Aluminum ||]  . . . .  333  Figure E.13 Gauge 13 - Thermal and thermal residual strains [Aluminum ||]  . . . .  334  Figure E.14 Gauge 14 - Thermal and thermal residual strains [Aluminum _L] . . . .  334  Figure E.15 Gauge 15 - Thermal and thermal residual strains [Aluminum _L] . . . .  334  Figure E.16 Gauge 16 - Thermal and thermal residual strains [Aluminum ||]  . . . .  335  . . .  335  Figure E.18 Gauge 18 - Thermal and thermal residual strains [Aluminum J_] . . . .  335  Figure E.19 Gauge 19 - Thermal and thermal residual strains [Aluminum ||]  . . . .  336  . . .  336  Figure E.21 Gauge 21 - Thermal and thermal residual strains [Aluminum _L] . . . .  336  Figure E.22 Gauge 22 - Thermal and thermal residual strains [Aluminum ||]  . . . .  337  Figure E.23 Gauge 23 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  337  Figure E.24 Gauge 24 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  337  Figure E.25 Gauge 25 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  338  Figure E.26 Gauge 26 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  338  Figure E.17 Gauge 17 - Thermal and thermal residual strains [Aluminum 45°]  Figure E.20 Gauge 20 - Thermal and thermal residual strains [Aluminum 45°]  List of Figures  xvii  Figure E.27 Gauge 27 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  338  Figure E.28 Gauge 28- Thermal and thermal residual strains [Boron/epoxy ||]  339  . . .  Figure E.29 Gauge 29 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  339  Figure E.30 Gauge 30 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  339  Figure E.31 Gauge 31 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  340  Figure E.32 Gauge 32 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  340  Figure E.33 Gauge 33 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  340  Figure E.34 Gauge 34 - Thermal and thermal residual strains [Boron/epoxy _L]  . .  341  Figure E.35 Gauge 35 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  341  Figure E.36 Gauge 36 - Thermal and thermal residual strains [Boron/epoxy _!_]  . .  341  Figure E.37 Gauge 37 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  342  Figure E.38 Gauge 38 - Thermal and thermal residual strains [Boron/epoxy J_]  . .  342  Figure E.39 Gauge 39 - Thermal and thermal residual strains [Boron/epoxy _L]  . .  342  Figure E.40 Gauge 40 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  343  Figure E.41 Gauge 41 - Thermal and thermal residual strains [Boron/epoxy JL]  . .  343  Figure E.42 Gauge 42 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  343  Figure E.43 Gauge 43 - Thermal and thermal residual strains [Boron/epoxy ||] . . .  344  Figure E.44 Gauge 44 - Thermal and thermal residual strains [Boron/epoxy _L]  344  . .  List of Symbols  xviii  List of Symbols Symbol  Description  Dimensions  a  Crack length  [mm]  A  Constant  [MPa]  b  Disbond length  [mm]  B  Constant  [MPa]  c  Geometry factor  [  C  Integration constant  d  B o l t diameter  e  Function  [  E  Young's modulus  [GPa]  f  Function  [  F  Force  [N]  F  Function  [N/mm]  F  Function  [  9  Integration constant  [mm]  G  Adhesive shear modulus  [MPa]  G  Patch shear modulus  [MPa]  G  Substrate shear modulus  [MPa]  G  Energy release rate  [ M P a m]  h  Integration constant  [mm]  H  Function  [  I  Current  [A]  K  Stress intensity factor  [MPa^/m]  Tranverse sensitivity factor  [  I  Length of steps  [mm]  hmRef.  5 m Lead wire reference specimen  [mm]  n  Number of steps  [  ]  N  Number of cycles  [  ]  P  A p p l i e d load per unit length  [N/mm]  r  Distance from crack tip  [mm]  r*p  Plastic zone radius  [mm]  R  Resistance  [0]  So  Strain gauge factor  [  c  f  a  p  s  K  t  ]  . [N/mm] [mm] ]  ]  ]  ]  ]  ]  List of Symbols  xix  Symbol  Description  Dimensions  SR  Stress ratio  [ ]  t  Thickness  [mm]  T  Direct stress resultant  [N/mm]  v  Displacement  [mm]  V  Voltage  [V]  W  Specimen width  [mm]  y  Axial coordinate of joint  [mm]  Y  Geometric correction factor  [ ]  z  Transverse coordinate of joint  [mm]  Z()  Complex stress function  [MPa]  a  Coefficient of thermal expansion  6  Constant  [ ]  AC  Function  [N/mm]  T  Temperature change  [°C]  e  Strain  [ ]  Adhesive shear strain  [ ]  7  Constant  [ ]  A  Elastic shear stress distribution parameter . [1/mm]  A  Characteristic crack length  [mm]  v  Poisson's ratio  [ ]  a  Normal stress  [MPa]  T  Shear stress  [MPa]  £  Range variable  [ ]  <f>  Constant  [ ]  9  Angle  [ °]  Subscript  Description  a  Adhesive  Al  Aluminum  AMRL  A M R L sandwich type specimen  B/Ep  Boron/epoxy  corr.  corrected  D  Double-doubler joint  A  Increment  7  A  [pefC]  List of Symbols  xx  Subscript  Description  Fourier  Discrete Fourier solution  G  Gauge  i  Range variable  I  Mode I  j  Range variable  LW  Lead wire  max  Maximum  min  Minimum  o  Reference  p  Patch  Preload  Preload  Ref.  Titanium silicate reference specimen  RS  Residual strain  SFT  Stress free temperature  s  Substrate  TS  Titanium silicate specimen  TSY  Transverse sensitivity  TSR  Thermal residual strain  r  Reinforced  R  Double-sided reinforcement  Shear — lag corr.  shear-lag corrected  u  Upper bound for short crack  TRS  Thermal Residual Stress  RS  Remote Stress  oo  Upper bound for long crack  _L  Perpendicular  ||  Parallel  Abreviations  Description  IAR  Institute for Aerospace Research  NRC  National Research Council of Canada  UBC  University of British Columbia  AMRL  Aeronautical and Maritime Research Laboratory  ARALL  Aramid Reinforced ALuminum Laminate  GLARE  GLAss Reinforced  Acknowledgements  xxi  Acknowledgements I gratefully acknowledge the HSP II scholarship of the German Academic Exchange Service (DAAD) as well as the University Graduate Fellowships from the University of British Columbia. I would like to thank the National Research Council of Canada for providing funding for this research project as well as the scientific support by Mr. Donald Raizenne. In addition, I would like to thank Dr. Douglas P. Romilly as well as my research committee members, Dr. Desmond Tromans, Dr. Reza Vaziri, Dr. Henry Vaughan and Mr. Dean Leonard, for their continuing efforts and consisting guidance. Thank you also to Dr. Alan Baker for sharing some of his knowledge of bonded composite repairs.  Furthermore I  would like to thank Jeff Aitchison, Rudolf Seethaler, Rob Thomson and Randy Clark for the helpful ideas and discussions.  The assistance of T. Benak and other staff members of the Institute for Aerospace Research (NRC) as well as the staff of the Mechanical Engineering Machine Shop at the University of British Columbia during specimen preparation is gratefully acknowledged. Special thanks to my parents as well as my wife Susan for their continuing support, patience and love.  1  Introduction  1  Chapter 1 Introduction  Throughout the past twenty years, there has been an increasing trend towards operating commercial and military aircraft structures beyond their original fatigue design life. To achieve this, cost efficient repair schemes must be developed that will both restore the strength of the damaged structure and prevent existing cracks from becoming critical. In 1973, researchers at the Aeronautical and Maritime Research Laboratory (AMRL) in Melbourne, Australia, first proposed the repair of cracked metal aircraft structures using adhesively bonded patches fabricated from composite materials [18].  In addition to the  repair of damaged structures, composite patches can be used as reinforcement to increase load carrying capability and improve fatigue resistance.  To date, a number of approaches involving analytical, numerical and experimental analysis have been undertaken to develop design schemes for composite repairs of metallic aircraft structures.  One of the important areas of continuing research is the analysis of fatigue  damage in composite repairs. The fatigue failure behaviour of a bonded composite repair is dependent on both the mean stress and the stress amplitude. Fatigue failure characteristics are furthermore complicated by the stress history prior to the repair and the stresses induced during the repair. Properly designed and installed bonded composite repairs can reduce or eliminate crack growth by reducing the crack opening displacement and subsequently  1 Introduction  2  the stress intensity factor range. Additional retardation of the crack growth results from the difference in plastic zone size prior to and after patching due to the reduction in the peak stress intensity factor. These beneficial retardation effects may, on the other hand, be offset by thermal residual stresses.  Thermal residual stresses are induced during the repair process as a result of the elevated temperature cure which is required to achieve suitable adhesive characteristics. The magnitude of the thermal residual stresses is dependent on the difference in the coefficients of thermal expansion between the metallic substrate and the composite patch material, the local heating and support conditions of the metallic structure and the temperature differential between the stress free temperature and the operating temperature. The heating and local support conditions in the metallic structure lead to a lower effective coefficient of thermal expansion (CTE) since the unheated surrounding structure will restrain the expansion of the locally heated repair section. In addition, the supporting structure of an aircraft frame will limit the out-of-plane bending and resulting curvature of a single-sided repair, which is also caused by the difference in C T E [18, 53].  Thermal residual stresses can be quite large depending on these local heating and support conditions, as well as material characteristics, thus affecting the desired reduction in the stress intensity factor significantly. It is therefore important to realize that thermal residual stresses will vary substantially between bonded composite repair test specimens which are heated uniformly without any restraint and real structural repairs which have significant restraint (see Figure 1.1 [14]).  The key is to understand the relationship between the  constraint conditions of the test specimen relative to those of the damaged aircraft structure such that the thermal residual stresses can be accounted for properly in the determination of the margin of safety.  1 Introduction  Repair Application  3  AMRL Specimen  Figure 1.1: Small scale test specimen for composite repairs  Precise knowledge of the stress state in the metal substrate after the patching process including thermal residual stresses is essential in order to carry out a detailed investigation of the fatigue damage initiation characteristics for bonded composite repairs. Thermal residual stresses will also affect the fatigue crack growth rates by changing the fatigue stress ratio for the metal substrate in the bonded repair section. Baker showed an approximate 10% change in the stress intensity by accounting for the change in the stress ratio from 0.1 to 0.6 due to thermal residual stresses [18]. Thermal residual stresses have often been quoted in the literature as a possible contributing factor without really knowing their true effect on the fatigue behaviour [18].  This research addresses this shortfall by investigating the magnitude of the thermal residual stresses in detail to establish the foundation for a fatigue damage initiation model and an improved fatigue crack propagation model. This research also examines the magnitude of temperature independent process induced residual strains from effects such as anisotropic shrinkage of the patch or adhesive, temperature gradients in the specimen during processing, and/or temperature gradients due to tooling and internal resin flow [50].  1 Introduction  4  Based on the need to establish the true stress/strain state after carrying out a bonded composite repair, the following research objectives were identified: 1. Determine the critical locations in a bonded composite repair; 2. Determine appropriate process specification, and manufacture suitable test specimens including an instrumented specimen; 3. Measure thermal residual stresses/strains and estimate temperature independent process induced strains; 4. Establish an appropriate thermal residual stress/strain model for bonded composite repairs, including an estimate for the stress distribution near the crack tip; 5. Determine the influence of disbonds on the stress intensity factor under thermal residual stress loading; 6. Establish the limitations of the thermal residual stress/strain model; These objectives were achieved in four stages: In the first stage, six A M R L sandwich type composite bonded repair specimens were manufactured to provide some of the specimens required for fatigue damage initiation testing. In addition, one test specimen was instrumented by placing 44 strain gauges at eight locations and within five different interfaces of the patched component (Chapter 3). Residual strains at ambient temperature (including both thermal residual strains and other process induced strains) were measured during the manufacturing process. In the second stage, the stress free temperature of the bonded repair was experimentally determined and the thermal residual strains measured as a function of operating temperature (Chapter 4). In the third stage, a theoretical analysis was carried out to determine the thermal residual stress and strain distributions in bonded repairs and reinforcements. This analysis also addressed the effect of symmetrical disbonds around the  1  Introduction  5  patched crack as well as the stress field ahead of the crack (Chapter 5). Finally, a detailed finite element analysis was carried out to assess the limitations of the theoretical analysis as well as providing a more detailed insight into the complex thermal residual stress and strain state of the A M R L sandwich type specimen (Chapter 6). A literature review (Chapter 2) is presented next to establish the state-of-the-art of composite repair research, thus providing the necessary background and highlighting the direction for the current work.  2 Literature Review  6  Chapter 2 Literature  2.1  Review  Introduction  The establishment of the damage tolerance approach in the airworthiness regulations allows aircraft operators to utilize their aircraft economically, well beyond their original fatigue design.  A n aircraft's life is now being determined by the economical balance between  revenue and maintenance costs. The increasing number of aging aircraft impose a growing challenge for aircraft maintenance facilities to provide safe and economical feasible repair schemes. The classical repair scheme for metal aircraft structures are riveted repairs. If a riveted repair is not feasible, the entire component must be replaced, which can be very expensive considering parts, labour and aircraft downtime. Chartering a replacement aircraft to meet operational requirements can be quite costly. Furthermore, riveted repairs often have a limited life since they introduce new potential fatigue crack initiation sites.  In 1973, the Aeronautical and Maritime Research Laboratory (AMRL) in Australia first proposed the repair of cracked metal aircraft structures using composite materials. This technology offers an excellent alternative to riveted repairs especially for fleet wide repairs. Not introducing additional fatigue crack initiation sites in a highly loaded structure is one of the key benefits of this repair scheme over riveted repairs. Unfortunately this technology also has its downsides.  The quality of the bond between the metal aircraft structure  2.1  Introduction  7  and the composite patch is highly depended on the process specifications and quality of workmanship provided by the technician carrying out the repair. A workshop organized by the Royal Australian Air Force addressing standardization of bonded repairs focused on this problem as part of the International Conference on Composite Materials ICCM-11 in 1997. A M R L has proven that these repairs can be carried out successfully as demonstrated by their experience with Hercules C130 wing plank repairs. While over 1000 repairs have been carried out on stiffeners of the C130, no crack growth has been observed in 15 years of service [45].  Most of the composite repairs have been carried out on military aircraft [45]. In recent years, Textron has promoted the application of bonded repairs to civilian aircraft by installing bonded doublers on Federal Express 747s for evaluation [20]. Examples of ongoing research to improve the understanding and confidence in bonded repairs is AMRL's work on assessment models for damage tolerance, such as the consequences of initial flaws in bonded repairs, impacts and damage growth [11, 13]. Furthermore, A M R L is working on a generic approach to bonded repairs since this is the key to offering a cost effective solution for a broader range of repair applications.  Development of smart patches which detect  disbond and damage growth is also underway. The research work presented in this thesis addresses the fundamentals for a fatigue damage initiation investigation for the cracked metal component in bonded repairs. Currently bonded repairs are evaluated based on their reduction in the crack growth rate of the damaged structure. Although this is a conservative approach, it neglects a significant portion of the component life. Knowledge of the time required to reinitiate the crack would improve patched structure life predictions and increase the confidence levels of certification authorities in this repair technique. This information becomes even more critical if repairs which have an initial crack length close to the critical crack length are carried out, such as for the F - l l l lower wing skin repair [97]. The first step in a life prediction is to determine the initial stress state of the bonded repair prior  2.2  Bonded Repairs Versus Riveted Repairs  8  to applying any external load and must include the thermal residual stresses within the patched structure. Proper experimental and theoretical determination of thermal residual strains is not only required to assess fatigue damage initiation, but also to allow for a more precise determination of the actual load experienced by the bonded repairs for the critical load cases which determine the feasibility of a repair. This chapter presents background information on patching technology, its advantages and disadvantages as well as the relevancy of the past research with respect to the current key issues of this repair method.  2.2  Bonded Repairs Versus Riveted Repairs  Currently riveted repairs represent the main repair scheme for metal aircraft structures. Their installation is generally easy and requires only standard equipment.  Additional  advantages are the possibility of subsequential disassembly as well as repair installation in an uncontrolled environment.  On the other hand, 'riveted repairs can substantially  degrade the fatigue life if extreme care is not taken' [35, 86]. Generally, riveted repairs can restore the static strength while the fatigue strength is often neglected.  Riveting  produces concentrated stresses at the bearing surfaces resulting in local stress risers and fatigue crack initiation sites [18]. In most cases the maintenance engineer does not have access to original loads to carry out a thorough analysis but must rely on the Structural Repair Manual (SRM) provided by the original equipment manufacturer showing approved methods and maximum repairable sizes [35].  The installation can also create problems  such as in-situ drilling during the riveted repair process causing internal damage to items such as hydraulic lines and electrical wiring. A poor mechanical fastening procedure may result in fretting damage or high residual stresses, thus encouraging fatigue cracking and/or stress-corrosion cracking [18].  2.2  Bonded Repairs Versus Riveted Repairs  9  An example of a poor repair was given by Fredell [35]. Figure 2.1 shows the repair to the lower wing skin of a Lockheed Electra. This particular repair was designed to reinforce the cracked section of the original blade-stiffened wing skin. Additionally, two riveted angled triplers inside the blade stiffeners were installed. Figure 2.1 [35] shows an alternating rivet pattern in the first and last rivet row. This particular rivet scheme lead to a higher bearing load in the remaining rivets in the end rows. The consequences of this riveting pattern in combination with a very stiff patch were new fatigue cracks after approximately 10000 flight hours. As a result, a complete fracture of the wing plank occurred. A previous inspection didn't reveal the new crack between location A and B since the design obstructed the inspectors view. This example shows some of the problems with riveted repairs if extreme care is not exercised in the design stage to avoid fatigue problems. In fact, one of the largest loss of life from a single aircraft accident was traced to an improperly performed structural riveted repair. In August 1985, 520 people died when the aft pressure bulkhead of a Japan First Row  3  Patch  O  OOUTBD.  {»Ao  Cracks  '«  Crack  Figure 2.1: Sketch of the Lockheed Electra lower wing skin  2.2  Bonded Repairs Versus Riveted Repairs  10  Air Lines Boeing 747 failed due to a mistake in the riveted bulkhead repair which caused a crack to form and propagate along the repair line [35].  While riveted repair schemes are more widely accepted, bonded repair techniques certainly offer much better fatigue characteristics since no additional rivet holes are introduced during the repair process. The high stiffness of the composites allows the implementation of thin patches thus offering aerodynamic advantage over riveted repairs especially in aerodynamically important areas such as the leading edge. Tapering and conformity to existing structures can be readily achieved in comparison to riveted repairs. For lightly loaded structures, adhesive joints are more efficient as the low density materials and reduced volume provide weight saving in areas where weight is critical (e.g.  control surfaces). The  orthotropic characteristic of the employed composites allows tailoring to the specific load requirement of a particular repair [18]. It should be noted that bonded repairs can also lead to structural failure with possibly catastrophic consequences if they are not designed and installed properly.  Once a bonded repair scheme is established, the implementation time and required labour are, in many cases, less than the required time and labour for a riveted repair thus making it easier to implement during regular maintenance checks. This can be of significant benefit to airlines since additional downtime of an aircraft is very expensive. Fleet wide repairs can offer significant savings as shown by the Hercules C130 upper wing plank repairs carried out by A M R L ($67 Million) as well as the C-141 weep hole repairs mainly carried out at the Warner Robins Air Logistics Center, where in total 770 repairs were performed out between September 1993 and December 1994 [83].  Bonded repairs also offer advantages with respect to the inspection of the underlying damaged structure. Eddy current can be used to monitor crack growth beneath boron/epoxy patches, i.e. the composite material most commonly used for structural repairs [18]. Crack  2.2  Bonded Repairs Versus Riveted Repairs  11  initiation and the growth of new cracks in riveted repairs are sometimes difficult to detect can lead to failures such as that which occurred in the Lockheed Electra example discussed previously [35]. One solution to this problem are 'fingered' inner doublers (see Figure 2.2 [86]) which combine long fatigue life with good damage tolerance and provide excellent external inspectability near the first row of fasteners. However, the implementation of fingered inner doublers certainly adds to the cost of the repair [35, 86].  OUTER PRIMARY DOUBLER (O.032 INCHES THICK)  Figure 2.2: 'Fingered' inner doubler for long joint fatigue life and external visual inspection  'On the other hand, bonded repairs should not be attempted unless stringent cleaning and processing steps can be adhered to within a controlled environment' [13]. This aspect was repeatedly emphasized during the workshop on 'Practical Adhesive Bonding for Performance and Durability: Standards and Standardisation' [11, 30, 31]. A n example of the consequences of poor bonding is the infamous Aloha Airlines flight 243, where the upper fuselage of the Boeing 737 separated in flight causing an explosive decompression [31]. Disbonding and corrosion was caused by applying the adhesive layer in a frozen state which lead to moisture attraction inhibiting the formation of the chemical bond to the adherends during the fuselage assembly process. Adhesive bond failure subsequently overloaded the  2.3  Analysis of Bonded Repairs  12  fasteners resulting in Multi-Site Damage (MSD) [31]. Since the capabilities of assessing the quality of an adhesive bond are limited, the training and certification of maintenance engineers and technicians remains a key issue for achieving durable bonded repairs [11, 31, 83]. One of the main disadvantages of applying composite repairs to metallic structures results from the mismatch in the coefficients of thermal expansion leading to thermal residual stresses if the bonded repair is cured at an elevated temperature [18]. Currently available room temperature curing adhesives such as Flexon 241 don't have the desired material properties, especially at elevated temperatures [13].  2.3  Analysis of Bonded Repairs  Having discussed the advantages and disadvantages of riveted repairs and adhesively bonded composite repairs, the next step is to review the available theoretical models for the design of composite repairs to estimate the stress intensity factor after patching as well as the stress distribution in the adherends and adhesive under remotely applied and thermal loading. L.J. Hart-Smith's one-dimensional approach to adhesive joints based on Volkersen's solution [96] represents the foundation for the analysis of bonded repairs. To improve the estimate of the load carrying capacity of a bonded joint his theory includes the mismatch in the coefficients of thermal expansion for the adherends in different joint and reinforcement types as well as the elastic-perfectly plastic behaviour for the adhesive. He used an iteration technique to solve for the stresses in the case of plasticity in the adhesive as well as for tapered or stepped joints. This solution, with the associated numerical problems due to the high stress gradients at the steps, was implemented in the program AE4I [42] and is currently in use.  Other more complex solutions for different joint types have been presented in the literature [4, 39, 58, 68, 74, 93, 94, 100]. The complexity of these models generally requires their  2.3  Analysis of Bonded Repairs  13  implementation in a sophisticated computer program. Unfortunately these codes are not widely available thus limiting the use of adhesive bonding technology.  Rose [18] used the basic one-dimensional linear-elastic solution as part of a two step model to derive an estimate for the stress intensity factor of the cracked metal substrate in a bonded composite repair. The first step is the determination of the stress at the prospective crack location. Rose applied an inclusion analogy to determine the stress field for a repaired section with an elliptical patch over an uncracked substrate as well as its surroundings. For complicated geometries, a finite element solution can be used to determine the stress at the prospective crack location [19] while for simpler repair geometries, a one-dimensional approach gives an excellent estimate. The second step is the determination of upper bounds for the stress intensity factor where Rose utilizes the basic one-dimensional linear-elastic model for adhesively bonded joints. [78, 79]. Rose also presented a thermal residual stress estimate due to adhesive curing for an isotropic patch accounting for both the restraint of the surrounding structure due to local heating as well as the influence of bending effects for a single-sided repair on the stress intensity factor [18]. Since it is not always possible for the reinforcement to cover the entire crack, he further discusses a partial reinforcement [18, 80]. This model is based on a distributed spring model which he extended to assess the efficiency of a repair as a function of load history when progressive deterioration occurs in the spring constant due to disbonding induced by cyclic loading, ingress of moisture or a combination of such factors [80, 81]. This model offers an efficient computational approach for keeping track of progressive deterioration of the bonded repair [81].  Rose's original closed form analytical model appears to still be the preferred choice for the initial assessment of a bonded repair. It shows the influence of the adhesive and adherend material properties and thickness on the stress intensity factor. Baker also used this closed form analytical model to estimate the influence of a disbond size on patch efficiency [13]. He proposed the definition of a damage tolerant zone in the center section of a bonded  2.3  Analysis of Bonded Repairs  14  composite repair where disbonding is acceptable, thus emphasizing the analysis of disbonds as a key issue with respect to certification and inspection. It is important to provide a tool which estimates the impact of disbonds with respect to the increase in the stress intensity factor. The inspector of a bonded repair needs to decide if the bonded repair should be rejected or if the aircraft can go back into service. Fredell [35] also implemented Rose's model in a computer code (CalcuRep  for Windows)  with further extensions to thermal influences. A peel model for this code is currently under development [35, 40]. The objective of CalcuRep is to cover the gap between complex finite element modeling and hand calculation.  It is aimed at aircraft maintenance engineers  rather than researchers allowing one to quickly study the effects of design variables and prepare a design. The complexity of a design might require a finite element analysis using the initial design data.  One of the missing components in this code is the estimation  of disbonds. Although the effect of thermal residual stresses is included in the code, it provides unnecessarily conservative results by using the curing temperature as the stress free temperature.  Baker presented a design procedure for minimum patch thickness which incorporates the Rose model [18]. This particular procedure is based on comparisons between the computed overlap length and the allowable length (Step 4), the peak shear strain in the patch and the allowable peak strain (Step 6), and the computed shear strain range in the adhesive and the allowable shear strain range (Step 8) as shown in Figure 2.3. The allowable peak strain (Step 5) for the patch is based on the allowable static strain corrected for strain concentration effects. The fatigue life or durability of the patch is assumed to be determined by the shear strain range (Step 7).  The allowable shear strain range is dependent on humidity and  temperature. Baker made no attempt at that time to design to a specified stress intensity or stress intensity factor range due to (i) the uncertainties in the estimate of AK^,  (ii) the  difficulty of accounting for the stress-ratio effects, (iii) the difficulty in allowing for overload  2.3  Analysis of Bonded Repairs  15  or crack-retardation effects and (iv) the large errors in the prediction of crack growth rates due to material variability and environmental influences [18]. Furthermore Baker didn't include disbond allowables in this initial design scheme.  -*) Initialization r*  Step 1 Step 2 Step 3  Yes Yes  n =  Step 4  n-1  Step 5  Yes  Step 6  Repair not Feasible  Step 7  Yes > AY!>  •  Step 8  Step 9  Yes  X  Warning of Limited Durability  Step 10  Repair Feasible  I  Option to Increment n  Step 11  Figure 2.3: Minimum patch thickness design scheme  Composite repairs have also been extensively investigated using numerical studies. Well known is the work by Ratwani, who considers an infinite plate with reinforcement [73]. Young, Cartwright, Dowrick and Rooke also authored a number of papers on model studies of repair patches using a strip patch modeling approach [33, 102, 103]. In 1979, G.L. Roderick investigated an analytical approach with an iterative solution for the determination of crack propagation in the aluminum structure and associated disbonding at the interface between the aluminum and the infinite patch using the Paris Law [75].  Although these  2,3  Analysis of Bonded Repairs  16  models have shown good results, the main problem with these approaches is portability. Most often the complexity of numerical solution schemes and the associated implementation work prevents these approaches from becoming a common modeling approach. In contrast, finite element approaches offering parametric studies as well as the investigation of complex geometries gained popularity due to the availability of good commercial packages and continuously increasing computing power. Thus finite element analysis approaches have become the commonly applied numerical scheme for the design and substantiation of bonded repairs [15, 18, 19, 24, 26, 46, 51, 52, 54, 55, 63, 83, 101]. Well known is the work from Jones and Callinan [24, 52, 54] who developed a two-dimensional finite element formulation for the adhesive layer in 1979. The adhesive element formulation was based on the displacement between the substrate and patch accounting also for the transverse shear stiffness in the adherends. Coupling the adhesive stiffness matrix with the composite patch stiffness matrix allowed them to use this bonded pair in conjunction with standard finite element routines. They also suggested appropriate modeling techniques for commercial F E M packages to include the transverse shear stress of the adherends in two-dimensional models. This scheme of using adhesive joint elements which account for the transverse shear deformation of the adherend was also used by other research institutions such as the Cranfield Institute of Technology [19]. Jones and Callinan also discussed the influence of local heating in combination with the supporting airframe on the magnitude of thermal residual stresses [53].  Since then the increase in computing power has allowed the use of three-dimensional finite element modeling to provide more accurate results [24], although care must be taken to ensure that the modeling is carried out correctly. Finite element analysis results are only as good as the assumptions used to create the model. Simplifications such as representing the taper in a composite patch by tapered elements or a normal stacking sequence instead of the actually used reversed stacking sequence may lead to misrepresentations depending  2.4  Test Specimens for Bonded Repairs  on the focus of the finite element analysis.  17 The important advantage of finite element  modeling especially with the available computing power and storage capacities over other numerical solution schemes is the possibility to transfer the modeling approach without the need to create a specialized code.  Rose's closed form analytical model is an essential key to obtaining a good initial guess for the design parameters thus accelerating the finite element modeling process. Not all design parameters (such as overlap length) can be readily determined by closed form analytical models or numerical approaches. Experimental testing is an important part of the determination of material behaviour and allowables as well as performance estimates for the bonded repair such as cycles to crack reinitiation, disbond growth rate and crack growth rate. It is generally assumed that the Paris law correlation still holds after the repair [35, 75, 81]. One of the main difficulties with experimental testing of bonded structures is the development of appropriate test specimens. It is important to realize the difference between results from a specific test specimen used to establish allowables and the performance of an actual repair and to account for them.  2.4  Test Specimens for Bonded Repairs  Baker's proposal [11, 12] for a generic certification approach gives an excellent overview of the required test specimens for bonded repairs. His approach includes testing of coupons, generic bonded joints, generic structural detail specimens and a representative structural detail specimen. Coupon tests are used to determine the basic adhesive and composite allowables.  Knock-down factors are evaluated for hot/wet conditions and off-optimum  manufacture. Generic bonded joints provide more valuable information since they simulate the actual loading conditions in a repair.  The amount of coupon testing can often be  reduced by generating some of the data from generic bonded joint tests. A double-doubler  2,4  Test Specimens for Bonded Repairs  18  joint type specimen has been commonly used as a generic bonded joint specimen [11, 18]. This type of specimen can provide the basic design data for static strength, threshold disbond growth, rate of disbond growth as well as be a platform to determine knockdown factors for hot/wet conditions, non-optimum manufacture and representative damage [11, 12]. Generic structural detail tests are used to validate the repair design approaches. Static strength, fatigue performance and damage tolerance are tested using a generic structural detail specimen such as the well known sandwich type specimen developed at the A M R L (see Figure 3.1) . This particular specimen reflects an attempt to include the influence of the supporting airframe structure in a test specimen for bonded repairs. The stiffening influence of the frame on the skin was implemented into the specimen design by bonding honeycomb between two cracked aluminum sheets. This honeycomb sandwich specimen minimizes the face-sheet curvature which occurs as a result of thermal residual stresses induced during the patching process. In addition, by using a symmetric configuration, this specimen reduces the out-of-plane deflection, which occurs on single-sided repairs due to the shift of the neutral axis under applied loading. This configuration has been proven to be representative of fighter aircraft having very stiff underlying structures. This type of specimen has also been used by the National Research Council of Canada [71], the University of British Columbia [1, 2, 3], the Aeronautical Research Laboratory/AIDC in Taiwan [67] as well as other institutions. Another type of generic structural detail test specimen, which was used by Textron Specialty Materials, is a single-sided bonded repair specimen using 0.8 mm to 2.5 mm thick 7075-T6 aluminum sheet with different patch thicknesses, ply orientations and lay-up sequences for which a large quantity of test data have been generated [20]. The final test specimen proposed by Baker is a representative structural detail test, which accounts for specific details of the repair and interrogates the differences to the generic structural detail test [11, 12].  2.4  Test Specimens for Bonded Repairs  19  In order to generate generic base data using the information obtained from previous tests as well as future tests, the loading conditions should be normalized with respect to geometry, material properties and manufacturing conditions. The actual load applied to a test specimen consists of three components: the remote loading, thermal residual stresses and process induced stresses. This research work addresses the magnitude of the process induced stresses and more significantly, the thermal residual stresses. It is critical to recognize that test specimens are generally free to expand during the curing process while the damaged section on an aircraft structure is typically highly restrained when heated only locally, thus reducing the 'effective' coefficient of thermal expansion. The determination of the 'effective' coefficient of thermal expansion of an aircraft component has been proven quite difficult. The analytical estimation used by A M R L [22] gave an effective coefficient of thermal expansion of 14.96 pe/°G  for the F - l l l wing skin repair while the experimental measurement  yielded an effective coefficient of thermal expansion of only 6 / / e / ° C . The difference in C T E to the unrestrained aluminum sheet, which would be applicable in a generic structural detail test specimen is 17 u-e/°C. Due to the generally lower effective coefficient of thermal expansion for the damaged metallic component in a real aircraft application, test specimens with no restraint provide conservative results under identical remote loading. It is, however, important to realize that depending on the degree of restraint in the damaged real structure, the thermal residual stresses might be quite substantial.  Accounting for  the thermal residual stresses correctly can then make the difference between rejecting or accepting a particular repair scheme.  Having established what type of specimens are needed to allow a generic approach to bonded repairs, an overview of some of the currently available experimental data applicable to Baker's proposed generic certification approach is presented next with the focus on experimental thermal residual stress and strain measurements.  2.5  2.5  Experimental Testing of Bonded Repairs  20  Experimental Testing of Bonded Repairs  Many design variables for adhesive joints have been evaluated during the P A B S T program (Primary Adhesively Bonded Structure Technology) leading to a significant gain in experience and knowledge of adhesively bonded joints which now serves as the basis for the design and fabrication of many bonded repairs [18, 44]. Recommendations for overlap length is one piece of valuable information derived from this program. For example, the overlap length for a double-lap joint should be a minimum of 30 times the thickness of the central member for subsonic aircraft made from aluminum in order to avoid creep. Hart-Smith also explains the design-philosophy behind this estimate for a double-lap joint with uniformly thick adherends [44]. The successful design of joints on the P A B S T fuselage, none of which have failed during four years of slow-cycle fatigue testing in a hostile hot/wet environment with the aluminum sheet stressed almost to yield, has increased the confidence in bonded structures significantly [44]. It should be noted that Baker recommends an overlap length of 6 times the load transfer length A (see Section 5.2.1) based on A M R L ' s experience with bonded repairs.  The A M R L sandwich type specimen has provided the link to the experimental verification of crack growth rates for bonded repairs with respect to the applied loading. A number of results for different process specifications for the patch application, different materials and thicknesses, a variety of stress intensity factor ranges and R-ratios as well as environmental influences are available for this particular specimen [11, 18, 67, 71].  Textron has also  provided a large quantity of experimental data for single-sided bonded repairs to create a generic body of test data.  Due to the lower effective coefficient of thermal expansion for the metal substrate of a real repair, it is beneficial to account for the thermal residual stresses in the absolute loading  2.5  Experimental Testing of Bonded Repairs  21  condition for the generic bonded repair specimens and then to estimate the thermal residual stresses for the actual repair based on the true adhesive behaviour combined with the proper boundary conditions (i.e., local heating and other reinforcing structural components in the vicinity of the repair).  The proper accounting of thermal residual stresses can be the  deciding factor for the feasibility of a bonded repair. Researchers at the Australian Aeronautical and Maritime Research Laboratory also investigated thermal residual stresses in bonded repairs. In 1978, they presented a paper on thermal residual stresses for C F R P - reinforced cracked specimens using X-ray diffraction procedures [17, 18]. Carbon/epoxy patches were used to allow X-ray penetration to the metal surface instead of the more common boron/epoxy patches.  These measurements  confirmed the high thermal stresses present in the bonded repair specimens. In addition to thermal residual stress measurements in the reinforcement, stress measurements were also taken around a machined crack with a tip radius of 0.2mm. The measured thermal residual stresses near the machined crack tip showed significant scatter (up to 19%) based on the remote thermal residual stress. The thermal residual stresses were interpreted in terms of stress intensity factors which appeared to be quite high and inconsistent with previous experimental observations by A M R L researchers on the growth of stress corrosion cracks in reinforced specimens [17]. The disadvantage of this particular measuring scheme for thermal residual strains near the crack tip is the need for a small diameter X-ray beam leading to significant data scatter in the results [17]. The measured remote thermal residual stress from the machined crack tip (40 mm ahead of a 10 mm crack) was reported up to 13% below the thermal residual stress measured in the center of a reinforced specimen giving an indication of the data scatter. Unfortunately, no data of thermal residual stresses versus temperature were presented, which would have given a better understanding of the elevated temperature behaviour of the adhesive and thus a confirmation of the theoretical relationship used to determine the thermal residual stresses.  2.5  Experimental Testing of Bonded Repairs  22  A M R L researchers also investigated the effective coefficient of thermal expansion for large aircraft components due to the surrounding structure and local heating during the repair process. Measuring the curvature of repaired panels with different restraining conditions showed the reduction of thermal residual stresses with increasing levels of restraint [16]. All of the thermal residual stress investigations carried out at A M R L were part of a thermal fatigue study.  These investigations lead to a number of recommendations by Baker to  minimize thermal residual stresses in a composite repair, and included the following. The heated area should be minimized to maximize the constraint offered by the surrounding structure. The adhesive should be cured at the lowest possible temperature. If feasible, the patch structure should be pre-stressed in compression during the patch application to a level which will partially or completely nullify the residual tensile stress on cooling.  Parallel to the research presented in this thesis, A M R L conducted a study of thermal effects on composite repairs as part of a repair substantiation program for primary structure on the F - l l l which was presented in 1996/1997 [22, 97]. In this study, a number of strain gauges were installed on the top of a boron/epoxy doubler on a representative wing section specimen. Thermal residual strains of-548 fie in the boron/epoxy patch and +531 fie in the aluminum skin were measured at 23°C with a cure temperature of 80° C [22]. Unfortunately no thermal residual strains versus temperature measurements were presented which would have allowed one to assess the reduction in thermal residual stresses by lowering the curing temperature. The key is recognition of the difference between the curing temperature and stress free temperature as well as assessing the validity of a linear-elastic relationship below the stress free temperature leading to a definition of an effective stress free temperature in this research. In addition to the thermal residual strain measurements, fatigue tests of bonded repairs were carried out at A M R L to investigate the performance of bonded repairs at temperatures of 83°C and 105°C. The experimental results with respect to crack growth indicated a lack of sensitivity to both temperature and loading frequency. Further  2.5  Experimental Testing of Bonded Repairs  23  experiments are being conducted to fully understand the effects of test temperature and loading frequency [22].  The Delft University of Technology, in a joint program with the US Air Force investigated G L A R E as a patch material due to its fatigue resistance and especially low coefficient of thermal expansion. The coefficients of thermal expansion in the longitudinal direction for G L A R E - 2 and G L A R E - 3 are 16.3 u-e/°C and 19.8 pe/°C  respectively, which are quite  close to the C T E of A l 2024-T3 with 23.45 /ze/°C. Considering the general restraint of an aircraft structure, a patch with a high C T E (such as G L A R E ) offers the benefit of crack closing forces rather than crack opening forces as caused by low C T E patches (such as boron/epoxy). This improvement, unfortunately, is traded off against significantly higher skin stresses (in comparison to a boron/epoxy patch repair) leading to possible skin cracking at the patch edge [35, 40, 95].  As part of this joint program, researchers at the Delft  University of Technology presented the results of a thermal residual stress investigation around different bonded fiber metal laminate repair patches on a Fokker F28 fuselage in 1997 [95]. The temperature and thermal residual strain distribution in both longitudinal and circumferential directions were measured at four to six locations in the repaired section and at approximately 20 locations outside the repair. The aim was to verify the temperature distribution during the cure as predicted by Rose's approach as well as to determine the thermal residual stresses introduced into the airframe and patch. While the temperature distribution was within 20% of the theoretical prediction, the thermal residual strains were only predicted to the correct order of magnitude. Bending was an important influencing factor due to the flexibility of the fuselage, making isolation of the thermal residual stress component a problem [95].  