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Damage tolerance of bonded composite aircraft repairs for metallic structures Clark, Randal John 2007

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D A M A G E T O L E R A N C E OF B O N D E D COMPOSITE A I R C R A F T REPAIRS FOR METALLIC STRUCTURES  by R A N D A L JOHN C L A R K  B . Eng., Carleton University, 1995 M . A . Sc., University of British Columbia, 2000  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 OF T H E REQUIREMENTS FOR THE D E G R E E OF  DOCTOR OF PHILOSOPHY  In  T H E F A C U L T Y OF G R A D U A T E STUDIES  (Mechanical Engineering)  The University of British Columbia A p r i l 2007 © Randal John Clark, 2007  Abstract This thesis describes the development and validation of methods for damage tolerance substantiation of bonded composite repairs applied to cracked plates. This technology is used to repair metal aircraft structures, offering improvements in fatigue life, cost, manufacturability, and inspectability when compared to riveted repairs. The work focuses on the effects of plate thickness and bending on repair life, and covers fundamental aspects of fracture and fatigue of cracked plates and bonded joints. This project falls under the U B C Bonded Composite Repair Program, which has the goal of certification and widespread use of bonded repairs in civilian air transportation. This thesis analyses the plate thickness and transverse stress effects on fracture of repaired plates and the related problem of induced geometrically nonlinear bending in unbalanced (single-sided) repairs. The author begins by developing a classification scheme for assigning repair damage tolerance substantiation requirements based upon stress-based adhesive fracture/fatigue criteria and the residual strength of the original structure. The governing equations for bending of cracked plates are then reformulated and line-spring models are developed for linear and nonlinear coupled bending and extension of reinforced cracks. The line-spring models were used to correct the W a n g and Rose energy method for the determination of the long-crack limit stress intensity, and to develop a new interpolation model for repaired cracks of arbitrary length. The analysis was validated using finite element models and data from mechanical tests performed on hybrid bonded joints and repair specimens that are representative of an in-service repair. This work w i l l allow designers to evaluate the damage tolerance of the repaired plate, the adhesive, and the composite patch, which is an airworthiness requirement under F A R (Federal Aviation Regulations) 25.571. The thesis concludes by assessing the remaining barriers to certification of bonded repairs, discussing the results of the analysis, and making suggestions for future work. The developed techniques should also prove to be useful for the analysis of fibrereinforced metal laminates and other layered structures. Some concepts are general and should be useful in the analysis of any plate with large in-plane stress gradients that lead to significant transverse stresses.  ii  Table of contents Abstract  ii  Table of contents  iii  List of tables  viii  List of figures  x  List of symbols  xv  Acknowledgments  xvii  Co-authorship statement  xviii  Chapter 1: Damage tolerance of bonded composite aircraft repairs  1  1.1 Introduction  1  1.2 Bonded composite repair technology  1  1.2.1 Assurance of structural integrity  4  1.2.2 Durability  4  1.3 Literature review  5  1.3.1 Repair experiments and failure modes  6  1.3.2 Certification and damage tolerance analysis  8  1.3.3 Mechanics of a repair  9  1.3.4 Damage mechanics  13  1.3.5 Critical analysis  16  1.4 Thesis objectives  20  1.5 Summary  22  1.6 References  22  Chapter 2: Bending of bonded composite repairs: an experimental and finite element design study  28  2.1 Introduction  28  2.2 Experimental study  32  2.3 Finite element modeling  37  iii  2.4 Validation  39  2.5 Residual strength and damage tolerance assessment  44  2.5.1 A l u m i n u m substrate fracture  45  2.5.2 Adhesive failure  46  2.5.3 Composite fracture and delamination  52  2.5.4 Synergy of aluminum and composite/adhesive fracture criteria  56  2.6 Discussion  57  2.7 Concluding remarks  62  2.8 References  63  Chapter 3: Extension of generalized plane strain plates with reinforced cracks  67  3.1 Introduction  67  3.2 The generalized plane strain plate  68  3.2.1 A cracked plate  71  3.2.2 Numerical solution  74  3.2.3 Geometry correction factors