Unfortunately, no strain measurements versus temperature  were published, thus no information was available to quantify the adhesive behaviour. It was commented that no increase in thermal residual strains was found for the 120° C  2.5  Experimental Testing of Bonded Repairs  24  post-cure over the 90°C pre-cure indicating that 3M's AF-163-2K adhesive might have a similar temperature independent behaviour as the F M 73 M used in this research work. Determination of the adhesive characteristics under thermal residual stress loading is an important key to proper prediction of thermal residual stresses. A simpler specimen such as a double-doubler joint specimen or the A M R L sandwich type specimen (which avoids the out-of-plane bending) is certainly more appropriate to determine this behaviour. Although the complexity of the bending in a flexible structure can be implemented using a finite element analysis for prediction of the performance of a bonded repair, knowledge of the proper material characteristics still remains the key to obtaining accurate results. While in general the focus for the analysis of bonded repairs is generally the reduction in stress intensity, other failure modes such as substrate yielding, adhesive shear, adhesive peel stresses, patch matrix peel, patch matrix shear and patch fiber in tension or compression should be carefully investigated [40, 101]. It should also be noted that the investigation should not be limited to just the patched section as the influence in the global load distribution due to patching may lead to failures in surrounding components. An important contribution to the experimental identification of failure modes was carried out by the National Research Council of Canada. They performed a fractographic investigation on a boron/epoxy repair for the CF116 which experienced compression induced fatigue cracking in the upper wing skin [37, 72]. The noted failure locations and modes were used to establish the locations for the strain gauges in this research effort. N R C also developed the total residual strain measurement technique which is employed in this research collaboration between the University of British Columbia and the National Research Council of Canada. This research led to an improved technique to measure thermal residual stresses versus temperature, which gives a better understanding of the overall adhesive behaviour.  2.6  Standards and Certification Issues for Bonded Repairs  25  Beyond the theoretical analysis and experimental verifications are certification issues and the need for standards for bonded repairs, some of which were discussed at a workshop of practical adhesive bonding for performance and durability in Brisbane in 1997.  2.6  Standards and Certification Issues for Bonded Repairs  One of the main problems with existing regulations is that they are inadequate to ensure that adhesive bonding is carried out correctly. This problem has lead to a history of failures, which has done damage to the credibility of the technology, increased maintenance costs and the loss of life. Regulations such as FA A FARs and JARs require that structures must demonstrate static strength and in some cases fatigue strength. None of these standards address the main cause of bond degradation i.e. hydration of the interface [31]. Poorly prepared surfaces can demonstrate short term strength and may even survive fatigue testing, but as the interface hydrates, disbonds gradually develop. Currently available NDI techniques allow bonds to be tested for disbonds and porosity, but they are not capable of inspecting for durability. Therefore existing regulations don't prevent inappropriate bonding practices from being used.  As a result of the history of adhesive bonding, a U S A F fail-safe criterion was proposed for the safety of flight components requiring design limit load to be carried under the assumption of a 100% disbond [60]. This type of criterion would make the A M R L F - l l l repair to primary structure with a remaining strength below design limit load impossible, thus limiting the used of bonded composite repair technology.  These deficiencies have lead to standards developed by the R A A F [82] and Wright Laboratories (Wright-Patterson Air Force Base) to manage adhesive bonding properly [31]. Managing adhesive bonding correctly includes the development of validated designs and processes, assuring the integrity of materials prior to processing, assuring the processes are  2.6  Standards and Certification Issues for Bonded Repairs  26  followed by trained personnel and non-destructively checking for disbonds in a properly controlled environment. It is not only important to train technicians but also engineers as well as ensuring engineering support for more complex problems [29, 31, 60]. It is essential to establish a better track record as demonstrated by organizations such as A M R L , the Wright Laboratories and NRC to expand the applications that benefit from the application of adhesive bonding.  Baker proposed a generic certification approach which requires standards to assure a low risk of failure due to inadequate bond durability [11]. The first step is the development of procedures to assure low risk failure due to bond environmental degradation. Significant research addressing the bond durability issue has been carried out at A M R L and has lead to improved surface treatments especially for field conditions [6]. Baker proposes a damage tolerant zone in the central region of the bonded repair where controlled crack growth and disbonding is allowed, and also defines a safe life zone in the tapered region to ensure that peel and shear strains are maintained below a damage threshold. The next step is to establish the mechanical property design allowables relevant to generic repair scenarios based on tests of both coupons and structural elements (generic bonded joints). Then the patching design procedures for generic scenarios based on design allowables are validated using generic structural details tests. The difference between the generic and actual repair has to be established by analysis and tests need to be conducted to interrogate only the expected differences. Limited tests on a specimen with representative structural details must be carried out for the final design of the repair based on the generic design process including knockdowns for variations of the actual repair. Furthermore the repaired structure or the representative component should be investigated for unexpected secondary stresses by analysis or strain survey [11].  Some relaxation of the certification requirements for the critical repair of very high cost components may be acceptable from the airworthiness perspective and may also be cost  2.7  Summary and Preview  27  efficient for the operator by using a 'Smart Patch' approach, where sensors monitor inservice damage [11].  2.7  Summary and Preview  This literature review gave a concise overview of the current important issues of adhesively bonded composite repairs. Lacking knowledge of the true stress state after patching is certainly a key issue due to limited knowledge of thermal residual stresses after an elevated temperature cure. Knowledge of the true stress state is important to develop and improve fatigue damage initiation and propagation models for bonded repairs. In order to create and use a generic data base, thermal residual stresses should be accounted for when assessing the performance of bonded repairs. Furthermore only little information is available about the adhesive behaviour at elevated temperatures. The question of the true benefit of lowering the curing temperature needs also to be addressed. In order to make adhesive technology more accessible, it is necessary to have standard analysis schemes which don't require specialized numerical codes. This also simplifies the certification procedure since it allows certification authorities to verify the repair design. The analysis of disbonds in the patch is another key issue in order to establish Baker's generic certification approach to bonded repairs. Furthermore a simple disbond model is necessary to assess partially disbonded patches in service. Further education and implementations of standards are needed to improve the current practice of adhesively bonded repairs thus increasing the number of successful applications and repairs. Making detailed process specifications and standards available is a key issue to limit inadequate repair installations. Additional research has to be conducted to analyse occurring failure modes and to subsequently improve design practices such as the use of a normal stacking sequence versus a reversed stacking sequence for the patch.  2.7  Summary and Preview  28  Thus in recognition of these problems, the following was performed in the current work. This research contributes analytical tools as well as fundamental experimental data to the proposed generic approach for bonded composite repairs. In order to contribute to the existing pool of data, the A M R L sandwich type specimen was chosen for the current research. The manufacturing process will be presented in detail (Chapter 3) to provide a precise record for the interpretation of future fatigue testing as well as presenting guidelines for the precise installation of embedded strain gauges in bonded repairs. Detailed thermal residual strain measurements and an estimation of the process induced strains will be presented in (Chapter 4) thus significantly improving the current understanding of the adhesive behaviour under thermal residual stress loading.  The presented measurement  approach in combination with the identified adhesive characteristics will allow one to predict thermal residual stresses in bonded composite repair specimens more accurately.  In addition to the experimental measurements of thermal residual strains, an improved Rose model will be presented (Chapter 5) which accounts for a symmetrical disbond around the crack.  This addresses the need for a simple tool to determine an appropriate size of a  damage tolerant zone. A new solution for the linear-elastic stress distribution in the taper region of a bonded repair is also provided (Chapter 5) using a closed form solution instead of the typical numerical solution, thus reducing the requirements for specialized codes. A stress field estimation ahead of the crack tip will be presented in Chapter 5 as a tool for the placement of strain gauges for a smart patch approach as well as a guide for placing strain sensors for experimental measurements. The stress field solution is based on a new concise solution of the classical fracture mechanics problem for the stress field around a center crack. Finally new considerations for the use of reversed stacking sequences versus normal stacking sequences for bonded repairs under given remote loading conditions will be discussed (Chapter 6) based on detailed finite element calculations.  3  A M R L Specimen Manufacturing Procedure  29  Chapter 3 A M R L Specimen Manufacturing Procedure  3.1  Introduction  The sandwich type A M R L specimen (see Figure 3.1) was chosen for the experimental determination of the stress state i n bonded repairs due to the incorporation of features characteristic of composite repairs to cracked aluminum aircraft structures. Another criteria was its frequent use i n the evaluation of the behaviour of bonded repairs w i t h respect to influencing factors such as surface preparation, curing conditions, material choice and loading conditions. A l t h o u g h many details and results of these test are not available through the literature, it was decided to use this specimen to provide other researchers who are also using this particular specimen w i t h valuable data. The thermal residual stress investigation i n bonded repairs on cracked metal structures represents an important part for a fatigue damage initiation study for this repair technology. It was decided not only to manufacture an instrumented specimen allowing one to measure the stress distribution in certain key locations, but also to manufacture an additional six A M R L specimens for future fatigue crack reinitation testing. T h e manufacturing of these additional specimens provided excellent training for the assembly of the much more sophisticated instrumented A M R L specimen. It is also important to realize that any experimental testing beyond the thermal residual stress measurements needed to be considered at this  3.1  Introduction  30  stage. Important choices included considerations such as the precrack loading conditions representing an influencing factor for crack reinitation, the change in the stress distribution with increasing crack length, strain variations through the thickness of the repair and typical failure locations.  The choice of non-destructive stress or strain measuring devices was based on the availability, expense, reliability as well as applicability to composites.  Strain gauges seemed  to be a good compromise for this particular specimen offering availability at reasonable prices, excellent repeatability if installed correctly, as well as good averaging capabilities for composites. One of the drawbacks is the difficulty in measuring strain gradients in the load transfer zones as well as close to the crack tip, which is of special interest in the metal substrate. A number of smaller gauges installed in these regions raises significant concern about changing the strain distribution due to the large number of lead wires. Additionally, strain gauges have a given thickness which will also lead to some change in the local strain distribution, especially if a number of gauges are installed at a particular location at different interfaces of the bonded repair. Larger gauges are required to measure the average strains in composites rather the local strain around a single fiber.  The National Research Council of Canada provided important support especially with respect to proper surface preparation. All specimens were assembled under the supervision of Don Raizenne at the Institute of Aerospace Research. This chapter describes the sandwich type A M R L specimen assembly process in detail thus giving a good foundation for the reproducibility of any experimental results as well as a good resource for any researcher wishing to manufacture this particular specimen. The outlined manufacturing procedure includes constant load amplitude pre-cracking, aluminum surface preparation, boron patch fabrication, assembly and specific aspects of the instrumented A M R L specimen construction.  3.2  3.2  A M R L Specimen Specifications  31  AMRL Specimen Specifications  The A M R L specimen consists of two edge-cracked 3.175 mm (0.125 in) thick, 2024-T3 aluminum face-sheets, which are bonded to an aluminum honeycomb core. 2024-T3 aluminum being a high strength, solution treated, cold worked and naturally aged aluminum alloy with 4.5% copper and 1.5% magnesium was chosen due to its widespread use in the aircraft industry. The edge cracks are repaired with semicircular boron patches. Figure 3.1 shows the A M R L sandwich type specimen.  This particular specimen, which was devel-  oped by the Aeronautical and Maritime Research Laboratories in Australia, reflects the attempt to include the influence of the supporting airframe structure in a test specimen for bonded repairs. The stiffening influence of the frame on the skin was implemented into the specimen design by bonding honeycomb between two cracked aluminum sheets. This honeycomb sandwich specimen minimizes the face-sheet curvature, which occurs as a result of thermal residual stresses induced during the patching process. In addition, by using a symmetric configuration, this specimen reduces the out-of-plane deflection, which occurs on single-sided repairs due to the shift of the neutral axis under applied loading. This configuration has been proven to be representative of fighter aircraft having very stiff structures.  In contrast, the stiffness of large commercial aircraft structures is generally  lower, which leads to a different behaviour of the repair. It should be noted, however, that most cracks will emerge from rivet holes or other stress concentration points, where the local stiffness is higher due to the local supporting frame.  For the manufacture of seven A M R L specimens, a total of fourteen face-sheets were machined to the desired shape.  All face-sheets were drilled and milled simultaneously to  ensure repeatability of the specimen geometry. Starter notches were machined to a depth of approximately 3.0 mm (0.118 in) using an E D M machine.  A copper shim having a  thickness of 0.25 mm (0.010 in) was used as the electrode. Due to erosion of the electrode, the final depth of the notches varied between 2.59 mm (0.102 in) and 2.94 mm (0.116 in). Figure A . l shows the drawing for the face-sheets complete with tolerances. The aluminum  3.3  Constant Load Amplitude Pre-cracking Procedure 178 mm  32 25.27 mm  Honeycomb Core 160 mm Rear Sheet EDM Notch Plus Precrack (a = 20 mm)  150 mm  460 mm 280 mm  Autoclave cocured 7 Layer BFRE Composite Patch with 1 Layer FM73, secondary bonded with 1 Layer FM73 Al Alloy 2024-T3 3.2 mm thick  Front Sheet  4>  Aluminum Spacer Block (12.70 mm thick)  <J>  ++  +  Aluminum Shims  Figure 3.1: A M R L specimen  spacer blocks (Figure A.2) were machined from A L 6061-T651.  Aluminum shims were  used to achieve a tight fit in the friction grips for pre-cracking. Tapered aluminum shims (Figure A.4) were bonded to the specimen for the fatigue testing to avoid fretting failure. Once prepared, the specimen face-sheets were precracked as outlined in the next section.  3.3  Constant Load Amplitude Pre-cracking Procedure  The ultimate goal of patching technology is to restore the static strength and the fatigue strength of a cracked structure. If the full fatigue strength of the test panel is restored, crack reinitiation should not occur at the maximum allowable fatigue load for the uncracked  3.3  Constant Load Amplitude Pre-cracking Procedure  33  aluminum sheet. A value of 138 MPa (20 ksi), which corresponds to the 'fatigue endurance limit' for A L 2024-T3 based on 5xl0  8  cycles using a rotating beam test [88], was chosen  as the characteristic maximum remote stress for the fatigue crack reinitiation testing. The literature review showed peak test loads for the A M R L sandwich type specimen range between 80 MPa (11.6 ksi) and 244 MPa (35.4 ksi) with the most common test condition of 138 MPa [12, 18], which does not include the thermal residual stresses. The experimental results of other researchers [13] indicated that at a load level of 138 MPa (20 ksi) for the A M R L sandwich type specimen, a reinitiation life of less than 100000 cycles can be found using a boron patch and F M 73M as an adhesive. Therefore this load level represents the upper limit for useful crack reinitiation testing. Any fatigue crack reinitiation testing will be dependent on the plastic zone size prior to patching. In order to minimize retardation effects, the largest allowable plastic zone size for the pre-cracking was determined by the plastic zone size due to thermal residual stresses after patching.  The literature review  suggested that the magnitude of the thermal residual stresses is dependent on the difference between the curing temperature and the ambient temperature, i.e. A T = 100°C  (212°F)  for a curing temperature of 121°C (250°F) [18]. The corresponding estimate for the stress intensity factor  due to thermal residual stresses is 7.44 M P a i / m (6.77 ksi\/in)- This  value was used as a limit for the stress intensity factor for pre-cracking. Any additional fatigue loading applied on top of the thermal residual stress for the patched A M R L sandwich type specimen gives a sufficient margin to minimize any retardation effects.  The A S T M E 399 'Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials' [7] was also referred to as a guideline for pre-cracking. The A S T M E 399 standard allows a maximum stress intensity factor of up to 80% of K  I c  for the initial portion of the  pre-cracking to avoid excessive crack growth. The stress intensity factor can be as high as 60% of Kic for the last 2.5% of the desired pre-crack length. Unfortunately, this standard does not address fatigue crack testing near the threshold stress intensity factor range. Precracking has to be carried out using a stress intensity factor range above the threshold stress intensity factor range since fatigue crack growth does not occur below this value.  3.3  Constant Load Amplitude Pre-cracking Procedure  34  The stress intensity factor caused by the induced thermal residual stresses is therefore a reasonable upper limit for pre-cracking of the face sheets for the A M R L sandwich type specimen. The face-sheets were fatigue pre-cracked from the machined 3 mm (0.118 in) notch to approximately 20 mm (0.787 in).  A load shedding technique, similar to the procedure  used by N R C , was employed to ensure that the plastic zone at the final crack length was realistically small [70, 71] and that the estimate for the stress intensity factor due to thermal residual stresses was not exceeded near the final crack length. Thus the mode I crack tip stress intensity factor range, A K T , was reduced throughout the pre-cracking sequence from a maximum of 9.4 to 6.5 MPav^m (8.6 to 5.9 ksi\/m) at the final crack length. The final peak stress intensity factor was 7.2 MPa-v/rEThe A S T M E 399 defines that the stress ratio should be between -1 and 0.1.  The pre-  cracking of one or, simultaneously two, face-sheets without the supporting honeycomb is not well suited to compressive loading. Buckling would be the limiting case or an elaborate fixture would be required to prevent it. Thus a R-ratio of +0.1, i.e. tension at all times, was employed for this application in order to avoid early fatigue failure at the grips. The available testing equipment allowed a pre-cracking frequency of 20 Hz which fell within the A S T M E 399 allowable pre-cracking frequency of up to 100 Hz in the absence of adverse environments [7]. Harris [77] obtained the Mode I stress intensity factor for a sheet with an edge crack and bending constraints to be: (3.1) where  5  (3.2)  Thus the stress intensity factor range is given by: (3.3)  3.3  Constant Load Amplitude Pre-cracking Procedure  35  It should be recognized that the Harris equation, which is based on an approximate solution by Neuber, may introduce an error up to 8% [77]. Although thin face-sheets are used for this specimen, plane strain conditions can be achieved by restricting the maximum applied load. A maximum load of 37500 N (8430 lb) for each face-sheet was chosen in order to maintain plane strain conditions throughout the precracking sequence. A further increase in the applied load would then lead to the change to plane stress and an increase in the plastic zone size.  Plane strain conditions can be  assumed when: K ^Face-sheet < 2.5  For 2024-T3 aluminum, the tensile yield stress a  ys  highest stress intensity factor for mode I, K  x  2  I max I  is given as 310 MPa (45 ksi) [21, 91]. The  , of 10.45 MPa-^/m (9.51  I m a x  (3.4)  ksi\An)  is reached  during the pre-cracking sequence at a crack length of 5 mm (0.197 in) and is significantly less than the critical plane strain fracture toughness for mode I of approximately 44 MPa-v/m (40 ksiVm) [47]. The remote stress range is given by: P 1  A  A a  —P  max  =  mm  /  0  \  r  -  wt  (3  5)  Irwin gives an estimation of the plastic zone including a plastic constraint factor for plane strain, which also takes into account that plane stress conditions are present at the specimen surface [23]. The plastic zone size is given as:  1 K  2  r* = TT^f P  (3-6)  Generally, the crack length should be modified in the case of large stress intensity factors to account for the plastic zone size [23]:  K = co AJTT (a + r*)  (3.7)  3.3  Constant Load Amplitude Pre-cracking Procedure  36  Since the plastic zone size for this particular pfe-cracking sequence is only approximately 1% of the crack length, the plastic zone size can be neglected for the determination of an effective crack length.  Table 3.1 shows the block loading sequence used for pre-cracking. Slight variations occurred for each specimen due to the difficulty in controlling the pre-cracking as a function of crack length.  Pre-crack  a  Sequence 1 2 3 4 5 6 7  a  c  p  p .  AKi  [mm]  w [ ]  [ ]  [N]  [N]  [MPaVm]  [mm]  3.0  0.019  1.125  37500  3750  7.256  0.036  5.0  0.031  1.130  37500  3750  9.407  0.060  5.0  0.031  1.130  30000  3000  7.526  0.039  7.5  0.047  1.136  30000  3000  9.267  0.059  7.5  0.047  1.136  25000  2500  7.723  0.041  10.0  0.063  1.142  25000  2500  8.967  0.055  10.0  0.063  1.142  20000  2000  7.174  12.5  0.078  1.149  20000  2000  8.067  0.035 0.044  12.5  0.078  1.149  17500  1750  7.058  0.034  15.0  0.094  1.156  17500  1750  7.778  0.041  15.0  0.094  1.156  15000  1500  6.667  0.030  17.5  0.109  1.163  15000  1500  7.245  0.036  17.5  0.109  1.163  12500  1250  6.037  0.025  20.0  0.125  1.170  12500  1250  6.494  0.029  x  max  1  min  Table 3.1: Pre-cracking load sequence  The load calibration for the M T S 810 load frame in conjunction with the M T S 406 controller was carried out using an optimum gain for a frequency range of 0 to 2 Hz. This particular gain setting used at 20 Hz led to an approximate 10 % over- and undershoot in the load response of the system. Due to the influence of the gain setting on the calibration, the command input of the computer was adjusted by changing the set point and span on the M T S 406 controller. A n initial span of 75% ensured that the specimen did not experience a higher load than intended for the pre-cracking sequence. A n oscilloscope was used to  3.3  Constant Load Amplitude Pre-cracking Procedure  Pre-crack Sequence 1 o Q  O  A *±  0  a  O  7  P x  max  [N]  p . 1  min  • [N]  XDCR2  m a x  37  XDCR2  m i n  MDT  m a x  MDT  [V]  [V]  [V]  [V]  m i n  37500  3750  3.049  0.305  3.049  0.305  37500  3750  3.049  0.305  3.049  0.305  30000  3000  2.440  0.244  2.440  0.244  30000  3000  2.440  0.244  2.440  0.244  25000  2500  2.033  0.203  2.033  25000  2500  2.033  0.203  2.033  0.203 0.203  20000  2000  1.626  0.163  1.626  0.163  20000  2000  1.626  0.163  1.626  0.163  17500  1750  1.423  0.142  1.423  0.142  17500  1750  1.423  0.142  1.423  0.142  15000  1500  1.220  0.122  1.220  0.122  15000  1500  1.220  0.122  1.220  0.122  12500  1250  1.016  0.102  1.016  0.102  12500  1250  1.016  0.102  1.016  0.102  Table 3.2: M D T load control input monitor the M T S 406 X D C R 2 signal. The load could be adjusted within an accuracy of 2.5%, which is essentially the accuracy of the oscilloscope. Table 3.2 shows the voltages, which were entered in the M D T computer program and adjusted in the M T S 406 Controller to control the block loading sequence. Due to the design of the M D T computer card for older IBM X T / A T computers, a frequency of 33.4 Hz was entered into the program to achieve a testing frequency of 20 Hz.  3.3.1  Experimental Set-up  A computer-controlled M T S 810 Model 318.25 (Figure 3.2) with a load capacity of 250 kN (55 kips) was used for pre-cracking of the A M R L specimens. The National Research Council of Canada provided a pair of friction grips, which were manufactured by IAR. Adapters were then made at U B C to fit the grips to the M T S 810 Model 318.25.  3.3  Constant L o a d A m p l i t u d e Pre-cracking Procedure  38  Each of the A M R L specimen face-sheets was prep a c k e d individually.  T h i s provided the oppor-  tunity to measure the crack growth rates without the interference of a second face-sheet. T h e installation of a face-sheet i n the friction grips required pre-loading of the face-sheet with the center locating p i n i n place. A tensile load of 2.5 k N (562 lb) was chosen to assure proper alignment without yielding the aluminum around the center locating pin. The required torque for the grip bolts was determined using the following relationship between torque and preload [84]: T = 0.20F  P  ,  Preload  A  (3.8)  Figure 3.2: M T S 810  The preload for the bolts can be calculated using the required load for the specimen and the appropriate friction coefficient for the weakest interface. In this case the critical interface is between the aluminum shims and the steel grips having the m i n i m u m friction coefficient of approximately 0.45 under clean surface conditions [66]. Therefore, based on equation (3.8), the m a x i m u m applied load of 37500 N (8430 lb) for one face-sheet requires a m i n i m u m bolt torque of 51 N m (451 inlb). In order to reduce fretting fatigue on the face-sheets and to account for the uncertainty i n the friction coefficient, a torque of 250 N m (2213 inlb) was applied for the pre-cracking. Two different techniques were employed to measure the crack length during the pre-cracking sequence. The first, a Fractomat Model 1078 was used to automate the pre-crack sequence. The Fractomat uses an 'indirect D C potential drop technique' to monitor crack growth. It utilizes a simple crack growth monitoring transducer known as 'Krak-gauge', which is applied in a similar manner as a conventional foil strain gauge (Figure 3.3). The overall  3.3  Constant Load Amplitude Pre-cracking Procedure  39  accuracy for measuring the crack length is 2% or better. The system has an integrated limit detector, which allows the interruption of the pre-cracking sequence as defined crack lengths are achieved.  The main advantages of the Fractomat system are [41]: • A high linearity between the potential output and crack length • High potential output (Volts) with very low current input (mA) • specimen geometry independent calibration • Independence from conductivity, size and material composition • Infinite resolution, continuous D C output • Suitability to fatigue testing • Suitability to automated testing  In addition to the Fractomat, a travelling Specimen  Krak-Gauge  microscope (Scientific Instruments, England) was used with a 5x Leitz eyepiece (Figure 3.2).  The travelling microscope  could be adjusted to achieve an accuracy of iO.Olmm ( ± 0 . 0 0 0 4 i n ) .  © AV  ©  Prior to mea-  surement, the specimen was loaded to the maximum allowable load of the particular  Figure 3.3: Indirect potential drop method  pre-crack sequence in order to enhance the visibility of the crack tip. While this optical setup is quite acceptable for pre-cracking, it is not suitable for monitoring crack growth in the patched specimen. However, due to the better accuracy of the travelling microscope, it was used as the reference for all crack length measurements.  3.3  Constant Load Amplitude Pre-cracking Procedure  3.3.2  40  Crack Growth Rates  The pre-cracking of the face-sheets offered the opportunity to record crack growth as a function of the number of cycles and thus allowed calculation of the crack growth rate. This information was used to select two face-sheets with similar crack growth rate to be combined into each A M R L specimen. The A S T M E 647 - 93, 'Standard Test Method for Measurement of Fatigue Crack Growth Rates'[8], was applied to the collected crack length versus number of cycles data.  The secant method was applied to determine the crack  growth rates according to:  dN ~  [  h  N -N i+1  [ t  ^  }  where a is the average crack length given by:  a = ^(a i + a,) !+  (3.10)  The average crack length a was used to calculate the stress intensity factor range using equation (3.3). Appendix B lists all of the results from the pre-cracking. The crack growth rate was not determined, if the effect of crack initiation or retardation due to the reduction of the load from a sequence change or an accidental overload was present. Some of the crack propagation rates versus stress intensity factor range diagrams (see Figure 3.4 and Figure B.l) show a change in the slope at small stress intensity factor ranges. This corresponds to the non-linear crack growth rate versus the stress intensity factor range for low stress intensity factors in a double logarithmic plot. It is recalled that the Paris law is only applicable for larger stress intensity factor ranges where the relationship between crack growth rate and stress intensity factor range is linear in a double logarithmic plot. The crack propagation rates versus the stress intensity factor ranges are shown for each face-sheet in Figure B . l through Figure B.14. Due to the overall research focus being on the fatigue crack reinitiation, the average crack propagation rate for the last 2.5 mm of the pre-crack length was considered to be the  3.3  Constant Load Amplitude Pre-cracking Procedure  41  deciding factor for combining two particular face-sheets. This growth range was chosen instead of just the final crack propagation rate in order to reduce possible error due to crack length measurement accuracy. The average crack propagation rate was thus determined for the last 2.5 mm which included the initial retarded crack growth after reducing the applied load. The retardation effect is not important to consider for combining two particular facesheets due to a relative comparison between the crack propagation rate of each face-sheet. Determination of the crack propagation rate had to be modified for face-sheet 1 due to an accidental overload at the beginning of the sequence. In this case, the crack propagation rate was determined after switching back to a maximum applied load of 12500 N (2810 lb) at a crack length of 18.65 mm (0.734 in). This ensured that a similar retardation effect was included in the determination of the average crack growth rate. The higher average stress intensity factor for this particular load block due to the larger initial crack length was neglected in this comparison.  0.200  o.ioo 0.080 0.060 M o  -—_  —•  ••  0.040  u  .1  0.020  •  :  ••••  TD  ~c3 0.010 0.008 0.006 7 3  •  *  0.004 • a < 17.5 mm • a > 17.5 mm  0.002 6.0  7.0  8.0 AK  9.0  [MPafm ] 1  Figure 3.4: Crack propagation rates for all face-sheets  10.0  3.3  Constant Load Amplitude Pre-cracking Procedure AMRL Specimen  Face-sheet  42  Patch  Average Crack Growth Rate (Sequence 7)  [ ]  [ ]  [ ]  Instrumented Specimen  2 7  13 14  14 6  1 2  8 1  3 4  12 3 4  5 6 7 8  1 2 3 4  13 9 10 11 5  5 6  9 10 11 12  LcvcleJ  0.0077 0.0077 0.0084 0.0089 0.0097 0.0100 0.0132 0.0132 0.0163 0.0147 0.0161 0.0162 0.0189 0.0180  Table 3.3: A M R L specimen component designation  Table 3.3 shows that face-sheets with very similar average crack growth rates were paired for the construction of the A M R L specimens.  The only exception is A M R L specimen 4,  where the crack growth rate varies by approximately 10% between the two face-sheets. Although face-sheet 4 had a higher average crack growth rate than face-sheet 9, A M R L specimen 5 was assembled using face-sheet 9 and 10 because they were adjacent in the original aluminum sheet. All pre-cracks showed some variation in cracking direction from the ideal straight extension of the E D M notch. Figure B.15 to Figure B.28 show the different cracks for all of the facesheets. Typically 2 or 3 small cracks would initiate at the E D M notch. After approximately 50000 cycles, one of these cracks would dominate and continue to grow faster. It was noted that crack initiation usually occurred at a slight angle from the E D M notch. This is visible in a number of pre-cracks, e.g. Figure 3.5, where the crack initially deviates significantly from the notch direction.  3.3  Constant Load A m p l i t u d e Pre-cracking Procedure  43  Figure 3.5: Pre-crack deviation (Face-sheet 3)  The effect of an accidental overload during the pre-cracking is also visible. The whitening at a crack length of approximately 12 m m (0.472 in) and 17.6 m m (0.693 in) in face-sheet 1 (Figure 3.6) reveals this clearly. W h i l e the crack tip was surrounded by a small resulting compressive stress field due to the overload, the adjacent material still remains i n a high tensile stress field. The load level was returned to the sequence load after the overload, which led to visible fatigue damage i n the clad layer in the form of whitening in the area which remained in a high tensile stress field. However, due to a dramatically reduced crack growth rate and increasing whitening, the load was temporarily increased to grow the crack out of the whitened area, then returned to the proper level.  Figure 3.6: Effect of pre-cracking overload (Face-sheet 1)  3.4  Aluminum Surface Preparation Procedure  3.4  44  Aluminum Surface Preparation Procedure  Surface preparation of the aluminum prior to patching is an extremely critical element of the composite repair procedure. It controls the durability of the repair, a property which is very difficult to verify. The Boeing wedge test [9] and the lap shear test [10] are typically used to verify proper surface preparation. If the surface was prepared properly, cohesive failure should occur.  Environmental testing is also used to give more insight into the  durability of the bond and thus assesses the quality of the surface preparation. Attention to the details of the surface preparation procedure is therefore critical.  Two different categories of aluminum surface preparation procedures can be used: tank and non-tank procedures. Tank processes such as phosphoric acid anodizing and chromic acid anodizing allow the treatment of complete parts by immersion. For repair applications, where the part is in many cases not removable, non-tank processes must be used. The phosphoric acid anodize non-tank (PANTA or PANTACS), the chromic-sulphuric acid etch (optimized Forest Products Laboratory (FPL) procedure, A S T M D2651), PASA J E L L 105 acid etch and reactive epoxy silanes process are four non-tank procedures which are currently used [18, 65, 72, 90].  The etching processes can cause potential problems such as corrosion between fasteners and the aluminum skin. Special care must be taken to isolate steel components, e.g. rivets, for the anodizing process. Anodizing is not possible if iron ions are present. The silane process was chosen for this investigation due to its simplicity and wide range of applications [72].  3.4.1  Clad Removal  Clad removal from the aluminum is the first step in the repair process. The clad material, forming a soft aluminum layer with a thickness of approximately 0.079 mm [21], provides  3,4  A l u m i n u m Surface Preparation Procedure  45  good corrosion protection for the aluminum skin. In the case of a composite repair it would result in a weak layer adjacent to the bond line. Unfortunately grit blasting is not very effective in removing the clad layer since the grit only tends to indent the clad layer rather than removing.  Wet 220 grit paper was therefore used to remove the clad material on both sides of the face-sheets as necessary, due to the additional bonding of the honeycomb, spacers and shims. A flat aluminum block (Figure 3.7) was used to remove the clad uniformly. A ' K i m t u f f W i p e r ' was placed under the face-sheet to avoid scratches. F i n a l grinding was done using wet 320 grit paper. T h i s ensured that no scratches from the 220 grit paper were visible after the final grit blasting step and reduced the grit blasting time required to achieve a uniform surface appearance. i  Figure 3.7: C l a d removal The face-sheet was then thoroughly cleaned under running water w i t h ' K i m t u f f W i p e r s ' until no trace of contamination was visible. A water break-free surface was achieved at this stage. The face-sheets were air dried over night. T h e face-sheets were then wiped w i t h M E K i n only one direction w i t h frequent  changes  of 'Kimtuff W i p e r s ' to avoid the spreading of any remaining contamination. The previous thorough cleaning under water reduced the required amount of M E K significantly and is therefore preferable from an environmental point of view. T h e face-sheets were handled with gloves and subsequently wrapped in Kraft paper to avoid possible contamination.  3,4  A l u m i n u m Surface Preparation Procedure  3.4.2  46  Grit Blasting  The grit blasting, which should be carried out i m mediately prior to the silane process, exposes the aluminum surface for the silane coupling agent. Attention to details in this step is essential to avoid contamination. The grit blast unit had to be cleaned prior to the application (Figure 3.8). New aluminum oxide 220 grit with a 75 micron d i ameter was used to exclude any effects from possible prior contamination of the grit. Nitrogen was used as the blast system gas to avoid the possibility of oil or water contamination from the regular air supply. The hose, which connects the nitrogen bottle w i t h the grit blasting unit, was  Figure 3.8: G r i t blasting set-up  only used for nitrogen. A pressure of 690 k P a (100 psi) was supplied by the nitrogen bottle for the grit blasting process. G r i t blasting was carried out using parallel passes, first along the longitudinal axis of a face-sheet, and then perpendicular to the axis until a uniform dull finish was achieved. B o t h sides of each face-sheet were grit blasted. After completion of the abrading process, the face-sheet was cleaned using a spray nozzle and nitrogen gas at 690 k P a (100 psi). Under no circumstances should compressed air, rags or solvents to be used to remove the residual abrasive. T h e face-sheets must be handled only with clean gloves before and after the grit blasting process, and wrapped in Kraft paper immediately after grit-blasting.  For representative testing purpose, the traveler coupon for the Boeing wedge test was also cleaned w i t h M E K and grit blasted using the same process.  3.4  A l u m i n u m Surface Preparation Procedure  3.4.3  Preparation Brush  and  47  Application  of  Silane  Coupling  Agent  Method  R-Si(OR') + 3H 0 3  RSi(OH)  OH  •  2  OH  •*  3  OH Aluminum Surface  R  3  OH  OH  HO - Si i R  O-sV  OH  hO-Si-Ri O |-0-Si-RO - Si - R i Aluminum  OH  h O H + HO - Si O - S i  i  RSi(OH) + 3 H 0 R '  OH  I R  Primer/Adhesive  Surface  Figure 3.9: Chemical bonding mechanism of the Silane treatment  The silane (7-Glycidpropyltrimethoxysilane) provides an improved coupling between the aluminum and the adhesive. T h e presumed mechanism by which this coupling agent functions in chemical bonding as shown in Figure 3.9 [69]. Skin contact should be avoided with the moisture sensitive silane due to possible severe irritation and mutation. The refrigerated silane was allowed to stand at room temperature for one hour prior to opening the container.  A 1% silane solution was re-  quired for the surface preparation procedure, thus  p  i  g  u  r  e  3  1 Q  . g  U  a  n  e  99 parts of deionized distilled water were mixed with one part silane by volume for the surface preparation procedure.  e  q  u  i  p  m  e  n  t  3.4  A l u m i n u m Surface Preparation Procedure  48  T h e mixing container was carefully cleaned and rinsed w i t h distilled water before use. The container used was polyethylene since distilled deionized water can react w i t h glass. In total, 250 m l of 1% silane solution was required for the preparation of the two sides of each face-sheet, i.e. 247.5 m l distilled deionized water and 2.5 m l silane. T h e silane was poured into a small polyethylene cup to avoid possible contamination of the silane and was measured w i t h a pipette (Figure 3.10), while a graduated cylinder was used to measure the volume of the distilled ionized water. T h e silane solution had to hydrolyse for one hour prior to use, thus the solution was agitated using a mixing agitator w i t h a mixing magnet in the sealed container.  Due to this time delay, preparation of the solution was carried  out prior to the grit blasting of the face-sheets to minimize the risk of contamination and oxidation of the a l u m i n u m surface [72].  The face-sheets and traveler coupons were placed at a steep angle to allow the silane to run down the aluminum surface. A honeycomb block placed i n a shallow glass or metal bowl allowed the entire surface to be accessed without one end sitting in the silane solution. A dispenser head for the mix-  F i  g  u r e  3.11: Silane coupling agent - brush method  ing container was used to spray the silane on the face-sheet  (Figure 3.11).  The silane  solution was then brushed downwards w i t h a clean, fine haired 25mm (1 in) wide paint brush. It is essential to brush in only one direction to drive any remaining contaminates from the grit blasting into the silane solution collecting bowl.  3.4  Aluminum Surface Preparation Procedure  49  A water-break free surface has to be visible during the silane treatment. Typically the silane treatment is continuously applied for ten minutes. In the case of the A M R L specimen, up to 4 face-sheets or 2 face-sheets and two sheets for a Boeing wedge panel were treated simultaneously.  Due to the increased number of parts, the silane treatment time was  increased to 15 minutes per side. After treating both sides, the face-sheets were air dried for approximately 20 to 30 minutes. The long air drying time was a result of the solution build up at the holes and the lower edge of the specimen. The face-sheets were heated to 8 0 ° C (175°F) for 30 minutes next. A small temperature controlled oven was used to heat dry up to 4 specimens (Figure 3.12). It is important that the surface is not touched to avoid any contamination. The primer must be applied within 1 hour after finishing the silane  Figure 3.12: Heat drying of the silane  treatment in order to ensure a lasting surface preparation.  3.4.4  P r e p a r a t i o n a n d A p p l i c a t i o n of B R 1 2 7 C o r r o s i o n I n h i b i t ing P r i m e r - S p r a y M e t h o d  The BR127 primer inhibits corrosion and offers superior durability and resistance to hostile environments within the bond line. It can also be used as a protective coating outside the repair area. In addition, the primer preserves the surface preparation for up to 6 months depending on the storage condition of the specimen. The refrigerated BR127 primer was  3.4  A l u m i n u m Surface Preparation Procedure  allowed to warm to room temperature prior to opening the container.  50 The primer was  then thoroughly mixed before use to achieve a yellow coloured homogenous mixture. The primer must also be agitated during the application to prevent the chromate particles from settling out of the solution.  Insufficient m i x i n g or lack of agitation results in a shiny,  nearly clear coating, while the properly agitated primer gives a matt light yellow finish after spraying [72].  The B R 1 2 7 primer should be sprayed to a dry primer thickness of 0.0025 m m (0.0001 in) w i t h a m a x i m u m thickness of 0.0075 m m (0.0003 in). The proper thickness of the primer was identified using the B R 1 2 7 primer colour chip w i t h the correct shade of yellow for optimal thickness during the spraying.  Due to space restrictions, only two face-sheets were primed at one time.  Figure 3.13 shows  the setup for the priming. The face-sheets were supported on clean aluminum bars on the outer three millimeters of the face-sheet. A rag was wrapped around the junction between Figure 3.13: Setup for B R 1 2 7 priming the cup and spray gun to avoid any accidental drops due to a possible leak in the seal. The primer was sprayed using 345 k P a (50 psi) nitrogen gas to avoid any possible contamination. The o p t i m a l thickness was achieved in a single pass w i t h the spray gun. Coverage of the whole face-sheet required three passes i n an ' S ' pattern w i t h a small overlap between the passes.  3.4 • Aluminum Surface Preparation Procedure  51  After priming the first side, the face-sheets were air dried for approximately 2-3 minutes, then turned over and primed on the second side. These two face-sheets were then placed on an additional set of aluminum bars, while two additional face-sheets or the two aluminum sheets for Boeing wedge test were primed in the same manner. After spraying, the facesheets were air dried for 30 minutes. The primer was then fully cured in a temperature controlled oven at 121°C (250°F) for 60 minutes, removed, cooled and wrapped in Kraft paper until assembly.  Table 3.4 outlines the full time sequence for the described specimen surface preparation.  Time  Task  1:00 pm  Setup for clad removal  1:30 pm 2:00 pm  Grinding face-sheet 1 with wet 220 and 320 grit paper Grinding face-sheet 2 with wet 220 and 320 grit paper  2:30 pm  Grinding face-sheet 3 with wet 220 and 320 grit paper  3:00 pm 3:30 pm  Grinding face-sheet 4 with wet 220 and 320 grit paper Cleaning all face-sheets under running water  Over night 7:00 am 7:05 am  A i r drying of face-sheets Taking BR127 primer out of freezer Preparing and cleaning work area  7:30 am  Preparing of the silane solution  7:45 am 8:30 am  Cleaning face-sheets with M E K Grit blasting face-sheets  9:10 am  Face-sheet cleaning with nitrogen blow  9:20 am  Starting silane brush procedure Silane air drying  9:50 am 10:20 am  Oven drying silane at 80° C  10:50 am  Face-sheet cool down  11:00 am  Priming face-sheets with BR127  11:10 am  A i r drying of BR127 primer  11:40 am  Curing of BR127 primer at 121° C  12:40 am  Cooling down face-sheets  13:00 am  Wrapping face-sheets in Kraft paper  Table 3.4: Schedule for surface preparation procedure  3.4  A l u m i n u m Surface Preparation Procedure  3.4.5  52  B o e i n g W e d g e Test  It was essential to verify the quality of the surface preparation.  Unfortunately the long  term performance of the surface preparation can only be evaluated to a limited degree using the available methods. The  Boeing wedge test gives  some indication of poor surface preparation. T h e details  Figure 3.14: Boeing wedge test specimens  for the Boeing wedge test are specified in A S T M D 1002 [9].  The  surface prep of the two  75.24 m m x 75.24 m m (6 i n x 6 in) A l 7075-T6 3.175 m m (0.125 in) thick plates for the Boeing wedge specimens was carried out exactly as on the A M R L face-sheets.  T w o lay-  Figure 3.15: Temperature and humidity chamber  ers of F M 7 3 M w i t h a density of 300 g / m (0.06 psf) were used as adhesive. In addition, 3 2  layers of 0.08 m m (0.003 in) thick teflon foil were used i n the top part of the panel as the starter notch. The edges of the panel were sealed w i t h high temperature tape to reduce adhesive bleed and to achieve a uniform bond thickness.  3.5  Boron Patch Fabrication Procedure  53  The panel was cured at 80°C (176°F) for 8 hours. Then the tape and the teflon foil were removed. The panel was cut with a bandsaw and milled to five 25.4 mm (1.0 in) wide specimens (Figure 3.14). The edges were discarded.  Next, the steel wedges were inserted into the notches causing crack initiation in the adhesive. The initial crack length was then marked under the microscope and measured again after 1 hour at room temperature.  No increase in crack length was noticed.  There was no  evidence of interfacial failure between the adhesive and the aluminum. To check the bond and surface preparation under severe environmental conditions, the specimens were placed in a temperature and humidity chamber (Figure 3.15) at 65° C (149°F) and 96% humidity for 1 hour. No crack growth was observed in this case. The specimens were then placed in the temperature and humidity chamber under the same conditions for additional 8 hours. No further crack growth was observed. One specimen was split open to check the failure mode.  Complete cohesive failure in the adhesive was found.  Based on this evidence it  was concluded that the surface preparation used on the A M R L specimen face-sheets was carried out properly.  3.5  Boron Patch Fabrication Procedure  The patch for the A M R L specimens consists of 7 layers of 5521/4 boron prepreg and one cocured layer of F M 73M. The cocured layer of F M 73M with a density of 300 g / m  2  (0.06  psf) penetrates up to two layers at the tapered end and therefore improves the properties of the 5521/4 boron/epoxy prepreg by replacing the boron prepreg epoxy system with a modern toughened epoxy.  3.5  Boron Patch Fabrication Procedure  3.5.1  C u t t i n g of B o r o n 5521/4 P r e p r e g and F M 7 3 M  54 Adhesive  Approximately 80.0 cm (31.5 in) of 15.2 cm (6.0 in) wide 5521/4 boron prepreg tape and 377.4 cm  2  (58.5 in ) of F M 73M film adhesive were required for each A M R L specimen, 2  i.e. 2 patches. The boron prepreg and the F M 73M had to unthaw in the unopened bag from the frozen storage to room temperature for 1 hour. If the bags are opened too early, moisture in the air will condense on the boron prepreg and the F M 73M film adhesive due to their surface temperatures being below the dew point. Even a very small amount of absorbed moisture will result in an increase in porosity leading to a loss in strength of the patch. As a further precaution, the boron prepreg and the F M 73M film adhesive were handled with gloves to avoid contamination [72].  The boron prepreg was unrolled with the white backing facing down (Figure 3.16). A perforated release film was placed on top of the tacky boron prepreg to allow easier template placing and removal. The nylon scrim cloth, which holds the boron fibers in place, faces the white backing paper.  All plies were cut with the scrim cloth facing down.  Seven  2 mm (0.079 in) thick aluminum templates with radii ranging from 57 mm (2.24 in) to 75 mm (6.73 in) in increments of 3 mm (0.12 in) were used for the precise cutting of the boron prepreg. The templates were cut 2 mm (0.079 in) wider along the straight edge due to the formation of a rounded edge of the patch during the curing process.  Cutting this  approximately 2 mm (0.079 in) wide strip off after curing ensured the proper thickness along the straight edge of the patch. One template at a time was clamped on the boron prepreg with a C-clamp to avoid any movement while cutting (Figure 3.16). It was important not to use excessive force during the clamping process to avoid damage to the prepreg. The straight edge of the template was lined up with the 0 ° direction of the prepreg. A roller type cutter (Figure 3.16) was used to cut along the perimeter of the template. The roller type cutter offered advantages over a regular pair of scissors or a razor blade due to the ability to shear the material with only limited pushing in cutting direction. This reduced the loss  3.5  Boron Patch Fabrication Procedure  55  of material due to distortion of the prepreg and thus allowed closer template spacing. The F M 73M ply, which is cocured with the boron prepreg plies, was cut using the template with a radius of 75 mm (2.95 in) in the same manner. The boron/epoxy prepreg and F M 73M film adhesive plies for each patch were then sealed in a polyethylene bag and stored below -18°C (0°F) until one hour before use.  The F M 73M film adhesive ply used to bond the precured patch on  to the cracked aluminum sheet was then cut using an additional template with a radius of 77.5 mm (3.05 in) and stored in a separate bag in the same manner.  3.5.2  L a y - u p of B o r o n 5521/4 P r e p r e g and F M 7 3 M  Adhesive  A reversed stacking sequence was chosen for the lay-up procedure of the patch to reduce peel stresses at the tapered end. In addition the reversed stacking sequence allowed easier placement of the plies during the lay-up. As a first step, a transparent template with a scale of 1:1 was taped on to a flat surface. If a laser printer is used for the template manufacturing, it is important to ensure that the dimensions are printed properly and not distorted or scaled. Time consuming program calibration may be necessary to achieve a good template to meet the specifications. One layer of perforated release film was placed over the template and secured (Figure 3.17). Then the perforated release film, which was placed during the cutting process, was removed from  3.5  Boron Patch Fabrication Procedure  56  the smallest boron prepreg ply. Using the template as a guide, the ply was placed carefully on the template with the white backing paper facing up. A teflon roller (Figure 3.17) was used to roll the ply down. This made the removal of the white backing significantly easier as well as it consolidated the lay-up. Otherwise the scrim cloth had the tendency to stick to the backing and got significantly distorted during the backing removal. A knife was carefully used to lift one corner of the backing. The remaining six boron prepreg plies were placed in the same manner. After placing the last boron ply, two layers of peel ply were cut to the dimensions of the largest boron prepreg ply using the same technique as for cutting the boron/epoxy prepreg. One peel ply was then carefully placed over the last boron prepreg ply. Perforated release film, which covered the Figure 3.17: Lay-up of the boron prepreg template on the layup table, was then cut around the patch. Next, the patch was turned upside down and the remaining perforated release film was removed. A single layer of F M 73M was then placed on the bottom side of the patch. Finally the second prepared peel ply was placed over the adhesive.  3.5.3  V a c u u m Bagging  of B o r o n  5521/4  Prepreg  and  F M  7 3 M  Adhesive  The seven ply boron patch and one layer of F M 73M were cocured prior to bonding to the aluminum face-sheet in order to reduce the complexity of the assembly procedure. This  3.5  Boron Patch Fabrication Procedure  57  cocuring step also allowed the trimming of the edges of the patch, especially the straight edge, which ensured that the proper patch thickness was achieved along the face-sheet edge. Figure 3.18 illustrates the vacuum bagging procedure prior to the autoclave curing.  Sealant Tape  Figure 3.18: Vacuum bagging procedure  The vacuum bagging procedure was carried out using a 6.35 mm (0.25 in) thick aluminum plate. One layer of non-perforated release film was placed on the aluminum to avoid any bonding to the aluminum as well as to facilitate easy removal of the patch after curing. The patch, consisting of one layer of peel ply on each side of the patch, one layer of F M 73M and seven layers of boron/epoxy prepreg, was then placed on the non-perforated release film. Two short fiberglass roving strands were positioned at the corners of the patch to allow for an additional breather path during the curing of the patch. A perforated layer of release film was placed on top of the patch to allow excessive resin and adhesive to flow away from the patch. The three consecutively stacked layers of fiberglass absorbed any epoxy flowing through the perforated release film. Two layers of thin breather cloth, which were extended beyond the patch for the placement of the vacuum fitting, were placed on top of the bleeder cloth to ensure full vacuum pressure over the entire patch as well as to remove any volatiles. A hole was then cut in the vacuum bag to mount the vacuum fitting. Next, the vacuum bag was positioned over the entire area and sealed with sealant tape around its edge.  3.5  Boron Patch Fabrication Procedure  58  It was important that none of the layers surrounding the patch had a wrinkle in the patch area as these can cause wrinkles in the cured patch leading to a reduction in strength. The bag was properly sealed and a vacuum of 100 kPa (29.5 in Hg) achieved. The bag was then carefully checked for leaks since the loss of vacuum during certain times of the autoclave cycle may result in a poorly processed laminate.  3.5.4  C o c u r e C y c l e of B o r o n 5521/4 P r e p r e g  and  F M 7 3 M A d -  hesive  The factory recommended cure cycle for the boron 5521/4 prepreg was used for the cocuring of the boron prepreg and the F M 73M (Figure 3.19).  -, 700 Temperature  - 600  o  <D 3 eS  -400  Pressure  -300  CU  e  •essui:e [kPa]  - 500  u  - 200  H  = - 100 Only Positive Pressure 0  10  20  30  40  50  60  70  80  90 100 110 120 130 140  0  Time [min] Figure 3.19: Recommended boron 5521/4 cure cycle  Prior to starting the cure cycle in the autoclave, the bagged patches were hooked up to the autoclave vacuum system. The vacuum system should be operating prior to hook up in order to avoid a loss of vacuum in the bag. After connecting the vacuum source, the vacuum control system was switched to the automatic mode. The elapsed time before the  3.5  Boron Patch Fabrication Procedure  59  vacuum system restarted gave a good indication about the quality of the vacuum bag. If this time period was at least 2 minutes, the autoclave was closed.  The curing cycle started with raising the pressure to 344.7 kPa (50 psi). Then the autoclave temperature was increased at a rate of 2.8 ° C / m i n (5.0 ° F / m i n ) . The 'Baron' Autoclave (see Figure 3.20), was operated in the manual mode for the curing cycle. The temperature set point was read-  Figure 3.20: Autoclave  justed every 2 minutes thus providing a fairly close temperature profile to the optimal ramp up. At 112.8 °C (235.0 °F) the vacuum was released. The temperature ramp up was finished at 121.1 °C (250.0 °F), which was maintained for 60 minutes.  After this hold period, the temperature was reduced at a rate of 2.8 ° C/min (5.0 ° F / m i n ) . The cooling rate decreased at approximately 65.6 ° C (150 °F) due reaching the maximum cooling rate of the autoclave. The pressure was dropped at approximately 50.0 °C (122.0 °F) after which the bagged patches were removed from the autoclave.  The aluminum  plate, on which the patches were bagged, served as an additional heat source to maintain cooling of the patches near the recommended rate. The vacuum bag was opened when the aluminum plate reached room temperature. Figure 3.21 shows the cocured patches on nonperforated release film after removing them from the vacuum bag.  The cured patch was trimmed to the desired size. The excessive adhesive was ground  3.5  Boron Patch Fabrication Procedure  60  off to the edge of the boron along the circular boundary. The straight edge of the semicircular patch was then cut using a diamond coated blade on a table saw. The resulting cured patch was found to be slightly curved on the bottom side due to the difference in the C T E between the F M 73M film adhesive and boron/epoxy prepreg in combination with adhesive shrinkage. Radial contraction of the boron fibers during the cooling is small in comparison to the epoxy used as a matrix for the F M 73M which resulted in the slightly curved shape. Prior to bonding the patch to the face-sheet, the bottom peel ply of the patch was removed and grit blasted with clean 220 aluminum oxide grit using nitrogen as the blast system. Afterwards the surface was cleaned with a nitrogen blast using a spray nozzle. The grit blasting removed a significant amount of the cocured film adhesive. The patches were then wrapped in Kraft paper until bonding to the face-sheets.  3.5.5  Ultrasonic Inspection of the Cocured Patches  All boron/epoxy patches were inspected for flaws by ultrasound. Amplitude C-scans were carried out using the ARIUS system at the Institute for Aerospace Research (NRC) in Ottawa. A 15 Mhz focused transducer was used in pulse-echo mode. The patches were immersed in water and placed on two brass bars. One bar was placed along the straight edge while the other one was positioned parallel to the first supporting the tapered end. It  3.5  Boron Patch Fabrication Procedure  61  was important to place the center part of the semicircular patch as flat as possible to avoid angled reflection of the signal causing a decrease in the signal amplitude.  Figure 3.22: C-scan of a cocured boron/epoxy patch  F u l l scale amplitude is indicated by white, w i t h decreasing amplitude indicated by red, yellow, green, blue, and black, which is the m i n i m u m amplitude.  Figure 3.22 gives an  example of a patch C-scan. T h e center area of the patch shows white, red and yellow. The tapered end shows a significant change i n colour. T h i s change is due to the angle of the taper, which reduces the amplitude of the signal received by the probe. T h e patch taper itself is not flat, which leads to an additional change in the amplitude of the signal shown by the periodical change between yellow and green. A few very small defects, indicated in blue, are visible in the tapered section. These are usually steeper local gradients in the taper right next to a ply drop off or possibly tiny air bubbles which are enclosed at a ply drop off. Changes in colour can also be noticed close to the supporting bars due to partial signal reflection off the bar.  A p p e n d i x C shows the N D I image of all cocured boron/epoxy patches. O n l y the C-scan of patch 1 (Figure C . l ) reveals a small defect left of the center of the patch. V i s u a l inspection  3.6  A M R L Specimen Assembly Procedure  62  also revealed a small area of discolouration, which may be related to poor bonding or the use of a damaged ply.  3.6  A M R L  3.6.1  Specimen Assembly  Fixture  Procedure  Design  The design of the fixture for the A M R L Specimen (Figure A.5) was dictated by the process specification:  • Temperature cycle from room temperature to 121 °C (250 ° F ) • Unrestricted thermal expansion of the face-sheets • Vacuum bagging • Autoclave pressure of 344.7 kPa (50 psi)  The large temperature change and associated thermal strains of the face-sheets were taken into account by manufacturing the entire fixture from aluminum. Thus equal thermal strains in the face-sheets and fixture were achieved. Alignment of the two face-sheets was maintained during the cure cycle by using aluminum center alignment bolts. Due to the local thickness change of the specimen in the patch area, the typical aluminum support plate used in the autoclave process was not applicable for this specific specimen. Thus the fixture required complete vacuum bag enclosure. This also ensured proper application of positive autoclave pressure on the patch and the honeycomb/face-sheet interface. Support in the grip area of the face-sheets was required to avoid bending of the face-sheets and possible damage to the honeycomb. This was achieved by machining the fixture to the exact thickness of the honeycomb.  The parallel sides of the fixture combined with  the tight machining tolerances required tapping the face-sheets with wooden block and a hammer to remove the cured specimen. A small gap was left between the honeycomb  3.6  A M R L Specimen Assembly Procedure  63  and the fixture to avoid accidental bonding between the fixture and the face-sheets. The risk of bonding the face-sheets to the fixture was significantly reduced by avoiding vacuum or positive pressure on the adhesive film between the honeycomb and face-sheet.  Only  the honeycomb/face-sheet interface itself was properly stressed through the vacuum and autoclave pressure. The film adhesive without any pressure did not flow significantly. The fixture was also designed with rounded corners and dimensions exceeding the facesheet area to avoid any possible ruptures of the vacuum bag at the sharp face-sheet corners during the curing process. From a manufacturing point of view, it was preferred to cut the honeycomb after the curing process along the edges due to the difficulty in precisely machining the honeycomb. Therefore the edges of the honeycomb needed sufficient protection to avoid damage during the curing process. This led to the two part design of the fixture, which enclosed the honeycomb completely and allowed for removal of the specimen after curing. The exposed honeycomb edges exceeding the face-sheet width were covered with thin aluminum sheets which had the same contour as the face-sheets on one side. As a further precaution, the alignment bolts (Figure A.6) were designed with tapered heads to avoid punctures in the bag.  3.6.2  Preparation of the  Honeycomb  The fixture design allowed large tolerances for the different sides of the honeycomb, which makes the cutting process significantly simpler.  A reasonably precise cut without any  significant distortions was achieved by clamping the honeycomb between two aluminum plates and cutting slowly with a sharp knife. Figure 3.23 shows the setup.  The honeycomb was solvent flushed with M E K prior to the bonding process [90]. The honeycomb core was then oven dried at 50°C (122°F) for 30 minutes. After drying, the honeycomb core was protected by wrapping it in Kraft paper.  3.6  A M R L Specimen Assembly Procedure  64  Figure 3.23: Setup for honeycomb cutting  3.6.3  Assembling the A M R L Specimen  The assembly procedure for the A M R L specimen is directly linked to the fixture design. The first part of the assembly deals with alignment of the two face-sheets and the honeycomb core. The second step is concerned with the proper installation of the patches.  During  this process it is essential to avoid any contamination of the clean surfaces of the patch  Figure 3.24: A M R L specimen assembly (Face-sheet)  3.6  A M R L Specimen Assembly Procedure  65  Figure 3.25: A M R L specimen assembly (Face-sheet mounting) and face-sheets. As the initial step, the adhesive films, which bond the honeycomb to the face-sheet, were firmly rolled onto the face-sheets using a teflon roller. The use of a heat gun helped to keep the adhesive sticking to the face-sheet.  Then a small strip of high  temperature tape was placed approximately 0.5 mm (0.02 in) adjacent to the adhesive as a small barrier towards the grip area (Figure 3.24).  Figure 3.26: A M R L specimen assembly (Honeycomb mounting)  3.6  A M R L Specimen Assembly Procedure  66  Then the first face-sheet was placed on the fixture and precisely located using the center alignment pins (Figure 3.25). Next, the backing of the F M 73M film adhesive was removed and the honeycomb placed on top of the adhesive.  A vacuum bag foil was placed over  the honeycomb and a teflon roller was used to press the honeycomb against the adhesive film and face-sheet. The fixture design provided light clamping of the honeycomb which eliminated any possible lateral movement (Figure 3.26). After installing the honeycomb, the second face-sheet was placed on the fixture after removing the film adhesive backing. The nuts for the center alignment bolts were then placed and tightened (Figure 3.27).  Figure 3.27: A M R L specimen assembly (2. Face-sheet mounting)  At this point, the area between the face-sheet and the fixture was covered with thin aluminum sheets (Figure 3.28), which had the same contour on one side as the face-sheet. High temperature tape was applied to the top of the aluminum sheet to avoid accidental bonding during the curing process. Each aluminum sheet was held in place by two pieces of high temperature tape. One layer of F M 73M film adhesive with a radius of 77.5 mm (3.05 in) was carefully marked in the center . The center point was then placed at the edge  3.6  A M R L Specimen Assembly Procedure  67  Figure 3.28: A M R L specimen assembly (Aluminum sheets) of the face-sheet over the pre-cracking notch. The adhesive film was rolled down with the teflon roller while carefully applying heat using a heat gun (Figure 3.29).  In order to achieve a nice fillet and avoid adhesive loss through bleeding, high temperature tape pieces were pre-cut using the 77.5 mm (3.05 in) template (Figure 3.30). The tape  Figure 3.29: A M R L specimen assembly (FM 73M film adhesive)  3.6  A M R L Specimen Assembly Procedure  68  was placed as close as possible to the adhesive and firmly pressed down. T h i s allowed to formation of a half circular barrier around the adhesive.  The  precured boron patch was then centered on the adhesive and firmly pressed down.  A second layer of high temperature tape was placed along the circular edge of the patch, therefore partially covering the patch, the adhesive and the previously placed high temperature tape on the face-sheet (Figure 3.30). In addition, high temperature tape was placed on the 3.175 m m (0.125 in) wide edge of the face-sheet and then folded onto the patch to seal in the adhesive. M u c h care was taken to firmly attach the tape to the t h i n edge. Vacuum sealant tape was then placed along this edge to ensure that the autoclave pressure was applied to this surface. T h i s procedure proved to be very successful for avoiding adhesive bleeding. T h e bleeding only occurred through the precrack notch. T h e patch was securely held i n place by two or three additional pieces of high temperature tape across the patch (Figure 3.31).  Figure 3.30: A M R L specimen assembly (High temperature tape barrier)  3.6  A M R L Specimen Assembly Procedure  69  Figure 3.31: A M R L specimen assembly (Patch fixation) The patch mounting procedure was then identically carried out on the other side of the face-sheet. The whole fixture was subsequently covered in one layer of perforated release film and one layer of bleeder cloth. In addition, one layer of breather cloth, which extended beyond the fixture on one side by approximately 15 cm (5.9 in) for the placement of the vacuum fitting, was wrapped around the fixture and affixed with high temperature tape.  Figure 3.32: A M R L specimen assembly (Vacuum bag)  3.6  A M R L Specimen Assembly Procedure  70  The covered fixture was then placed in a vacuum bag. Folds of the vacuum bag were avoided in the patch areas (Figure 3.32). The vacuum bag was then connected to the autoclave system.  The cure cycle for the A M R L specimen was based on the F M 73M film adhesive. This cure cycle is similar to the one for the boron prepreg except that the vacuum was maintained throughout the curing process (Figure 3.33). The cure cycle started by raising the pressure to 344.7 kPa (50 psi). The temperature was then increased at a rate of 2.8 ° C / m i n (5.0 °F/min).  The 'Baron' Autoclave was operated in the manual mode for the curing cycle.  The set point was adjusted every 2 minutes during the ramp up and down. The temperature ramp up was finished at 121.1 °C (250.0 °F), which was maintained for 60 minutes. After this time period, the temperature was ramped down at a rate of 2.8 ° C/min (5.0 ° F / m i n ) . The cooling rate was slowed at approximately 65.6 °C (150 °F) due reaching the maximum cooling rate of the autoclave. The pressure was then dropped at approximately 45.0 °C (113.0 °F) and the bagged A M R L specimen removed from the autoclave.  The vacuum  i 700  Temperature  - 600  o  - 400  Pressure  - 300  <D &,  a  - 200 - 100 Maintain Vacuum 40  50  60  70  80  90  100 110 120 130 140  Time [min] Figure 3.33: Cure cycle for the A M R L specimen  •essui-e [kPa]  - 500  u  (X  3.6  A M R L Specimen Assembly Procedure  bag was opened when the fixture and A M R L specimen reached room temperature.  71 For  future work it is recommended to drop the vacuum after the i n i t i a l temperature ramp up (see Figure 3.19) to avoid the possibility of 'adhesive boiling' at the honeycomb/face-sheet interface since this interface doesn't experience positive pressure.  The top part of the fixture and the center alignment pins were unscrewed and removed. The wide b o t t o m section of the fixture was clamped i n a vice and a wooden block w i t h a hammer was used to tap the specimen out of the fixture. Touching the grip area of the specimen was avoided i n order to maintain adequate surface preparation for gluing the spacers and shims. T h e grip area was then covered w i t h Kraft paper. Small scratches on the fixture, which occurred close to the inner edge of the fixture, were removed using 220 wet grinding paper.  The shims (Figure A . 4 ) and spacers (Figure A . 2 ) were bonded to the specimen using the room temperature curing Hysol E A 9396 Q T System. T h e spacers were machined to a tight tolerance specification to ensure proper fit. The adhesive was spread i n a t h i n layer on each bonding surface prior to inserting the spacer. Alignment was achieved using the center alignment pins and a p i n i n one of the bolt holes on each side. Special care was taken to achieve a uniform bondline between the face-sheet and the shims. Piano wires with a diameter of 0.25 m m (0.01 in) were placed in the bondline. P i a n o wires were chosen due to their uniform thickness and allowed better adhesive flow throughout the mounting process. T h e face-sheet section outside the grip area was masked off w i t h tape to avoid undesired adhesive spill. The wires were cut and placed first on one side of the specimen in the grip area. T h e n the adhesive was spread slightly thicker than the wires. The shims were aligned w i t h the two pins on each grip side and firmly pressed down. A n y excessive adhesive was then removed. Tape was placed along the tapered section of the shims and onto the face-sheets. T h i s avoided adhesive being squeezed out between the face-sheet and  3.6  A M R L Specimen Assembly Procedure  72  the shim when the grip section was clamped onto a 25.4 m m (1 in) thick flat aluminum plate after the same process was repeated w i t h the opposite side. T w o 12.78 m m (0.5 in) thick a l u m i n u m plates w i t h slightly larger holes than the alignment pins were placed on top of the shim, which were facing up, to ensure uniform clamping pressure. T h e clamping pressure was applied by four C-clamps. Excessive adhesive was removed after the clamping.  The required thickness of 25.27 m m (0.995 in) i n the grip area was achieved by m i l l i n g half of the excessive amount on each side off. T h e mounting on a thick a l u m i n u m plate ensured that one side was perfectly flat i n both grip areas, which simplified the mounting on the m i l l i n g table.  Flatness of the face-sheets was also verified prior to milling.  N o cutting  fluids were used to avoid degradation of the patches and bonds. In addition the patch was protected against chips during the m i l l i n g process.  3.6.4  Ultrasonic Inspection of the  Boron  Patch  Repair  Submersion of the A M R L specimen for ultrasonic inspection was not chosen due to the possibility of water getting into the honeycomb, which could lead to deterioration of the bond if a l l of the water could not be fully removed after the N D I inspection. Instead, a Panametrics E p o c h III M o d e l 2300 ultrasonic system including a Nova N D T Instruments (D11P) hand-held probe was used i n combination w i t h a diagnostic ultrasound scanning couplant. T h e probe was fitted w i t h a plexiglas delay shoe for signal damping and noise reduction at the interface.  T h e instrument settings were chosen based on the signal reflection of the back surface of a 3.175 m m (0.125 in) a l u m i n u m plate. The reflected signal of the composite repair remained approximately at 80% i n comparison to the reflecting signal for the plain aluminum plate. T h e taper area was also carefully checked. A s the probe was moved slowly towards the tapered end, the signal moved to the left on the time axis indicating a thinner specimen. In  3.7  Instrumented A M R L Specimen Construction  73  addition, the signal decayed due to the angle of the taper. It is important that the transition of the signal is smooth.  If a local disbond is present, the signal would be significantly  reduced or even disappear.  The A M R L specimens did not show any evidence of local  disbonds in the patched area.  3.7  Instrumented A M R L Specimen Construction  One A M R L specimen was instrumented with strain gauges to provide experimental verification of the strain distribution in the patch and the cracked aluminum. The intent was to measure the strain distribution in the patched area as well as the strain distribution through the thickness of the repair in several critical locations. To accomplish this task, 350 fl strain gauges were chosen in single element and 90° rosette configurations since high resistance gauges offer the advantage of reducing heat effects and therefore providing better stability.  The locations of the strain gauges were selected based predominantly on previously reported failure locations in the literature [13, 37, 70]. Strain gauges were therefore placed near the crack tip and at the tapered end. In addition, strain gauges were located in the center area of the patch close to the beginning of the taper. This gave a good reference location due to the reduced influence of the crack and taper on the magnitude of the strains at these locations. Figure A.11 shows most of the gauges in a superimposed drawing. Two reference gauges were mounted on the aluminum outside the patch. This allows for the normalization of the strains in comparison to a remote strain to allow for better applicability for other repair applications.  A number of strain gauges were installed at each of the above described locations. The gauges were placed between the different layers through the patch thickness and on top of  3.7  Instrumented A M R L Specimen Construction  74  the patch. The through thickness locations were based on the critical points of the load transfer, where failures have been reported [13, 37, 70]. Therefore gauges were typically mounted on the aluminum, on top of the adhesive and on top of the first boron/epoxy layer. In addition some gauges were mounted on top of the patch. Unfortunately the strain gauges do not allow measurement of the shear strain in the adhesive. A main concern regarding placing strain gauges through the thickness of the patch is the presence of a local increase in thickness. For this reason the number of gauges was limited to four through the thickness of the repair. The gauge on top of the patch has limited influence by increasing the local stiffness only slightly, however the gauges embedded in the patch do increase the thickness of the patch introducing some error in the measured strains at the desired location. This error will be discussed in detail in conjunction with the finite element results in Chapter 6.  In order to minimize the increase in thickness, special attention was given to the types of gauges, their orientation, lead attachment and placement procedure. The 90° rosette configurations, where the gauges were arranged side by side, were a thinner choice for biaxial measurements than a stacked rosette configuration. These gauges were placed on the face-sheet and on top of the patch. Single element gauges were chosen for the placement on top of the adhesive and on top of the first boron/epoxy layer in order to reduce the number of lead wires in the patch. The measurement of biaxial strains on the aluminum was favoured due to the easier placement of the wires in the adhesive layer. All lead wires were preferably aligned with the boron fibers to minimize strain distortion. All gauges were placed with the lead wires initially facing different directions to allow for the smallest possible disturbance of the strain distribution especially along the fiber direction. The 34 gauge (0.16 mm or 0.006 in) lead wires, having a slightly larger diameter than the boron prepreg thickness of 0.13 mm (0.005 in), were attached with minimal solder to allow the wires to be easily embedded in the boron prepreg. The lead wires, which were connected to the gauges on the face-sheet, were entirely covered in the adhesive layer (approximately  3.7  Instrumented A M R L Specimen Construction  0.25mm (0.010 in) thick) and thus didn't add additional thickness.  75 Only the lead wires  embedded in the adhesive or placed on top of the patch were oriented perpendicular to the loading direction and fiber orientation close to the gauge. Figure A. 11 shows the complex arrangement of lead wires in the patch. The concern of changing the strain distribution significantly by using a large number of gauges and the associated leadwires led also to the decision not to place a row of very small strain gauges in the critical load transfer zones especially around the crack tip. Note that strain gauges mounted on composite material require a certain minimum size to get averaged strain readings across a number of fibers rather than measuring the local distribution around a fiber.  In addition, two strain gauge rosettes were placed on the back of the aluminum face-sheet. These particular gauges offer the option to measure destructively the residual strains in the aluminum. This allows a comparison with the measured residual strains using the non-destructive methods as described in Chapter 4, if the specimen is not required for any additional measurements.  3.7.1  Instrumented  Boron  Patch Fabrication Procedure  Manufacturing of the instrumented cocured boron/epoxy patch was divided into three steps: 1. Preparing the gauges, 2. Lay-up of the patch including placing the gauges and 3. Curing the patch.  Only single element strain gauges (CEA-13-062UW-350) were placed in the patch. Unfortunately, these gauges were unavailable with sufficiently long preattached leads. Therefore  3.7  Instrumented A M R L Specimen Construction  76  150 mm (5.91 in) long lead wires were soldered prior to placing the gauges on the prepreg material. Special care was taken to minimize the solder build up at the tabs. Stripper X was used to remove the polyimide cover on the wire. The exposed solder tabs with the attached copper wire required a protective coating due to the conductivity of the tungsten core of the boron fibers. M-Coat D was chosen due to its excellent electrical isolation properties. The white colour offered easy verification of complete coverage.  The solder joint  and coating were inspected under the microscope. The coating was allowed to cure in 24 hours at room temperature.  The placement of the gauges and terminals was based on an accurate template of the desired gauge locations. The template was fixed on an aluminum plate and covered with a clear plastic sheet. In addition, a perforated release film was placed on top. A wooden frame was built around this setup. The next step dealt with marking the x and y locations of the gauges on the wooden frame. Nails were then placed in the frame in order to attach thin wires (38 gauge), which crossed at the center of each gauge (Figure 3.34). Then the F M 73M film adhesive layer was placed. Earlier trials showed that the handling of wires after the patch curing process is extremely difficult and in most cases led to damage to the wire or its coating. Therefore all wires including the terminals were embedded in the larger F M 73M adhesive layer. The terminal had to be cleaned after the curing process. The only the drawback to this procedure was that the face-sheet was now slightly reinforced through the larger adhesive layer. The transparency of the F M 73M film adhesive allowed placement of gauges and terminals using the template underneath. Prior to placing gauges on the adhesive, all terminals were placed at their predefined locations. Once a gauge was placed on the adhesive, removal was extremely difficult. It also proved to be very difficult to place the wires along the desired straight lines. Due to the viscoelastic behaviour of the adhesive prior to curing, only limited pulling of the wires was permitted to straighten them out. Excessive tension led to movement of the gauge. The wires adhered only temporarily  3.7  Instrumented A M R L Specimen Construction  77  to the adhesive after pressing them down with a flat spatula. Then the wires were cut to the correct length and the coating removed. Soldering to the terminal had to be done quickly due to the significant change in viscosity of the adhesive with temperature, i.e. the adhesive began to flow.  After placing the desired six gauges on the film adhesive, the first boron/epoxy prepreg ply was placed. The wire crossings were used to position the additional four gauges on this boron/epoxy ply at the proper locations. The lead wires were placed and connected as beFigure 3.34: Placement of the strain gauges in the patch fore. Then the remaining six boron/epoxy plies were positioned. The gauges on the top of the patch were not installed at this point to avoid any additional distortion in the boron/epoxy prepreg. A peel ply was added on each side. The peel ply on the top ensured acceptable handling qualities of the extended adhesive layer of the precured patch which protected the lead wires during the mounting procedure on the cracked face-sheet. The bagging and curing of the lay-up was carried out identically to the procedure outlined previously for the regular patches with the exception that the adhesive which exceeded the boron/epoxy prepreg was covered with high temperature tape to avoid excessive bleed. It should be noted that the placement of the 10 gauges required approximately 8 hours. The patch was inspected after the autoclave curing process with both X-ray (Figure 3.35) and C-scan (Figure C.14) techniques. The X-ray film shows the precise placement of the  3.7  Instrumented A M R L Specimen Construction  78  gauges on top of each other while the C-scan did not indicate the gauges clearly. In addition, the C-scan showed some distortion of the prepreg caused by the lead wires.  After t r i m m i n g the edges of the patch, the gauge locations were acurately measured on a milling table.  These coordi-  nates, which differed by a maxi m u m of 0.5 m m (0.02 in) from the desired values, were then used for the placement of the gauges on the face-sheet  and  ,,, , , m, , on top of the patch. T h e draw-  Figure 3.35: X - r a y of the instrumented patch ^  ings for the gauge locations were then modified to reflect the true positions.  3.7.2  Instrumented  A M R L  Specimen Assembly  Procedure  The first step prior to assembling the instrumented A M R L specimen was the mounting of the six 90° rosettes (CEA-13-062UT-350) on the the aluminum face-sheet, which were later covered by the patch. In addition, two residual strain gauge rosettes (EA-13-062RE-120) were installed on the backside of the face-sheet. The terminals for these gauges were placed on the a l u m i n u m just outside the extended adhesive layer of the patch. T h e strain gauge adhesive system had to withstand the 121 °C (250 °F) curing temperature of the specimen, thus M - B o n d 610 was chosen as an appropriate adhesive. O n l y two gauges were bonded simultaneously due to the difficulty of using spring clamps i n the patch area.  T h e two  component, solvent-thinned, epoxy-phenolic adhesive was cured at 121 ° C (250 °F) for 2.5  3.7  Instrumented A M R L Specimen Construction  hours according to its specifications.  79  The face-sheet surface was protected by a vacuum  bag sheet against accidental touching and other possible contamination.  The lead wires were positioned as per Figure A.7 and Figure A.12. The wires were held in place using M-Coat D. The residual strain gauges on the backside of the face-sheet were coated with 3 layers of M-Coat A. The clear M-Coat A was chosen to allow for proper placement of the drill for the possibility of destructive residual strain measurements. The solder tabs with the attached wires were protected with M-Coat D.  The honeycomb which extended past the face-sheet edges also required modification to avoid cutting through the lead wires on the backside of the face-sheet during the autoclave cycle. The honeycomb was carefully indented, which led to small distortions. Small strips of F M 73M were placed in the indented section of the honeycomb with the exception of the gauge area to achieve some degree of bonding between the face-sheet and the honeycomb in this modified area.  The larger rectangular F M 73M film adhesive was placed carefully over the gauges on the topside of the face-sheet. The F M 73M layer exceeded the patch including the large rectangular cocured adhesive film by 2.5 mm. The gauges on the backside of the face-sheet were not covered with the film adhesive as these sections were carefully cut out of the film adhesive prior to the placement. The remaining assembly and curing procedure for the bonding of the patches, aluminum face-sheets and the honeycomb of this instrumented specimen were identical to the previously outlined procedure.  In total, 28 gauges were installed at this stage. After removal and clean-up, spacers and shims were installed in the grip region using the room curing adhesive H Y S O L E A 9396. The last step in the instrumented specimen preparation process was the installation of eight, two-element ( 9 0 ° ) strain gauges on top of the instrumented patch and outside the repair  3.7  Instrumented A M R L Specimen Construction  80  section (see Figure A. 10). A maximum of two gauges were installed at any one time using M-Bond 610, resulting in the need for four curing cycles. To limit the influence of thermal cycling, the strain gauge glue was cured at 100°C. A final post-cure was then carried out at 107°C , which was the highest temperature anticipated for the thermal experiments. This completed the installation of the 44 strain gauges.  The described manufacturing process is a significant achievement in itself, which wouldn't have been possible without the help from NRC especially Don Raizenne. One of the biggest challenges was the collection of all details required to build this specimen type. of the details are not revealed in the literature to the extent necessary.  Many  N R C provided  missing details with respect to the surface preparation procedure. No jig specifications were provided in the literature either. The designed jig is characterized by simplicity and low manufacturing cost. Installing the large number of strain gauge required patience and skill as well as extensive planning to minimize the impact on the strain distribution. It is important to keep in mind that any mistakes in the manufacturing process, such as poor surface preparation leading to a poor bond or disbonds, is detrimental for any kind of experimental work.  Having outlined the construction of an instrumented A M R L specimen in detail, the following chapter will deal with the challenges of carrying out thermal residual strain measurements and the total residual strain measurement.  The total residual strains are a  combination of mainly thermal residual strains and some processed induced strains such as adhesive shrinkage. Their measurement was integrated in the instrumented A M R L specimen assembly procedure based on measuring the strain state in the patch and face-sheet prior and after assembling the specimen.  4  Residual and Thermal Residual Strain Measurements  81  Chapter 4 Residual and Thermal Residual Strain Measurements  4.1  Introduction  This chapter discusses the challenges involved in measuring residual strains induced by the elevated temperature cure in bonded repairs. A significant amount of new experimental data is provided giving the opportunity to verify theoretical estimates of the residual stress and strain state, as well as establishing the foundation for a fatigue damage initiation model for bonded repairs, where knowledge of the initial stress/strain state of the repair is important. Although thermal residual strains are the primary focus of this research, residual strains due to the elevated temperature cure of any bonded repair consists of both thermal and process induced residual strains, the latter resulting from effects such as anisotropic shrinkage of the patch or adhesive, temperature gradients in the specimen during processing, and/or temperature gradients due to tooling and internal resin flow [50]. While thermal strains are affected by subsequent changes in temperature, process induced residual strains may not be affected.  The residual strain measurement procedure, which was carried out during the assembly stage of the instrumented A M R L specimen, is outlined first. This particular method was only carried out for measuring the total residual stresses at room temperature, while the thermal residual stresses were continuously measured from the stress free temperature (101.2°C) to extremely low temperatures representing typical aircraft operating temper-  4.2  Residual Strain Measurement Procedure  82  atures (—56.5°C). The stress free temperature was experimentally measured rather than assuming that the curing temperature corresponds to the stress free temperature as is commonly done. The results emphasized the importance of this measurement. The continuous measurement of the thermal residual strains revealed additional characteristics about the adhesive at elevated temperatures giving a better understanding for the estimation of thermal residual strains. Process induced strains were assessed based on the difference between the measured residual strains and the thermal residual strains. In order to simplify the estimation of thermal residual strains for other material combinations especially with respect to the utilized adhesive, a suggestion for a simple test specimen rather than using the complex A M R L specimen is presented focusing on the behaviour of the adhesive near the stress free temperature.  4.2 4.2.1  Residual Strain Measurement Procedure A p p r o a c h a n d E x p e r i m e n t a l Set-up  To determine the magnitude of the residual strains introduced during curing requires knowledge of the strain state both prior to and after completion of the thermal curing cycle.  The strain state prior  to the curing process was obtained using two Vishay 2310 signal conditioning amplifiers,  Figure 4.1: Initial patch strain measurement  4.2  Residual Strain Measurement Procedure  83  one each for the 120 0 and 350 Q quarter bridge configurations.  At this point, 120 0  and 350 Q precision resistors, respectively, were used in the quarter bridge to zero the two signal conditioning amplifiers. The precision resistors were -then removed and replaced by the mounted strain gauges, one gauge at a time, with the indicated resistance change (i.e. strain) being recorded (see Figure 4.1). Since the cocuring process of the one layer of F M 73M with the seven boron/epoxy plies had introduced residual stresses in the patch itself (as indicated by the slightly curved patch), the patch was vacuum-bagged against a flat aluminum plate for the initial strain recordings to provide a repeatable and representative measurement condition both pre- and post-cure. While this flat patch geometry was representative of the pre-curing setup in the autoclave, it should be noted that the peel ply on the bottom of the patch was not removed for this measurement to protect the bonding surface from contamination. After curing, this measurement condition (including temperature conditions) was maintained and the strain measurement recording procedure repeated.  4.2.2  Data  Analysis  The residual strains induced during the curing cycle were then determined by calculating the difference in the pre- and post-cure strain measurements.  The residual strain values  were then corrected for the resistance of the lead wires [28]:  e C  =  L W corr.RS  £ r s  1 — /?  (4.1) /  where  R  I  ,  1  ^  a  2-RLOIIK LW+2-Rshort LW  (4.2)  Ra  In addition the residual strain values of the strain gauge rosettes were adjusted for transverse sensitivity as follows [61].  4.2  Residual Strain Measurement Procedure  84  For 90° rosettes parallel and perpendicular to the fiber direction:  ,  e =  |lLWcorr.R (  ~  1  S  U  Q t \ \ ) ~ # t | | € ± W c o r r . RS i -i TS TS K  ~ "o^t-l.)  1  L  H T S Y corr. RS  1 -  , , V * - " /  A ||A x t  t  and  ,  _  e  -LLWcorr.R (  ~  1  S  ^O^tx) ~ ^tXe|[ i TS TS  -i-TSY corr. RS  corr. RS (  1  L  W  ~  ^^tll)  , , V ^ ' V  1 - A i|A _L t  t  For gauges 17 and 20 (see residual strain rosettes in Figure A.12):  , G17 SYcorr.  fc  T  e  G- 1 r r .RS Rs( - '7L Wc o corr.  _  1  1  —  KQ )  V 0  L W  RS  t  X1  ^tG17  -^tG17(eG18  L W  (1 —  corr.Rs(  1  ^ Q - ^ t G l s ) (1 ~  t  ^tG17(£G19 wcorr.Rs(  ~  1  L  (1  ~  if G18-KtG19)(l.  —  (4.5)  tGU  ^ o - K ' t G i g ) (1 ~  — -^tG18-KtG19)(l  ^tG19)  K ) -^tGls)  — KtGn)  and  .^OLWcorr.Rsl 1 TS 1  e  G 2 0 S Y c o r r . RS T  _  ^O^tG2())  1 — A 20 tG  _  •^tG2o(eG21 wcorr.Rs( - ~ :1  L  ^o-^tG2l)(l  -  K 2) tG2  (4.6)  (1 — ^ 0 2 1 - ^ 0 2 2 ) (1 — -KtG2fj) _  -^tG2o(eG22 wcorr.Rs(  1  L  (1 — K lK 2)(l iG2  tG2  ~  ^Q^tG22)(l ~  -^tG2l)  — K 2o) tG  where  u = 0.285 0  (4.7)  Correcting for the transverse sensitivity of the single element gauges was considered unnecessary as the maximum error introduced was calculated to be less than 4//e due to the low residual strains perpendicular to the fiber direction. The error estimate is based on  4.2  Residual Strain Measurement Procedure  85  the change in the measured residual strains due to transverse sensitivity from the rosettes mounted on the aluminum and on top of the patch. An additional small error in the strain gauge readings was introduced by the applied hydrostatic pressure (0.1 M P a (29.5 inHg)) during the initial strain measurement of the boron/epoxy patch. The magnitude of the error for the gauges installed in the boron/epoxy prepreg was estimated using Hooke's law:  e\\ = ^((-p)-2v\\{-p))  (4.8)  and  = ~((-p)  - 2u (-p)) ±  (4.9)  Using the material properties provided in Appendix D, the estimated strain parallel to the fiber orientation was calculated to be — 0.3fie and —3.8/xe perpendicular to the fiber orientation. The total residual strains were not corrected for this small error due to the lack of a suitable instrumented boron/epoxy specimen with a flat bottom surface for experimental verification. In order to experimentally determine the magnitude of these strains, the instrumented A M R L specimen was vacuum-bagged. Strains between 0 and 3 pe were measured in the boron/epoxy patch.  4.2.3  Results  Table 4.1 lists the results for the residual strain measurement.  The residual strains are  also shown in Figure 4.16 at their measured locations. Having measured the total residual strains, the next step was the experimental determination of the thermal residual strains as a function of temperature which exclude any process induced strains. The stress free temperature was required for the material combination used in the A M R L specimen in order to carry this measurement out.  4.2  Residual Strain Measurement Procedure Location  Gauge Orientation No.  86  Gauge  Initial  Final  Resistance  Strain  Strain  Residual Corrected Strain  Reading Reading n  [n]  1  90  350  608  Face-sheet (Front)  2  0  350  Face-sheet (Front)  3  0  Face-sheet (Front)  4  Face-sheet (Front)  Residual Strain  [fie]  M  580  -28  -28  871  868  -3  -3  350  981  1674  693  693  90  350  1092  951  -141  -147  5  0  350  1413  1926  513  513  Face-sheet (Front)  6  90  350  246  107  -139  -143  Face-sheet (Front)  7  0  350  1227  1560  333  334  Face-sheet (Front)  8  90  350  606  488  -118  -121  Face-sheet (Front)  9  90  350  158  -6  -164  -168  Face-sheet (Front)  10  0  350  1677  2163  486  487  Face-sheet (Front)  11  90  350  1221  1176  -45  -50  Face-sheet (Front)  12  0  350  1335  1932  597  596  Face-sheet (Back)  17  45  120  1767  1938  171  168  Face-sheet (Back)  18  90  120  1587  1383  -204  -211  Face-sheet (Back)  19  0  120  1170  1722  552  554  Face-sheet (Back)  20  45  120  2244  2409  165  165  Face-sheet (Back)  21  90  120  2112  1854  -258  -263  Face-sheet (Back)  22  0  120  730  1164  434  437  Patch (Adhesive)  23  0  350  1458  1686  228  228  Patch (Adhesive)  24  0  350  1665  975  -690  -690  Patch (Adhesive)  25  0  350  1656  926  -730  -730  Patch (Adhesive)  26  0  350  1494  660  -834  -835  Patch (Adhesive)  27  0  350  1266  450  -816  -817  Patch (Adhesive)  28  0  350  2121  1443  -678  -678  Patch (1 B/ep Ply)  29  0  350  1278  540  -738  -738  Patch (1 B/ep Ply)  30  0  350  1782  1059  -723  -724  Patch (1 B/ep Ply)  31  0  350  1686  910  -776  -777  Patch (1 B/ep Ply)  32  0  350  1473  651  -822  -823  [ ]  [ ]  Face-sheet (Front)  st  st  st  st  Table 4.1: Residual Strains  Note: The 0° orientation corresponds to the fiber orientation.  