for a cracked plate  76  3.3 Crack bridging: a line-spring model  80  3.4 Discussion  85  3.5 Concluding remarks  86  3.6 References  87  Chapter 4: Transverse shear and pressurization in the bending of reinforced cracks  88  4.1 Introduction  88  4.2 Basic plate bending models and fracture  90  4.3 A n 8 -order plate bending model  92  4.4 Hyper-singular integral equations  95  4.5 Bending of a cracked plate  99  th  4.6 Comparison to three dimensional finite element results  109  4.7 Bending of reinforced cracked plates  114  4.8 Discussion  120  4.9 Concluding remarks  122  iv  4.10 References  i22  Chapter 5: Linear coupled bending and extension of a bonded repair  124  5.1 Introduction  124  5.2 A methodology for damage tolerance assessment  128  5.2.1 Stress in the crack plane  128  5.2.2 Characterization of crack-bridging springs  129  5.2.3 Long-crack limit behaviour  133  5.2.4 Interpolation for arbitrary crack lengths  135  5.3 A line-spring model for crack-bridging  138  5.4 Comparison to finite element models  141  5.4.1 Stress intensity and crack face displacements  141  5.4.2 Spring stiffness and nominal stress determination  146  5.5 Results and discussion  148  5.6 Concluding remarks  149  5.7 References  150  Chapter 6: Elastic analysis of hybrid bonded joints and bonded composite repairs  153  6.1 Introduction  153  6.2 Background  155  6.3 Linear bonded joint formulation  157  6.3.1 Solution and boundary conditions  161  6.3.2 Thermal strains  165  6.3.3 Joint failure  166  6.4 Analysis of a bonded composite repair 6.4.1 Nominal stresses  169 •  171  6.4.2 Spring compliance and adhesive stresses  173  6.4.3 Crack bridging analysis  176  6.5 Results and discussion  180  6.6 Concluding remarks  182  6.7 References  182  v  Chapter 7: Nonlinear mechanics of hybrid bonded joints and composite repairs  184  7.1 Introduction  1  7.2 Lap joint experiments  186  7.3 Bonded joint finite element model  189  7.4 Stress intensity calculation: the Rose model  196  7.5 Comparison to a 3D F E model and mode separation  200  8  4  7.5.1 M o d e Separation  202  7.5.2 Adhesive and Composite Failure  207  7.6 Discussion  211  7.7 Concluding Remarks  213  7.8 References  214  Chapter 8: Results, contributions, and future work  216  8.1 Introduction  216  8.2 Patch testing and finite element analysis  217  8.3 Plate models and linear coupled bending and extension  219  8.4 Hybrid bonded joints and orthotropic patches  221  8.5 Geometrically nonlinear bending  222  8.6 Contributions and recommendations for future work  223  8.7 Concluding Remarks  228  8.8 References....  229  Appendix A : Bonded joint specimen construction A . 1 Description of specimen types  231 231  A . 2 Parts and materials  232  A . 3 Panel construction  245  A.3.1 Composite manufacture  ,  245  A.3.2 A l u m i n u m surface preparation  248  A.3.3 Laminated panel construction  250  Appendix B : Instrumented repair specimen  259  B . l A M R L specimen geometry  259  vi  B.2 Instrumented specimen  259  Appendix C : Test data  264  Appendix D : Engineering plate models  280  D . 1 Plane stress extension  282  D . 1.1 A cracked plate in plane stress  283  D.2 Plane strain extension  286  D.2.1 A cracked plate in plane strain  287  D.3 Plane stress bending  288  D.3.1 Bending of a cracked plate under plane stress conditions D.4 Plane strain bending  289 292  D.4.1 A cracked plate under plane strain bending conditions D.5 Shear-deformable plane stress bending D.5.1 A shear deformable cracked plate under plane stress conditions D.6 Shear-deformable plane strain bending D.6.1 A shear deformable cracked plate under plane strain conditions D . 7 References  293 294 296 298 299 301  Appendix E : Material properties  302  E . l Boron/Epoxy 5521/4 Laminate  302  E.2 A l u m i n u m 2024-T3 (A-Basis)  303  E.3 Adhesive F M 7 3 M  304  vii  List of tables Table 2-1: Material properties and dimensions  37  Table 2-2: Possible composite failure modes for unidirectional patch  53  Table 3-1: Material properties used to test the G P S model  76  Table 4-1: Material properties used to test the G P S model  102  Table 5-1: Material properties and dimensions  142  Table 6-1: Material properties and dimensions  170  Table 7-1: Material properties and dimensions  190  Table 7-2: Thermal residual elastic strain energy (J/m)  200  Table 8-3: Steps for repair design, application, and certification  225  Table A - 4 : Coupon specimen materials  233  Table A - 5 : A l u m i n u m surface preparation  254  Table A - 6 : Wedge