4.3  Stress Free Temperature Measurement Procedure  4.3  87  Stress Free Temperature Measurement Procedure  Determination of the thermal residual strains at any temperature requires knowledge of the stress free temperature at which no thermal residual strains are present. In order to determine the stress free temperature for the current application, an additional identical patch was bonded to a 152.4 mm x 152.4 mm aluminum alloy sheet using one layer of F M 73M. After curing and returning to room temperature the residual strains caused a significant curvature in the aluminum sheet. To determine the stress free temperature, the patched aluminum test sheet was heated until it returned to its initial straight condition in the fiber direction.  This measurement was carried out using a Blue M industrial oven equipped with a Microtrol controller. The patched aluminum test sheet was supported with a flat steel bar and a pin (see Figure 4.2).  A halogen light was used to clearly indicate the presence of any gap  existing between the curved aluminum sheet and the flat steel bar.  Heat  Figure 4.2: Stress free temperature measurement setup  Both the oven and aluminum sheet temperatures were monitored using thermocouples. The temperature measurement of the aluminum was carried out by clamping a thermocouple  4.4  T h e r m a l Residual Strain Measurement Procedure  88  between a silicon rubber pad and the aluminum sheet using a spring clamp. The welded joint of the thermocouple was flattened to the wire thickness prior to clamping. A t 101.2°C, with both the specimen and oven i n thermal equilibrium, the single-side patched aluminum sheet was straight along the supported edge. The temperature was held for over 30 minutes without monitoring any changes in the shape. This experimental set-up had an estimated potential error of 1.5 °C based on the stated assumptions (i.e. potential resolution error of 0.5°C + potential temperature measurement error of < 1 ° C using listed device specifications). Thus the stress free temperature for the boron/epoxy patch-aluminum sheet combination was deemed to be 101.2 ± 1.5 °C.  4.4  Thermal Residual Strain Measurement Procedure  4.4.1  Approach  To quantify the thermal residual strains ( T R S ) within the A M R L specimen, it was decided to heat the specimen to the  stress free  temperature  ( S F T ) . However, accurate determination of the thermal residual strains i n the bonded composite repair required the measurement of not only ther-  F i  S  u r e  4  -  3 :  T i t a n i u m silicate bar w i t h installed gauges  mal strains in the A M R L specimen, but also measurement of the thermal strains for an unbonded separate aluminum and a boron/epoxy material specimens as required by equa-  4.4  Thermal Residual Strain Measurement Procedure  89  tions (4.10) and (4.11). The thermal strains in the boron/epoxy were measured in the fiber direction as well as the transverse direction. In addition, the thermal strains in the aluminum were distinguished between those measured in rolling direction and those perpendicular to the rolling direction. It should be noted that the fiber orientation in the patch corresponds with the rolling direction of the aluminum.  ^AITRSIT  ^B/Ep  TRS I T  e  AllT=SFT  e  B/EplT=SFT  C  AMR.LIT=SFT  C  A 1 IT  ^AMRL IT=SFT  C  A M R L  C  B/Ep I T  C  (4.10)  IT  (4.11)  A M R L I T  To allow separation of the true thermal strain from the indicated strains, one identical gauge of each installed strain gauge type was mounted on a titanium silicate bar (see Figure 4.3) having a near zero coefficient of thermal expansion (i.e. 0.03 ± 0.03 • 1 0 ~ / ° C ) as shown 6  in Figure 4.4 [62].  7.0  I  6.0 •3  5.0  ,E  4.0  Titanium Silcate  —  (0  55  To  3.0  g  2.0  Z  LO  a  0.0 -1.0 -60  -40  -20  20  40  60  80  100  120  Temperature [°C]  Figure 4.4: Thermal expansion characteristics of titanium silicate  By accounting for the small thermal strains of the titanium silicate, the actual thermal strains at the strain gauged locations on the aluminum, boron/epoxy as well as on the A M R L specimen are given by equation (4.12):  : ™ Thermal TU „ „iIT IT Specimen  : /t„ j;„„*. i J Specimen (Indicated ec  |T  \ ^T T S"  " --'-- ' ^ (Indicated) IT /  T  , R  ,  V  ^^ T S R T  (Ref, )  I T )  (4.12)  4.4  Thermal Residual Strain Measurement Procedure  4.4.2  90  Experimental Setup  The measurements of the thermal residual strains were carried out inside a Blue M industrial oven equipped with a Microtrol controller. The instrumented A M R L specimen was supported on a frame consisting of steel rods and two small aluminum sheets. Thin teflon tape was used in the con-  F i  S  u r e  4  -  5 :  A M R L specimen wrapped in breather cloth and aluminum foil  tact areas between the specimens and the frame to reduce friction. To minimize thermal shock effects, the instrumented A M R L specimen together with the titanium silicate bar and the lead wire resistance measurement specimens were wrapped in breather cloth and covered with a thin aluminum foil (see Figure 4 . 5 ) . The same protection was provided for the instrumented aluminum and the boron/epoxy specimens.  This setup led to the very small scatter found in the  recorded data. The temperature of the A M R L specimen and the titanium silicate bar were both measured using bonded thermocouples. A Schlumberger Orion datalogger utilizing a high stability, dual current technique allowed simultaneous acquisition of temperature, wire resistance and strain gauge data. To avoid any potential heating effects due to power dissipation at the strain gauge sites, only gauges within a single interface (i.e. ply, layer, etc.) were measured during any single test. The difference in thermal conductivity and mass between the A M R L and titanium silicate specimens caused a significant difference in the rate of temperature change. Therefore it was  4.4  Thermal Residual Strain Measurement Procedure  91  decided to establish thermal equilibrium between the specimens and the oven temperature at specific temperatures, i.e. 2 1 ° C (room temperature), 5 0 ° C , 8 0 ° C and 107°C for the elevated temperature cycle (see Figure 4.6).  120  c £  100  3  to k_ a> a E  .»  I•c a> E  I I I Regular Temperature Cycle High Heating/Cooling Rate Cycle  C  B B/  80 60  < 3 a>  40  tn _i c c s  20  /  )'  A/  A  E  F  Q.  <  100  200  300  400  500  600  700  800  900  Time [min]  Figure 4.6: Regular and high heating/cooling rate cycles  Measuring to temperatures slightly above the stress free temperature showed that the strain change with temperature remained essentially constant in this region. The strain measurements were carried out both during the heating and cooling portions of the thermal cycles. Approximately two hours were required to achieve thermal equilibrium at each of the specified temperatures.  To reduce testing time, the oven temperature was initially  raised above each equilibrium temperature by 20° C and then returned to the equilibrium temperature when the A M R L specimen was close to reaching this temperature. Due to the indication of stress relaxation by an open hysteresis while holding at elevated temperatures, this measurement procedure was repeated for 12 gauges to check if the measured strains were repeatable. In addition, data was also obtained using an approximate 2 to 3 fold increase in heating and cooling rates (see Figure 4.6) to investigate the influence of heating and cooling rates on the hysteresis caused by stress relaxation.  4.4  Thermal Residual Strain Measurement Procedure  4.4.3  92  Data Analysis  Figure 4.7 shows the indicated strains with temperature for the regular and the high heating/cooling rate cycles for gauge 3 which was mounted on the aluminum and aligned with the fiber direction.  Aluminum Temperature [°C] 60  80  120  •o ura u T C3  Figure 4.7: Indicated strains (Gauge 3) for different temperature cycles  It should be noted that Figure 4.7 shows compressive strains instead of the expected tensile strains for the aluminum which are caused by using an aluminum temperature compensated strain gauge on the boron/epoxy restrained aluminum sheet, i.e. the strain gauge material expands more than the restrained aluminum sheet. The true thermal strains determined in the data analysis are tensile strains.  The indicated strain versus temperature for this gauge location remains roughly linear up to approximately 8 0 ° C .  Between 8 0 ° C and 9 0 ° C , a slope change occurs to a near zero  strain versus temperature gradient. The initial change of indicated strain between ambient temperature and approximately 80 °C reflects the combined expansion of the bonded repair, while above 90 °C a reduction in shear modulus combined with the visco-elastic properties of the F M 73M film adhesive allow nearly unrestrained expansion of the patch  4.4  Thermal Residual Strain Measurement Procedure  93  and the aluminum. Studies by Jurf and Vinson [56] indicated that the creep compliance of F M 73M changes significantly above 8 0 ° C . The visco-elastic behavior of F M 73M is also clearly visible at 80°C in Figure 4.7 (see location B-B' and D-D'), where stress relaxation occurred while holding the specimen at this temperature for an extended period of time (see Figure 4.6).  Hysteresis develops mainly in the temperature range between 80 and 90°C.  The indicated strains between ambient temperature and 80° C appear to be only slightly affected by the visco-elastic behavior. The strain measurements obtained for higher heating and cooling rates (see Figure 4.7) also show the development of hysteresis between 80 °C and 9 0 ° C with the difference that no drop in the indicated strain was present at 80°C. This measurement case represents more closely the cooling of the specimen from the curing temperature to room temperature at the end of the autoclave cycle.  Due to these observations it was decided to correct the measured data for stress relaxation while holding the specimen at an elevated temperature, e.g. shifting point B' on to B and D' on to D. This procedure is also shown in Figure 4.10. This procedure will not eliminate stress relaxation effects entirely, which will also occur during heating and cooling periods the A M R L specimen, but it allows better comparison between measurements.  Figure 4.8: Origin and stress relaxation corrected indicated strains (Gauge 3)  4.4  Thermal Residual Strain Measurement Procedure  94  Essential for the determination of the thermal residual strains is the strain measured at the stress free temperature (101.2°C). This particular strain represents the reference value for zero thermal residual strain. Up to three measurements for the heating and cooling cycles were carried out for each strain gauge. Due to the agreement of the indicated strains in the temperature range between ambient temperature and approximately 80° C for the repeated measurements, the data from the heating and cooling segments were separated and the indicated strains from the cooling cycle were shifted to Ofxe at ambient temperature. The result of correcting for stress relaxation while holding at elevated temperatures and shifting of the measured strains during the cooling cycle is shown in Figure 4.8. It should be noted that the resulting curve is now much closer to the strain measurement with the high heating and cooling rate (see Figure 4.7).  Aluminum Temperature [°C] 120  .700 J  1  1  1  1  '  1  Figure 4.9: Superposition of all corrected indicated strains (Gauge 3)  This procedure has the advantage that the hysteresis effect is projected to the stress free temperature, where now a certain scatter between the heating and cooling cycle as well as repeated measurements is visible (see Figure 4.9). The scatter of the corrected indicated strains at the stress free temperature appears rather large, but it is only 2.5% in comparison to the true thermal strain at this temperature.  4.4  Thermal Residual Strain Measurement Procedure  95  Patch Temperature [°C] 150 100 00 CO 0) O) 3  n (5  50 0 120  -50 -100 -150 -Indicated Strain Data -Origin and Stress Relaxation Corrected Data •High Heating/Cooling Cycle Data  -200 -250  Figure 4.10: Indicated strains for elevated temperature range (Gauge 38) The amount of stress relaxation while holding at an elevated temperature is not only dependent on the time but also on the location and orientation within the patch. The largest change in indicated strain was found at gauge 38 at the tapered edge of the boron/epoxy patch oriented transverse to the fiber direction (see location B-B' and C-C'). It is interesting to note that no stress relaxation occurred during the cooling cycle while holding at 8 0 ° C (see location D-D'). The previously outlined method of adjusting for stress relaxation and shifting the cooling cycle to 0/xe at ambient temperature shows acceptable agreement between the indicated strain measured using the regular temperature cycle and the high heating/cooling rate cycle.  Although the hysteresis seems quite large when evaluating  Figure 4.10, the true thermal strain at this location is close to 1900/xe at the stress free temperature, i.e. the hysteresis is less than 6% of the largest thermal strain for the origin and stress relaxation corrected measurement.  Only negligible strain hysteresis response were found for the strain gauges mounted either inside or on top of the boron/epoxy patch for the gauges oriented in the fiber direction. Figure 4.11 shows the uncorrected indicated strains for gauge 24 for three repeated measurements.  4.4  Thermal Residual Strain Measurement Procedure  0 0r-  a  20  40  Patch Temperature [°C] 60  96  80  100  120  "100 -  ~  -200 -  g,  -300 -  3  <S -400  c '5 -500 -  in •o -600 o> 8 -700'•o £  -800 -900 -  Figure 4.11: Indicated strains (Gauge 24) Due to the low thermal conductivity of the titanium silicate, long holding periods were required to achieve accurate readings at the equilibrium temperatures. In-between these equilibrium points, large hystereses are present (see Figure 4.12).  Titanium Silicate Temperature [°C] 140  -3000 J  1  1  1  1  1  1  Figure 4.12: Indicated thermal strains of titanium silicate (Gauge 45)  Only the indicated strains at the equilibrium temperatures of the gauges mounted on the titanium silicate bar were used in the subsequent analysis due to these large hystereses. A second order polynomial regression analysis which uses different polynomials depending on  4.4  Thermal Residual Strain Measurement Procedure  97  the location on the data curve was employed to approximate the indicated thermal strains for the entire temperature range.  Generally the indicated strains for the low temperature range (see Figure 4.13) show only minor hysteresis effects in comparison to the elevated temperature range.  Figure 4.13  displays the uncorrected indicated strains for gauge 38 which has a large hysteresis for the elevated temperature range. The data analysis of the low temperature range includes a shift of the indicated thermal strains during the heating cycle to O/ie at 21°C. This projects any hysteresis to the lowest temperature.  -60  -50  -40  -30  Patch Temperature [°C] -20 -10  0  10  20  30  01 — cuU OxJxf—  *rUU  i  ^00-  Figure 4.13: Indicated strains for low temperature range (Gauge 38)  All recorded strain measurements were subsequently corrected for errors due to the resistance change with temperature of the lead wires connecting the strain gauges to the terminals (referred to subsequently as short lead wires), the lead wire resistance itself, the gauge factor change with temperature and the transverse strain sensitivity. A specific order for correcting errors was maintained to avoid introducing unnecessary small errors. As a first step, a polynomial regression using a localized regression scheme [92] was carried out to determine the indicated strains at defined temperatures allowing subsequent mathematical operations. Next the recorded strains were converted to the ratio of the resistance changes  4.4  Thermal Residual Strain Measurement Procedure  98  to the gauge resistance: A-^indicated  S  G  ^indicated  (4.13)  Three lead wire configurations providing thermal compensation were installed between the terminal blocks mounted on the specimens and the data acquisition system (referred to subsequently as long lead wires).  Using a two lead wire system between the terminals  and strain gauges introduces a significant error due to the lack of thermal compensation in the circuit. In order to correct the measured data, the resistance change per °C and per mm of lead wire length between the strain gauge and terminal blocks was obtained from two lead wire resistance specimens.  The resistance values for both a 5.01 m long  and a 0.01 m long lead wire specimen were measured as a function of temperature using four wire resistance measurement setups. These resistances values were determined only after stabilizing at specified temperatures within the measurement range. Employing the difference of these two measured resistances in the analysis provides the resistance for a 5 m long lead wire section and compensates for possible small thermoelectric effects. Due to the approximately linear relationship of the resistance change with temperature for a metal conductor within the investigated temperature range [57],  R\T = # | T = I ° C ( 1 + « ( T - 21°C)) 2  (4.14)  the resistance change versus temperature was determined using a linear regression analysis (see Figure 4.14). The resistance change per mm of lead wire and °C was determined to be:  AiWtLW AT  3.46-^-  k Ref. Short LW m  with a standard error of 0.036 / x O / ° C m m .  (4.15)  ° C mm  Note that the resistance change of a 120 f2  strain gauge with a gauge factor of 2 is 240 p£t for an applied strain of 1 [it. For some gauges having up to 230 mm long iead wires and a measurement temperature range between  4.4  Thermal Residual Strain Measurement Procedure  99  -466-  E E  300  a  200  zi.  1  100  -100  -80  -60  -40  20  -100  40  60  80  100  120  -200 +  CO  -300  <  JP Technologies W-07-A Wire Regression Analysis  -466-  AT [°C] Figure 4.14: Lead wire resistance change with temperature —56°C and 101.2°C, it was found that the introduced error is significant (up to 1000 u-e) and cannot be neglected. The 350 fl gauges have a significant advantage over the 120 fl gauges in this respect. The superimposed resistance change of the two short lead wires on the strain gauge resistance change is:  RG  +  2^ hortLW f  AR Short LW RLong LW + 2R•Short LW  f l s h o r t L W  A T  S  A i *5mRef. Short LW  (T - 21°C) V  -RG + -RLongLW + 2-Rshort LW  :  '  (4.16)  Thus the ratio of the corrected resistance change due to the resistance change of the lead wire to the resistance of the strain gauge including the lead wires is: L\R  G  AR, G indicated  Short LW corr.  RG + i?LongLW + 2i?ShortLW  R  T  G  + -RLongLW + 2i?ShortLW  (4.17)  Ai?short LW RG + -RLongLW + 2-RshortLW  Adjusting now for the total lead wire resistance gives the ratio of resistance change to gauge resistance corrected for all lead wire errors:  •G LW corr.  A f i p Short LW corr. ^ G + ^ L o n g LW+2-Rshort LW  1 — PAMRL  (4.18)  4.4  Thermal Residual Strain Measurement Procedure  100  where -ftLong LW _|_ 2 R  PAMRL =  L  1  ,  W  J t L o n LW S  2  ,  S h o r t  g  L  W  (4.19)  G  2 K h o r t LW S  V  Ra  '  Note that only the resistance for one long lead wire is used in this equation due to the three wire system employed between the terminals and the data acquisition system. The resistance for both short lead wires is required based on the two wire system between the terminals and the strain gauges.  Although the three lead wire system connecting the terminals to the data acquisition unit offers temperature compensation, the absolute resistance of the lead wire increases with temperature. In equation (4.19) only the lead wire resistance at room temperature was used and thus introduces a small second order error. In order to maintain temperature compensation, special attention was brought to the arrangement of the long lead wires at the oven exit. A l l long lead wires were cut to an identical length (2.25m) and arranged to have an identical length inside the oven.  Based on the information provided by the strain gauge manufacturer, the gauge factor was adjusted with respect to the temperature based on the following:  SG|  t  =  5  G  |  T  =  2  4  o  c  + ^ ( T - 24°C)  (4.20)  The strain at the strain gauged location was then determined using:  j  ^Specimen I x —  DQ|  i X  f  V  ^Specimen LW corr.  r> -KG  A 7? RQ  )  (4-21)  T  Note that the indicated thermal strains from the titanium silicate specimen were corrected for the true thermal strain of the titanium silicate. The next step included the correction for transverse sensitivity thus giving the true thermal strains. The transverse sensitivity  4.4  Thermal Residual Strain Measurement Procedure  101  correction for the rosettes is carried out using (4.3) through (4.6) identically to the correction used for the total residual strains. The transverse sensitivity correction for single element gauges mounted inside the boron/epoxy patch or at the interface between the film adhesive and the boron/epoxy were corrected using the transverse strains measured on top of the patch. This approach is reasonable due to the similar magnitude of thermal strains measured on the aluminum sheet and on top of the patch perpendicular to the fiber orientation. The maximum difference in thermal strains between the aluminum and the transverse direction of the boron/epoxy was 175 /ie at — 56.5°C thus resulting in a possible error in the transverse thermal strains of only 2fie. Finally the thermal residual strains for the gauges mounted on the aluminum and on the boron/epoxy are as follows:  ^ A L T R S I T  ^ B / E P T R S I T  ^Al I T = S F T ^ B / E  P  _  I T = S F T  C  —  A M R L IT=SFT  C  ~~  A M R L I T = S F T  C  —  A I I T  ^ A M R L I T  ^ B / E P I T  " ^ ^ A M R L I T  (4.22) (4.23)  Error estimates indicate that the maximum variation due to hysteresis during these tests was 31 lie with an average of only 9 lie, which does not include stress relaxation effects while holding at elevated temperatures during the experiments. It was noted that an error of 1% in the gauge factor leads to a change of up to 8 tie in the thermal residual strain while the maximum uncertainty of 10 mm in the length of the lead wires connecting the strain gauges with the terminals will result in an error of only 4 fie. A n error in the measured resistance change with temperature of the lead wires connecting the strain gauges to the terminals has a significant effect on the thermal strain of the 120O gauges. A n error of up to 30 /ie was observed for a 6% change in the resistance change with temperature. The maximum error for the 350Q was only 5 jie due to the higher gauge resistance and the shorter lead wires. Errors regarding the absolute resistance of all lead wires can be neglected. An error of 2 ° C in the stress free temperature was found to have only a small effect (approximately 1/ze).  4.4  Thermal Residual Strain Measurement Procedure  4.4.4  102  Results  Figure 4.15 shows the location of the installed strain gauges as well as their numbers. The complete record of the corrected results obtained for the thermal and thermal residual strains is presented in Appendix E . The thermal strains were plotted in addition to the thermal residual strains to show the relative magnitude between them. Figure 4.16 shows the thermal residual strains based on the temperature difference between the measured stress free temperature (101.2°C) and ambient temperature (21 °C).  The more severe  case with respect to thermal residual strains for an operating temperature of — 56.5°C is summarized in Figure 4.17, which is equivalent to the altitude range between 11050m and 20000m based on the ICAO standard atmosphere [49].  The thermal residual strains presented in Figure 4.16 and Figure 4.17 are averaged results based on data obtained from the heating and cooling cycles as well as repeated measurements. Holding for extended periods of time at elevated temperatures caused detectable strain change due to stress relaxation in the bonded repair, especially perpendicular to the fiber direction (on average < 20 pe with maximum of 90 pe at the tapered end in 55 minutes at 80 °C). All of the displayed data were corrected for any stress relaxation which occurred while holding at elevated temperatures  Appendix E contains plots of the measured thermal strains as well as thermal residual strains for all strain gauges. As an example, the typical thermal residual strain characteristic versus temperature is discussed for gauges mounted at one strain gauged location. Figure 4.18 shows the thermal residual strains versus the temperature at specimen location 'B' (see Figure 4.16) close to the crack tip. The curves show the thermal residual strains in both the longitudinal and transverse directions of the specimen for the aluminum top surface and for the top surface of the boron/epoxy patch in the fiber direction.  Figure 4.15: Gauge locations  4.4  Thermal Residual Strain Measurement Procedure  104  Figure 4.16: Thermal residual and total residual strains at ambient temperature due to an elevated temperature cure  4.4  Thermal Residual Strain Measurement Procedure  105  Figure 4.17: Thermal residual strains at — 56.5°C due to an elevated temperature cure  4.4  Thermal Residual Strain Measurement Procedure  106  The thermal residual strains for the aluminum in the longitudinal direction decrease nearly linearly between —56.5°C and approximately 8 3 ° C . Above 9 0 ° C the change in the thermal residual strain with temperature is small, i.e. the boron/epoxy patch and the aluminum expand nearly at their unrestrained rates.  1500 Aluminum (Longitudinal) —  1000  Aluminum (Transverse) Boron/Epoxy (Longitudinal)  3  c 'ra  500  -500  -1000  -1500 -60  20  40  120  Temperature [°C]  Figure 4.18: Thermal residual strain versus temperature  The thermal residual strain for the aluminum in transverse direction is nearly constant between —56.5°C and —20°C and then slowly increases up to approximately 15°C. The thermal residual strain gradient remains approximately linear beyond this point up to 83°C. The thermal residual strains for the aluminum in the transverse direction remain near zero above 90°C.  The thermal residual strain in the fiber direction versus temperature curve for the top surface of the boron/epoxy is initially linear from — 56.5°C to approximately 6 0 ° C . At this temperature the slope of the curve changes to a slightly steeper gradient before the slope changes significantly at approximately 83°C. Above 8 7 ° C the residual thermal strain in the boron/epoxy changes only slightly with trends corresponding to those of the aluminum.  4.4  Thermal Residual Strain Measurement Procedure  107  The cause for the change of the thermal residual strain gradient with temperature of the boron/epoxy patch above approximately 60 °C is not conclusive based on the obtained measurements.  It should be noted that for other strain gauged locations in or on the  boron/epoxy patch, the slope above 60 — 65°C changes in a slightly different fashion. Figure 4.19 shows a superposition plot of the gauges mounted in the central section of the patch, i.e. locations B, C , D and H. Location A and G being near the free edge of the patch were excluded due to their higher sensitivity to creep.  All plots for the thermal  residual strains versus temperature for the boron/epoxy except for gauge 25 seem to merge between 60°C and 70°C before some spread is visible at higher temperatures. A possible cause might be the matrix behaviour of the boron/epoxy prepreg with temperature which will also experience a change in shear modulus as well as increased creep with high temperatures. The situation is additionally complicated by the penetration of F M 73 into the boron/epoxy, resulting in the presence of two epoxies with different thermal properties in the patch. It should be noted that the thermal residual strain versus temperature plot for gauge 25 shows a nearly constant offset (see Figure 4.19) in comparison to the general bottom envelope of the other curves. Since the thermal residual strain versus temperature gradient change near 60°C seems both location and load dependent, as well as of reasonable small magnitude, the slight non-linearity at elevated temperatures has been neglected with respect to the development of a predictive model for thermal residual stresses. Additional research is required to determine the thermal behaviour of boron/epoxy under shear load transfer at elevated temperatures. The focus of this research work is on the development of an improved, simple to use predictive tool as well as experimentally established guidelines for the estimation of thermal residual stresses allowing composite patch designers to include this effect as part of a generic design and analysis procedure for composite repairs on metallic structures.  Some discrepancy between the measured and predicted thermal  residual strains should be expected due to this simplification.  4.4  Thermal Residual Strain Measurement Procedure  108  Temperature [°C]  Figure 4.19: Thermal residual strain versus temperature for strain gauges mounted in or on the boron/epoxy patch at locations B, C, D and H The thermal residual strain versus temperature for the transverse direction on top of the boron/epoxy patch at location B is nearly constant between — 56.5°C and 60°C. The curve slowly approaches zero thermal residual strains with increasing temperature.  Only the  strain gauges oriented in the transverse direction of the specimen near the free edges show a more linear behaviour as well as an increase in magnitude (see Figure E.38 and Figure E.44). This behaviour can be attributed to the high thermal residual strains in fiber orientation in combination with in-plane bending at these particular locations.  The overall small  magnitude of transverse thermal residual strains can be attributed to the small difference in the coefficients of thermal expansion between the aluminum and the boron/epoxy in transverse direction, i.e. around 2 (j,e/°C.  It should be noted that earlier published results [2, 3] show small deviations (on average 7 jie) in comparison to the data shown in Figure 4.16, mainly due to the implementation of additional experimental results of residual strain measurements for the temperature range between 21 °C and - 5 6 . 5 ° C .  The resistance change with temperature for the lead wires  connecting the strain gauges with the terminals changed by 6% based on a regression analysis over the entire temperature range. The largest change of up to 30 pe occurred  4.5  Discussion and Estimation of the Process Induced Strains  109  for the thermal residual strains for the rosettes mounted on the backside of the aluminum. For most of the other strain gauges, the change is significantly less as indicated by the average given above. In addition, the analysis was improved by using a localized polynomial regression analysis [92] instead of the localized linear interpolation, an implementation of the true thermal strains of the titanium silicate as well as distinguishing between the thermal strains for the aluminum in rolling and transverse directions.  4.5  Discussion and Estimation of the Process Induced Strains  The thermal residual strains at ambient temperature for the aluminum within the center section of the repair (specimen locations B , C , D and H in Figure 4.16) vary in fiber direction of the patch between 491 fj,e and 686 //e with an average of 541 fie (standard deviation of 14%). The residual strains due to the elevated temperature cure are very close to this result with an average of 546 fj,e (standard deviation of 16%), which indicates that the process induced strains present are essentially negligible in this direction. In contrast, the average residual strain due to the elevated temperature cure measured in the aluminum perpendicular to the fiber direction of the patch at these same locations (-164 [ie with a standard deviation of 44%) is significantly higher in compression than the thermal residual strain average of -111 pe (with a standard deviation of 20%) indicating that process induced strains are substantially higher in this direction. Unfortunately, the possibility of residual strain relaxation during the first few thermal cycles cannot be investigated here.  The thermal residual strains for the boron/epoxy patch at these four locations decline from an average of -638 fie at the patch adhesive interface to -572 fj,e on top of the patch (standard deviations < 6%).  It is expected that the local increase in patch thickness  due to the placement of strain gauges is one of the influencing factors in the measured through thickness strain gradient.  The finite element analysis in Chapter 6 provides a  4.5  Discussion and Estimation of the Process Induced Strains  110  better understanding of the implications of placing a number of strain gauges at a particular location. It is important to note that the initial strain state of the boron/epoxy patch cocured with one layer of F M 73M was not identical to the seven ply boron/epoxy specimen which was used for the thermal residual strain measurement thus not allowing identification of any process induced strains. The peel ply was removed after the initial measurement and the bottom of the patch grit blasted, both of which will have an effect on the initial strain state. The cause for the very large discrepancy in the measured thermal residual strains and residual strains due to the elevated temperature cure in the boron/epoxy at specimen location A has not been identified. A measurement error cannot be ruled out.  The maximum thermal residual strains found at the strain gauged locations at room temperature were close to 690 yue at the aluminum/adhesive interface (see specimen location B ) , which is approximately 15% of the aluminum yield strain and should therefore be carefully considered for the determination of the margin of safety of a bonded repair. The thermal residual strains in the boron/epoxy exceeded -740 \xt (see specimen location G) which may be a concern for repairs under compression loading. These thermal residual strains are increasing to 1500/xe for the aluminum in the fiber direction at low temperatures (associated with high altitude flight), which is about 33 % of the yield strain. The thermal residual strains for the boron/epoxy reach — 1720/^e at — 56.5°C (location G) while the peak value —1808/ue of for the boron/epoxy is reached at location A .  In summary, the results provided here for a bonded repair utilizing F M 73M indicate that using the difference between the curing and ambient temperature as input to a thermal residual strain model may substantially over-estimate the thermal residual strains present at ambient temperature in the repaired structure. Approximately 855 pe are predicted using a conventional closed form solution for the uncracked metallic component of a reinforced specimen [18] based on a temperature differential between curing temperature and ambient  4.6  Effective Stress Free Temperature  111  temperature, which is nearly 60 % higher than the measured average of 541 jie for the cracked aluminum within the center section of the A M R L specimen repair. Unfortunately, the more general question of the real benefit of a lower temperature cure (80 °C) for F M 73M cannot be answered at this point since more detailed experiments are required to show the thermal residual strain versus temperature behavior. Very recently Walker [97] provided some basic information about the thermal residual strains measurement at A M R L on the F111 wing skin test specimen. Unfortunately no plots for the thermal residual strain versus temperature behaviour are available for this at 80°C cured specimen.  Furthermore, the  effect of temperature on crack growth retardation in the aluminum must also be considered in any comparison to assess the actual benefits of lower curing temperatures.  Finding this significant difference between the measured thermal residual strains and the basic one-dimensional analysis, an improved experimentally determined 'effective stress free temperature' is discussed as a reference temperature instead of the commonly used curing temperature in the following section.  4.6  Effective Stress Free Temperature  The experimental results suggest that the thermal residual strains can be effectively estimated using a linear relationship for the temperature range of interest. Non-linear effects are mainly limited to temperatures above 80°C, where the thermal residual strains are close to zero. The temperature range above approximately 70 — 80° C should preferably be avoided as an operating range for real bonded repairs utilizing a curing temperature of 121°C due to a significant reduction in the shear modulus of the adhesive. Current research at A M R L [22, 97] is focused on the performance of bonded repairs at temperatures above 80°C. It should be noted that the A M R L has been investigating dynamic loading at frequencies of 0.5 Hz and 5 Hz, while the thermal residual strain measurements and related  4.6  Effective Stress Free Temperature  112  adhesive characteristics were carried out quasi static in comparison. Based on the experimental evidence that the adherends expand with temperature at nearly their unconstrained rate above 87°C to 90°C, the definition of an 'effective stress free temperature' based on a linear regression analysis for the temperature range between 21°C and 70°C (Figure 4.20) allows one to employ elastic theory, which permits easy modification of the current models and avoids the need for determination of visco-elastic properties and shear modulus characteristics at elevated temperature for currently used film adhesives.  Note that only the  shear modulus at the operating temperature is required assuming elastic behaviour for the adhesive.  Using the outlined approach, an effective stress free temperature of 85.8°C with a standard deviation of 2.1°C was determined using all gauges oriented in fiber direction except gauges near the free edge. The small standard deviation reflects the quality of the measurement as well as the strong nature of investigated effect.  1500  20 40 Temperature [°C]  60  8Q  I  100  120  ffecnve Stress Free Temperature  Figure 4.20: Effective stress free temperature  The difference between the measured stress free temperature and effective stress free temperature is characterized by the significant change in the adhesive character with temperature. A large nearly linear change in thermal residual strains with temperature is dominant  4.6  Effective Stress Free Temperature  113  up to approximately 80°C. The significant reduction in shear modulus around 80°C seems to be the major cause for the small gradient of the thermal residual strains with temperature above 80°C. In addition, significant strain relaxation, especially at the tapered edges, is occurring above 80° C. Note that the stress free temperature measurement was held for approximately 30 minutes at 101.2°C which should have allowed for strain relaxation to occur. Similarly the A M R L specimen was held at 80° C and 107° C for extended periods of time during which strain relaxation was measured. More experimental work needs to be carried out to improve the understanding of the adhesive behaviour under thermal residual stress loading at elevated temperatures close to the curing temperature. Research at A M R L is currently being carried out to investigate F M 73 adhesive characteristics above 80°C especially under dynamic loading. This research has established that a linear-elastic approach to thermal residual strains below 80° C is acceptable for the utilized material combinations using an effective stress free temperature instead of the commonly used cure temperature.  A more generic specimen might be desirable since the sandwich type A M R L specimen is not the ideal specimen for determining adhesive behaviour due to its complexity. In order to eliminate any non-linear behaviour of the composite patch, a tungsten patch should be chosen instead of the boron/epoxy patch. Tungsten also has a coefficient of thermal expansion of 4.5 P P M / ° C (i.e. very similar to the boron/epoxy) and is available in a 1 mm thick foil. A 6.35 mm thick aluminum plate can be used as the center part of the specimen as outlined in Figure 4.21.  Tungsten FM 73M Aluminum FM73M Tungsten Figure 4.21: Effective stress free temperature test specimen  4.6 Effective Stress Free Temperature  114  The adhesive behaviour can be monitored in the autoclave by bonding strain gauges to the adherends prior to assembling the test specimen.  Due to isotropic adherends, the  number of strain gauges can be reduced to one gauge mounted on the aluminum and one on the boron/epoxy. The gauges should face the film adhesive on different sides of the aluminum core therefore reducing the disturbance of the strain distribution in the adhesive. Curing the strain gauge glue at 150°C will allow proper strain readings during the entire autoclave process.  In order to measure thermal residual strains, identical strain gauges  need to be placed on an unrestrained aluminum and tungsten piece. Additionally identical strain gauges are required on a titanium silicate specimen to determine the true strains in the adhesively bonded specimen. This measurement setup should allow a more refined determination of the effective stress free temperature than presented here for the A M R L specimen.  If measurement during the actual cure (autoclave, vacuum bag or positive pressure) of the specimen shown in Figure 4.21 is not possible, the measurement setup as used for the A M R L specimen can be employed. In this  second single sided specimen is  required to determine the stress free temperature. The use of a tungsten plate instead of the boron/epoxy is also suggested in this case to avoid influences from the composite  material (see Figure 4.22).  Tungsten FM73 Aluminum  Figure 4.22: Stress free temperature test specimen  The establishment of an 'effective stress free temperature' presents the link to an improved theoretical model for thermal residual strains. It is now possible to use a linear-elastic  4.6  Effective Stress Free Temperature  115  model, which should correlate significantly better than previous models due to a better understanding of the adhesive behaviour. Note that not only the theoretical approach (as discussed in the following chapter) but also any previously developed model will benefit from the thermal residual strain measurements carried out as part of this research work leading to the proposed effective stress free temperature. The attention to details in the experimental work has proved again to be the key to good results.  5  Theoretical Analysis of Thermal Residual Stresses in Bonded Repair Specimens  116  Chapter 5 Theoretical Analysis of Thermal Residual Stresses in Bonded Repair Specimens  5.1  Introduction  This chapter presents a closed form analytical approach for the the stress distribution in bonded repair specimens. In order to be cost effective, data from small test specimens rather than from full scale tests are used to predict the behaviour of real repairs. However, while bonded repair applications on aluminum aircraft structures are usually characterized by significant restraints, bonded repair specimens may differ substantially due to having an unrestrained substrate during the curing cycle. The effect of thermal residual strains is significant in patching applications on unrestrained substrates, where the effective coefficient of thermal expansion of the substrate varies largely from the patch. The objective of the current work is to provide a closed form analytical approach to allow the designer to relate data from small test specimens to real applications. It will be evident that the the use of test data from unrestrained specimens is conservative, but the key here is to determine the true stress intensity factor under which the specimen is tested and therefore give a better estimate of the margin of safety for a real repair.  First, a one-dimensional linear-elastic analysis is presented for bonded reinforcements and double-doubler joints with uniform thickness based on Hart-Smith's work [42]. A significant  5.1  Introduction  117  achievement in the current research was the development of an algorithm which allows the direct determination of the elastic stress distribution for a tapered patch without the usual requirement for a specialized numerical code, thus making this approach more accessible. A correction for the shear-lag in the adherends as presented by A M R L researchers [18, 52] was included to improve Hart-Smith's one-dimensional analysis.  The Rose model was then used to determine the stress intensity factor due to thermal residual stresses. The model, as presented by Rose [78, 79], was modified to determine stress intensity factor for the A M R L specimen due to the thermal residual stresses for a highly anisotropic patch. A two-dimensional approach was taken to estimate the error of the simpler one-dimensional model. Additionally the model was modified to account for an edge crack.  In order to improve the theoretical predictions near the crack tip, a stress field model was introduced for the determination of the stresses in the crack tip region. One of the key achievements is the development of a concise solution for the stress field of a center crack in an infinite plate which can be used as a predictive tool for bonded repairs. The complex stress field approach solves this classical fracture mechanics problem in a unique way replacing the requirement to use a large number of boundary conditions to determine a sufficient number of coefficients for an accurate result, or the more widely employed approximations with their limitations.  An additional contribution to the analysis of bonded repairs was achieved by developing a more generalized version of the Rose model, which accounts for disbonds based on Baker's initial approach to the problem [13]. The model can account for all disbond lengths including a complete disbond. This extension shows most clearly the severity of thermal residual stresses and should become an important tool for the initial assessment of bonded repairs.  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 118  The generalized Rose model maintains the simplicity of the original Rose model as one of its features. This approach was further explored by accounting for the tapered edge of the patch, which was achieved by using the direct solution for tapered patches developed in the first part of this chapter. An excellent model which characterizes the impact of thermal residual stresses on bonded repairs especially for test specimens was developed by combining all these aspects.  A  detailed finite element analysis as presented in Chapter 6 will generally be required to study the stress distributions for most repairs in more detail, while the presented theory based on the work of L . J . Hart-Smith and researchers at A M R L gives a good estimate on the feasibility of the repair.  5.2  One-dimensional Analysis of T a p e r e d Double-doubler and  Joints  Reinforcements  One of the problems associated with adhesive bonding is the complexity of the stress analysis of utilized joints and reinforcements, and has lead to the development of specialized computer programs addressing a number of joint configurations [4, 39, 58, 68, 74, 93, 94, 100]. Unfortunately these codes are not widely available thus limiting the use of adhesive bonding technology.  In an effort to reduce the dependence on these numerical solutions, this  section contains a linear-elastic analysis of a number of joint and reinforcement configurations based on Hart-Smith's one-dimensional approach without the need of such programs thus giving a basic tool for the determination of the stress distribution in bonded repairs. A double symmetric bonded repair can be reduced to sections which can be represented as an idealized double-doubler joint, i.e. where the patch bridges the crack, or a double-sided reinforcement in order to get a reasonable estimate of the stress distribution.  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 119  Hart-Smith extended the classical elastic solution for double-lap joints first published by Volkersen [96] in 1938. Volkersen's solution, as well as the similar one by de Bruyne [32], included only adherend stiffness imbalance but not thermal mismatch. Hart-Smith's [42] derived relationships accounted for both adhesive plasticity and thermal mismatch thus allowing a better prediction of the load carrying capacity of adhesively bonded joints. He also analyzed the shear stress distribution in both scarf and stepped-lap joints [43] commonly used in aircraft structures. The numerical solution for stepped-lap joints proved to be difficult due to the non-uniformities in the load transfer especially iri the outer three steps of this joint type [43].  One of the achievements presented in this section are the easy-to-use linear-elastic solutions for tapered double-sided reinforcements and tapered double-doubler joints as well as stepped-lap joints. The derivation of the adhesive shear stress and the adherend normal stress distribution for uniform thickness double-doubler joints and double-sided reinforcements based on Hart-Smith's one-dimensional approach is reiterated in detail thus providing the foundation for the investigation of tapered double-sided reinforcements and tapered double-doubler joints. In this analysis, stepped-lap joints are treated as tapered doubledoubler joints with changing thickness of the substrate. A correction for shear-lag in the adherends is also given, which can be important when using composite adherends with a low shear modulus.  It is important to note that an elastic-plastic analysis such as given by Hart-Smith [42] is required for the determination of the load bearing capacity of a joint or repair, while for many problems such as patch design within the elastic limits or the determination of thermal residual stresses and strains due to the curing of the adhesive at elevated temperatures, the one-dimensional linear-elastic theory will generally be sufficiently accurate.  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 120  5.2.1  G e n e r a l A p p r o a c h for D o u b l e S y m m e t r i c Joints a n d forcements  with Uniform  Rein-  Thickness  The linear-elastic stress analysis for double-sided reinforcements and double-doubler joints having uniform thickness substrate and patch materials (see Figure 5.1) represents the foundation for the evaluation of double symmetric tapered reinforcements and double-doubler joints. Note that the double-doubler joint can be represented by two double-lap joints. The presented derivation is an extension of Hart-Smith's elastic solution [42]. The equilibrium  z r —  1 ft  1  (a)  1  (b)  Figure 5.1: Reinforcement (a) and double-doubler joint (b) with uniform substrate and patch thickness  equations and boundary conditions, including stiffness imbalance and thermal mismatch of the adherends, are derived using the free body diagram in Figure 5.2 in conjunction with the chosen coordinate system as shown in Figure 5.1. For these configurations, the two differential equations for the horizontal force equilibrium are given by: d7>)  and  T(w) = 0  dy  a  dT ) s(;/  dy  + 2r (y) = 0 a  (5.1)  where T (y) = a- (y)t p  p  p  and  T ( ) = cr (y)t s  y  s  s  (5.2)  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 121  The compatibility equations for the adherends and the adhesive can be expressed as:  dy  Eptp  dv (y)  T (y)  s  Cs(l/) =  (5.3)  s  +  dy  a AT  (5.4)  s  (5.5)  T*(y) = (r 7a(y) ~ — (Up(w) - V (y)) a  a  Note that the Young's modulus along the axial coordinate of the joint or reinforcement is used in the case of composite materials. Differentiating equation (5.5) with respect to y and substituting equation (5.3) and (5.4) yields: dr ( ) _ G a  tf  dy  t  ±  a  ( dvpjy) _  dv ( )\  V dy  dy  t& \Eptp  a  y  )  (« -a )AT s  Et  .  p  s s  The differential equation governing the adhesive shear stress distribution is derived by differentiation of equation (5.6) and substitution of the equilibrium equations (5.1) thus obtaining: d r (j/) 2  a  <iT (y) dT (y) dy dy dy \ Et p  G  :  dy  2  s  1 +  ta. V Eptp  R H  Et  r (y) . a  s s  T (y) p  * (y)  A  a  y  7  L  ^a(y) A y •  T (y) s  t (y) a  A  dT (y) T (y) ^^-Ay dy s  s  +  y  \ t (y) A y a  T (y) p  Figure 5.2: Force equilibrium in a bonded joint  (5.7)  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 122  The elastic shear stress distribution parameter A is then defined as:  £  (-Epip  a  E ts)  ^ ^  s  where the characteristic load transfer length of a joint is defined as A  - 1  . Using this para-  meter, equation (5.7) is rewritten as: d r (2/) 2  a  - A r (y) = 0 .  (5.9)  2  a  dy  2  The general solution for this characteristic differential equation is: ( ) = A sinh(Ay) + B cosh(Ay) .  T&  (5.10)  y  The unknown coefficients A and B are then evaluated using boundary conditions of the stress resultants T ( ) p  equations (5.1).  and T ( ), which are given through integration of the equilibrium  y  s  y  f A B T (y) = / r (»)dy = - cosh(Ay) + — sinh(Ay) + C p  a  /  (5.11)  p  IA 2,B -2r (j/)dy = — — cosh(Ay) - — sinh(Ay) + C a  s  (5.12)  The two integration constants C and C can be determined using the overall force equis  p  librium: P = 27>) + T (y) = 2C + C s  P  (5.13)  s  and the first derivative of the shear stress r (y) with respect to y, which can be equated to a  equation (5.6):  d^O  =  A  X  c  o  s  H  X  y  )  h{\y)  +  BXfAn  (5 14) _ Ga. / T (y) _ T (y) _ ^ p  ta. \ E/ptp  s  _ ^  ^rp^  Et  s s  Substituting the relationships for the stress resultants (5.11 and 5.12) and equation (5.13) into this equation yields: AXcosh(Xy)  + BXsmh(Xy)  = ^ (-J— + - = M (Acosh(Xy) *a^ \ &ptp / + —- (——— + ^ | (j ^a \Eptp EtJ ( P . . . _\ P  s s  +  Bsmh(Xy))  (5 15)  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 123  Solving for C gives p  rfr- + (a a )AT = ^ _/( I/ptp Ests J s  C  P  p  (5-16)  +  while C can be determined using the overall force equilibrium equation (5.13). s  p  2{a -  a )AT  s  C =^  p  1  s  y Eptp  .  (5.17)  Et J s s  Note that these two integration constants are independent of the actual boundary conditions and thus apply for both double-doubler joints and double-sided reinforcements. Using these derived relationships, the equations associated with each joint type are thus generated based on their specific boundary conditions.  5.2.2  Double-sided Reinforcements  with Uniform  Thickness  The boundary conditions for the stress resultants of a symmetric double-sided reinforcement having a constant thickness substrate are evaluated at the center of the substrate and at the edge of the patch as follows: Boundary condition at the center r  8  („) = T (-„)  (5.18)  8  Boundary conditions at the edges T (y=z) = 4 - cosh(AZ) + ^ - sinh(AZ) + C = 0 A A  (5.19)  A TD T (y=-o = — cosh(-AZ) + — sinh(-AZ) + C = 0 . A A  (5.20)  R  p  and R  R  R  p  Since the boundary condition in the center as given by equation (5.18) can only be fulfilled if B  K  = 0  (5.21)  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 124  equations (5.19) and (5.20) representing the boundary condition at the edge of the patch must be identical due to symmetry and thus allows determination of the second coefficient A  R  for all y:  The shear stress and adherend stress distributions for a constant thickness double-sided reinforcement can then be written as:  T  :  . ^ ; ^ f \ ^ / _j |_ _ 2 \ cosh(At) V Eptp Egts J a  w  =  p  Et  + (a -  a )AT  s  cosh(Ay)\  p  s s  cosh(At') J  »,,_o I£  +  (  a  ^  \E t p  n  a  p )  A  r  Etj s s  \ Eptp  Ests J  -l<y  <l.  l  ( 5  .  2 3 )  (5.24)  cosh(Ay) /  (5 25)  where (5.26)  Having determined the shear stress and adherend stress distributions for constant thickness double-sided reinforcements, the stress distributions for double-doubler joints can be derived in a similar manner.  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 125  5.2.3  Double-doubler Joints with Uniform  Thickness  The boundary conditions for the stress resultants of a double-doubler joint are evaluated at the center of the substrate and at the edges of the patch as follows:  Boundary condition at the center  T (y=o) =  —+ C = 0  D  s  (5.27)  S  (Note that this boundary condition assumes that the two substrate sections don't come into contact, which might occur under remote compression loading or if (a — a )AT s  p  > 0. Under  these two conditions the double-doubler joint would act like a double-sided reinforcement.) Boundary condition at the positive edge  (y=i) = — cosh(AZ) + — sinh(A/) + C = 0 A A  (5.28)  p  The boundary condition in the center (equation (5.27)) can be used to solve for A : U  A° = ±C .  (5.29)  S  The second coefficient B° can now easily be determined by substituting A° in equation (5-28)): 2 tanh(AZ)  sinh(AZ) '  '  k  '  The shear stress and adherend stress distributions for the positive section of a uniform thickness double-doubler joint are now expressed as: A / 4 " % - « p ) A r W (*V \E ptp  A  P , + ^sts E S T S  + ^r) EtJ  I /  s s  - «P)  V Eptp  A  Et J s  s  T  cosh(Ay)\ V  —  tanh A0; (  (  \ cosh (Ay)  sinh(A/)  5  3  i  )  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 126  ^  l ^ ~  2  (  a  s  ~  p )  a  A  T  \ (,  ,  M  s , sinh(Ay)\ (5.32)  + 2  ^  + (a - a ) A T \ sinh(Ay) s  p  y Eptp  sinh(A/)  Et j s s  (5.33) [ i f e + (a, - Q ) A r \ P  /  _ sinh(Aj/)\  The stresses for the negative section of the double-doubler joint can also be generated using additional boundary conditions: °(-v)  T  = - °(y) r  (5-34)  crpi-v) = <7p(y)  (5.35)  o-si-v) = Osiv)  ( -36)  0 < y < I.  (5.37)  5  where  5.2.4  Interface  C o n d i t i o n s b e t w e e n Steps for T a p e r e d Joints  and  Reinforcements  The relationship governing the stresses in both tapered double-doubler joints and doublesided reinforcements as well as stepped-lap joints can now be derived as an extension of the basic theory for double symmetric joints and reinforcements with uniform thickness. All of the equations ((5.1) - (5.17)) derived in Section 5.2.1 are valid for any step. For each step it is assumed that the elastic moduli for the substrate and patch are identical, while the thickness and length of the substrate, patch and adhesive, as well as the shear modulus  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 127  of the adhesive can be individually specified for each step. These parameters are indexed with the step number in the subsequent analysis. The stepped-lap joint can be treated as a special case of a single side of a double-doubler joint with changing substrate thickness. The step specific indexed coordinate systems for the reinforcements and investigated tapered joints are shown in Figure 5.3 and Figure 5.4. Note that origins of the positive and negative coordinates of the outer steps are separated by the inner step lengths for tapered joints and reinforcements.  In order to reduce computational time, the system of equa-  tions was developed only for positive coordinates. The stresses for negative coordinates are provided based on symmetry. Each step adds two interface conditions to the center and edge boundary conditions. These additional conditions are based on the horizontal force equilibrium of the patch and the shear stress:  T . ( P  fh±l  fk  =  j +1  c o s h  Aj  A  Aj+i  +1  w>+1  =o) = T .(„.=j.)  (5.38)  p  ( . /.) + | i sinh(A^) + ( C . - C . ) A  r.  p  j + 1  (»  Aj  i + 1  = o ) = r».(» .=i .) J  Aj Aj+i  Aj  p  + 1  (5.39) (5.40)  J  Aj i +  where 1 < j < n - 1.  (5.42)  Generally these interface conditions are combined with the boundary conditions at the outer edges (y  n  = ±l ) n  and at the center (yi = 0) and solved numerically. The chosen  approach in this work is based on rewriting the interface conditions between steps so that A  n  and B  n  are functions of Ai and B\. ^  = IAA^  A  n  Ai  + IAB^ Ai  +AC  A  (5.43)  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 128  and  A  = IBA^ + JBB^ + A C Al Al  n  B  •  (5.44)  In order to evaluate f , IAB, IBA and /BB, a substitution scheme based on equations AA  (5.39) and (5.41) is utilized. Using j = 1 in equation (5.39) and (5.41) provides the desired form of equation (5.43) and (5.44) for n = 2. This particular form is achieved for additional steps by evaluating equations (5.39) and (5.41) for j = 1 and substituting them in equations (5.39) and (5.41) with j = 2 thus giving the relationship between A , A\ and Bi as well as 3  between B , A\ and B as follows: 3  x  ^  = ( cosh(AiZi) cosh(A Z ) + ~r- sinh(AiZi) sinh(A £ ) J "T^ V A /A 2  A3  2  2  2  2  X  + (sinh(AiZi) cosh(A Z ) + ^ cosh(A / ) sinh(A / ) ) ^ V 2 J Ai + ( ( C - C ) cosh(A Z ) + (C - C ) ) 2  2  1  1  2  2  (- ^ 5  45  A  P l  P 2  2  Y^- = (^cosh(Ai/i)sinh(A l ) A3  p >  P2  +  2 2  \ A3  2  A  2  A3  sinh(AiZi) cosh(A / ) ] —^ / Ai 2  2  + ( —^ sinh(AiZi) sinh(A / ) + y ^ cosh(AiZi) cosh(A / ) J y^ VA3 A A / Ai 2  2  2  2  2  (5.46)  3  + ((C -Cp )^sinh(A / )). P l  2  2  2  This substitution scheme is continued until j = n — 1 and equations (5.43) and (5.44) are generated. It should be noted that with each additional step the number of terms in f A B , / B A „ J IBB„, n  and AC  ACA  U  BN  fA, A  N  approximately double. Furthermore, each of these  terms is multiplied by up to two additional coefficients for each step.  Repeating patterns of (Aj/Aj+i), sinh(Aj^) and cosh(Aj^) in each of the terms which are combined to f  ,  AAn  f  ,  ABn  JBA , U  / B B „ , ACU„ and AC  BN  can be found. Three functions  (equations (5.49) - (5.51)) were derived for these patterns returning a value of 1 if the particular coefficient (Aj/Aj+i), sinh(AjZj) or cosh(Aj^) exists, otherwise a value of 0 is returned. These functions were based on an expression which is 0, unless S is a multiple of  5.2 2  7 + 1  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 129 , thereby becoming 1. This relationship is: 7+1 ' l + c o s ^ T r )  (5.47)  Yl  e(7,<5) =  {=1  Alternatively the discrete Fourier solution shown below can be used: _ e  Fourier('>'' ) <5  t^  1 + cos(£7r)  T1  1  ^7 + 1  —  ^  /  (5  (5.48)  ,,  These expressions can be shifted by a value of ( in the 6 domain by replacing 5 with 5—( in the previous equations. The desired patterns of l's and O's were then achieved by adding the shifted basic expressions for certain values of C leading to e (~/,6), e\(-y,s) and eAc(7><5)sc  n  7-1  3-2  7+1  e sc (7,<5) ,7-1  C=l+2  n  .7+1  1 4- cos  (^)  (5.50)  ?=i  7  7+1 eAC(7,5) =  (5.49)  7+1  C=l+2  1 + cos  1 4- cos (5.51)  Yl  These functions are then used as an exponent for the (Aj/Aj+i), sinh(Aj^) and cosh(Xjlj) coefficients or as a factor for the change in the integration constants from one step to the next, i.e. {C  PJ  - C . ). p  + 1  The terms /AA, IAB, IBA and f  BB  can actually be expressed using a single function  with different values of (j). The function AC{4>) ACB-  f(<f>)  can similarly be used to express AC A and  Thus equations (5.43) and (5.44) can be rewritten as:  = / ( ^ - i j - r + /(0=O)-T^ + A C V = o ) ^1 Ai  (5.52)  = / ( ^ " - i - l ) - - ^ + /(0=2"- )-^- + A C ( * = 2 » - ' )  (5.53)  ^ A  1  n  where 1  A  Ai  Ai  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 130  It can be proven that the functions /((/>) and AC{<t>) are:  /<*>=£ n n-1  6»=1  l  \ j_  \ e (j,26+4>) x  A  A •3+1  j=l  (5.54)  • (sinh(A / )) - ' +*) e  J  (j  28  7  • (cosh(A /j)) ^- +<*) 1-esc  29  i  n-1  £  *cU,o+*)(C .-C )  e  p  =1 Lj=i L ~l  n  r /  Pj+1  x  (^+7) n  ni(  s i n h  (5.55)  (Axy)  X=3 (sinh(Aj-Zj))  e cO,9+0) S  n-1  II | (cosh(A J )) x  l-e  J C  (x,fl+0)  x  X=7  (coshtA^)) -^ '^ 1  0  Equations (5.52) through (5.55) represent the required additional equations to determine the stress distributions for the various stepped or tapered joints and reinforcements.  5.2.5  Tapered  Double-sided  Reinforcements  Figure 5.3: Tapered reinforcement The boundary conditions for a tapered double-sided reinforcement as shown in Figure 5.3 are similar to the conditions for a constant thickness double-sided reinforcement as given  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 131  by equations (5.18) through (5.20) but must address the presence of multiple steps (i.e. n) as shown below.  Boundary condition at the center  Ti S  (</i) —  T  (5.56)  (-J/J  sl  Boundary condition at the positive edge (y = ln) n  T {y =i ) = ^  cosh(A„Z ) + ^  K  v n  n  n  sinh(AJ ) + C  n  n  (5.57)  =0  Pn  n  A  A N  The boundary condition in the center can only be fulfilled if:  sr=o.  (5.58)  A can be determined by substituting equation (5.52) and (5.53) into equation (5.57) and X  then simplified using ^  R  _ - A i (C  P n  + AC(^=o)cosh(A Z ) + AC(0=2"-i)sinh(A l )) n  n  n  ra  /(4>=-i)cosh(A / ) + /(</,=2"- -i)sinh(A / )  1  1  n  Having determined A  R  n  n  n  and B , seeded iteration with the seed vector (A* B ) R  R  employed to determine the remaining coefficients A  and B*  R  +1  T  is then  based on equation (5.39)  +1  and (5.41):  A  'A  B  R  K  R  cosh(\jlj) + ^-smh{X l ) j  j  + (C . - C . ) p  p  + 1  (5.60)  A* sinh(Aj/j) + B cosh(AjZj) R  Thus the stresses for the positive sections of the reinforcement are:  T %>°) a  =  i_ (  A  A:  R  sinh(Ai2/i)  +  B*~ cosh(A;y;)  (5.61)  B  (5.62)  o- .(vi>o) = — I - r - cosh(Ait/i) 4- - p sinh(Ai2/i) + C . p  p  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 132 1 / 2A 2B o\.{yi>o) = — ( — c o s h ( A ^ ) R  R  R  \ sinh(A y ) + C . J i  l  s  (5.63)  where 1 < i < n .  (5.64)  The stresses for the negative sections of the reinforcement can then be determined using the additional boundary conditions:  rli-Vi)  -r*( )  =  Vi  V-v<) = Vsi-Vi)  = <MiO  (5-65)  ( - ) 5  66  (5.67)  where 0<yi<li.  (5.68)  Having determined the shear stress and adherend stress distributions for tapered doublesided reinforcements, the stress distributions for tapered double-doubler joints can be derived in a similar manner.  5.2.6  Tapered Double-doubler Joints  Figure 5.4: Tapered double-doubler joint  The boundary conditions for a tapered double-doubler joint as shown in Figure 5.4 are similar to the conditions for a constant thickness double-doubler joint as given by equations  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 133  (5.27) and (5.28) modified to account for multiple steps (i.e. n) as shown below. Boundary condition at the center T  D sl(j/l  2A° - ^+ C ,=0 A  =o) =  (5.69)  S  Boundary condition at the positive edge A° T  U p  n  {y =i ) n  =  n  ~  B°  cosh(A /„) +  sinh(A Z ) + C  n  n  n  n  = 0  Pn  (5.70)  n  A  A  The boundary condition i n the center can be used to solve for A°: A*i = ^C B  x  .  s,  s  2  (5.71)  can now be calculated by substituting equation (5.52) and (5.53) into equation (5.70)  and then simplified using equation (5.71): A x ^ - (/(0=-i)cosh(A Z ) + / ( ^ » = 2 " - - i ) s i n h ( A / ) )  _  3  1  n  n  n  n  /(</>=o)cosh(A„Z„) + / ( 0 = 2 » - ) s i n h ( A / )  1  1  n  A i (C  n  + ACw>=o)cosh(A / ) + A C ( 0 = 2 " - ) s i n h ( A / ) )  (5.72)  1  Pn  n n  n  n  f(<p=o) c o s h ( A / ) + / ( i = 2 ' - ) s i n h ( A / „ ) i  n  n  A seeded iteration w i t h the seed vector (A B ) 1  coefficients A°  +1  and B °  + 1  Aj+i  A  u  x  1  ?  T  n  is also used to determine the remaining  : i~ coshiXjlj)  + -jj- s i n h ( A ^ )  +  (C . - C . ) p  p  +1  (5.73) A° sinh(Aj^) + B°  cosh(Xjlj)  Thus the stresses for the positive sections of the tapered double-doubler joint are: T a . G ^ o ) = A sinh(A y ) + B i  i  i  (5.74)  cosh(A j/;)  {  n  1 (A B ffp.(y >o) = T — ( - r - cosh(Aiyi) + -f- sinh(Ajyj) + C,Pi 4  ^p. \  1  1  i  A  2A A,; cosh(Aj|/j)  (5.75)  i  A  2B - - sinh(Ajyi) + C A,-  (5.76)  1  s  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 134  The stresses for the negative sections of the double-doubler joint can then be determined utilizing the additional boundary conditions: M-iO = -  where  r  a \ )  (5-77)  V - * )  =  (  PA-Vi)  = <Mw*)  ( - )  0 < yi < h .  5  5  -  7  8  )  79  (5.80)  The presented one-dimensional analysis provides a good estimation for the stress distribution in the discussed joints and reinforcements. Shortfalls in the one-dimensional analysis include the neglect of peel stresses and shear-lag in the adherends. The shear-lag in the adherends can be included in the results of the one-dimensional analysis by carrying out the following analysis.  5.2.7  Correction for Shear-lag in the Adherends  The importance of including shear-lag effects of the adherends when determining the gap opening displacement between the two substrate sections of a double-doubler joint was shown by researchers at the Aeronautical and Maritime Research Laboratory (AMRL) in Australia [18, 52]. The one-dimensional analysis shows significant error for joint configurations with low shear modulus adherends such as composites. Boron/epoxy has a shear modulus of 7.24 G P a versus 27.2 GPa for 2024-T3 aluminum. This correction to the onedimensional analysis is provided for double-sided reinforcements and double-doubler joints with uniform thickness, but it can also be readily adapted for tapered/stepped joints or reinforcements.  It is reasonable to assume that the shear stresses, which develop in the adhesive, are continuous across the adhesive-adherend interfaces. In addition, equilibrium requires that  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 135  the shear stress  r{ ,z) y  be zero at the free surface as well as at the plane of symmetry. Due to  the general t h i n adherends, i t can be assumed that the shear stress  r{ ,z) y  is linear through  the substrate a n d patch thickness. T h e assumed distribution is shown i n Figure 5.5 [52].  Patch k +t 2  Adhesive  a  k 2  I Substrate *> X(y,z)  t (y) a  Figure 5.5: D i s t r i b u t i o n of the shear stress through the thickness of a double-sided joint or reinforcement Note that the nomenclature of the previously determined shear stress i n the adhesive r (y) a  is kept. In general the shear stress i n a bonded joint or reinforcement  r(y,z)  ,dv{ ,z) y  T(y,z) = Gj(y,z) « Gdz  is given by: (5.81)  Evaluating the shear stress for the patch gives: 1 r (y,z) P  T h e in-plane displacement  ft  dv { ,z) v  = — y-^ + i a + *p - zj  v (y,z) p  Vp{y,z)  y  (5.82)  Gp—^-  r (y) = a  is then determined through integration  -I  dv ( ,z) p  y  -dz  dz  z  (5.83)  it*  Gptp \ 2  +  t  &+  ~2>  tp  Ta(2/)  +  k  (  y  )  w i t h the integration constant h{y) being determined as a function of the midplane displacement  u (j/,z=^+ta+^r) p  of the patch, i.e. 1  '  T  fu — 4- t  Gpt \2 p  4- — 2  + t & +  /  4  +  2  +  4  t  p  J ™  (5.84)  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 136  Substituting h(y) into equation (5.83) and evaluating the in-plane displacement v ( ,z) p y  at  the patch/adhesive interface yields:  Vpiy^f+ti)  =  U p f o ^ + t a + k )  3 t -77-r (») .  -  (5.85)  a  O Up  The in-plane displacement v ( ,z)  at the substrate/adhesive interface can be determined in a  s y  similar manner. Based on Figure Figure 5.5, the shear stress at any point in the substrate is evaluated as:  r (y,z) s  2z = —  dv ( ,z)  ^  r („) = G a  %  s  y  - ^ -  .  (5.86)  The in-plane displacement of the substrate v { ,z) can also be determined through integration s y  as:  {  *'  2  (5.87)  = 7 r r a ( y ) + 9(y) U I r  S  S  where the integration constant g( ) can be assessed as a function of the midplane displacey  ment v (y,z=o) of the substrate, i.e. s  (5.88)  9(y) = v ( ,z=o) . s  y  Substituting g( ) in equation (5.87) and evaluating the in-plane displacement y  v { ,z) s  y  at the  substrate/adhesive interface gives:  w (v,*=£) =  v ( ,z=o)  s  s  y  - 7^-r (») .  (5.89)  a  Applying the assumption of constant shear stress through the adhesive yields:  r (y) a  Q = -p  (y (. ,z=$+tj p  y  -  v (y,z=$)) a  .  (5.90)  The relationship between the midplane displacements of the adherends and the adhesive shear stress can be obtained by substituting equations (5.85) and (5.89) into equation (5.90)  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 137  to give:  Ta(tf)  =  (5.91)  This result can easily be incorporated into the basic one-dimensional analysis by modifying the elastic shear stress distribution parameter A. If equation (5.5) is replaced by equation (5.91), the elastic shear stress distribution parameter A for a constant thickness doublesided reinforcement or a double-doubler joint yields:  (5.92)  The remaining one-dimensional analysis is unaffected by accounting for the shear-lag effect in the adherends. Thus the elastic shear stress distribution parameter Aj for tapered or stepped joints and reinforcements is obtained simply by indexing the appropriate variables.  5.2.8  Results  This section describes the application of the derived equations associated with the various joint and reinforcement types for a specific set of composite patch and metal substrate material properties representing cross sections of the sandwich type A M R L specimen. The honeycomb which is used in this particular specimen to eliminate out-of-plane bending is neglected in this case and the two face sheets are combined as the substrate in the analysis. The stress distributions for uniform thickness adherends versus tapered edges of the patch are then directly compared. Boron/epoxy 5521/4 was chosen as patch material, aluminum 2024-T3 as substrate and FM73M as film adhesive with the following properties:  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 138  Boron/epoxy E  5521/4  (0°-Orientation):  = 210 G P a  p  Aluminum E  FM73M  p  = 4.61  pe/°C  G = 27.2 G P a  a  p  = 23.45 pe/"C  2024-T3:  Film  G  a  p  = 72.4 G P a  s  G = 7.24 G P a  s  Adhesive  = 844 MPa .  &  The applied loads for the reinforcements and the double-doubler joint were chosen identically as thermal loads representing thermal residual stress loading of the A M R L specimen after the curing cycle. The temperature differential is given as the difference between the effective stress free temperature and ambient temperature. Note that the shear modulus at ambient temperature was used in the subsequent calculations. Applied  Load:  AT = - 6 4 . 8 ° C  P = 0 m m  Double-sided reinforcement - U n i f o r m versus tapered To compare the effect of tapering for a double-sided reinforcement, the following geometric parameters were employed. Double-sided  reinforcement  t = 6.35 mm s  with uniform thickness:  t = 0.25 mm a  t = 0.924 mm p  I = 75 mm Double-sided L h  l  t  reinforcement  = 6.35 mm l<i<7 =  57 mm  with tapered edges  U d  = 0.25 mm  l<i<7  h<t<7  =  t  D  P  3  = 1.056 mm—i-0.132 mm  l<*<7  mm  Equations (5.23) through (5.25) as well as equation (5.92) were utilized to calculate the stress distribution for double-sided reinforcements with uniform thickness. To determine  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 139  the stress distribution for tapered double-sided reinforcement, Equations (5.16), (5.17), (5.49)-(5.51),(5.54),(5.55), and (5.58) - (5.68) were applied.  Figure 5.6 through Figure 5.8 show the stress distributions for the double-sided reinforcements with uniform thickness as well as with a taper at the patch edges. High shear stresses occur at the outer edges of the reinforcement while they drop to zero in the center of the reinforcement as shown in Figure 5.6.  The taper leads to a significant reduction in the  shear stress at the edge of the patch. This reduction in the adhesive shear stress at the tapered edge at the expense of a higher compressive stress in the patch is clearly indicated by comparing Figure 5.6 and Figure 5.7. The peak compressive stress in the tapered patch is reached in the outer step and has a magnitude of approximately twice the compressive stress reached in the center of the uniform thickness patch (see Figure 5.7).  The sharp  jumps in the stress distribution for the tapered patch are caused by the change in patch thickness. The thermal load leads to significant tensile stresses in the substrate by applying the patch as indicated in Figure 5.8. The patch taper leads to a reduced stress gradient in the substrate when approaching the outer edge.  Figure 5.6 also indicates a shortfall of the one-dimensional analysis. The shear stress should be zero at the free edge. Although a more complex analysis will be able to predict zero  /  20.0  i  flPa]  15.0 -  «± CO  10.0 -  5.0 -  ft ft u.u -5.0 -10.0 -15.0 -20.0 -  7  -75.0  1  Tapered Edge Reinforcement Uniform Thickness Reinforcement 1  1  r —- 1 — i — i — i — i — I — i — i — i — i — I — i — i — i — i — I — i — i — i — i — I — i — i — i — i —  -50.0  -25.0  0.0  25.0  50.0  75.0  Distance from the center [mm]  Figure 5.6: Adhesive shear stress in double-sided reinforcements with uniform thickness and tapered edges  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 140  -Tapered Edge Reinforcement •Uniform Thickness Reinforcement —i—i—i—i—I—i—i—i-250.0 -75.0 -50.0  H—I-  -I—I—I—h-  -25.0  0.0  -f25.0  -t—i—i—i—i—i-  50.0  75.0  Distance from the center [mm]  Figure 5.7: Normal stress in the patch of double-sided reinforcements with uniform thickness and tapered edges  -Tapered Edge Reinforcement •Uniform Thickness Reinforcement 0.0 4—i—i—i—i—I—i-  -75.0  H—i—I—i—i—i—i—I—i—i—i-  -50.0  -25.0  0.0  -+25.0  H  1 1  50.0  1  1  1 75.0  Distance from the center [mm]  Figure 5.8: Normal stress in the substrate of double-sided reinforcements with uniform thickness and tapered edges shear stress at the free edge and the peak shear stress being reached approximately 1/2 adhesive layer thickness from the edge, the peak shear stress is predicted quite accurately by the one-dimensional analysis.  Double-doubler joint - U n i f o r m versus tapered To compare the effect of tapering for a double-doubler joint, the following geometric parameters were employed.  Double-doubler 4 = 6.35 mm / = 75mm  joint with uniform thickness: £ = 0.25 mm a  t  p  = 0.924 mm  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 141  Double-doubler £ „„ Sl  joint with tapered edges  = 6.35 mm  t  l<i<7  ^  = 0.25 mm  a  1<><7  = 57 mm  h<i<i  =  i  = 1.056 mm-z-0.132 mm  D  1<'<7  3mm  The stress distribution for a double-doubler joint with uniform thickness was found using equations (5.31) through (5.37) in combination with equation (5.92). To determine the stress distribution for tapered double-doubler joints, equations (5.16), (5.17), (5.49)(5.51),(5.54),(5.55) and (5.71) - (5.80) were employed. The stress distributions for double-doubler joints with uniform thickness as well as tapered edges of the patch are presented in Figure 5.9 through Figure 5.11. Peak shear stresses are reached at the joint center and at the edge of the patch (see Figure 5.9) for the applied thermal loading condition. The advantage of the tapering is clearly shown by the reduction in shear stress at the tapered edge. The high shear stress at the joint center is unaffected by the taper. The peak stress in the patch for a uniform thickness joint is reached between the joint center and edge while the peak compressive stress for a tapered patch is reached within the outer step of the patch for this particular loading condition (see Figure 5.10). As for the tapered reinforcement, the stress distribution shows sharp jumps due to the thickness change of the patch.  Figure 5.11 shows that the highest tensile stress in the  substrate is reached between the outer edge and the center of the joint while zero tensile 20.0 -, 15.0 -  To EQ. a  10.0 • 5.0  Tapered Double-Doubler Joint  /  Uniform Double-Doubler Joint  / /  ;  0.0 -5.0  ;  -10.0  ;  -15.0 • -20.0 - —l—l—l—l—1—l—l—l—l—1—l—l—l—l— —i—i—i—i—I—i—i—i—i—I—i—i—i—i— -75.0  -50.0  -25.0  0.0  25.0  50.0  75.0  Distance from the center [mm]  Figure 5.9: Adhesive shear stress in double-doubler joints with uniform thickness and tapered edges  5.2  One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 142  Distance from the center [mm]  Figure 5.10: Normal stress in the patch of double-doubler joints with uniform thickness and tapered edges stress is predicted at the joint center and at the edge. The taper leads to a reduced stress gradient in the substrate when approaching the outer edge. It should be emphasized that the discussed results are specific for the given loading condition. The critical location of the shear stress and the stress in the adherends will be dependent on the applied combination of thermal and mechanical loads.  The presented analysis provides an effective tool to determine the stress distribution in double-doubler joints and double-sided reinforcements. This analysis scheme can now be used to get a first estimate of the stress distribution in bonded repairs applications on cracked structures. The repaired section can be divided into two regions, one containing the 50.0  Distance from the center [mm]  Figure 5.11: Normal stress in the substrate of double-doubler joints with uniform thickness and tapered edges  5.3  A p p l i c a t i o n of the Rose M o d e l for T h e r m a l Residual Stress L o a d i n g  143  crack and the other being essentially a reinforcement. T y p i c a l l y the composite material will be oriented perpendicular to the crack giving a clear line at the crack tip which represents the boundary between the two regions. The region containing the crack is treated as a double-doubler joint, while the second region is analysed as a double-sided reinforcement. In the application of one-sided bonded repairs to cracked a l u m i n u m aircraft structures it is generally assumed that the supporting substructure is sufficiently stiff to prevent out-ofplane bending thus allowing the repair to be treated as a double symmetric case. T h e one-dimensional model does not account for the stress concentrations at the ply dropoffs i n the tapered region which w i l l lead to a localized error i n the stress and strain prediction for the patch.  A d d i t i o n a l l y minor errors w i l l be introduced i n the adhesive  and substrate. T h e finite element analysis on the other hand w i l l overpredict this stress concentration due to modeling this ply drop off as a sharp corner. It is generally prohibitive to model the true microstructure of a composite due to the size of such a model. In order to improve the one dimensional analysis, the influence of the crack t i p stress field has to be included. Therefore the next step is the determination of the stress intensity factor of the patched crack followed by an estimation of the resulting stress field i n the cracked substrate.  5.3  Application of the Rose M o d e l for Thermal Residual Stress Loading i n Unrestrained and Uniformly Heated Specimens  Dr. L . F . R . Rose developed a model for the estimation of the stress intensity factor reduction due to patching. T h e basic model, which consists of two stages, is well explained in his original papers [78, 79].  T h e first stage determines the stress at the prospective crack  location for a bonded reinforcement over the undamaged structure.  T h e second stage  estimates two upper boundaries of the stress intensity factor based on the stress at the  5.3  Application of the Rose Model for Thermal Residual Stress Loading  144  prospective crack location. The stress intensity factor reduction for any given crack length is then determined using an interpolation between these two upper boundaries.  5.3.1  Stage I: Stress at the Prospective Crack Location  In addition to the determination of the stress at the prospective crack location due to remote stress loading using the inclusion analogy [18, 78], Rose discusses the determination of thermal residual stress in an uncracked substrate below an isotropic reinforcement as well [18]. The efforts with respect to thermal residual stresses are focused on accounting for the restraining effect of the surrounding structure. In this particular investigation the analyses considered the thermal residual stress at the prospective crack location of a restrained isotropic circular plate with an isotropic reinforcement [18] with identical Poisson's ratios (v = v = z/ ). The bond was assumed to be rigid. Note that this particular solution s  p  as given by equation (5.93) overestimates the thermal residual stress by approximately a factor of l/{v — 1) for a highly orthotropic patch with no restraints and uniform heating. The thermal residual stress for an unrestrained isotropic plate and double-sided reinforcement with identical Poisson's ratios u = u = v and uniform heating/cooling using Rose's s  v  approach is: 2(a s  a )AT p  (5.93)  Additional work using the Rose model was carried out by R. Fredell who presents a twodimensional solution for the thermal stress and thermal residual stress distributions near the prospective crack location incorporating the anisotropic characteristics of the reinforcement [35]. This approach deals with the thermal stress and thermal residual stress at the prospective crack location for a fuselage section in which the expansion of the skin is restrained at the stringer and frame locations. In applying the approach as presented in [35],  5.3  Application of the Rose Model for Thermal Residual Stress Loading  145  care needs to be exercised with respect to the employed coordinate system and orthotropic material definitions of the patch.  The sandwich type A M R L specimen, which can be idealized as an unrestrained repair with a uniform temperature distribution throughout the curing process, represents a special case of Fredell's approach. Neglecting that it is only a partial reinforcement across the width and length of the specimen, the thermal residual stress at the prospective crack location can be easily determined as follows.  The stress and strain relationship for the isotropic substrate is given by: e  Sx  =  s  =  e  - u a ) + a AT  (5.94)  u a ) + a AT  (5.95)  s  -=-(<7  8y  -  s  5y  s  Sx  s  where E =E s  Sx  =E  and  Sy  u =u s  Sx  =u  Sy  .  The stress and strain relationships for the orthotropic reinforcement with E  (5.96)  Py  u  Pyx  >E  Px  and  being the major Poisson's ratio: 1  e  p  - K c - ^ p J + a p x A T  1 = jr-(-v v  y  Pyx  Px  + a ) + a AT Py  Py  (5.97) .  (5.98)  To simplify the estimation of the thermal residual stress, a rigid bond is assumed based on ° the small load transfer length in comparison to the overlap length (1/A <C I) such that e  s  x  = e  and  Px  e  Sy  = e  Py  .  (5.99)  The equilibrium requirements for a double-sided reinforcement then yields: e t  + 2a t  =0  (5.100)  v t  + 2a t  =0.  (5.101)  Sx  Sy  s  a  Px p  Py p  5.3  Application of the Rose Model for Thermal Residual Stress Loading  146  Substituting the stress-strain relationships (equations (5.94) through (5.98)) into the rigid bond assumptions (5.99) and employing the equilibrium conditions to eliminate the stress components in the patch yields the following linear system of equations.  SX  S x  (-^PX^P  Ests) E ts)  (-Epy^p  E ts)  (^py^p  S y  \^E t  Sy  a  t^  s  E ts)  Py p  Px  ^  ^  ^  ^  ^  ^  s  s  t^  s  s  Py  s  The solution of this linear system of equations represents the thermal residual stresses in a rigidly bonded reinforcement, i.e. (<*  ~  ^  +  T  h. ( (—k 2  K a  Sy  -  \  A  P  E  X  *  I __2_\ £  P  S  +  A)  1  ~  " p  )  A  +  r  \s  P y  tp  ^  £ t s  S y  2 I  l  _2_\  +  -Es*S  \^£p tp x  ( /  i  +  _2_)  _  Et  \^Py*p  s  s  ( "py«  J  Q  4  )  y  /  + —^  Py  ( I  1  I  - ^ P J A T ( j ^ + | t ) + K - * )AT ^— h  It)  L j . U ( ^ L + i )  V^pytp  W  y  ^) .  (5.105)  2ji_A  +  yEp tp  Et  y  s  s  J  Using the following material property characteristics (which can be observed for the typical material combination of unidirectional Boron/epoxy 5521/4 and aluminum 2024-T3)  ^  . > V 2v  Ep tp  EtJ  y  <  s s  (_L_  \E tp Px  (a. - a p J A T ( -?f  +  V-C'Pytp  +  M (_L_ + JL) EtJ  \E tp  s s  ( 5  Et  Py  j « (a. - c* )AT (  + ^Fr )  Py  V^PX^P  A ^ S /  ,„  6 )  s s  (5-107)  &sts  The thermal residual stresses can be expressed as: „  2(o. - O p J A T  2(a -a )AT s  Px  f-^- + — 1 £>s*s/ \ ^ P X * P  a  t y(^— + E—) ^px'p tJ s  as  Sy  (—— + yEpyt  s  „ 2( = .  - a )AT P y  +jfc)  s  p  •  2  (5.108)  Ests  (5-109)  Note that equation (5.109) is identical to the result from the one-dimensional solution at the center of a double-sided reinforcement with 1 / A < I . The loss in accuracy by reducing the two-dimensional analysis to a one-dimensional problem for the given material combination  5.3  Application of the Rose Model for Thermal Residual Stress Loading  147  is well below 1% for the thermal residual stress in the substrate along the fiber direction (i.e. o~ ) which is the main concern with respect to the performance of the bonded repair. In s  addition, the one-dimensional analysis of a double-sided reinforcement allows one to easily include the influence of the adhesive and the associated load transfer zone using equation (5.25). Thus the stress in the fiber direction at the prospective crack location in the center of the reinforced region due to the elevated temperature cure is given by:  (5.110)  5.3.2  Stage II: Upper Bounds for K  v  Having determined the stress at the prospective crack location a , the crack is introduced 0  in the second stage of the analysis. The load which was transmitted across the crack is now partly transferred into the patch and partly redistributed in the plate. The relative magnitudes of this redistribution primarily depend on the crack length, i.e. the longer the crack, the more load is transferred into the reinforcement [78]. The stress intensity factor K  r  for a crack with a crack length a in a bonded repair is  estimated by interpolation between two upper bounds, K  u  respectively. A n upper bound for K  r  and  for short and long cracks  for short cracks can be derived on the assumption  that no load is transferred into the reinforcement [23]:  (5.111)  Rose gave the equations for a center crack, where Y = 1, while the A M R L specimen contains an edge crack. Based on the assumption that the edge crack is small in comparison to the width of the specimen, the correction factor for this geometry is Y = 1.12.  5.3  Application of the Rose Model for Thermal Residual Stress Loading  148  Rose showed that the relationship between energy release rate G and stress intensity factor  K also holds for a reinforced cracked structure, i.e. that the body force field corresponding to the shear tractions does not affect the relation between G and K [78]. The crack in the isotropic substrate will not always be under plane stress conditions as it might be expected in a repair of an aircraft skin. The crack in a bonded repair under only thermal residual stress loading will generally be under plane strain conditions while combined with high external loading it will be under plane stress conditions. The relationships between the energy release rate and the stress intensity factor are given for the crack in the isotropic substrate under plane stress and plane strain conditions as:  G = — G = — E (1 - vl)  (5.112)  (Plane stress)  (5.113)  (Plane strain)  Thus the corresponding energy release rate Gu can be determined using equation (5.112) or (5.113). This upper bound will be close to the correct value for short cracks and represents an asymptote with first order contact in the limit a —» 0 for Kx of an infinite patched plate with a crack [78]. Note that the energy release rate Gu is linearly increasing with the crack length, although Gu cannot exceed the energy release rate Goo f °  r a  semi-infinite crack.  Rose showed that the limiting value for the energy release rate G^ can be adequately estimated using one-dimensional modeling which ignores the stress variation across the thickness [79]. This upper value for the energy release rate can be estimated by the change in potential energy when introducing a cut in the substrate of a double-sided reinforced plate converting the specimen to a double-doubler joint. Thus Goo is the work extracted (per unit thickness and per unit crack growth) on the stress a0 at the prospective crack location relaxed to zero through the displacement v on both crack faces [79]: 0  (5.114)  5.3 Application of the Rose Model for Thermal Residual Stress Loading  149  The stress at the prospective crack location a was determined using equation (5.3.1) in 0  stage I of the Rose model using a one-dimensional model of a double-sided reinforcement. The displacement v as a result of introducing a cut in the double-sided reinforcement can 0  be easily determined using superposition. According to Figure 5.12, the displacement v is 0  P  P  G (y=0)=a s  o  Figure 5.12: Illustration of the superposition principle  given as the displacement of one joint or crack face for a pressurized cut in a double-doubler joint. The general analysis as given in Section 5.2.1 has to be slightly altered to account for the different boundary conditions. The compatibility equations for the adherends and the adhesive are now expressed as: dv (y) _ T (y) p  €p(»)  C (») 8  r (y) = a  p  (5.115)  dy =  dy  GVMy) «  (5.116)  Et  s s  ^(vpiy) - v ( )) B  y  (5.117)  Differentiating equation (5.117) with respect to y and substituting equation (5.115) and  5.3  Application of the Rose Model for Thermal Residual Stress Loading  150  (5.116) yields: C  dT (2/) a  a  / dv (y)  dv (y)  p  s  t V dy  dy  C  a  a  dy , T Et  / 7 » \Eptp  ( 5  aiy)  -  1 1 8 )  s s  The characteristic differential equation remains identical. The two integration constants C and Cp can be determined using the overall force equilibrium s  2T (») + T (y) = 2C + C = 0 P  s  P  s  (5.119)  and the first derivative of the shear stress r (j/) with respect to y, which can be equated to a  equation (5.118) as shown below: - AX cosh(Ay) + BAsinh(Ay) _ C t  a  (Tp  T  {y)  y Eptp  a  s  (  ^  ( 5  '  1 2 0 )  Et  s s  Substituting the relationships for the stress resultants (5.11, 5.12) and equation (5.119) yields AXcosh(Xy)  + BXsinh(Xy)  = % (-J— + J (Acosh(Ay) + Ssinh(Ay)) i A \Eptp EtJ a  s s  ta, \ Eptp  19-n  Et J s a  Solving for C and subsequently for C gives p  s  Cp = C = 0 . a  (5.122)  The boundary conditions for the stress resultants of an internally pressurized doubledoubler joint are evaluated at the center of the sheet and at the edge of the patch as follows. Boundary condition at the center  T („=o) = -a t s  0  s  2A = - —  (5.123)  5.3  Application of the Rose Model for Thermal Residual Stress Loading  151  Boundary conditions at the positive edge A B Tp(y=i) = - T - cosh(AZ) + — sinh(A<!) = 0 A A P  P  (5.124)  The boundary condition in the center as given by equation (5.123) is fulfilled if: A  P  = A(T |  (5.125)  0  p  p  The second coefficient B can now easily determined by substituting A in equation (5.124):  '  B  =  -  ^  S  S  A  ?  (  '  5  1  2  6  )  Thus the shear stress distributions for an internally pressurized double-doubler joint can , w w iw \ „ *s cosh(A / - y)) r (y) = -XG . , — 2 sinh(Xl)  be written as:  a  where  0  ^ 1 0 7 ^ (5.127)  N  0 <y< I  (5.128)  The displacement v at the center of the joint can be evaluated using equation (5.117). 0  Note that the displacement for the patch is zero at this location. In order to determine the displacement v also for the case where shear-lag in the adherends is included, equation 0  (5.91) has to be used. Both cases can be readily combined by using the stress distribution parameter A and the elastic moduli and thickness of the adherends instead of the shear moduli and thickness of the adhesive. t V O  =  V S I Y = 0 )  =  a  at  P  -G-J*  {  0  Y  =  0  )  =  ^  a  {  (  1  E  ^  2 \  +  EIJ ) s  1 X^MM)  (  5  '  1  2  9  Thus the upper limit for the energy release rate can be expressed for plane stress and plane strain conditions as follows [18]. Plane stress conditions Yo 2  Coo = v0v0 -  2  — ^ T T A -E's  (5.130)  )  5.3  Application of the Rose Model for Thermal Residual Stress Loading  152  where A =  ^ (  1  2 7 ^ )  +  Atanh(AZ)  ( 5  '  1 3 1 )  Plane strain conditions Ya Goo = <J v = - = - ^ 7 r A ( l - u ) 2  2  (5.132)  2  0  0  E  s  where A  _  *Y*(1 - v ) (* 2  +  Atanh(A0  2E t ) p p  ( 5  '  Based on the validity of equations (5.112) and (5.113), the stress intensity factor  1 3 3 )  for  this upper bound is: # o o = V sGoo = Ya VirA.  (Plane stress)  E  0  K<x> — \ T—-—~T\ C i)  V  1 _  Ya V^A  =  (5.134)  (Plane strain) .  0  (5.135)  v  Note that the differentiation between plane stress and plane strain conditions for G^ was introduced due to the definition of A in K^. Furthermore, Rose showed that this result is also the first term in the asymptotic expansion of K  x  for A <C a. He gives an analytical  approximation by interpolating between the asymptotes K  u  and K^, which is generalized  for different crack shapes below [18]. Rose verified the validity of K  T  using the numerical  results from other researchers such as Ratwani, Jones and Callinan [78], i.e.  °\la~TK  I =  Ya  ( 5  -  1 3 6 )  with the corresponding crack extension force K G = - f (Plane stress) E K G = - f (1 - v ) (Plane strain) . E 2  T  (5.137)  s  2  2  r  s  (5.138)  s  The influence of the adherend shear stiffnesses can be readily implemented by using Ashear-iag as given in equation (5.92) instead of A. The characteristic crack length A can be derived from physical parameters of the repair without reference to the crack length a  5.3  Application of the Rose Model for Thermal Residual Stress Loading  153  or the remote applied loading as long as the crack length is small in comparison to the extent of the specimen. Otherwise the specimen width to crack ratio has to be accounted for as typically done in metal fracture specimens.  Due to the long overlap length in the  case of the A M R L specimen, the influence of the taper can be neglected for an investigation focused around the crack without a loss in accuracy. Thus the characteristic crack length, including accounting for the shear stiffness of the adherends, can be determined under typical thermal residual stress loading as A = 4.0 mm using the A M R L sandwich type specifications.  This crack length justifies the previously made assumption that the  characteristic crack length in comparison to the specimen width is small. The introduced error in the geometric correction factor Y is less than 0.25% within the valid range of a between 0 mm < a < A for K . u  If A gets large, the geometric correction factor becomes a  function of the crack length in which case A needs to be solved numerically. Figure 5.13 shows the variation of K  T  with crack length as well as the two asymptotes K  for the investigated A M R L specimen.  u  K  r  has reached 91% of  and  at a crack length of  7 -r  a [mm] Figure 5.13: Estimation and upper bounds for the stress intensity factor  a = 20 mm as shown in Figure 5.13. This clearly shows the usefulness of determining the boundary value K^.  Based on the estimation of K , the patched 20 mm long edge crack r  5.4  Estimation of the Stress Field around the Crack Tip  154  is acting like an unpatched edge crack with a crack length of: EC  K  patched = ^ 1  x =  3-4 mm  (5.139)  Having determined the stress intensity factor for the patched specimen, it is now necessary to find the corresponding stress distribution in the substrate. This will allow comparison of the experimentally measured strains with results of the theoretical analysis.  5.4  Estimation of the Stress Field around the Crack Tip  An estimation of the stress field of a patched cracked plate can be obtained by calculating the stress field for a uniaxially loaded center crack in an infinite plate using the stress intensity factor estimation K  T  of the patched crack with the load a  0  at the prospective  crack location as given in equation (5.3.1) for the thermal residual stress due to an elevated temperature cure. While this approach gives a good estimation of the stress field ahead of the crack, it is not suitable to predict the stress field behind the crack tip and close to the crack faces since localized load transfer effects due to the stress singularity at the crack tip are not addressed.  Another approximation in this approach is the use of the center  crack solution for an edge crack problem. The boundary conditions for the stress along the fiber direction, which is of main concern, are met, while the boundary conditions for the stress perpendicular to the fiber orientation are not properly fulfilled. The boundary value problem is additionally complicated by the use of an effective smaller crack length ^patched in comparison to the real crack length a.  Currently available solutions for the two-dimensional stress field around a crack in an infinite plate are typically based on an infinite power series using a complex variable approach as discussed by Westergaard [98], Sih [85] and Liebowitz [59], or an eigenfunction expansion with an infinite number of terms as devised by Williams [99]. Most researchers neglect the higher order terms and include only the singularity term, which describes the stress  5.4  Estimation of the Stress Field around the Crack Tip  155  2a  I  J  I  I  I  I I  Figure 5.14: Biaxial loaded center crack in an infinite plate field close to the crack tip quite well. Unfortunately these approximated solutions show significant error when estimating the far field stresses for a crack in an infinite plate. The chosen approach for the stress field solution for crack patching is based on Westergaard's [98] complex variable analysis. In previous work, Sih [85] corrected Westergaard's original analysis accounting for a missing integration coefficient. Eftis and Liebowitz [34] gave the resulting Mode I stress field equations for biaxial loading (Figure 5.14 [23]) using only the singularity term as:  'xy  Note that neither a  x  Ki 9 . 9 . 39 , cos - I 1 sm -2 sm — 2 V^rr 2 . 9 . 39 Ki 9 —== cos - 1 + sm - sin — 2 2 yphtr 2 . 9 9 39 , : sin - cos - cos — . v^r7 2 2 2  - (1 -  0  (5.141) (5.142)  nor <r converge to the remote stresses ka y  (5.140)  k)a  0  or a  0  respectively. A n  improved stress field solution using the complete series solution for the stress field was  5.4  Estimation of the Stress Field around the Crack Tip  156  given by Theocaris and Gdoutos [38, 89] as shown below: 2r\" 0 — cos - I 1 a J 2  °~o  [fv ^  + d-V \2a)  2  cos  •(=)"' /2r\~  — Va J  2  .  9 .  cos  T.  2r  O  a  xy_  r  n n  [  1  39  "  5  (2a)  r\«  0  0 2  Ay/2  9 cos 2  - (1 -  2  x  0  • 2^  sin  [ l - 3 - 5 - - - - ( 2 n - l ) ] ( 2 n + 3) 2 ("+ )(n + l)!  k)a  2  (5.144)  (2n + l)0 (2n + l) . ( 2 n - l ) 0 — — + - — - — - sin — sin 9  0  +  (5.143)  1  (2n + l) . ( 2 n - l ) 0 . ' — - — - sin - — - — — sin 9  "  [ y ^  2  J l - 3 - 5 - - - - ( 2 n - l ) ] ( 2 n + 3) ' 2 ("+ )(n+l)!  9 /  \2a)  c o s - l l + sin  4V2  cos 1 + sin - sin — 2 \ 2 2  +  •0  l V  (2ra + l)0 —  5  3  sin - sin — 2 2  9 . 9 30 cos - sin - cos 2  D1- D  (n+l)  2  4X/2  (;)  n 0 . 0 cos - sin 2 2  l - 3 - 5 - - - - ( 2 n + 3) 2(2"+3)(  n +  l)?  (5.145)  .71 =  . (2n - 1)0 sin0 sin  Figure 5.15 presents the results using these approximations for the stress in the loading direction for a center crack with a = 4.22 mm and a  0  = 40.43 M P a along 0 = 0°. The  approximate series solutions as found in the literature are compared to the correct elastic solution which is derived in this section.  The one term approximation using equation  (5.141) shows a significant error of 53% at a distance of 2a from the crack tip. The error reduces to 18% if a two term approximation is used. Excellent agreement with respect to the correct elastic solution up to a distance of 2a from the crack tip can be observed when a larger number of terms is used.  Unfortunately, the series approximation is not applicable for locations beyond a distance of 2a from the crack tip.  The determined stress diverges rapidly beyond this point as  5.4  Estimation of the Stress Field around the Crack Tip  157  o-i  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  0  2  4  6  8  10  12  14  16  18  20  r [mm]  Figure 5.15: Approximate crack tip stress field solution shown in Figure 5.15. This approximate solution is inadequate for the estimation of the stress field in the substrate of a bonded repair due to the small effective crack length ^patched-  Significant interest is currently focused on 'smart patches' where crack growth is  monitored using strain sensors, however knowledge of the stress distribution is essential for their proper placement.  Figure 5.15 indicates that the approximate solution would give  false stress values only a short distance away from the crack tip. Therefore it is important to use a stress field solution which more accurately (or better) fulfills the remote stress conditions. The concise solution discussed in this section, which satisfies all of the required boundary conditions for a center crack in an infinite plate, is achieved by carrying out the full complex variable analysis without using the approximation of including only the singularity term (1/y/r) or an infinite power series substitution.  5.4.1  Theoretical Derivation  Sih's [85] correction of the original Westergaard approach [98] is derived using the complex representation of the general solution of the equations based on plane theory of elasticity as given by Muskhelishvili [64]. If the external loads are symmetric with respect to the  5.4  Estimation of the Stress Field around the Crack Tip  158  x-axis along which the crack is situated, the general complex representation of the stress field yields [34, 85]: CT = Re[Z(z)}  - ylmfij^]  X  a  y  - A  +y l m [ ^ J ]  = Re[Z(z)}  + A  (5.146) (5.147)  dz r  ^  x y  AZIZ),  = -yRe[-^-±]  .  (5.148)  Westergaard's [98] derivation did not include the constant A due to an oversight in MacGregor's work [34]. The boundary conditions for a center crack in an infinite plate using the coordinate system shown in Figure 5.14 are [85]: =0  (5.149)  r (-2a<x<o,i/=o) = 0  (5.150)  O- (-2a<x<0 /=0) y  >2  xy  0x(\A +2/ ->oo) 2  2  ka  =  0  (5.151)  o (^x + 2 ->oo) = a  (5.152)  0.  (5.153)  2  y  y  0  Txy(\A +2/ ">-oo) 2  2  =  Sih gave the corrected Westergaard type stress function, which fulfills the above stated boundary conditions as [85]:  where  v^  >  A =(l-k)^.  (5.155)  The derivative of Z{z) with respect to z can be easily determined as: dZ(z) dz  aa  2  0  (z  (5.156)  + 2az)-2  2  In the second step of the analysis, the real and imaginary part of Z(z) and the derivative dZ(z)/dz have to be determined. It is convenient to switch to polar coordinates using: z = re  ie  (5.157)  5.4  Estimation of the Stress Field around the Crack Tip  159  where r = y/x  4 y  2  e  (5.158)  2  = cos0 4 i s i n 0 .  ie  (5.159)  Thus Z(z) can be rewritten as:  z = -^  +  z  a)  7  M  ^{z + a) - a a (re + a) 2  2  ' 2 a  V  ie  -  0  vV e « 2  i2  +  2 a r e  (!-«)•  ie  2  a (re  ie  0  y/(r  2  _ a (re 0  Q  + a)  cos 20 4 2ar cos 0) 4 i(r sin 20 4- 2ar sin 0) 2  ie  ^  -J^Q^  4- a) -^(r cos 20 4- 2ar cos 9) - i(r sin 29 4 2ar sin 0) 2  y/(r  2  2  cos 20 4 2ar cos 9) 4 (r sin 20 4- 2ar sin 0) 2  2  2  In order to allow the separation of Z(z) in real and imaginary parts as well as to reduce the size of the expression, r*(r,e) and 9*(r,e) are defined as:  r*( ,9) = {(r  2  r  cos 20 4- 2ar cos 9)'  2  4- (r  2  sin 20 4 2ar sin 0) ] < 2  (5.161) = [r 4 4 a V 4 4ar cos 0]* 4  3  r cos 20 4- 2ar cos 0 2  0 (r,9) = arccos( v  r*(r,e)  ).  '  2  (5.162)  v  ;  Note that the arccos() is used to maintain a unique solution for 0 < 0 < ir. Dealing with quarter symmetry, it is important to keep in mind that r is limited to:  — cos 0  for  | < 0 < 7r 2  (5.163)  Based on this expression it is resolved that 9*{r,e) reaches its maximum value of 7r along the line x — —a, y > 0 and at the crack face. The parameter 9*(r,e) has its minimum value of  5.4  Estimation of the Stress Field around the Crack Tip  0 along the line x > 0, y = 0. Using the relationships for o- (re  + a)r*(r,8)e- ^ '  ie  l  0  Zi(z) =  {r e)  2  <  "  ,  i  "  v  M  )  ,  = (^  r*(r,9)  ^ - '  +  and 0*(r,e),  Z(z)  yields:  ,a 1 — A; — ' 2 u  r*(r,e)  = " « f e '  160  "  9  0  M  "-  cos(  ) - (  )  +  1  - * ' f  7^r \ " - ) {  9  {  e))  -<i-*)£ (  r 1 a 1 \ — — sin(0 - -0*(r.*)) + sin(-0*(r,«)) \r*(r,e) 2 r*(r,e) 2 /  + i a[ 0  .  Finally Re[.Zf»] and Im[Z(z)] can be expressed as: Re[Z<«)] = o  cos(9 - \»-(,,))  Q  Im[Z( ] = a 2)  0  \T  (r,6j  + _ | - c o s ( i t l ' <,,.)))  sin(0 - ± 0 > , * ) ) + Z  sin( T (r,9)  The evaluation of the real and imaginary part of dZ(z)/dz dZ(z) dz  Vr^))) Z  (5.165)  .  (5.166)  J  is carried out in a similar fashion:  aa  2  0  ~  . (z + 2az)i ,2 o ar 2  n  (r e 2  + 2are )*  i2B  i6  aa  2  0  [(r cos 29 + 2ar cos 9) + i(r sin 29 + 2arsin9)] 2 2  ^  2  a a [(r 2  cos 20 + 2ar cos0) — i(r sin 20 + 2arsm0)]t  2  2  0  (r cos 20 + 2ar cos 0) + (r sin 20 + 2arsm0) ] 2 2  2  2  2  «2  -<?o— "  r*(r,9)  -i\0*{r,0) e2  T  d  Thus the real and imaginary part of dZ( )/dz are: z  Re[^]  = - a  Im[^]  = a  0  0  ^ c o s ( ^ M ) )  ^ s i n ( ^  M  )  ) .  (5.168) (5.169)  '  (5 164)  5.4  Estimation of the Stress Field around the Crack Tip  161  Having determined the real and imaginary parts of Z( ) and dZ(z)/dz,  the stresses a ,  z  and r  xy  x  a  y  can be expressed in a concise form as:  v  <7 (r,9) = cr x  0  Lr*(r,e)  2  v  '  9  v  r 0  T y(r,9) = < J X  0  1  a  ,  2  a  9  r  r*(r,0) 2  r  v  2  (5.170)  1  , sin0sin(-0*(r,e)) - (1 - k)  cos(0 - -0*(r,o)) -\  Lr*(r,0)  +  T*{T,B)  3  err a (r,e) = a  a 1 — cos(-0*M))  1  cos(0 - -0*(r,0)) H  cos(^0*(r,e))  a  (5.171)  r*(r,<9)  sin0sin(-0*(r,e)) ;  , sin0cos(-0*(r,6»))  lr*(r,9)  3  v  2  (5.172)  '  where r*(r,9) and 0*(r,6») are given by equation (5.161) and (5.162) previously.  As previously discussed, 9 and r are limited to 0 < 9 < IT and 0 < r < —a/ cos(0) for TT/2 < 0 < 7T due to the use of 0*. Symmetry can be readily used to determine the full stress field around the crack. As in the simplified Westergaard solution (equation (5.140)(5.142)),  the Mode I stress intensity factor Ki can be introduced by replacing the crack  half length a with:  IK  2  a  (5.173)  2  =  TT  a  2  The solution for the stress field can also be expressed as a function of x and y with the coordinate system origin at the crack tip (Figure 5.14) by replacing r and 0 using:  r =  \J x + y 2  2  9 = arccos(—.  (5.174)  X  \A + 2  ) = arcsin(— ^ ). y/x + y' 2  (5.175)  5.4  Estimation of the Stress Field around the Crack Tip  162  Thus x+a  1  y  . , 1 „*  \  o~ (x,y) — C o — — - COs(-0*(x,„)) + Sm(-e*(X,y)) V (x,y) Z T (x,y) Z  (5.176)  x  gy sin(-0*(x,„)) - (1 - k) r*(x, y 2  y  x+ a  1  y  .  1  (5.177)  0~y(x,y) — a — — - c o s ( - 0 * ( x , ) ) + - 7 7 — s i n (-0*(x,»)) r*(x,») 2 r*(z,w) 2 y  0  r W "  +  T~xy{x,y) — CTQ  ^ W  C  O  S  (  (  i '  M  )  (5.178)  2 ^ ™ \  where r*(x, ) = {(x - y + 2ax) + {2xy + 2ay) }^ 2  2  2  (5.179)  2  y  ,x — y + 2arr, 2  2  9*(x,y) = arccos(-  (5.180)  -)  Note that x and y are limited in this case to —a < x < 0 0 and 0 < y < 0 0 due to the use of 9*. This solution simplifies slightly by moving the origin of the Cartesian coordinate system to the center of the crack. Furthermore, it is easier to determine the full stress field around the crack since symmetry conditions are now applied with respect to the center of the crack. The stress field solution with the origin placed at the crack center is given by:  o~ ( c,yc) — <y———cos(]-9*(xc,yc)) r*(x ,y ) 2  +  x  x  0  c  sin( J-(9*(x,y))  V c  r*(x ,y )  c  c  c  2  c  c  (5.181)  - - ^ ^ s i n ( | ( ? W ) ) - ( l - A : ) r*( ,y )  2  a  Xc  0~y(x ,y ) — a c  c  c  ,1  Xr 0  Vc + — r*(x , )  cos(-9*(x ,y )) 2  r*(x , )  c  v  c yc  ay  c  . /3.  2  c  Sin(-0*(x ,yc)) 2 v  c yc  c  (5.182)  .  t  . 3 sin - f l W ) r*(x ,vc) 2  +  3  c  (5.183)  ^COS(-0 (xc,yc)) T y{x ,yc) — 0~ — ir*{x , y 2 X  c  0  c yc  where r*(x y ) = {(x - y - a ) 2  C)  c  2  c  2  9*( ,y ) = arccos^ Xc  2  c  2  2  0  V c  c  r*(x , r c yc  +  (2x y ) ]^ 2  c  c  (5.184)  2  ° ).  (5.185)  5.4  Estimation of the Stress Field around the Crack Tip  163  Note that x and y are limited for the case of the coordinate system placed in the center c  c  of the crack to 0 < x < oo and 0 < y < oo due to the use of 9*. c  c  In addition to the analyzed stress field for symmetric loading conditions, the displacement field can be solved using the same approach by utilizing the displacement equations provided by Sih [85].  Furthermore, the screw symmetric problem can also be solved in a similar  fashion.  5.4.2  Results  The derived system of equations for a , a x  dependence.  y  and r  x y  does not readily indicate the ( l / \ / r )  By substituting r* and 9* back into the equations for a , a  dependence on (a/r)  x  a  y  and r , the x y  is revealed, where a represents different fractions. The singularity  characteristic at the crack tips can then be readily shown by plotting the stress field. A double-sided reinforced infinite plate with a center crack is used to present an estimation for the sandwich type A M R L specimen. Using equations (5.182) through (5.185), the stress distributions for uniaxial remote tension of a  0  = 40.43 MPa applied perpendicular to a  C C  center crack with the half crack length of a  p a t c h e d  = 4.22 mm are presented in Figure 5.16  through Figure 5.18. The crack length for the stress field estimation is based on determining an equivalent center crack length for a given stress intensity factor. Therefore the crack length was determined using the stress intensity factor K = 4.66 MPa^/m given by Rose's r  model for the sandwich type A M R L specimen containing an edge crack with a length of a = 20 mm and thermal residual stresses on the order of a = 40.43 MPa caused by cooling 0  from an effective stress free temperature of 85.8°C to ambient temperature (21°C).  Note  that the singularities are not perfectly displayed due to the employed discrete mesh. The singularities can be better illustrated with decreasing mesh size. The stress distribution a  x  parallel to the crack is plotted in Figure 5.16. The solution reveals that along the crack  5.4  Estimation of the Stress Field around the Crack Tip  164  -10  -10 10 Figure 5.16: Stress parallel to a crack (—4.22mm < x < 4.22mm, y = Omm) in an infinite plate with uniaxial loading of 40.43 MPa perpendicular to the crack face, < 7 drops to — a . X  0  The crack tip singularities are clearly present and the boundary  condition of no stress in the x-direction far away from the crack is satisfied.  The stress  distribution of cr as presented in Figure 5.17 fulfills the boundary conditions of a stress free y  crack face at an applied remote stress of a . Here also the singularities are very pronounced 0  at the crack tips. The shear stress is shown in Figure 5.18 with the singularities visible. Note that the z-axis scale has been changed to emphasize the singularities. Other features to note are that zero shear stress is attained both at the symmetry lines and far away from the crack tips.  The presented analysis offers an estimate of the stress field in the isotropic substrate for bonded repairs based on Rose's model. The results show the stress distributions for the sandwich type A M R L specimen under thermal residual stress loading. In order to compare  5.4  Estimation of the Stress Field around the Crack Tip  165  -10  Figure 5.17: Stress perpendicular to a crack (—4.22mm < x < 4.22mm, y = 0mm) in an infinite plate with uniaxial loading of 40.43 M P a perpendicular to the crack the experimentally measured strains with the theoretical predictions, the corresponding strains from the predicted stress field have to be determined. It is important to account for the stresses perpendicular to the fiber orientation close to the crack tip, although they can be neglected further away due to their small magnitude. If only the thermal residual stresses due to cooling from an effective stress free temperature to ambient temperature or operating temperature are considered, the thermal residual strains along the fiber direction being the main concern are given by: 1 / TRS = -— la — vo  TRS  e  y  s  e™ = -^-cr™ E S  y  y  S  ,  TRS\ x  1  .  .  i  •  \  / r-  (close to the crack tip)  (5.186)  (far away from the crack tip) .  (5.187)  s  Having derived an approximation for the thermal residual strain field for a patched crack,  5.5  Comparison between Closed Form Theoretical and Experimental Results  166  -1010  Figure 5.18: Shear stress around a crack (—4.22mm < x < 4.22mm, y = Omm) in an infinite plate with uniaxial loading of 40.43 M P a perpendicular to the crack it is now possible to compare the theoretically derived thermal residual strains with the experimentally measured values at the strain gauged locations.  5.5  Comparison between Closed Form Theoretical and Experimental Results  Two practically important temperatures were chosen for a comparison between the experimentally measured thermal residual strains and the theoretically determined thermal residual strains: ambient temperature (21°C) and the temperature at cruising altitude (-56.5°C).  The theoretical evaluation of the thermal residual strains was based on the  5.5  Comparison between Closed Form Theoretical and Experimental Results  167  experimentally determined effective stress free temperature of (85.8°C) as discussed in the previous Section 4.6 giving a temperature differential of — 64.8°C and —142.3°C.  One of the concerns is the use of the adhesive shear modulus in the theoretical estimation of the thermal residual stresses and strains. The shear modulus of the adhesive changes significantly from near zero during the initial part of the curing process to approximately 905 MPa for the low temperature at cruising altitude. A significant change in shear modulus for F M 73M can be identified from the experimental measurements occurring, when the slope of thermal residual strains versus temperature changes significantly at a temperature of approximately 85° C. The creep rate associated with the experimental testing sequence might be the reason for the variation to the glass transition temperature. This theory is supported by the measured creep which occurred during testing for extended periods at elevated temperatures as well as by the viscoelastic measurements carried out by Jurf and Vinson [56] .  A concern arises when considering which shear modulus should be used to determine the thermal residual stresses and strains at a specific temperature. Many researchers utilize an incremental approach in the determination of thermal residual and other process induced stresses and strains in which the shear modulus gradually changes from near zero at curing temperature to the shear modulus corresponding to the temperature at the end of the process [24, 50]. Although this seems to be a good model for the initial curing phase due to the change in cross-linking between the molecules, it is important to note that this is not an elastic modeling approach. This model assumes that a temperature change leads to an incremental stress/strain change caused by the mismatch in the C T E of the adherends which is independent of the actual stress and strain state. Curve A in Figure 5.19 shows the shear stress behaviour at the edge of a long reinforcement with four temperature changes utilizing this type of approach. Alternatively an elastic model can be assumed which is represented in Curve B.  5.5  Comparison between Closed Form Theoretical and Experimental Results  168  Figure 5.19: Shear stress versus shear strain for the adhesive layer of a reinforcement during the cooling cycle A simple experiment can be used to investigate the true behaviour of the adhesive.  A  double-doubler joint consisting of identical materials for the adherends can be loaded using a dead weight at elevated temperature but below a temperature where creep is significant during the cooling cycle from the curing temperature. Measuring the displacement as a function of temperature should reveal if an elastic approach is reasonable.  If the non-  elastic behaviour is true such as shown in curve A , no change in displacement should occur with change in temperature due to a constant load. A similar experiment with increasing temperature can be carried out.  Here it seems very likely that the reduction in shear  modulus will lead to an increase in the displacement.  The experimental thermal residual strain measurements using the A M R L specimen are unfortunately not conclusive enough to allow for determination of the appropriate shear modulus. Most gauges were not placed in the region of very high adhesive shear stress gradients with the exception of location A and to a limited extent location B . Because of the integrating character of strain gauges and the gauge size involved in the thermal residual strain measurement using the sandwich type A M R L specimen, no clear indication can be made from the measured data as to which of the above described models should  5.5  Comparison between Closed Form Theoretical and Experimental Results  169  be used to justify the adhesive shear modulus change in the analysis. The true behaviour, which might very well be a combination of both models depending on temperature, needs to be determined in future experiments. The attempt made in this research work is to simplify the prediction of thermal residual stresses to an elastic model by using definitions such as an effective stress free temperature to create readily available engineering tools. Based on the speculated outcome of the above described experiment, the shear modulus change is assumed elastic for this theoretical evaluation, i.e. the adhesive shear modulus is evaluated at the desired temperature. Based on the elastic assumption, adhesive shear moduli of 844 M P a and 905 M P a are used at 21°C and —56.5°C, respectively. Unfortunately, the one-dimensional analysis is not suitable to predict through thickness changes in the adherends. Thus, in order to compare the theoretical estimates with the experimental measurements, average through thickness values of the thermal residual strains for the adherends were used. One of the disadvantages of using a through thickness average value is the neglect of errors caused by embedding strain gauges in the bonded repair with the associated small local thickness change. These thickness variations can result in bending stresses thus affecting the indicated strains. The finite element analysis gives a better tool to determine the cause of through thickness strain variations such as shear-lag, out-of-plane bending or embedding strain gauges. It should also be recalled that the one-dimensional analysis focuses on the fiber direction being the main loading direction with respect to thermal residual strains and generally of main concern due to their magnitude. It is estimated that the locations of the strain gauges were measured to an accuracy of approximately ± 0 . 2 5 mm. This error is accounted for in the subsequent analysis. No other uncertainties such as the moduli of the adherends are included in this comparison.  5.5  Comparison between Closed Form Theoretical and Experimental Results  170  A rough estimation for the thermal residual stress perpendicular to the fiber direction in the center region of the patch can be achieved using equations (5.104) and (5.105) in conjunction with equations (5.100) and (5.101). It becomes evident that significant in-plane bending is present by checking the strain gauge readings of locations E and F, thus making identification of the transverse strains below 140fie at the strain gauged locations using this type of analysis unsatisfactory. In addition, the influence of the crack creates significant additional transverse stresses in the crack tip region. The finite element analysis, being readily able to include multi-dimensional influences, is more suitable for the investigation of the transverse stresses and strains.  Table 5.1 shows the comparison between theoretically determined thermal residual strains in the fiber direction and the experimental measurements at ambient temperature. Table 5.2 presents a comparison at the typical operating temperature of aircraft structures (—56.5°C). The theoretical evaluation is carried out first using the one-dimensional model for double-sided reinforcements and double-doubler joints. Additionally the thermal residual strains are determined using the Rose model combined with the presented stress field estimation for applicable gauge locations.  Excellent agreement between the theoretical analysis and the experimental measurements for the thermal residual stresses in the aluminum at locations C , D and H are achieved. The fracture mechanics analysis predicts higher thermal residual strains at the strain gauged locations than the experimental measurements with a maximum difference of 35 fie at ambient temperature and 81 fie at —56.5°C. The measured thermal residual stresses in the aluminum at location B are approximately 25% higher than the strains predicted by the fracture mechanics analysis. A possible explanation for such an increase would be a large disbond.  Using the analysis presented in Section 5.6, a disbond length of 2b w 60 mm  would be required to increase the thermal residual strain to such a high level. A disbond of such an extent would also increase the thermal residual strains in other locations, espe-  5.5  Comparison between Closed Form Theoretical and Experimental Results  Location  A  B  C  D  G  H  Material Al  Gauge  Experimental  No.  Measurement  2  5 [it  B/Ep  23  -607 lie  B/Ep  43  -665 pt  Al  3  696  Ate  B/Ep  24  -598 /j,e  B/Ep  30  -593 ue  B/Ep  33  -532 //e  Al  12  B/Ep  25  -688 /xe  B/Ep  31  -627  B/Ep  42  -585 \xe  Al  10  508  Al  22  528 /Me  B/Ep  28  -638  Ate  B/Ep  29  -641  Ate  B/Ep  40  -568 fie  Al  7  566 jj,e  327  Ate  /xe  Ate  B/Ep  27  -720 fie  B/Ep  37  -748  //e  Al  5  532  >ne  Al  19  B/Ep  26  -632  B/Ep  32  -647 fie  B/Ep  35  -622  521 fie  Experimental Measurement Average 5 pe -636  Ate  696 fie -574  A/e  566 fie  Fracture  1-D  Mechanics  Analysis  Analysis  84 ± 1 1 -696 ± 89  Ate  N/A  Ate  N/A  488 ± 3 A<e -579 ± 4 Ate 558 ± 1  171  Ate  518 ± 12  Ate  N/A 585 ± 2  Ate  N/A  -633  A^e  -661 ± 1  Ate  518  Ate  541 ± 1  Ate  -616 pe  -641 ± 1  Ate  N/A  550 ± 1  327  //e  134 ± 1  Ate  N/A  -734  Ate  -1038 ± 1  Ate  N/A  527  Ate  558 ± 1  Ate  -634  Ate  -661 ± 1  562 ± 1  Ate  Ate  Ate Ate  N/A  Table 5.1: Comparison of the thermal residual strains at 21°C between experimental results, the one-dimensional analysis and the fracture mechanics analysis cially location C where the thermal residual strains would increase to approximately 660 Ate, which is certainly not the case. The thermal residual strains are predicted at location C quite well assuming no disbond. Therefore a disbond can be ruled out as a likely cause. Neglecting that the patch covers only 47% of the specimen width leads to some inaccuracy in the estimation of the stress at the prospective crack location. The stress estimate at the  5.5  Comparison between Closed Form Theoretical and Experimental Results  Location A  B  C  D  G  H  Material Al  Gauge  Experimental  No.  Measurement  2  38 fie  B/Ep  23  -1469 lie  B/Ep  43  -1808 fie  Al  3  1504 //e  B/Ep  24  -1262 /ie  B/Ep  30  -1258 fie  B/Ep  33  -1152 /ie  Al  12  1215 lie  B/Ep  25  -1456 /ie  B/Ep  31  -1343 /ie  B/Ep  42  -1293 lie  Al  10  1145 /ie  Al  22  1154 fie  B/Ep  28  -1399 /ie  B/Ep  29  -1386 /ie  B/Ep  40  -1282 fie  7  758 /ie  B/Ep  27  -1602 lie  B/Ep  37  -1719 /ie  Al  5  1161 /ie  Al  19  1115 lie  B/Ep  26  -1371 /ie  B/Ep  32  -1417 lie  B/Ep  35  -1399 /ie  Al  Experimental Measurement Average 38 fie -1639 /ie  1-D Analysis 184±23/xe -1528 ± 195 lie  172 Fracture  Mechanics Analysis N/A N/A  1504 fie  1065 ±6 lie  -1224 /ie  -1261 ± 7 /ie  N/A  1215 lie  1212 ± 1/ie  1268 ± 3 fie  -1364 /ie  -1435 ± 1/ze  1150 /ie  1177 ± 2 /xe  -1356 fie  -1395 ± 2/ie  N/A  758 /ie  291 ± 1 /ie  N/A  -1661 fie  -2261 ± 1 /ie  N/A  1138 /ie  1210 ± 1/ie  -1396 /ie  -1434 ± l^e  1122 ± 2 6 /ie  N/A  1194 ± 1 /ie  1219 ± 1/ie  N/A  Table 5.2: Comparison of the thermal residual strains at — 56.5°C between experimental results, the one-dimensional analysis and the fracture mechanics analysis prospective crack location can be improved over the one-dimensional analysis by carrying out a finite element analysis of the A M R L specimen without a crack. The adhesive does not need to be modeled in this particular analysis since virtually no load transfer occurs at the prospective crack location.  The finite element analysis of the A M R L specimen as discussed in the following chapter  5.5  Comparison between Closed Form Theoretical and Experimental Results  173  gives additional insight into the complex stress distribution in the vicinity of the crack thus giving a better understanding of the cause for the high tensile residual strains in the aluminum at location B . The stress and strain distributions as caused by the crack tip stress intensity factor are affected by the load transfer into the patch due to the crack as shown in the transverse shear stress plot for the A M R L specimen (see Figure 6.32). This influence is not accounted for in the fracture mechanics approach. The predictions at location G are not accurate either (Ae = 193 /Lie for the aluminum). The explanation for the discrepancy is given by the gradient of the thermal residual strains in the fiber direction through the tapered section. to be expected.  Additionally, in-plane bending has  The finite element model gives an excellent prediction for the thermal  residual strains in the aluminum in this particular region. The thermal residual strains at the tapered location A are predicted reasonably well.  After accounting for position  uncertainty of the strain gauge, the thermal residual stresses are well predicted for the boron/epoxy. The thermal residual stresses in the aluminum are somewhat overpredicted (84fie at ambient temperature and 184//e at an operating temperature of —56.5°C). The prediction of the thermal residual strains in the boron/epoxy patch at location B , C , D and H are quite good. The average discrepancy is only 20 fie at ambient temperature and 46 pe at an operating temperature of —56.5°C.  The thermal residual stress field in the bonded repair will change with crack growth when the test specimen is subjected to fatigue loading. In addition to crack growth, disbonds around the crack face are quite common. Initial flaws such a inadequate surface preparation can also lead to large disbonds. The influence of disbonds on the stress and strain distribution can be quite large. A more generalized Rose model, which can account for disbonds, is therefore developed in the following section and represents one of the major achievements of this research work.  5.6 5.6  Generalization of the Rose Model for Partially Disbonded Patches Generalization  of  the  Rose  Model  for  Partially  174 Disbonded  Patches  The accurate prediction of the effect of a disbond on crack propagation in the metallic substrate requires elaborate calculations [5, 73, 76, 81]. Baker [13] presented the development of a useful estimate of the influence of disbond size by adapting the Rose model [79] giving a ratio between the stress intensity factor for a repair without a disbond and the stress intensity accounting for a symmetric disbond. The approach in this section expands on Baker's initial formulation by adding an additional term to improve the determination of the crack opening displacement as well as accounting for an imbalance in the stiffness of the patches and substrate and their thermal mismatch. The approach is first carried out for uniform thickness patches and then extended to tapered patches. The analysis presents a significantly improved estimate for the change of the stress intensity factor in the presence of a disbond.  Figure 5.20: Illustration of the superposition principle for a composite repair with disbonds  5.6  Generalization of the Rose Model for Partially Disbonded Patches  175  As discussed in Section 5.3, the stress at the prospective crack location in the presence of a disbond has to be determined first using the stage I analysis of the Rose model. The previously outlined method to determine the stress at the prospective crack location for a repair without a disbond utilizing a double-sided reinforcement can be used for a disbonded patch only if the two patches and substrate have identical stiffnesses and equal coefficients of thermal expansion.  The reinforcement would be modeled from the free  edge to the edge of the disbond. The requirement of identical stiffness and C T E of the adherends fulfills the requirement for zero shear stress at the edge of the disbond thus being equivalent to the center of a reinforcement without a disbond. The requirement for zero shear stress at the edge of the disbond can also be fulfilled for specific combinations of adherend stiffnesses and C T E mismatches. In any other case, it is necessary to derive a one-dimensional model, which takes the disbond as a boundary condition into account. The half crack opening displacement v as a result of introducing a cut in the substrate of 0  the double-sided reinforcement can then be assessed using superposition. Superposition can be used to determine the joint opening displacement v* in the presence of a disbond which is required to determine the upper bound presented in Section 5.3.  similar to the approach  Figure 5.20 shows the superposition scheme with symmetric  disbonds.  5.6.1  Double-sided Uniform  Thickness Repair with  Disbonds  Stage I: Stress at the prospective crack location The first step of the Rose model requires the determination of the stress at the prospective crack location. Figure 5.21 shows the free body diagram for half of a reinforcement with a disbond. In order to be able to use the system of equations derived for a reinforcement without a disbond, two coordinate systems were introduced.  5.6  Generalization of the Rose Model for Partially Disbonded Patches  176  The y - coordinate starts at the end of the disbond, while the y* - coordinate originates in the center of the reinforcement.  The following approach is directed toward deriving  the influence of a disbond as a boundary condition. The disbond generally results in a shear stress in the adhesive at y* = b in comparison to zero shear stress in the center of a reinforcement without a disbond. The length changes in the patch and substrate due to both a direct stress resultant and a temperature change in the disbonded area and are given by:  (5.188)  Eptp AL = eJ)  ^b + a ATb  .  s  (5.189)  -E's 4  Using the definition of the elastic shear stress distribution parameter A and the force equilibrium given in equation (5.13), the shear stress at y = 0 can be expressed as: A  Rd  r (»=o)  2  {v (y=o) - V (y=0))  (5.190)  (Alp -  (5.191)  P  a  y Eptp  A  s  Et J s  s  2  \ Eptp  Al ) s  Et J s  s  =x*b( ;<-^; - i {as  a  (5.192)  AT  T  y Eptp AT  s  s  AT  po  l  'so  L  po  l  l  l  po so  po  \  11  M  ^  -• y*  Et J  1  ^  •y  Figure 5.21: Free body diagram for a reinforcement with a disbond  5.6 Generalization of the Rose Model for Partially Disbonded Patches  177  The shear stress expression simplifies even further using the integration constant C as p  given in equation (5.16).  r (,=o) = A 6 ( T Rd  2  a  R p  d o  -C )  (5.193)  p  A proper boundary condition can now be established by substituting T direct stress resultant for the patch T  Rd p  and r ( =o). The Rd  y  can be evaluated using equation (5.11) as  P o  • Rd  ,Rd T; o  =  O ) = A — + C .  T^.Rd ; V  (5.194)  p  The shear stress r (y=o) is then determined using equation (5.10). Rd  r  R a  V o ) = 73  (5.195)  Rd  The boundary condition for the disbonded reinforcement is given by substituting equation (5.194) and (5.195) into equation (5.193) thus yielding:  B  = A Xb  Rd  Rd  .  (5.196)  Note that this boundary condition for b = 0 is identical to the boundary at the center of a double-sided reinforcement without a disbond. The second boundary condition for the double-sided partly disbonded reinforcement is evaluated at the edge of the reinforcement (y - l) as: . Rd  „ Rd  T \y=i) = — cosh(AZ) + - — sinh(Af) + C = 0 p  where y4.  Rd  A  can be determined by substituting B  Rd  A  Rd  =  p  A  in equation (5.197) yielding:  — icosh(AZ) + 6sinh(A0 Wf  s  +  + («s -  (5.197)  (5.198) « )AT P  (icosh(A0+6sinh(A/)) '  (5.199)  5.6  Generalization of the Rose Model for Partially Disbonded Patches  The second coefficient B  D  R  178  and then be calculated using equation (5.196): Cp  d  ,  ~ ^ c o s h ( A 0 + isinh(A0  =  ( 5  -  2 0 0 )  (5.201) + Having determined A  and B  ( ^ c o s h ( A / ) + Isinh(A/))  , the stress distributions for the adhesive and adherends  can be readily determined. The stress at the prospective crack location in the presence of a disbond is given as: 2A ^ Rd  o  0  = - I  =  2  1- Cs  (5.202)  , E t, + ("s - « ) A T \ s  (_k  t S  |_ _2__\  \ ^P*p  / cosh(AZ) + 6Asinh(A(!)  &sts )  (5.203)  /  -2(a -a )AT  p  a  Stage II: U p p e r bounds  1  P  p  K  T  After determining the stress at the prospective crack location o~ , the crack is introduced d  into the second stage of the analysis. The analysis is, for the most part, identical with the approach given in Section 5.3.2. The main difference is the contribution of three components to the half crack opening  rather than one. The strains in the disbonded sections of the  patches and substrate lead to increased displacements, in addition to the crack opening due to shear in the adhesive.  An upper bound for the stress intensity factor for short cracks  assuming no load transfer  into the reinforcement is given as [23]:  K* = Yo~ y/ira . d  (5.204)  5.6  Generalization of the Rose Model for Partially Disbonded Patches  The corresponding energy release rate G  u  179  can then be determined based on plane stress  or plane strain conditions:  (1 — u ) 2  (Plane stress)  (5.205)  (Plane strain) .  (5.206)  This upper bound will be close to the correct value for short cracks and represents an asymptote with first order contact in the limit a - » 0 for K* of an infinite patched plate with a crack [78]. Note that the energy release rate G  d  length, although  u  is linearly increasing with crack  cannot exceed the energy release rate G^ for a semi-infinite crack. The  limiting value for the energy release rate G^ is again modeled using the one-dimensional approach which ignores the stress variation across the thickness [79]. This upper value for the energy release rate is estimated by the change in potential energy when introducing a cut in the substrate of a double-sided reinforced plate with disbonds, therefore converting the specimen to a double-doubler joint with disbonds centered around the joint. Thus as given in equation (5.114) is the work extracted (per unit thickness and per unit crack growth) on the stress at the prospective crack location cr to relax to zero through the d  displacement v* on both crack faces [79]. The stress at the prospective crack location o* was determined in equation (5.203) as stage I of the Rose model using a one-dimensional model of a double-sided partially disbonded reinforcement. The displacement v% as a result of introducing a cut in the double-sided partially disbonded reinforcement can then be easily determined using superposition as shown in Figure 5.20. According to Figure 5.20, the displacement v* is given as the displacement of one joint or crack face for a pressurized cut in a partially disbonded double-doubler joint. The general analysis of the shear stress in the adhesive as given in Section 5.3.2 is still applicable in contrast to the partially disbonded double-sided reinforcement.  5.6  Generalization of the Rose Model for Partially Disbonded Patches  180  Thus the shear stress at the edge of the disbond for an internally pressurized partially disbonded double-doubler joint can be written as:  ^  =-  A J m -  x  ( 5  '  2 0 7 )  The displacement v due to the shear displacement in the adhesive can be evaluated using d  ol  equation (5.117). Therefore the displacement v  d  d  o\U f V  * ~  1  2 UP'P S  ol  is given by equation (5.129) as:  2 E tJ  +  s  1 Atanh(AJ) '  (  5  ,  2  °  8  )  As stated above, it is necessary to include the displacements due to the strain in the disbonded section of the patch and substrate. These additional displacements are given as:  d  v  d om  =  fb  (5.210)  U  which sum to provide the total half crack opening displacement as: l = l\ + %n + %ni  v  (5.211)  v  2  \E tp  E tJ  Atanh(AZ)  \E t  E tJ  \Atanh(A0  v  3  2 Et p  ^ 4  2  p  p  s  p  E  s  (5.213)  Thus the upper limit for the energy release rate can be expressed for plane stress and plane strain conditions as follows [18]. Plane stress '2 , dd Y a„ Gl = v v = ^ T T A ~E.s z  d  0  where A  = ^  (  1  +  (5.214)  0  2^)(n^5(Ao  + 6  l  (5  -  215)  Plane strain Gl  = o'X = ^ T A ( 1 - vj)  (5.216)  5.6  Generalization of the Rose Model for Partially Disbonded Patches  181  where  Based on the validity of equation (5.112) and (5.113), the stress intensity factor  for  this upper bound is: = y/EsG^  = ]  J  =  =  y<7„ VTTA  Y  a  o ^  (Plane stress)  (  Plane  s t r a i n  (5.218)  ) •  (- ) 5  As before, this result can be used as the first term in the asymptotic expansion of K  r  219  for  A < a. The Rose model suggests an analytical approximation by interpolating between the asymptotes K  d  and  which is generalized for different crack shapes below [18] K t - Y ^ ^  (5.220,  with the corresponding crack extension force K = -±(Plane stress) E* ^ = ^-(1 - v ) (Plane strain) . d2  G  (5.221)  d  2  G  d  (5.222)  2  s  The accuracy of the analysis can also be improved by accounting for the shear stiffness of the adherends. This can be readily implemented by using  Ash r-iag ea  as given in equation  (5.92) instead of A. The results of this analysis scheme are presented for three different cases:  • Thermal residual stress loading ( A T = — 64.8°C)  • Remote stress loading (0^ = 40 MPa)  • Combined thermal residual stress and remote stress loading ( A T = —64.8°C, aoo = 40 MPa)  5.6  Generalization of the Rose Model for Partially Disbonded Patches  182  Note that the stress intensity factors are determined for all cases using the plane stress solution to allow for a better comparison between the three cases. Plane stress conditions are typically present under a combination of high external loading and thermal residual stresses in a real application. The material and geometric data used is this analysis are given below: Substrate Material Young's modulus  E = 72.4 GPa  Shear modulus  G = 27.2 GPa  Coef. of Therm. Expan.  a = 23.45 ^  Thickness  t = 6.35 mm  Edge crack length  a = 20 mm  Correction factor  Y = 1.12  a  s  s  s  Patch Material Young's modulus (||)  E  Shear modulus  G  Coef. of Therm. Expan.  a  Thickness  7J  Length  lp = 75.0 mm  p  = 210 G P a 7.24 G P a  r  = 4.61  p  ^  = 6.35 mm  P  Adhesive Shear modulus  C  Thickness  7j  The stress intensity factors K^,  a  a  = 0.844 GPa = 0.25 mm  K  d  and K* were determined for each of the loading  conditions for the entire range of b, i.e. from no disbond (6=0 mm) to completely disbonded (6=75 mm). Note that K* is dependent on the chosen crack length and will therefore shift up or down for different crack lengths. K* indicates for which disbond length is dominant.  or K*  5.6  Generalization of the Rose Model for Partially Disbonded Patches  183  20 - r  I  --  17.5  15 - • --  12.5  (0 Q.  s  10 - -  7.5 - -  5 -; 2.5 - -  0 --  35  40  b [mm]  Figure 5.22: Stress intensity factor dependence on disbond length for a bonded repair under thermal residual stress loading ( A T = —64.8°C) Figure 5.22 through Figure 5.24 show some interesting features when looking at large disbonds. The stress intensity factors drop to zero for the case of thermal residual stress loading when approaching a complete disbond, which should be expected. This effect seems to be dominant only when the remaining bonded length is approximately equal to the load transfer length, which in this case is 6.5 mm. The opposite behaviour can be seen for the  1  20  I  17.5  ra  Q.  s  15  + +  12.5  10  K  7.5  K  5 2.5  0  -f10  15  20  25  30  35  40  45  50  55  +  60  65  70  75  b [mm]  Figure 5.23: Stress intensity factor dependence on disbond length for a bonded repair under remote stress loading (O-QO = 40 MPa)  5.6  Generalization of the Rose Model for Partially Disbonded Patches  184  35 40 b [mm] Figure 5.24: Stress intensity factor dependence on disbond length for a bonded repair under combined thermal residual stress and remote stress loading ( A T = — 64.8°C, (Too  = 40 MPa)  case of a remotely applied stress (Figure 5.23). The stress intensity factor  converges  to infinity, which is to be expected due to using a pressurized double-doubler joint model, i.e. the substrate would separate. K* reaches its limiting value of an unpatched specimen, which is also properly reflected by K*. The behavior of the stress intensity factors for the combined loading case (Figure 5.24) also satisfies expectations.  is again converging  to infinity caused by the remaining remote stress after the complete disbond, while K* is again limited by the unpatched scenario. K* reflects the change between the limiting stress intensity factors well.  Rose's patching efficiency definition [78] leading to zero patching efficiency for the unpatched crack and 1.0 for elimination of the crack i.e. a reduction in stress intensity to 0 M P a i / m shows the problem of disbonds clearly:  RK =  1  K:  K(b=h).  (5.223)  This particular definition cannot be used for the case where only thermal residual stresses are present, i.e.  K*(b=i ) p  equal to zero.  But the definition reflects the patch efficiency  5.6  Generalization of the Rose Model for Partially Disbonded Patches  185  quite well for partially disbonded patches with a loading combination of thermal residual stresses and remote stresses, which is most common for test specimens (Figure 5.25). Under the given boundary and loading conditions, the patching leads only to a reduction in the absolute stress intensity factor to a disbonding length of approximately 19.1 mm. Larger disbonds represent a worse case than the unrepaired specimen due to the thermal residual stresses. Note that this particular disbond length shifts to a higher value if the crack length increases. For a crack length of 50 mm the disbond can be up to approximately 58.7 mm before patching will lead to an increase in the stress intensity factor.  The influence of  thermal residual stresses on the stress intensity factor for real bonded repair applications with disbonds is generally not as severe as shown here for an unrestrained specimen due to the lower effective coefficient of thermal expansion of the substrate. The patching efficiency is much closer to the case of remote stress loading (Figure 5.25).  1x  Remote Stress  0.8 +.  Thermal,Residual Stress & Remote Stress  0.6 +  0.4-k 0.2 + 0  -0.2 + -0.4  -L  5  10  15  20  55  60  65  70  '5  b [mm] Figure 5.25: Patch efficiency for partially disbonded patches  The results show how readily an idealized disbond can be included in the analysis.  In  comparison to the basic analysis carried out in Section 5.3, the revised results for the determination of an upper bound for the stress intensity factor including disbonds have  5.6  Generalization of the Rose Model for Partially Disbonded Patches  186  only the additional disbond length term b. If b = 0, the original solution without disbonds is obtained. In this respect, the presented solution is a more generalized estimation for the performance of bonded repairs and represents one of the main achievements of this research work.  The solution can be improved further by accounting for plastic strain in the adhesive although the influence of the adhesive shear deformation towards the crack opening displacement is reduced from 100% of the total crack opening displacement for a fully bonded repair to only 10% for a repair containing large disbonds using the fully elastic analysis. Even for a significant increase in the shear displacement by accounting for plastic strain, the change in the stress intensity factor will be relatively small making the elastic solution an excellent first order approximation. Additional experimental work as well as higher order analytical work is required to determine the overall performance of the method.  5.6.2  Double-sided Tapered Repair with  Disbonds  The stress intensity factor change with disbond growth can also determined for a repair involving a patch with a tapered edge. The direct closed form method for determining the stress distribution for tapered reinforcements and double-doubler joints was presented in Section 5.2. This method can be easily adapted for analyzing the stress at the prospective crack location using a partially disbonded reinforcement model and the shear stress at the edge of the disbond in a pressured double-doubler joint. Note that this approach is only valid until the disbond reaches the last step adjacent to the edge. Then the model reduces to a simpler analysis similar to the uniform thickness patch but must include the thickness change of the patch within the disbonded region. Using this two step model, the influence of a taper on disbond tolerance can be investigated for thermal, mechanical and  5.6  Generalization of the Rose Model for Partially Disbonded Patches  187  Figure 5.26: Free body diagram for a tapered reinforcement with a disbond a combination of both load types. The model is fully elastic and can be extended using a numerical approach for elastic-plastic behaviour in the adhesive.  Stage I: Stress at the prospective crack location Determination of the stress at the prospective crack location is carried out in a manner similar to that of the uniform thickness repair with a disbond. In the case of a tapered edge, the boundary condition for determining the stress at the prospective crack location must consider the change in thickness over the partial disbond. Figure 5.26 shows the free body diagram for this particular case. Note that there is no need for an interface condition between steps when only the last step l remains bonded, thus simplifying the analysis. n  In order to use the previously developed system of equations for tapered reinforcements, the notation for the different steps in the disbonded section is indexed 2-eL..-1,0,1 where 1 indicates the identical geometry as the first section of the remaining bonded area and d is the number of disbonded steps. The stress at the prospective crack location for a disbond with a fully bonded last step is determined in the first part of the stage I analysis of the Rose model.  5.6  Generalization of the Rose Model for Partially Disbonded Patches  188  As before, the change in length in the patch and substrate can be evaluated using: i  i  k=2-d  k=2-d  (5.224) Rd  Al  s  = eb = T s  +  S1  P  k  Pfc  a ATb  (5.225)  s  where b=  J2  (5.226)  k=2-d  Using the definition of the elastic shear stress distribution parameter A and the force equi librium given in equation (5.13), the shear stress at yi = 0 can be expressed as: A?  Rd T  ai  (Wi=°)  (VI/I=O)  =  A?  s  (A/ - A/ ) p  Af6  (5.227)  - w (yi=o))  (5.228)  s  _,Rd fc=2-d  (5.229)  "Pi  yE t  P1 P1  EtJ  yE t  s s  Pl Pl  Et J s s  The shear stress expression can be simplified by defining the constant H as  H =  (5.230)  and using the integration constant C from Section 5.2 as given by equation (5.16): p  Rd r  ai  =o) =  {yi  (5.231)  \ b(T™H-C ) 2  Pl  A proper boundary condition can now be established by substituting  and  r™( =o). yi  The direct stress resultant for the patch can be evaluated using equation (5.11). . Rd  Rd T  PI  A P I (wi=°) = ~\7' Rd  = T  1  +  C  ^  (5.232)  5.6  Generalization of the Rose Model for Partially Disbonded Patches  189  The shear stress is determined using equation (5.10) Rd  r  „  „ R d  ( =o) = B  a i  Vl  (5.233)  1  thus the boundary condition yields:  B™ = A^XibH  + C Xlb(H  - 1) .  Pl  The additional equation to solve for A  and B  Rd  (5.234)  is given by the boundary condition at  Rd  the edge, thus:  T (y =i ) = Y± cosh(A„Z„) + Pn  n  sinh(A„/ ) + C  n  An  n  = 0.  Pn  (5.235)  A  n  Using equation (5.52) and (5.53) developed in Section 5.2, the boundary condition at the edge yields: . Rd A  R d /(0=o) cosh(A„Z„) 4- / O ^ " - ) sinh(A / ) 1  n  \  -Bi  =  n  /(</»=-i)cosh(AJ ) + / ( ^ = 2 ' - - i ) s i n h ( A / ) i  1  n  n  n  AC(0=o) cosh(A Z ) + AC(</»=2"- )sinh(A i ) + 1  -Xi  n  n  n  n  (5.236)  C  Pn  /(0=-i)cosh(A / ) + / ( • / » = 2 " - - i ) s i n h ( A / „ ) 1  n  = -B™F  n  n  - XiF  f  (5.237)  c  where _  F  /(^=o) cosh(AJ ) + / ( 0 = 2 " - ) s i n h ( A „ l ) 1  w  n  n  2  =  n  n  n  AC(^=o) cosh(A Z ) + A C c ^ " - ) sinh(A / ) + w  g  1  1  F  3  /(0=2"- -i)sinh(A / )  /(*=-i)cosh(A Z ) +  f  w  w  n  C  Pn  /(0=-i)cosh(A / ) +/(,/,=2"- -i)sinh(A ^) 1  n  n  n  Thus A™ and B™ can be determined by solving the linear system of equations given by the boundary conditions at the edge of the disbond and at the patch edge, i.e. .Rd _ A  2  c  ~  l  5  A i F + X bC F (H  l  ~  Pl  1+  X  l  b  {  - 1)  XibFfH  1 + XibFfH  •  ( 5  -  2 4 1 )  5.6  Generalization of the Rose Model for Partially Disbonded Patches  190  Finally, the stress at the prospective crack location for a disbond leaving the last step fully bonded can be determined using:  -t {  l  8  +  +Csi  X FH lb  f  J •  -  (5  243)  The stress at the prospective crack location for a disbond extending underneath the last step of the patch can be easily determined using the analysis for uniform patch thickness as well as the analysis presented for the tapered patch excluding the last step. For this particular case the index of the tapered analysis is 1 = i — n. Equation (5.234) and (5.235) give the two equations required to determine A  as  Rd  cosh(AiZi) + ifAi6sinh(Ai/ )  1  1  '  1  '  '  Thus the stress at the prospective crack location for a repair where the disbond extends underneath the last step of the patch can be evaluated using equation (5.242) yielding d 0  1 /  2C +2A 6C (ff-l)sinh(A M P l  t\  P l  1  1  \  1  cosh(A i ) + H"A 6sinh(AJ )  a  S t a g e II: U p p e r b o u n d s  1  J  1  1  ^^j'  '  {0  0 )  K  r  Having determined the stress at the prospective crack location cr^, the crack is introduced in the second stage of the analysis. The analysis is, for the most part, identical with the approach for double-sided uniform thickness repairs with disbonds given in Section 5.6.1. The half crack opening v consists again of three components. Crack opening displacements 0  are caused by the shear strain in the adhesive and the strains in the disbonded sections of the patches and substrate.  An upper bound for the stress intensity factor for short cracks  and the corresponding  5.6  Generalization of the Rose Model for Partially Disbonded Patches  energy release rate G  u  191  assuming no load transfer into the reinforcement can be evaluated  using equations (5.204), (5.205) and (5.206) as presented for the case of partially disbonded uniform thickness repairs. The second upper bound for the stress intensity and energy release rate as given already by equations (5.218), (5.219), (5.214) and (5.216) is estimated by the change in potential energy when introducing a cut in the substrate of a double-sided reinforced plate with disbonds, thereby converting the specimen to a double-doubler joint with disbonds adjacent to the joint.  The stress at the prospective crack location a  was determined in equation (5.243) and  d  (5.245) depending on the disbond size in stage I of the Rose model using a one-dimensional model of a tapered, double-sided, partially disbonded reinforcement. The displacement v  d  as a result of introducing a cut in the tapered, double-sided, partially disbonded reinforcement can be easily determined using superposition as shown in Figure 5.20.  According to Figure 5.20, the displacement v is given as the displacement of one joint or d  crack face for a pressurized cut in a partially disbonded double-doubler joint. The general analysis of the shear stress in the adhesive as given in Section 5.3.2 is still applicable in contrast to the partially disbonded double-sided reinforcement. In the case of accounting for the influence of the tapered edges of the patch, the stress analysis for tapered patches as outlined in Section 5.2 can be applied. It is again necessary to distinguish between disbonds extending underneath the last step of the patch and those having a shorter disbond length. Tapered bonded repairs with disbond length not extending underneath the last step are considered first. The boundary conditions for this case are given below.  Boundary condition at the disbond edge: Pd  2A[  (5.246)  5.6  Generalization of the Rose M o d e l for P a r t i a l l y Disbonded Patches  Thus A  x  192  yields:  t  ^  = M i f  (5-247)  Boundary condition at the edge: A T*\y =i ) = - f - c o s h ( A l ) +  B  An  An  Fd  n  n  n  T h e fundamental relationships between A  Fd  sinh(AJ ) = 0  n  (5.248)  n  and A™ as well as between B  Pd  Pd  and B  Pd  are  given i n equations (5.52) and (5.53). For pressurized joints it turns out that AC(<p) is zero thus simplifying the relationships to: pd  Pd  pd  - f - = /(«=-D-^- + /(*=o)-^An  (5-249)  M  Al  B A B - f - = /(*=2-i-i)-i- + / *= »-i)-L- . Pd  Pd  A„  (  Ai  Pd  2  (5.250)  Ai  Substituting equations (5.247), (5.249) and (5.250) i n the edge boundary condition equation yields the constant B  Pd  p  as: 4/(0=-i)cosh(A„Z ) +/(^=2"- -i)sinh(A / )  „,,,  1  d  d  n  2  n  7l  /(^=o)cosh(A Z ) + / ( ^ = 2 " - ) s i n h ( A / ) 1  n  Having determined the two constants A  n  n  and B ,  Pd  n  the shear stress at the edge of the  Pd  disbond is given by:  7-^=0) = ^ _ _  (5.252)  d  t f(<p=-i) cosh(A Z ) + f{<f>=2^-\) s i n h ( A / ) s  w  w  n  ° 2 /r>=o)cosh(A J ) + / ( ^ 2 " - ) s i n h ( A Z ) = - ^ A i | j r a  l  n  (5 253)  1  n  n  n  The corresponding half crack opening displacement v  d Ql  n  (5-254)  is then calculated using equation  (5.8) and (5.117):  0 1  2  (i? £ P l  P l  £ £ ) A]Ff s  s  ^  ^  5.6  Generalization of the Rose Model for Partially Disbonded Patches  193  The additional displacements due to strains in the substrate and patch are given as:  oIl  u  2 d  7  (5.256)  ^—' E t  k=2-d  "  p  lpk  (5.257)  oIII  which are then used to calculate the total half crack opening displacement as follows:  d  d d «oi + o i i +  =  d om  v  1 2  \E  2  +  "'Pi  P1  d , 2  (5.258)  v  -^s^s  1 \£ p 7j  1  d  1  d  i _ J _ _L ° a  ta  i. ,  1  a  o.  (5.259)  2  ,  1  \  p i  (5.260)  + E'gt,  As previously discussed, it is necessary to determine the half crack opening for a disbond extending underneath the last step of the patch separately. The half crack opening due to the shear displacement in the adhesive can be determined using equation (5.208) from the uniform thickness patch analysis while the displacements due to the strain in the substrate and patch are given in equation (5.256) and (5.257). Thus the total half crack opening displacement for a disbond extending underneath the last step of the patch is given by:  \ crj o"s :  Et  Pl Pl  + E^t  s  A i tanh(Ai_Zi)  (  1  _|_ 2 \  (5.261)  The upper limit for the stress intensity factor under plane stress conditions can be expressed for the partially disbonded tapered repair as  (5.262)  5.6  Generalization of the Rose M o d e l for P a r t i a l l y Disbonded Patches  194  where A is given for disbonds not extending underneath the last step of the patch as  1  A =  vrF  Eat a  1 +  2  •iF  V 2£ t  + t  P1 P1  (5.263)  (_1_ + _2_)  V  \ ^ P l ^Pl  Et J s s  and for disbonds extending underneath the last step of the patch as  A  1  /  s s  2E t  TTY  2  i  Et Pl Pl  SJ U* ** *0 k  1  +  \  k  Ai tanh(Aiii)  (  I  _| 2_\  (5.264)  A factor of 1/(1 — f ) needs to be included i n A in the case of plane strain conditions. Using 2  and factor  as asymptotes, as suggested by the original Rose model, the stress intensity and the corresponding energy release rate  for different crack lengths are then  derived by interpolation using equation (5.220) and (5.221). It is noted that the accuracy of this analysis was improved by accounting for the shear stiffness of the adherends by using Ashear-iag (equation (5.92)) instead of A. The results of this analysis scheme are presented for the same three cases as presented in Section 5.6.1, i.e.: • T h e r m a l residual stress loading ( A T = — 64.8°C) • Remote stress loading (O-QO = 40 M P a ) • C o m b i n e d thermal residual stress and remote stress loading ( A T = —64.8°C, (Too  = 40 M P a )  The only difference between the uniform thickness patch analysis and the tapered patch analysis is the thickness change of the patch near the edge. A l l other material parameters remain unchanged.  5.6  Generalization of the Rose Model for Partially Disbonded Patches  195  The patch thickness is defined as:  Pl<i<7  = 1.056 mm — i 0.132 mm  (5.265)  with step lengths of: l = 57mm  and . I, = 3 mm . 2<i<7  x  (5.266)  Note that the above given length and thickness data are indexed for a specimen with a disbond length not exceeding the first step.  For larger disbonds, the index changes as  presented in Figure 5.26. The stress intensity factors K^, K  d  and K  d  were determined for  each of the loading conditions for the entire range of b from, i.e. from no disbond (b=0 mm) to completely disbonded (b=75 mm). Note that K  d  is dependent on the chosen crack  length and will therefore shift up or down for different crack lengths. The parameter K  d  shows for which disbond length  or K  d  is dominant. In order to highlight the change in  the behaviour of the stress intensity factors due to introducing a tapered edge, the results for the uniform thickness patch have been plotted in each graph. The observed behaviour of  20 -r  uniform patch  2.5 +  tapered patch  0 0  5  10  15  20  25  30  35  40  b [mm]  45  50  55  60  65  70  75  Figure 5.27: Stress intensity factor dependencies on disbond length for a bonded tapered and uniform thickness patches under thermal residual stress loading ( A T = -64.8°C)  5.6  Generalization of the Rose M o d e l for P a r t i a l l y Disbonded Patches  o -I 0  1 5  1  1  10  15  1 20  196  1  1  1  1  1  1  1  1  1  h  25  30  35  40  45  50  55  60  65  70  b [mm]  75  Figure 5.28: Stress intensity factor dependencies on disbond length for bonded tapered and uniform thickness patches under remote stress loading (a^ = 40 M P a ) partially disbonded uniform thickness patches is very similar to the results determined for partially disbonded tapered patches. It becomes evident that only a relatively small change in the stress intensity factor is introduced by designing a bonded repair w i t h a tapered edge. The stress intensity factors for the tapered patch changes less rapidly when approaching a complete disbond (Figure 5.27 through Figure 5.29).  T h e change i n the stress intensity  factors is quite small for a combined thermal residual stress and remote stress loading. T h i s is due to an increase i n K for t h remote loaded situation and a decrease i n K for the thermal residual stress loaded situation, thus leading to a partial cancellation of the individual effects.  A s done previously, the results can also be shown i n the form of patch efficiency plots (Figure 5.30) for the case of remote stress loading (noted R S ) and combined thermal residual stress and remote stress loading (noted R S & T R S ) using equation (5.223) as before. The small change i n the efficiency leading to a gentler transition when approaching complete disbonding is also confined to the tapered region. Realistic disbond sizes measure typically 2b «  1 1 m m w i t h large disbonds reaching 2b « 2 7 m m [13].  F r o m these experimental  5.6  G e n e r a l i z a t i o n of the Rose M o d e l for P a r t i a l l y D i s b o n d e d  Patches  197  observations it w i l l generally not be necessary to i n c l u d e the t a p e r i n the stress intensity analysis since using the same patch length w i t h uniform thickness will provide w i t h sufficient a c c u r a c y as l o n g as t h e d i s b o n d  region.  The  efficiency of a b o n d e d repair test s p e c i m e n w i l l still have a significant i m p r o v e m e n t  over  the unrepaired growth. poor  does not  s p e c i m e n due to the development  enter the tapered  predictions  of generally s m a l l disbonds  w i t h crack  L a r g e r d i s b o n d s w i l l most likely be the cause of very p o o r surface p r e p a r a t i o n  or  design.  If a m o r e a c c u r a t e s o l u t i o n is r e q u i r e d partially disbonded  tapered  for d e t e r m i n i n g  the stress i n t e n s i t y factor of  patch, the analysis can be improved  for plastic s t r a i n i n the adhesive.  by  accounting  in the case of p a r t i a l l y d i s b o n d e d  tapered  patches, the influence of the shear d e f o r m a t i o n of the adhesive t o w a r d s the crack  opening  d i s p l a c e m e n t is r e d u c e d  Note that  further  a  f r o m 1 0 0 % of the total crack opening  displacement for a  fully  b o n d e d repair to o n l y 1 0 % for large d i s b o n d s in the repair using the o u t l i n e d elastic analysis. E v e n for a significant increase i n the shear d i s p l a c e m e n t by a c c o u n t i n g for plastic s t r a i n ,  40  0  T  -I  1  1  0  5  10  F i g u r e 5.29:  1 15  1  1  1  1  1  1  1  1  1  20  25  30  35  40  45  50  55  60  b  [mm]  1 65  1  1  70  75  Stress intensity factor dependencies on d i s b o n d length for b o n d e d tapered u n i f o r m thickness patches under c o m b i n e d t h e r m a l residual stress a n d stress l o a d i n g ( A T  =  -64.8°C,  =  40 M P a )  and  remote  5.7  Summary and Discussion  198  1 -r  RS (uniform patch)  Figure 5.30: Patch efficiency for partially disbonded patches with tapered edges under remote stress loading (RS) and combined thermal residual stress and remote stress loading (TRS&RS) the change in the stress intensity factor will still be relatively small making the elastic solution an excellent first order approximation.  Based on these results, additional experimental work as well as higher order analytical work can be focused on uniform thickness patches rather than tapered patches to determine the overall performance of the method for estimating the stress intensity factor for partially disbonded patches in the crack vicinity.  5.7  Summary and  Discussion  The influence of thermal residual stresses on the stress intensity factor in bonded repairs was theoretically investigated and presented in detail. A number of improvements over previous theoretical models have been established.  One of the most significant achievements was  the stress field estimation in the isotropic substrate based on a new concise solution for the classic fracture mechanics problem of a center crack in an infinite plate. This stress field  5.7  Summary and Discussion  199  estimation is a useful tool for experimental work such as the placement of strain gauges near the crack tip. T h i s model can also be used for the development of 'smart patches', where strain sensing devices are used to detect crack growth.  Another significant advance is the disbond model for bonded repairs based on the Rose model which gives a tool for estimating the impact of disbonds on the stress intensity factor. T h e model considers all disbond lengths up to full patch lengths and has also been extended for tapered patches. T h e model indicates the severity of thermal residual stresses on unrestrained specimens with relatively short disbonds. Special attention should therefore be brought to real applications where the effective coefficient of thermal expansion of the substrate is only slightly lower than the unrestrained specimen and quite different from the patch material. T h e effectiveness of the bonded repair with respect to the reduction in the stress intensity factor is also dependent on the ratio of the thermal residual stresses compared to the applied loading. T h e importance of such a theoretical tool is emphasized by the necessity to assess what disbond size around the crack is permissible prior to rejecting a repair. Significant experimental testing will be required in the future to validate this type of model and gain confidence in its performance.  Additionally a new theoretical approach has been presented for determining the stresses in tapered joints and reinforcements as well as stepped joints. T h i s approach avoids the typical numerical solution scheme requiring specialized computer codes, which are not readily available. A n y mathematical spreadsheet such as M a t h C a d , M a t L a b or Excel can be used to determine the stress distribution in tapered or stepped joints with a limited number of steps which will be sufficient for many applications. Therefore, this theoretical approach provides engineers with a simple tool to understand the behaviour of adhesive joints better. Emphasis in this chapter was placed on providing an easy to follow derivation of the basic equations for the stress distribution in uniform thickness as well as tapered joints and reinforcements, thus giving the complete background rather than a 'black box'.  5.7  Summary and Discussion  200  The presented models have been evaluated for the thermal residual stresses and strains present in the A M R L sandwich type specimen. Generally good agreement has been found with the experimental measurements within the limitations of the model such as in-plane bending. The results of a detailed finite element analysis representing the next step in this research work will reveal the strength and weaknesses of these models in more detail than it was possible with the experimental measurements using strain gauges.  6  Finite Element Analysis of the A M R L Specimen  201  Chapter 6 Finite Element Analysis of the A M R L Specimen  6.1  Introduction  The finite element analysis represents an important part of this research work as it provides better insight into the stress distribution of bonded repairs. A significant number of research papers have been published discussing finite element modeling approaches for bonded repairs. Well known is the work from Jones and Callinan [24, 52, 54], who developed a twodimensional finite element formulation for the adhesive layer in 1979. The adhesive element formulation was based on the displacement between the substrate and patch accounting also for the transverse shear stiffness in the adherends.  Coupling the adhesive stiffness  matrix with the composite patch stiffness matrix allowed them to use this bonded pair in conjunction with standard finite element routines. The authors also suggested appropriate modeling techniques for commercial F E M packages to include the transverse shear stress of the adherends in two-dimensional models. This scheme of using adhesive joint elements which account for the transverse shear deformation of the adherend was also used by other research institutions such as the Cranfield Institute of Technology [19]. Since 1979 the increase in computing power has allowed the development of three-dimensional finite element modeling thus providing a more accurate results tool which can better ac-  6.1  Introduction  202  count for the transverse shear stress through the adherends and adhesive [24]. Additionally the three-dimensional representation is desirable as it allows for a better comparison with the experimental thermal residual strain data measured at a number of locations through the thickness. A two-dimensional approach utilizing plate elements for the substrate and patch cannot model the effect of the transverse shear stress at different through thickness locations in the adherents properly. Plate elements generally assume zero shear stress at the top and bottom surface. A three-dimensional approach was chosen based on the inadequacy of the two-dimensional approach to predict the strains properly at the strain gauged locations. In this model, 20 noded brick elements were chosen as the main element type in order to get a linear stress prediction capability sufficient to model the transition zones with high stress gradients, such as the taper and crack region with less elements.  Submodeling can be a useful option for modeling details of bonded repairs, such as the crack region or the tapered section, depending on the available computing power. Such an approach was recently presented by the Y . Xiong and M.D. Raizenne [101]. Similarly, Callinan uses a number of models with refinements in different critical regions [24]. These approaches are certainly an excellent option when plastic deformations are investigated or the computing power is limited. A Pentium 11-266 P C with 128 M B R A M and 12 G B of swap/storage space in conjunction with the finite element program MSC/Nastran for Windows was available for this research, thus allowing creation of a linear-elastic model of the A M R L sandwich type specimen in sufficient detail without the need for submodeling.  Due to the complexity of bonded repairs, a number of verification models were created to investigate each critical section in detail. These models allowed one to determine a sufficiently fine mesh ensuring confidence in the results while reducing the overall degrees of freedom as well as the required solution time for an entire model of the A M R L sandwich type specimen significantly.  6.1  Introduction  203  The following material properties were employed for the verification models and the A M R L model: B o r o n / e p o x y 5521/4 Laminate: E  = E  E  = 210 G P a  Px  Py  = 25 G P a  Vz  G  = G  G  = 1 GPa  Pxy  Pxz  v  Vxy  = 7.24 GPa  Pyz  = 0.025  = 0-17 a  —a  Px  Q  — 25.87 / / e / ° C  Pz  P w  =4.61 /xe/°C  2024-T3 A l u m i n u m : E  Sx  G  Sxy  = E  =  Sy  = G  V  3  3  = 23.29 / i e / ° C  Sz  5 y  = 27.2 GPa  Sxz  = >« = ° -  = a a  = G  Syz  "*»v =  = 72.4 GPa  = 23.45  nt/°C  F M 73M: E  = E  ax  = E  ay  G  axy  = G  ayz  "a  = Va  «a  = Oi  xy  x  yz  ay  = 2.19 G P a  az  = G  axz  = "a  -  xz  — a  a  z  =  = 844 MPa 0.3  50 / i e / ° C  Note: The y-direction is aligned with the boron fiber and aluminum grain orientation.  6.2 6.2  Verification Models  204  Verification Models  In total, five verification models were created. The initial model representing a uniform thickness reinforcement was used to determine the mesh size in regions of high shear stress. This model served as a basic reference case due to the availability of a good theoretical model for comparison. A tapered reinforcement was chosen as a second model to optimize the required number of elements in the tapered region. Both normal and reversed stacking sequence were investigated to assess the importance of detailed representation of the actual structure. Next, a model of the substrate in the cracked section was investigated. The focus of this model was placed on the stress intensity factor prediction. The mesh for the crack region had to be sufficiently fine along the crack face to predict the adhesive shear stress properly once the adhesive and patch were added to the model, i.e. the next step. The results of this model were then compared to the developed closed form approximation for the stress field around a patched crack. The A M R L specimen exhibits additional in-plane bending due to the partial patching across the specimen width which are not considered in the theoretical approach making this patched crack model an assessment tool for the theoretical crack tip stress field model.  The half circular geometry of the patch also complicated the finite element analysis. It was required to check if the aspect ratios found for the tapered reinforcement were acceptable for a circular patch due to the change in the material orientation with respect to the element orientation. Finally the entire A M R L specimen was modeled using the mesh specifications derived by the different verification models.  The number of elements required to get an accurate  representation of the A M R L specimen was so large that an attempt to verify the mesh size in the full model was too time consuming and resources were quickly exhausted.  6.2  Verification Models  6.2.1  205  Constant Thickness Double-sided Reinforcement  This first model was intended to confirm the input data into MSC/Nastran as well as compare the results with the proven one-dimensional model of L . J . Hart-Smith. In order to avoid any stress concentration effects, the model had identical areas for the substrate, adhesive and patch. In-plane quarter symmetry as well as symmetry through the constant thickness model were used. The material thickness were selected to represent the inner section of the A M R L specimen: 3.175 mm thick aluminum face sheet, 0.25 mm thick adhesive layer and the 0.924 mm thick seven layers of boron/epoxy.  This investigation  was also useful for determining the required element size for the reinforced crack region in the A M R L specimen since the shear stress distribution away from the crack tip can be approximated by the shear stress distribution in a double-doubler joint. It should be noted that the shear stress distribution near the center of a double-doubler joint under thermal loading is identical to the shear stress distribution near the free edge of reinforcement considering a sufficiently long overlap length. The aluminum, adhesive and patch were modeled initially using two elements through the thickness of each material. Five elements were used across the half-width of 12.5 mm. The specimen half-length was 75 mm. The element size was reduced from 5 mm in the center of the specimen with zero shear stress to 0.25 mm, 0.0625 mm and 0.03125 mm respectively along the length to the specimen edge with the highest shear stresses. The last four elements adjacent to the free edge had the same element length when an element length of 0.25 mm and 0.0625 mm was used. Eight elements with equal length were chosen for the model with an element length of 0.03125 mm. The chosen element lengths were based on fractions of the adhesive thickness between t and t /8 a  a  covering Callinan's recommendation of a maximum  element length of rj /5 near the edge [24]. A temperature change of — 64.8°C was applied a  as a loading condition representing the cooling stage from the stress free temperature to room temperature.  6.2  Verification Models  206  Figure 6.1 shows the shear stress in the mid-plane of the adhesive along the centerline and close to the end of the specimen for the three F E M analyses and Hart-Smith's theoretical solution. The theoretical analysis shows excellent agreement up to 0.5 mm away from the free edge. This one-dimensional approach cannot account for the drop to zero shear stress at the free edge. The coarse F E M model shows the highest shear stress (21.4 MPa) 0.25 mm away from the free edge while the two finer meshes show a peak shear stress (20.0 MPa) at approximately 74.85 mm from the center of the specimen. It should be noted that the one-dimensional analysis predicts nearly the same value (19.7 MPa). Changing from an element length equivalent of 25% to 12.5% of the adhesive thickness results in only minor differences in the peak shear stress and its location. 22 20 18 16 14 12 |  8  " H  6  4 2 0  -1-D Theory  -2  -0.25 mm  -4  - 0.0625 mm  -6  -0.03125 mmm  -8 74.125  74.25  74.375  74.5  74.625  74.75  74.875  y [mm]  Figure 6.1: Adhesive shear stress for a uniform thickness reinforcement using element lengths between t  a  and t /8 a  The mesh with an element length of 12.5% of the adhesive thickness shows a shear stress of -5.0 M P a at the free edge while the finer mesh drops to -6.1 MPa. Both meshing schemes show a significant error which is not surprising considering the large stress gradient near the free edge. The shear stress prediction for an element length equivalent to 1/4 of the adhesive thickness shows the better result for this particular finite element verification model.  6.2  Verification Models  207  In addition, a model with a larger number of elements through the thickness was tested. The mesh with the smallest element length of (12.5%) t was used with 6 elements through a  the thickness for each material. Figure 6.2 shows the variation in shear stress results for 2 and 6 elements through the thickness.  22 20 18 16 14 12 ^  10  Q-  8  a  - 2 Elements - 6 Elements  74.125  74.25  74.375  74.5  74.625  74.75  74.875  75  y [mm]  Figure 6.2: Adhesive shear stress for a uniform thickness reinforcement with a different number of elements through the thickness  The position of the peak shear stress shifted by only 0.04 mm and the peak stress was reduced by 0.5 MPa. The shear stress at the free edge was less than 0.1 MPa, i.e. nearly the theoretical value of 0 MPa. The larger number of elements through the thickness would be preferred near the free edge, but it would increase the total number of elements since a transition should not be made in sections with high stress gradients.  Based on  the importance of predicting the peak shear stress and its location, it was decided to use an element length equivalent to 25% of the adhesive thickness.  Generally, two elements  through the thickness were used for each material except in the aluminum near the crack where four elements through the thickness were employed to improve the prediction of the crack opening.  6.2  Verification Models  6.2.2  208  Tapered Reinforcement with a Reversed Stacking  Sequence  A tapered reinforcement was chosen as the second verification model. It was important to verify that a sufficient number of elements has been used for each step to determine the thermal residual strains in the adherends and the adhesive shear stress properly. The smallest number of elements was desirable to reduce the overall degrees of freedom for the A M R L sandwich type model. In the past, tapered edges have been modeled using tapered elements or a normal stacking sequence, not necessarily representing the real layup sequence [19, 24]. Specially tapered elements reduce the required number of elements but they do not represent the true structural details of the actual taper. As part of this verification model, a normal stacking sequence was compared to the actual reversed stacking sequence of the A M R L specimen (Figure 6.3) showing the importance of proper structural representation and design choices under given loading conditions.  Reversed Stacking  Normal Stacking  Figure 6.3: Reverse and normal stacking sequence  The reversed stacking sequence is significantly more complex to model than the normal stacking sequence if the change in material orientation in the thickness direction is included. The size of the adhesive fillet was measured using a precured patch where the peel ply was not removed.  The fillet shows up as a smooth region versus the typical fabric texture  of the peel ply. The size of the fillet varies with respect to the location along the outer edge in the case of a half circular patch as used for the A M R L sandwich type specimen. The fillet length is approximately 1 to 1.25 mm for boron/epoxy if the taper is aligned with the fiber orientation, while it is only 0.25 mm where the fiber orientation is parallel  6.2  Verification Models  209  to the taper. The carrier fabric of the boron/epoxy prepreg causes a small fillet where the fibers are perpendicular to the taper direction while the larger fillet is a result of the stiffness of the boron fibers where the fibers are parallel to the taper orientation. In order to reduce the complexity and the required number of elements to account for the fillet properly, a triangular fillet with a length of 1 mm was chosen regardless of the location in the patch. This approach can be justified by the relatively long distance from the locations of the locally thicker adhesive to the regions of high shear stress in case where the fiber orientation is not parallel to the taper orientation, thus not affecting the results in a critical area.  Figure 6.4 through Figure 6.9 show results of the convergence study which was carried out for the reversed stacking sequence. The shear stress in the mid-plane of the adhesive as well as the thermal residual strains at the top and bottom surface of the patch and substrate were evaluated. The effect of smaller elements as well as more elements through the thickness are shown in these figures.  Figure 6.4: Transverse shear stress at the adhesive mid-plane in the tapered region for different F E M meshes  6.2  Verification Models  210  T y p i c a l l y seven elements through the thickness of the boron/epoxy were chosen to simplify the modeling while the adhesive and aluminum were represented by 2 elements through the thickness as shown in Figure 6.5.  LiiJL._llfg=aa,  Figure 6.5: Detailed view of the finite element mesh i n the tapered section-reversed stacking  The number of elements was doubled to study the element number dependence through the thickness. These figures show the extreme cases of this convergence study for element length along the taper w i t h the smallest element size of both 0.125 m m and 0.03125 m m at the ply drop-off. Furthermore, the number of elements was reduced along the taper for the  Figure 6.6: T h e r m a l residual strains at the top surface of the patch in the tapered region for different F E M meshes case of seven elements through the thickness of the boron/epoxy, two elements through the adhesive and two elements through the aluminum thickness. T h e smallest element length  6.2  Verification Models  211  Figure 6.7: Thermal residual strains at the bottom surface of the patch in the tapered region for different F E M meshes was 0.125 mm in comparison to the curve for the model with twice as many elements through the thickness.  The labels in the plots identify the number of elements for each  component and the smallest element size at the ply drop-offs.  600 500 400 300 Ji  200 100 - - - - 7 B/en-2 FM73M - 2 Al-0.03125 mm  0  14 B/ep - 4 FM73M - 2 Al - 0.125 mm 7 B/ep - 2 FM73M - 2 Al - 0 125 mm  -100 -200 48  51  54  57  60  63  66  69  72  75  78  y [mm]  Figure 6.8: Thermal residual strains at the top surface of the substrate in the tapered region for different F E M meshes  6.2  Verification Models  212  7 B/ep - 2 FM73M - 2 Al - 0 03125 mm 14 B/ep - 4 FM73M - 4 Al - D.125 mm 7 B/ep - 2 FM73M - 2 Al - 0 125 mm  I 48  51  54  57  60  63  66  69  72  75  78  y [mm]  Figure 6.9: Thermal residual strains at the bottom surface of the substrate in the tapered region for different F E M meshes All curves in Figure 6.4 through Figure 6.9 show good agreement, especially the thermal residual strains at the aluminum substrate bottom (see Figure 6.9).  Location A in  Figure 6.6 shows the effect of reducing the number of elements along the taper in some non-critical regions. The smaller element length at the ply drop-offs emphasizes the singularities in the thermal residual strains in the boron/epoxy (see location B in Figure 6.7). The mesh with seven elements through the patch and two elements through the thickness of the adhesive as well as the substrate and an element length of 0.125 mm at the ply drop-off was chosen to model the tapered region of the A M R L specimen based on the good agreement with the finer meshes.  A comparison between the normal and reverse stacking sequence was carried out next. In addition, the results of the one-dimensional analysis are plotted. Figure 6.10 shows how well the one-dimensional analysis can predict the shear stresses in the mid-plane of the adhesive especially for the normal stacking sequence.  The general trend is predicted properly, even  for the reversed stacking sequence but the finite element analysis shows clearly the influence of local increases in the adhesive thickness. The peak adhesive shear stress is overpredicted  6.2  Verification Models 14.0  213  I  13.0  Reverse Stacking  12.0  i  Normal Stacking  11.0  /A  ///  1 -D Theory  10.0 9.0  A  8.0  a" 2 £  7.0 6.0 5.0  /  4.0 3.0  J  2.0 1.0  .  /  J  A  \  TI  \J  f  A  \\i  /  V  \  V  0.0 -1.0 -2.0 48  51  54  57  60  63  66  69  72  75  \> 78  y [mm]  Figure 6.10: Transverse shear stress at the adhesive mid-plane for reversed and normal stacking sequence by the one-dimensional analysis, which is caused by extending the adhesive layer past the last boron/epoxy layer. Figure 6.1 shows how well the one-dimensional theory predicts the peak adhesive shear stress if the patch and adhesive end at the same location. It should be noted that the mid-plane does not show the highest shear stress using the reversed stacking sequence. The singularities at the ply drop-offs lead to an increase in the shear stress close to the patch/adhesive interface.  Figure 6.11 and Figure 6.12 show the sensitivity of the thermal residual strains in the patch towards the choice of stacking sequence.  A large discrepancy between the finite  element results and the one-dimensional analysis is visible. The magnitude of the thermal residual strains predicted by the one-dimensional analysis compares only reasonably well to the strains determined in the F E M analysis at the patch/adhesive interface with a patch utilizing the normal stacking sequence.  The large spikes in the finite element results for the thermal residual strains using either stacking sequence are caused by the stress singularities at the ply drop-offs. This effect is not included in the theoretical one-dimensional analysis. A significant reduction of over  6.2  Verification Models  214  200  Figure 6.11: Thermal residual strains at the top surface of the patch for reversed and normal stacking sequence 500 fie in the peak compressive strain through the thickness of the patch is evident for the reversed stacking sequence. In addition the location of the peak compressive strain also shifts from near the ply drop-off at the bottom of the patch to the end of the adhesive fillet at the top surface of the patch. For the normal stacking sequence for the patch lay-up,  Figure 6.12: Thermal residual strains at the bottom surface of the patch for reversed and normal stacking sequence  6.2  Verification Models  215  the reduction in compressive thermal residual strains through the thickness of the patch is even more significant (up to 700/ie) for this particular loading case. Figure 6.13 and Figure 6.14 show the results for the thermal residual strains in the substrate. The effect of the stacking sequence on the aluminum substrate is not significant especially at the bottom side of the 3.175 mm thick aluminum sheet where the deviation in thermal residual strains between the normal and reversed stacking sequence is less than 15 fie. The result of the one-dimensional analysis is also quite close to the F E M analysis with generally less than 30 fie variation regardless of the stacking sequence at this location. It should also be noted that the thermal residual stresses in the substrate don't drop to zero at the patch edge due to the extended adhesive layer. The effect of the reversed stacking sequence on the thermal residual strains in the substrate are more pronounced at the substrate/adhesive interface. The thermal residual strains are reduced in the regions of increased adhesive thickness while they are in excellent agreement in the mid-section of a tapered step.  600 500 400 300  \  J r 200 100 Reverse Stacking  0  Normal St.acking 1 -D Theon 1  -100 -200 48  51  54  57  60  63  66  69  72  75  78  y [mm]  Figure 6.13: Thermal residual strain at the top surface of the substrate for reversed and normal stacking sequence  6.2  Verification Models  216  600  Figure 6.14: Thermal residual strain at the bottom surface of the substrate for reversed and normal stacking sequence One of the most important findings are the high peel stresses in the adhesive due to the reversed stacking sequence (see Figure 6.15). The main cause is the orientation change of the upper plies passing over a ply drop-off resulting in peel forces. The peel stresses develop at different locations depending on tension or compression in the patch. The peak peel stress in the adhesive mid-plane occurs at the ply drop off in case of compressive stresses in the 10.0  8.0  il  ll  6.0  /  4.0  J  2.0 0.0  /  -2.0 -4.0 -6.0 -8.0  r  51  54  57  /  / / / / / / / / / / / / / / 7/ / / / / \ if  -10.0 48  Jt  60  i  63  66  69  / /  I  \c I  72  75  78  y [mm]  Figure 6.15: Peel stresses in the adhesive mid-plane for the reversed stacking sequence  6.2  Verification Models  217  patch (see location A in Figure 6.16) while for tensile stresses the peak occurs at location B . Tensile loading of the bonded repair will therefore generally change the location and reduce the magnitude of the peel stresses introduced by the thermal residual stresses, while compressive loading will reverse the peel stresses and thus may contribute to the failure of the bonded repair. A fractographic investigation by the National Research Council showed a change in failure mode at the ply drop using a reversed stacking sequence. This failure mode change was not present when using a normal stacking sequence [37]. Based on the findings in the finite element analysis, peel stresses can be associated with this failure mode change.  t  t  A  B  Figure 6.16: Critical location for peel stresses in the adhesive mid-plane for the reversed stacking sequence  Two conclusions can be drawn from the prediction of high peel stresses: The stresses and strains in a tapered patch with a reversed stacking sequence should not be analysed by approximating the taper using a normal stacking sequence nor a tapered element. Additionally special attention should be paid to the peel stress in the tapered region when using a reversed stacking sequence under compressive loading of the bonded repair.  The stresses in the substrate and in the adhesive are generally the main concern for bonded repairs. Most crack patching applications arise from cracking in sections loaded in tension. The one-dimensional analysis predicts the thermal residual strains for the aluminum substrate and the adhesive shear stress very well. Small discrepancies arise due to specific de-  6.2  Verification Models  218  sign choices, such as extending the adhesive layer beyond the patch. Patch design choices such as the stacking sequence become far more important for reinforcements or cracked sections of components loaded in compression based on the high peel stresses found in the taper using a reversed stacking sequence.  6.2.3  Crack  M e s h Verification  The crack mesh verification model was the first step in generating an appropriate finite element model for the cracked region of the A M R L specimen.  This model deals only  with the metal substrate due to the availability of a closed form solution allowing the verification of the finite element results rather than a convergence study. A large number of crack meshes were investigated. One of the main difficulties in this investigation was the small element length requirement perpendicular to the crack in order to allow for the proper determination of the adhesive shear stresses and thermal residual stresses in the adherends. This requirement was complicated by the lack of a crack tip element in MSC/Nastran for Windows. For this reason three approaches were investigated. First, a finite element mesh was generated using only parabolic elements. In a second approach, additional bulk data cards were submitted as part of the analysis sequence creating MSC/Nastran C R A C 3 D crack tip elements which are not supported in the user interface. The third approach utilized the typical quarter point method where the mid-side nodes of the elements surrounding the crack tip are moved to the quarter point location.  This third approach was ruled out due to the number of warnings issued by MSC/Nastran with respect to the determination of the stresses, i.e. crack tip stress reached 10000 GPa. The MSC/Nastran user guide noted that this approach might also lead to an early abortion of the calculation. Another significant disadvantage of moving the mid-side is the connec-  6.2  Verification Models  219  tivity to the bonded patch. If the mid-side node in the patch is moved to the quarter point location, the stress distribution will be highly distorted. The other option is to make a transition in the aluminum or adhesive which will lead to uncertain answers due to the unknown element behaviour. Significant verification work would be required to study such element behaviour.  The advantage of using the regular parabolic elements is a better spatial stress distribution. The shape of the constant stress contour lines are closer to the theoretical estimation especially for elements with high through thickness aspect ratios (25:1). In order to get a good spatial distribution with the MSC/Nastran C R A C 3 D elements, a through thickness aspect ratio of 6:1 was required, thus using significantly more elements through the thickness. The use of the C R A C 3 D elements is significantly more complex and the implementation is time consuming. In addition, the elements cannot be visually checked and results cannot be displayed although they were available in the T06 output file. It should also be noted that the computed stress intensity factor by M S C / N A S T R A N varied by approximately 13% depending on the use of the full modeling option or the symmetry option. The listed element stresses for these two modeling options varied as well based on different natural coordinates. The stresses, strains and displacements for the surrounding elements were compared and found identical despite the differences in the output of the crack tip element. The stress intensity predictions given by M S C / N A S T R A N seemed rather low in comparison to extrapolation of stress intensity factor plots based on the crack opening displacement. The theoretical stress intensity factor for this particular verification model can be calculated using [23]: (6.1) where Y = 1.12-0.23  +  - « • " ( £ ) '  (6.2)  6.2  Verification Models  220  and was determined to be K  r  = 2\.\2MPa^/m  for a specimen containing a 20 mm long  crack and a width of 50 mm under 40 MPa of remote tensile loading. This theoretical value was used to assess the quality of the finite element crack tip mesh by comparing it to the stress intensity factor calculated from the finite element results. The finite element model length was chosen to be 100 mm. It was decided to use a sheet thickness of 1.5875 mm to achieve plane stress conditions under the chosen remote loading. The choice of plate thickness also improved the aspect ratio of the elements in the crack tip region to approximately 1:6 by utilizing symmetry through the thickness and with respect to the crack. Two elements were placed to represent the half sheet thickness. The crack opening displacements determined by the finite element analysis were used to calculate the stress intensity factor as given by [25]:  K  Note that the equation for K  2TTE IFEM  (6.3)  r 4  is shown for plane stress conditions. Figure 6.17 shows  IpEM  the stress intensity factors calculated for the finite element crack opening displacements utilizing the different crack tip modeling approaches. The results for the stress intensity 23.5 23 22.5 \/—  22 21.5 21 QL  20.5 20  •or  f  19.5 19  *  18.5  •  Crack tip element (Quarter point node)  — A — No crack tip element (Parabolic element)  18 17.5  Crack tip element (CRAC3D)  ,i  o  ;  ^  5  f3  10  12  14  16  18  r [mm]  Figure 6.17: Stress intensity factors for different crack tip elements  2  6.2  Verification Models  221  factor based on the crack opening displacements are within 1% of each other for the different crack tip modeling approaches for locations which are more than 10% of the crack length away from the crack tip. Larger deviations occur closer to the crack tip. The MSC/Nastran C R A C 3 D element shows a spike after the second element from the crack tip. The reason for this spike is possibly the different displacement formulation used by MSC/Nastran for the crack tip element in comparison to the parabolic elements. The quarter point approach results also in a spike in the stress intensity factor at the interface between the crack tip element and the adjacent parabolic elements. The model with only parabolic elements shows a gentle decrease in the stress intensity factor close to the crack tip. Chan et.al.'s [25] approach of using a linear extrapolation is applied rather than calculating K  I  F  B  M  just from the crack tip elements. Good linear sections of the stress intensity factor  versus r are found from approximately 15% to 50% of the crack length. The extrapolated stress intensity factors are within 0.5% of the theoretical estimation for the three modeling procedures representing excellent results.  This can be largely contributed to the small  element size of approximately 0.3% of the crack length in the crack tip vicinity.  The stress in the loading direction was investigated as a second parameter to check the performance of the crack tip modeling. Figure 6.18 shows the stress in the loading direction (a ) ahead of the crack along the through thickness center line. Negative coordinates refer y  in this case to the cracked section.  The nodal stress for the C R A C 3 D element are not  provided in the post-processor thereby not allowing nodal stress averaging with the adjacent elements. Generally this is not a concern with a fine finite element mesh, but the large stress gradient present at a crack tip it will distort the results. Therefore the nodal stresses in the elements next to the crack tip element were not displayed.  It is apparent that the stresses for the model with the C R A C 3 D element are higher ahead of the crack tip than the results given by the two other modeling approaches. It should  6.2  Verification Models  222  3000 2800 2600 2400 2200 2000 1800 „  1600  £  1400  ~  1200  0  1000  X  Crack tip element ( C R A C 3 D )  — • — Crack tip element (Quarter point node)  -  — A —  j  No crack tip element (Parabolic element)  800  v  -&  600 400  /  200 0  w  w ML  M  w A  s-x^  w A  M.  w  k  t  1  4_  v  J\—  ^  -200 -400 2  -1 75  -1 .5  -1 25  -1  -0 75  -0 .5  -0 25  0 r  0. 25  05  0. 75  1. 25  1 5  1. 75  2  [mm]  Figure 6.18: Stress perpendicular to the crack (a ) for different crack tip elements y  be noted that this difference only becomes evident for stresses exceeding the yield stress of the material (310 MPa) very close to the crack tip ( « 0.35 mm). The modeling approach using the quarter point method and no crack tip element have excellent agreement ahead of the crack tip except that the stresses for the quarter node element approach generate extremely large values (> lOOOOGPa) in the crack tip element while the parabolic element adjacent to the crack tip reaches nine times the yield stress of the material.  A point of verification is that the stress along the crack face should be zero. All modeling approaches reach this value within 1-1.5% of the total crack length from the crack tip. The parabolic element shows the highest over shoot in the element adjacent to the crack tip, which is not surprising considering the steep gradient and the element formulation.  It should be noted that the crack tip displacements the in x-direction (ahead of the crack tip) using the C R A C 3 D element were not satisfactory. The crack front through the thickness appeared jagged and possibly unreliable.  6.2  Verification Models  223  Having investigated these different modeling approaches, it was decided to use parabolic elements at the crack tip. A l l three methods can be used to determine the stress intensity factor quite accurately utilizing Chan's [25] extrapolation technique.  The quarter node  method was rejected due to the difficulties in connectivity to the patch. The C R A C 3 D element approach showed generally good results but required a tedious input as well as a larger number of elements through the thickness to avoid distortion of the in-plane stress field. Additionally the jagged displacements found along the crack front were unrealistic. The behaviour of the parabolic elements was quite good due to the small element size in the crack tip vicinity. The parabolic element also handled higher aspect ratios through the thickness better. The only short fall was the approximately 10% overshoot in comparison to the peak stress along the crack front. Having established a good mesh scheme using parabolic elements for the crack region of the substrate, the adhesive layer and the patch were added to the model as shown in Figure 6.19.  6.2.4  Patched  Crack  Model  The patched crack verification model was chosen to evaluate the theoretical crack tip stress field model for bonded repairs. The previous crack mesh verification model provided confidence in a proper mesh for the substrate.  A constant thickness patch of 0.924 mm as  well as the 0.25 mm thick adhesive layer were added to the crack tip verification model after increasing the thickness of the substrate to 3.175 mm thereby representing the center section of A M R L sandwich type specimen. The honeycomb used in the A M R L specimen is neglected since the theoretical model doesn't account for it. A uniform thickness patch over the entire substrate was chosen to avoid any additional bending due to partial coverage of the patch across the width of the model. The model length of 100 mm was chosen long enough to avoid an interaction between the load transfer zone at the free edge of the model and the high stress region around the crack tip. No mesh refinement was carried out near  6.2  Verification Models  224  the free edge due to the focus on the stress distribution in the vicinity of the crack tip. Two elements were used through the thickness of the adhesive, while four elements through the thickness of the seven ply patch were applied. The first layer of elements was used to represent the boron/epoxy layer next to the adhesive. The three remaining element layers for the patch always represented two boron/epoxy plies each. The mesh for the aluminum substrate was left identical to the previous verification model. Figure 6.19 shows the finite element mesh used in this investigation:  One of the most difficult decisions to be made in this finite element approach was the modeling of the adhesive along and across the crack. Modeling the adhesive continuously across the crack implies that no crack opening occurs at the adhesive/substrate interface based on no crack opening in the initial unstressed state. In order to model the adhesive continuously across the crack using a discrete modeling approach such as finite elements, a small crack opening displacement must be present and the adhesive must be extended past the crack face towards the symmetry line of the repair. A plasticity analysis can then be carried out to get a proper result for the crack opening displacement in the substrate.  The approach taken in this research work is based on a discontinuous adhesive layer across the crack.  Its advantage is that the analysis can be carried out using a linear-elastic  Figure 6.19: Finite element mesh for the patched crack verification model  6.2  Verification Models  225  approach thus reducing the computational requirements. Additionally it presents an upper bound approach for the crack opening displacement.  Failure of the adhesive in tension  on top of the crack is a realistic possibility. Planned fractographic investigations of the region around the crack after fatigue failure will hopefully give a better indication about the true failure sequence. Unfortunately, papers dealing with the finite element modeling of this critical region have not discussed this problem in detail. The approach taken in this research work is similar to finite element models of bonded repairs developed by other researchers using spring elements to represent the adhesive [19]. A spring model implies that the adhesive is discontinuous across the crack. These models have been successfully used to predict the crack growth rate in bonded repairs.  The finite element stress distributions for a , a and r x  y  xy  along 9 = 0° and 90° are presented  and compared to the closed form theoretical analysis in Figure 6.20 through Figure 6.25. All graphs show the theoretical results for plane stress and plane strain conditions. The stress intensity factor caused by the thermal residual stresses alone will generally lead to plane strain conditions for the crack analysis while combined with other external loading plane stress conditions at the crack tip are common. The finite element results for the substrate are displayed at the bottom and top surface of the aluminum face-sheet. The substrate bottom surface is the critical case due to the larger crack opening displacement. The adhesive restricts the crack opening displacement at the top surface of the substrate to a significant degree. The difference in the crack opening displacement between the top and bottom surface of one face-sheet at the edge of the A M R L specimen is approximately 22% which compares well with the 19% given by Baker for the F E M analysis of a simple double-doubler joint [18].  Figure 6.20 shows excellent agreement with the theoretical stresses in the loading direction along 9 = 0° for plane strain conditions and the finite element results for the substrate bottom. Note that the substrate bottom coincides with the through thickness symmetry  6.2  Verification Models  226  plane due to neglecting the honeycomb. The difference between the fracture mechanics analysis for plane stress and plane strain conditions is approx. 5% for stresses below 150 MPa.  Significant difference is visible between the finite element results for the top and  bottom surface of the substrate.  Good agreement between the closed form theoretical  analysis and the F E M analysis for the symmetry plane can be attributed to accounting for the shear-lag in the adherends. The discrepancy in the stresses at locations more than 17 mm from the crack tip is due to modeling an edge crack in the F E M analysis and using the center crack assumption in the theoretical approach.  The estimation of the stress intensity factor for the substrate using the finite element results imposes some difficulties. Typically the crack opening displacement is used to estimate the stress intensity factor for a cracked sheet. The questions that arises is, can this approach be applied to bonded repairs? Inspecting the stress field behind the crack tip (6 > 90°) for a bonded repair reveals differences in comparison to the typical stress field of a crack in an unreinforced sheet due to the load transfer into the patch rather than through the remaining 150  100  CO  I  50  o  II  —  0 Fracture Mechanics Theory (Plane Strain) Fracture Mechanics Theory (Plane Stress)  -50  FEM Substrate Symmetry Plane FEM Substrate Top  -100 -20  -15  10  -10  15  20  25  Figure 6.20: Stress in loading direction (a ) for the patched substrate along 9 = 0° y  30  6.2  Verification Models  227  uncracked section. This finding makes the application of the crack opening displacement method based on an unreinforced sheet questionable. On the other hand, the stress field observed ahead of the crack tip for a bonded repair shows the typical stress distribution. It seems therefore more appropriate to use a stress extrapolation method to determine the stress intensity factor for different through thickness locations in the substrate. This approach is also supported due to the stresses ahead of the crack tip being the driving force for the crack rather than the crack opening displacement. The stress extrapolation method can be based on the usual one term approximation if data points are selected close to the crack tip. The corresponding stress intensity factor for a given stress in loading direction at a distance r from the crack tip along 9 = 0° can be determined using: Kr = V27rra  (6.4)  y  Alternatively the correct elastic solution for a center crack in an infinite plate can be used to determine the stress intensity factor. In addition to the local stress <r , the remote stress y  cr has to be provided: 0  The extrapolation of both equations yield stress intensity factors which are within 0.25% of each other. Using equation (6.5), the stress intensity factor is determined as 5 . 0 2 M P a m /  v  for the symmetry plane of the substrate and 2.72 MPa-^/m for the top surface. The determination of the stress intensity factor using the crack opening displacement method yields 5.25 MPa^/rn for the symmetry plane and 4.61 MPa-^/rn for the top surface of the substrate. Comparing the results between these extrapolation methods, a large discrepancy (70%) is present at the top surface, while the stress intensity factor matches quite well for the symmetry plane of the substrate (5%). This clearly shows the problem using the existing stress and displacement extrapolation methods for the stress intensity factor. One option for the determination of the average stress intensity factor for the substrate could be established using the change in strain energy with an increase in crack length. Using Rose's proof,  6,2  Verification Models  228  the stress intensity factor can be calculated applying the standard relationship between the energy release rate and the stress intensity factor. Unfortunately the maximum stress intensity factor at the symmetry plane of the substrate cannot be established. More research is needed to identify the proper determination methods of the stress intensity factor for bonded repairs using finite element results.  Additionally these stress intensity factors can be compared to Rose's one-dimensional model. Assuming plane strain conditions, the upper bound  yields 5.11 MPav^n. The  stress intensity factor K can be calculated as 4.66MPa /m by applying the interpolation T  scheme between K  u  and K^.  x  It is at first surprising that K is approximately 7% below the r  highest stress intensity factor at the symmetry plane of the substrate. Verification work carried out by an A M R L researcher [18] shows similar findings. Their F E M analysis of double-doubler joints shows higher opening displacements of up to 13% between the two substrate pieces for comparison to the one-dimensional analysis. The opening displacement is directly related to the stress intensity factor  in the Rose model. This discrepancy  between the one-dimensional analysis and the F E M analysis can be attributed to the threedimensional effects which are accounted for in the solid finite element model. In addition, the shear-lag correction for the adherends in Rose's model uses a linear relationship. In reality, a quadratic approach is more likely thus introducing some error in the theoretical analysis.  The stress in loading direction along 9 = 90° is quite well predicted close to the crack tip (r < 2.5 mm) and shows good agreement with the finite element results for the symmetry plane of the substrate (see Figure 6.21). The F E M results show higher stresses for r > 2.5 mm. This can be explained by the load transfer into the patch close to the crack rather than redistribution through the remaining section, and shows clearly the limitations of the developed theoretical approach. The theoretical approach provides good results ahead of the crack tip but the model does not account for the local load transfer into the patch.  6.2  Verification Models  229  150  I  I  I  I  I  • Fracture M e c h a n i c s T h e o r y (Plane Strain) Fracture M e c h a n i c s T h e o r y (Plane Stress)  125  • F E M Substrate S y m m e t r y P l a n e F E M Substrate T o p  Q.  s O  75  O  50  at II  '°  25  2.5  7.5  10  12.5  15  17.5  20  22.5  25  27.5  r [mm]  30  Figure 6.21: Stress perpendicular to crack (a ) in the patched substrate for 9 = 90° y  Thus care should be exercised when using the model close to the load transfer zone. Note that the finite element results are only shown for r < 12.5 mm due to the lack of nodes along that line. Unfortunately MSC/Nastran for Windows does not offer postprocessing along arbitrary lines.  The stresses in the x-direction are also quite well predicted by the closed form analysis especially close to the crack tip as shown in Figure 6.22 and Figure 6.23.  It should be  noted that the finite element analysis accounts for the small difference in the coefficient of thermal expansion between the patch and substrate in the x-direction. This influence was neglected in the theoretical analysis. The predicted stress parallel to the crack using the theoretical model is in-between the F E M results for the top and bottom surface of the substrate along 0 = 0° and 9 = 90° and close to the crack tip. stresses for a  x  The drop into compressive  along 9 — 90° near the crack tip is also properly indicated. The remote  stresses are also in good agreement. The finite element analysis generally predicts higher stresses for 2 mm < r < 10 mm. The stress a  x  along the crack face cannot really be  6.2  Verification Models  230  Fracture M e c h a n i c s T h e o r y (Plane Strain) Fracture M e c h a n i c s T h e o r y (Plane Stress) F E M Substrate S y m m e t r y P l a n e F E M Substrate T o p  in the patched substrate for 9 — 0°  Figure 6.22: Stress parallel to the crack (a ) x  predicted using the theoretical model. The theoretical model is based on a center crack and an effective crack length for the bonded repair which is only a fraction of the real crack.  150 - Fracture M e c h a n i c s T h e o r y (Plane Strain) 125  - Fracture M e c h a n i c s T h e o r y (Plane Stress) - F E M Substrate S y m m e t r y P l a n e  100  F E M Substrate T o p  cs  Q.  75  o II  5  50  25  -25 2.5  7.5  10  12.5  15  17.5  20  22.5  25  27.5  r [mm]  Figure 6.23: Stress parallel to the crack (a ) x  in the patched substrate for 9 = 90°  30  6.2  Verification Models  231  The theoretical prediction for the in-plane shear stresses, r , xy  show excellent agreement  with the F E M analysis. The shear stress along 9 = 0° must be zero. Figure 6.24 shows clearly the error in the F E M analysis which must be expected at the singularity considering the use of parabolic elements and the high stress gradient near the crack tip. Figure 6.25 shows again just slightly better agreement between F E M results for the substrate bottom and the theoretical approach.  150 —  125  Fracture M e c h a n i c s T h e o r y (Plane S t r e s s / P l a n e Strain)  — F E M Substrate S y m m e t r y P l a n e —  F E M Substrate T o p  100 co  Q. 75 O II  S  50 25  -25 -20  -15  -10  5  10  15  20  r [mm]  Figure 6.24: Shear stress (r ) xy  25  30  in the patched substrate for 9 = 0°  In general the developed closed form theoretical model for the stress field in a bonded repair gives a good estimate for the stresses in the substrate considering the limitations of the model. The most important limitation is imposed by the load transfer across the crack from the substrate into the patch. The load transfer itself is accounted for in the reduction of the stress intensity factor but it is not accounted for in the stress field prediction. Thus the stress field estimate especially for the stress in loading direction, is certainly not well applicable for 9 > 80° to 90° due to the local load transfer as well as modeling an effective crack length instead of the real crack length with the restraint of the patch.  6.2  Verification Models  232  150  _  125  Fractu re M e c h a n i c s T h e o r y (Plane Strain)  -  Fractu re M e c h a n i c s T h e o r y (Plane Stress)  100  F E M S ubstrate S y m m e t r y P l a n e 75  - - - -  reivi ouDSiraie i o p  [MP  'a  50  o  25  O o> II  CO  1r  0 -25  if -50 -75 100 2.5  7.5  10  12.5  15  17.5  20  22.5  25  27.5  30  r [mm]  Figure 6.25: Shear stress (r ) xy  in the patched substrate for 6 = 90°  The stress field ahead of the crack tip is predicted quite well although it doesn't account for the localized load transfer into the patch around the stress singularity. In this case the load transfer is linked to the stress increase due to the singularity thus limiting the distortion of the stress field as predicted for a cracked sheet without a reinforcement. The reduction in the stress intensity factor due to the load transfer caused by the stress rise ahead of the crack tip is accounted for in Rose's model. The finite element results also showed that the transverse shear stress ahead of the crack tip is quite low i.e. less than 1 MPa.  The through thickness variation of the in-plane stresses is not addressed in the stress field approach.  The Rose model accounts for the shear-lag in the adherends using a linear  relationship thus showing generally better agreement with the stresses at the substrate bottom, which is the through thickness symmetry plane of the model. Some improvement should be expected if a higher order approach is taken for the transverse shear distribution. A significant improvement in the stress field prediction should be achieved if proper stress functions are used which address the interaction with the patch. The developed theoretical  6.2  Verification Models  233  model can be used for preliminary design work and experimental work such as the placement of strain gauges.  6.2.5  Half Circular Patch Model  The half circular patch model is the last requirement prior to modeling the sandwich type A M R L specimen. The objective is the establishment of the number of elements needed along the perimeter of the patch. The model consists of a two ply patch with one step using the reversed stacking sequence. Only two plies were chosen to reduce the modeling time. The substrate is modeled with the same in-plane dimensions as the longest ply, i.e. 75 mm. No crack was introduced in this model and symmetry conditions were applied to reduce the model to a 90° patched section. To check for convergence, runs of 8, 16 and 32 elements were used along the 90° patched section.  Figure 6.26 shows the mesh used for the half circular patch. verification model with 32 elements along the circular perimeter. Figure 6.27 shows the through thickness meshing  FH44444IIIIIIIII  I I I I I I l ± H  Figure 6.26: F E M mesh for circular patch verification model  6.2  Verification Models  234  scheme at the tapered edge. The two bottom element layers represent the aluminum. The adjacent two element layers are used to model the adhesive. In addition, the fillet in the tapered step has adhesive material characteristics. The top element layer to the right of the ply drop-off and the two top layers left of the ply drop-off represent the boron/epoxy composite layers. The adhesive fillet can be better seen in Figure 6.34.  Figure 6.27: Detailed view of the tapered edge in the circular patch verification model  The transverse shear stress r  yz  at a distance of 0.125 mm away from the free edge was  chosen as the convergence parameter. Figure 6.28 shows the results of this convergence study. The highest shear stress occurs for 9 = 90° as expected, while it is nearly zero for 6 = 0°. Less than a 0.2% difference in the shear stress prediction is visible between 20 and 90 degrees using different number of elements along the perimeter. The error increases to approximately 1% for smaller angles which is an excellent result.  Based on these results, 20 elements were chosen along the perimeter of the A M R L specimen patch. In order to account better for free edge effects, the element size was reduced for four elements near the free edge. The same refinement was used along the symmetry line at y = 0 mm to ensure accurate results for the outer taper step with the shortest fiber length. Element sizes 13% larger than the presented case of 16 elements for the 90° section were chosen for the section between 6° < 9 < 78°.  6.3  6.3  A M R L Sandwich Type Specimen Model  A M R L  Sandwich Type  Specimen  235  Model  By utilizing the results of the previously discussed refinement models, a finite element model for the A M R L specimen can now be created without the need of further mesh refinements. Figure 6.29 shows the final mesh for the A M R L sandwich type specimen using double-symmetry. A representation of the honeycomb was added in order to get a better comparison with the experimental data. Although the honeycomb limits the out-of-plane bending, it does not eliminate it. The honeycomb was modeled as an isotropic material with an elastic modulus equivalent to the compressive modulus of the honeycomb in thickness direction (1.517 GPa [48]) and a coefficient of thermal expansion identical to the C T E of the aluminum face-sheet. Although the assumed in-plane elastic modulus of the honeycomb is only 2% of the aluminum face-sheet elastic modulus, it is still somewhat higher than the true moduli. A n attempt to use significantly lower values for the in-plane modulus  6.3  A M R L Sandwich Type Specimen Model  Honeycomb = 6.35 mm  -H  hAluminum = 3.175 mm  Figure 6.29: F E M mesh for A M R L sandwich type specimen  236  6.3  A M R L Sandwich Type Specimen Model  237  of the honeycomb caused numerical problems, i.e. pivot ratio of the stiffness matrix was not within the specifications of the finite element solver. Accounting for the honeycomb in the finite element model leads to changes of up to 9% in the thermal residual strains at the strain gauged locations. It should be noted that the maximum out-of-plane deflection does not occur in the center of a patch but near the tapered region for this particular honeycomb. The A M R L sandwich type specimen is modeled using 18806 elements with 254670 degrees of freedom. The purpose of the finite element analysis was to provide a better understanding of the detailed stress and strain distributions while the experimental work gave a good foundation for the analysis assumptions.  This goal has been fulfilled as illustrated by the following  results. Figure 6.30 and Figure 6.31 show the thermal residual strain fields for the mid-plane of the substrate and the top layer of the boron/epoxy patch in longitudinal direction giving examples for the complexity of the stress and strain fields in this particular application. Additionally, the transverse shear stress in the mid-plane of the adhesive caused by the mismatch in the coefficients of thermal expansion in the adherends is shown in Figure 6.32 since it is one of the important design variables. The peel stresses caused by the thermal loading are presented due to their importance with respect to disbonding. The top surface of the adhesive layer was chosen for this representation based on the magnitude of the peel stress at this interface. Finite element results are only presented for the cooling cycle from the effective stress free temperature to ambient temperature, i.e. A T = 64.8°, since the stress and strain distributions don't change assuming the coefficient of thermal expansion remains constant. The measured changes in these material coefficients with temperature are not large enough to cause any significant changes in the stress and strain distributions.  Figure 6.30 shows the thermal residual strains parallel to the fiber orientation in the midplane of the aluminum face-sheet. The stress field ahead of the crack tip shows the typical stress field solution of a center crack in an infinite plate. The stress field gets more dis-  6.3  A M R L Sandwich T y p e Specimen M o d e l  0.  10.  20.  30.  40.  50.  60.  70.  80.  238  90.  100.  110.  120.  130.  140.  150.  160.  Figure 6.30: T h e r m a l residual strains parallel to the fiber orientation (y-direction) for the A M R L sandwich type specimen i n [fie] at the mid-plane of the substrate torted w i t h increasing distance from the crack tip as shown by the 500 fie contour line in Figure 6.30.  The load transfer from the substrate into the patch near the crack face  occurs approximately parallel to the crack and does not show the typical distribution of an unpatched crack where the load in carried by the remaining section ahead of the crack. The bending influence of the specimen is clearly visible for x > 117 m m . A n increase in the strains outside the patched region is found along the center line of the specimen although its magnitude is not critical.  6.3  A M R L Sandwich T y p e Specimen M o d e l  0.  10.  20.  30.  40.  239  50.  60.  70.  80.  x  Figure 6.31: T h e r m a l residual strains parallel to the fiber orientation (y-direction) for the A M R L sandwich type specimen in [fie] i n the top-surface of the patch  The thermal residual strain distribution for the top layer of the boron/epoxy patch as presented in Figure 6.31 shows high strain gradients i n the tapered region and near the crack. The thermal residual strain distribution i n the tapered section along the y-axis i n Figure 6.31 is very similar to the results of the tapered reinforcement verification model as presented in Figure 6.11. M o v i n g from the y-axis to the x-axis, the thermal residual strain distribution changes initially just slightly but past 45° more significant changes occur in the  6.3  A M R L Sandwich Type Specimen Model  240  tapered region. The high gradients are significantly reduced as the symmetry line of the patch is approached. Due to the large element size in tangential direction, some inaccuracy in the tangential extent of contour lines should be expected. The convergence study for the circular patch showed that the chosen element size leads only to relatively small changes in the transverse shear stress distribution near the free edge.  Large thermal residual strain changes can be identified near the crack face, where the load is transferred from the substrate into the patch. The compressive thermal residual strains decrease from approximately -700 fit (y = 22.5 mm) to -200 fie (y = 8 mm). Within 2 mm of the crack face, an additional increase in compressive stresses can be found at the top surface of the patch. Note that at the bottom surface of the patch, tensile strains are present leading to a high through thickness strain gradient in the patch over the crack.  Figure 6.32 shows the transverse shear stress r  yz  in the mid-plane of the adhesive. The load  transfer zone from the substrate into the patch extends approximately 30 mm perpendicular from the crack face. The transverse shear stress drops to near zero at the crack face fulfilling the boundary condition. It is important to note that the transverse shear stress ahead of the crack tip and up to 6 angles of 80° measured from the crack tip against the x-axis is nearly zero, i.e. below 1 M P a for this particular load case. This substantiates the assumption of negligible interaction between the patch and substrate ahead of the crack made for the estimation of the stress field. The increase in transverse shear stresses past approximately 80° shows the limitations in the stress field approach in the cracked section since the local load transfer is not accounted for in the theoretical approach. The transverse shear stress increases up to 15.5 M P a in the tapered section near the free edge of the patch, as also shown in Figure 6.10 for the tapered reinforcement verification model. Note that very low transverse shear stresses are present in the 2.5 mm section extending beyond the patch. The overall highest transverse shear stress is found close to the crack faces at the free edge  6.3  A M R L Sandwich Type Specimen Model  Figure 6.32: Transverse shear stress r  yz  241  for the A M R L sandwich type specimen in [MPa]  in the mid-plane of the adhesive of the specimen. The transverse shear stress reaches nearly -20 MPa in the mid-plane of the adhesive at this location leading to the typical disbond pattern as presented by Baker [13].  One of the most important findings of the finite element analysis were the high tensile peel stresses in the adhesive at each tapered step caused by the reversed stacking sequence for the patch. The most critical locations are at the interface between the adhesive and the  6.3  A M R L Sandwich Type Specimen Model  0.  10.  20.  30.  Figure 6.33: Adhesive peel stress a  z  40.  242  50.  60.  70.  80.  for the A M R L sandwich type specimen in [MPa] at  the adhesive/patch interface patch. Figure 6.33 shows the peel stress distribution and the critical locations in red on the top surface of the adhesive layer. Note that this view includes the adhesive fillets at the taper steps. A detailed isometric view of two tapered steps along the free edge of the specimen (R = 66 mm and R = 69 mm) is shown in Figure 6.34. This view also shows the modeling of the adhesive fillets well. Note that only one element is used through the  6.3  A M R L Sandwich T y p e Specimen M o d e l  Figure 6.34: Detailed view of the adhesive peel stress a  243  z  for the A M R L sandwich type  specimen i n [MPa] for the ply drop-offs at R = 66 m m and R = 69 m m thickness.  Increasing the number of elements does not necessarily improve the accuracy  since additional elements in the adjacent composite may lead to a misrepresentation.  A  more accurate determination of the stress and strain field would require modeling the actual fibers and m a t r i x of the composite material. The highest peel stresses i n the taper are found at the t i p of the adhesive fillet i n the current representation (see Figure 6.34). T h e overall  6.3  A M R L Sandwich T y p e Specimen M o d e l  244  highest peel stresses are found near the crack face at the interface between the adhesive and patch.  Figure 6.35: Fractographic investigation patch/adhesive interface at the ply drop-off  The high peel stresses for the reversed stacking sequence at the steps i n the tapered section could be the cause for the change i n failure mode at a ply drop-off documented by the N a t i o n a l Research C o u n c i l of C a n a d a [37]. Figure 6.35 [37] shows the finding i n the fractographic investigation. T h i s particular repair was subjected to high compressive loading which increases the peel stresses at the ply drop-off in addition to the peel stresses caused by the elevated temperature cure. Most bonded repairs are subjected to tensile loading which acts to reduce the peel stresses caused by the thermal residual loading, thus reducing the risk of failure initiation at this site. Based on this finding, it should be clear that the design of the tapered region of composite patches must account for the stacking sequence induced stresses at the tapered edge under combined thermal residual and applied loading.  Table 6.1 shows the finite element results for the thermal residual strains at the strain gauged locations. T h e finite element results were obtained by averaging over the strain  6.3  A M R L Sandwich Type Specimen Model  245  gauge area using the contour plots for the thermal residual strains. By zooming in on the strain gauge area or sections of it, contour lines were plotted for every 2 fie (or less) thereby allowing the determination the average thermal residual strains quite accurately. A n error of less than 20 fie is estimated for locations where the strain distribution varies in a highly non-linear fashion, e.g. locations A and G (see Figure 4.15). A n error of less than 5 fie is estimated for locations with nearly linear thermal residual strain gradients, e.g. locations B - F and H .  Generally good agreement between the finite element results and the experimental measurements has been achieved considering the assumptions made for the numerical modeling as well as the limitations of the experimental measurements. The two important assumptions made in the finite element analysis are 1.) a uniform thickness adhesive layer exists under the patch except for the fillets in the tapered section and 2.) zero thickness of the strain gauges. It is known that some adhesive bleeding occurred at the free edge of the bonded repair, through the crack and starter notch during the curing process, however this is not believed to have had any significant effect on the stress distribution. Additionally, a local reduction in the adhesive thickness should be expected near the tapered edge. It is also known that the thickness of the patch was also been altered by embedding up to three strain gauges at one location. The placement of the gauges themselves were carefully planned to minimize any effect, but some increase in thickness cannot be avoided. It is assumed that the two gauges placed at the adhesive interface to the adherends were embedded into the adhesive thus reducing the increase in overall thickness. This has certainly had some limited effect on the local adhesive thickness which may also explain some of the discrepancy between the experimental and finite element results at locations with high transverse shear stresses, such as location A .  The thermal residual strains for the aluminum substrate are quite well predicted for location A . The maximum discrepancy is only 31 fie, a good result considering the high strain  6.3  A M R L Sandwich Type Specimen Model  Location  A  Material  Al  X  1  -24 tie  7 fie  7 /ze  B/Ep  II II  23  -746 tie  -607 ize  43  -889 /ze  -665 /ze  -636 /ze  1  44  129 tie  176 tie  176 tie  Al  II  3  606 tie  696 /ze  696 /ze  Al  J.  4  -174 txe  -130 /ie  -130 tie  B/Ep  II II II  24  -623 tie  -598 /ze  30  -618 ue  -593 /ze  33  -594 tie  -532 /ze  B/Ep  X  34  34 fie  -94 /ze  -94 /ze  Al  II  12  512 / i e  566 /ze  566 fie  Al  X  11  -43 tie  -78 /ze  -78 /ze  B/Ep  II II II  25  -681 /ze  -688 /ze  31  -682 tie  -627 fie  42  -693 /ze  -585 /ze  B/Ep  J_  41  116 /ze  -105 /ze  Al  II  10  496 tie  508 /ze  22  519  528 /ze  Al Al  X  09  -164 fie  -139 fie  Al  X  21  -164 / i e  -112 /ze  B/Ep  II  28  -682 / i e  -638 lie  B/Ep  II II  29  -679 lie  -641 /ze  40  -677 /ze  -568 /ue  B/Ep  X  39  5 tie  -95 /ie  Al  II  13  36 /ze  96 /ze  Al  X  14  -32 / i e  Al  II  16  Al  X  Al Al B/Ep  -574 /ze  Fracture  1-D Analysis 84 ± 11/ze N/A - 6 9 6 X 8 9 /ze N/A 488 ±  3 fie  N/A -579  ±4/ze  N/A 558 ± 1 fie  Mechanics Analysis N/A N/A N/A N/A 518 ± 12 fie N/A N/A N/A 585 ± 2 /te  N/A  N/A  -633 fie  - 6 6 1 ± 1/ze  N/A  -105 /ze  N/A  N/A  518  / i e  541 ± 1/ze  550 X 1 / i e  -126 /ie  N/A  N/A  -616 / i e  - 6 4 1 ± 1 tie  N/A  -95 lie  N/A  N/A  96 /ze  N/A  N/A  -32 fie  -32 /ze  N/A  N/A  -87 lie  -96 /ze  -96 fie  N/A  N/A  15  35 /ue  49 /ze  49 /ze  N/A  N/A  II  7  317 tie  327 tie  327 tie  X  8  -77 /ze  -62 /ze  -62 /ze  II II  27  -866  37  -860 /it  B/Ep  X  38  130 /ue  235 /ze  Al  II II  5  484 fie  532 tie  19  477 /ze  521 /ze  B/Ep  B/Ep  H  Average 5 /ze  B/Ep  G  Measurement  5 tie  B/Ep  F  Measurement  Experimental  -24 tie  B/Ep  E  Experimental  2  B/Ep  D  No.  FEM  II  B/Ep  C  Gauge  Al  B/Ep  B  Orient.  246  Al  l i e  -720 tie -748  / z e  Al  X  6  -130 /ze  -117 /ze  Al  X  18  -132 fie  -91 /ze  B/Ep  II II II  26  -739 / i e  -632 /ze  32  -739 fie  -647 tie  35  -740 ize  -622 / i e  X  36  31 /Lie  30 lie  B/Ep B/Ep B/Ep  -734 /ze 235 tie 527 /ze  134 ± 1 / i e N/A - 1 0 3 8 ± 1/ze N/A 558 ± 1 fie  N/A N/A N/A N/A 562 ± 1 fie  -104 lie  N/A  N/A  -634 / i e  - 6 6 1 ± 1/ze  N/A  N/A  N/A  30 / i e  Table 6.1: Comparison of the thermal residual strains at 21°C between finite element results, experimental results, the one-dimensional analysis and the fracture mechanics analysis gradient and the uncertainty in the shear modulus behaviour with temperature. A discrepancy of up to 25% between the finite element thermal residual strain prediction and the experimental measurements is present for the boron/epoxy patch at this location. Possible  6.3  A M R L Sandwich Type Specimen Model  247  causes are the position inaccuracy of the gauges, unaccounted changes in the local adhesive thickness especially in a high strain gradient area as well as local thickness increase due to the embedded gauges. The error for position inaccuracy is certainly large ( « 90 fie) as presented in Table 5.1. The uncertainty in the shear modulus behaviour with temperature might also be a contributing factor.  The finite element analysis provided improved insight as to why high tensile thermal residual strains for gauge 3 mounted on the aluminum face sheet in fiber orientation at location B were measured. The transverse shear distribution showed that the fracture mechanics model should only be applied to an angle 9 of approximately 80° or less to limit the influence of the local load transfer for this particular load case. Gauge 3 is located in an area with increasing transverse shear stress. The increasing transverse shear stress indicates a distortion of the stress and strain distribution of an unreinforced crack which was assumed in the theoretical fracture mechanics analysis. In addition to the local increase in transverse shear stress in the adhesive, the placement of the strain gauges at the adhesive interfaces leads to a thinner adhesive layer, which will also affect the local stress distribution in comparison the uniform thickness adhesive layer assumed in the finite element model. The thermal residual strain in the aluminum face-sheet at gauge 4 is reasonable well predicted with an discrepancy of only 44 fie (25%). The thermal residual strains measured in the fiber direction in the boron/epoxy patch near the adhesive interface shows excellent agreement with the finite element results with a discrepancy of only 4%. The thermal residual strains for the gauge mounted on top of the patch show larger differences between the numerical and experimental result which is likely caused by the local increase in thickness due to embedding strain gauges. This local section experiences some bending which decreases the strains in fiber orientation and increases the strains perpendicular to the fiber orientation under the given thermal residual loading conditions.  6.3 A M R L Sandwich Type Specimen Model  248  The quality of the finite element prediction at location C is quite similar. The gauges placed in or on top of the patch show again the effect of the local increase in thickness. The finite element analysis and the experimental measurement agree very well at the bottom surface of the patch (within « 1%). The thermal residual strains for gauge 31, which is mounted on the first boron/epoxy ply, show a discrepancy of 8% which is quite high for this particular location. A probable cause is the thickness of the underlying gauges including the solder connections. The highest difference was found for the thermal residual strains in the transverse direction on top of the patch. Two factors may have contributed to this result. The local increase in thickness leads to bending stresses under the applied tension due to the transverse mismatch in the coefficients of thermal expansion. Furthermore, the experimental strain measurements in the transverse direction show significantly more creep than the gauges in the fiber orientation at this location i.e. the fiber direction.  60 fie vs.  10 fie in  It should be noted that the measured thermal residual strains are  significantly smaller perpendicular in comparison to parallel to the fiber orientation thus making the inaccuracy due to creep more significant in the transverse direction. The thermal residual strains for the aluminum in fiber direction show a difference of just over 9% and perpendicular to the fiber orientation a discrepancy of 35 fie is found.  The thermal residual strains are very well predicted for the aluminum in fiber direction at location D . The discrepancy between the prediction and measurement is less than 12 fie. The change through the thickness of the aluminum was predicted by including the honeycomb in the finite element analysis. The thermal residual strains in the transverse direction show larger discrepancies of 25fie and 52 fie. This larger difference might be due to a combination of the long lead wires whose effect was numerically corrected and the lower resistance of this gauge. The difference between the predicted and measured thermal residual strains for the boron/epoxy in fiber orientation shows again the effect of the local increase in thickness. The discrepancy for the two gauges close to the bottom of the patch  6.3  A M R L Sandwich Type Specimen Model  249  is approximately 6% increasing to 16% at the top. The thermal residual strains for the boron/epoxy in transverse direction show a similar trend as discussed for location B and C. The thermal residual strains measured at location E outside the patched area show a significant discrepancy of 60 fie in longitudinal direction between the F E M prediction and the experimental measurement while the transverse direction indicates perfect agreement. A possible cause might be the influence of the grip arrangement which was not modeled in the theoretical analysis. Furthermore, a small difference in the coefficients of thermal expansion might be present between the A l 6061-T6, which was used for the reinforcements and spacers in the grip section, and the A l 2024-T3 used for the face-sheets. The experimental measurements and finite element prediction for the thermal residual strains at location F show only a discrepancy of less than 14 fie, which is excellent. The magnitude certainly shows the in-plane bending generated by the thermal residual stresses.  The thermal residual strains in the aluminum for the longitudinal and transverse direction at location G show excellent agreement between the finite element results and the experimental measurements (< 15 fie). This is a significant improvement over the onedimensional analysis previously used to get an estimate for the thermal residual strains. The finite element analysis also provides a better prediction of the thermal residual strain distribution for the boron/epoxy patch but still overpredicts their magnitude.  Possible  causes include the uncertainty in the shear modulus assumptions with temperature, the presence of creep especially for the transverse direction (150 fie), or possibly local increases in thickness as well as inaccuracy in the gauge position.  The finite element predictions don't correspond to the experimental measurements as well as expected at location H . The through thickness change in thermal residual strains for  6.3  A M R L Sandwich Type Specimen Model  250  the aluminum in the longitudinal direction is well predicted due the inclusion of the honeycomb in the finite element model, but they disagree in the absolute magnitude by about 10%. The thermal residual strain in the transverse direction on top of the face-sheet is well predicted (discrepancy of only 13 fie) but for the bottom surface they disagree by 41 fie. Possible explanations are the long lead wires and the low resistance of the strain gauge installed at the bottom surface of the face-sheet combined with the lower measured strain. The discrepancy in the thermal residual strains between the experimental measurements and finite element results for the boron/epoxy patch is between 12 and 16%. Again the local increase in thickness due to embedded gauges is probably one reason for these disagreements.  Other factors include misalignment of gauges, local reduction the adhesive  thickness and the simplification of not modeling the grip section thus affecting the in-plane bending behaviour of the specimen.  Overall, the thermal residual strains are well predicted for the aluminum in both longitudinal and transverse directions. The average discrepancy between the finite element results and the experimental results are only 36 fie and 27 fie, respectively. Although these magnitudes are small, it should be noted that the finite element analysis underpredicted the thermal residual strains for the aluminum in longitudinal direction which is not particularly desirable from a design viewpoint.  One recommendation would be to increase the  effective stress free temperature slightly such that a conservative finite element prediction is achieved.  The thermal residual strains determined by the finite element analysis for  the boron/epoxy are, on average, 88 fie higher than the experimental results. The most important cause is the local increase in thickness by embedding strain gauges, which is also applicable to the transverse thermal residual strains for the boron/epoxy patch. The average discrepancy of even 100 fie in transverse direction can also be attributed to creep. The through thickness variation of the thermal residual strains in the patch will be influenced by the particular placement of each gauge as well as the solder connection to the lead wires.  6.3  A M R L Sandwich Type Specimen Model  251  Considering the influencing factors which were discussed in this section, good agreement between the experimental results and the finite element results has been achieved. These measurements suggest that for future experimental work, only a single gauge should be embedded at any given location. The finite element analysis revealed that significant through thickness effects are only present very close to the crack and in the tapered section. The use of strain gauges is limited in these regions due to their size. Very small gauges may be applied to the metallic substrate but are inadequate for composites since they would be affected by the local interaction between fibers and matrix. The other important achievement is the actual embedding of gauges in a bonded repair with reliable connections. Embedded gauges offer a good option for 'smart patches' since the gauge is protected from environmental influences. A M R L started to investigate the feasibility of 'smart patches' [36] by monitoring surface strains of the patch using an A M R L sandwich type specimen. The challenge is to find a location for the strain gauge where the gauge installation doesn't lead to disbonds around the gauge under fatigue loading while at the same time can indicate disbonds around the crack tip and/or crack propagation. A possible solution would be the placement of a gauge underneath the top layer of the patch. The optimum in-plane location needs to be determined by a number of finite element representations which model the crack growth and disbond growth as observed by A M R L [13].  Their surface strain  measurements should give an excellent verification tool as well as an excellent initial guess for an appropriate location.  7 Discussion and Conclusion  252  Chapter 7 Discussion and Conclusion  7.1  Introduction  As outlined in Chapter 1, the main goals of this research were to identify the magnitude of thermal residual stresses in bonded composite repairs and to establish an appropriate thermal residual stress/strain model.  These research goals were achieved through four  stages: The first stage involved manufacturing six A M R L sandwich type composite bonded repair specimens to provide some of the specimens required for fatigue damage initiation testing in addition to one test specimen which was instrumented by placing 44 strain gauge at eight locations and at up to five different interfaces. Residual strains at ambient temperature combining thermal residual strains and other process induced strains were measured during the manufacturing process.  In stage two, the stress free temperature  of the bonded repair was experimentally determined and thermal residual strains were measured as a function of operating temperature. For stage three, a theoretical analysis was carried out to determine the thermal residual stress and strain distributions in bonded repairs which also addressed the effect of symmetrical disbonds around the crack as well as the stress field ahead of the crack. As final stage four, a detailed finite element analysis was carried out to assess the limitations of the theoretical analysis as well as providing a more detailed insight in the complex thermal residual stress and strain state of the A M R L sandwich type specimen.  7.2  A M R L Specimen Manufacturing Procedure  253  The conclusions and contributions of this research work are highlighted in the following sections.  7.2  A M R L  Specimen Manufacturing  Procedure  The objective of this first stage included the selection of a suitable test specimen for thermal residual stress or strain measurements as well as for future fatigue damage initiation testing. Appropriate process specifications had to be determined as part of the manufacturing procedure in order to create durable and representative specimens. The instrumentation locations had to be determined based on critical failure locations as reported in the literature.  The conclusions and contributions of this portion of the research are outlined below: • A representative specimen was selected, constructed and instrumented for determining the residual strain distribution in a bonded repair. The sandwich type A M R L specimen (see Figure 3.1) was chosen for the experimental determination of the stress state in bonded repairs due to its ability to represent real aircraft structures and frequent use in the evaluation of bonded repairs. Improving the understanding of the stress distribution in the A M R L specimen is valuable for all researchers interpreting experimental results obtained from this specimen. • A n effective manufacturing process for the A M R L sandwich type specimen has be