test procedure  255  Table A - 7 : Composite parts fabrication  256  Table A - 8 : H S - D L J specimen assembly  257  Table A - 9 : C L S specimen assembly  258  Table B - l : Strain gauge types  260  Table C - l : Strains for connector A l  264  Table C-2: Strains for connector A 2  265  Table C - 3 : Strains for connector A 3  265  Table C-4: Strains for connector A 4  266  Table C - 5 : Strains for connector B l  266  Table C-6: Strains for connector C I  267  Table C-7: Strains for connector C 2  267  Table C-8: Strains for connector D I  268  Table C-9: Strains for connector E l  268  Table C-10: Strains for connector F  269  viii  Table C - l 1: Strains for connector A l  270  Table C - l 2 : Strains for connector A 2  270  Table C-13: Strains for connector A 3  271  Table C-14: Strains for connector A 4  271  Table C - l 5 : Strains for connector B l  ,  272  Table C-16: Strains for connector C I  272  Table C - l 7 : Strains for connector C 2  273  Table C - l 8 : Strains for connector D I  273  Table C-19: Strains for connector E l  274  Table C-20: Strains for connector F  •  274  Table C-21: Strains for connector A l  275  Table C-22: Strains for connector A 2  275  Table C-23: Strains for connector A 3  276  Table C-24: Strains for connector A 4  276  Table C-25: Strains for connector B l  277  Table C-26: Strains for connector C I  277  Table C-27: Strains for connector C 2  278  Table C-28: Strains for connector D I  278  Table C-29: Strains for connector E l  279  Table C-30: Strains for connector F  279  Table E - l : Material properties for 5521/4 boron/epoxy  302  Table E-2: Material properties for aluminum 2024-T3 (A-basis)  303  Table E-3: Material properties for adhesive F M - 7 3  304  ix  List of figures Figure 1-1: Bonded (left) and mechanically fastened (right) repairs  3  Figure 1-2: Load transfer regions  11  Figure 1-3: T w o step method  11  Figure 2-1: A bonded repair and a mechanically fastened repair  29  Figure 2-2: Experimental apparatus  33  Figure 2-3: Strain gauge locations  34  Figure 2-4: Bending under tension (left) and compression (right)  35  Figure 2-5: Adhesive failure surface  35  Figure 2-6: Interfacial failure surface  36  Figure 2-7: The crack face after pre-cracking (left) and after 175,000 cycles (right)  36  Figure 2-8: Finite element mesh, (a) the crack tip and (b) the entire model  38  Figure 2-9: Three-dimensional view of the cracked region  39  Figure 2-10: Strain distribution in undamaged repair  40  Figure 2-11: Adhesive elements in the region of the crack, showing the disbond  40  Figure 2-12: Strain distribution, with honeycomb  41  Figure 2-13: Variation of strains with applied stress, with honeycomb  42  Figure 2-14: Strains, without honeycomb  43  Figure 2-15: Variation of strain with applied stress, without honeycomb  44  Figure 2-16: Membrane and bending stress intensity factors  45  Figure 2-17: Adhesive stresses  47  Figure 2-18: Adhesive yield criteria and finite element model adhesive stresses  48  Figure 2-19: Comparison of adhesive fracture data and yield criteria  51  Figure 2-20: Composite stresses  55  Figure 2-21: Proposed classification of repairs by D T substantiation requirements  59  Figure 2-22: Process for 'Quadrant D ' repairs  61  Figure 3-1: Distribution of plate displacements and stresses  69  x  Figure 3-2: Stress intensity for a through crack  77  Figure 3-3: Stress intensity difference plotted against normalized crack length  77  Figure 3-4: Crack face displacement plotted against crack length  79  Figure 3-5: Difference in crack face displacement plotted against crack length  79  Figure 3-6: A reinforced crack in a plate  80  Figure 3-7: Stress intensity results from the line-spring model  81  Figure 3-8: Normalized stress intensity plotted against crack length  82  Figure 3-9: Comparison of stress intensity predictions  83  Figure 3-10: Crack face displacement for various spring stiffnesses  84  Figure 3-11: Comparison of crack face displacement predictions  85  Figure 4-1: Displacements and stresses for the 8th-order plate bending model  92  Figure 4-2: Stress intensities for shear deformable plane stress bending  102  Figure 4-3: Stress intensity difference plotted against normalized crack length  103  Figure 4-4: Crack face displacements for shear-deformable plane stress bending  104  Figure 4-5: Difference in crack face displacements plotted against crack length  105  Figure 4-6: Stress intensity for 8th-order plate bending formulation  106  Figure 4-7: Crack face displacement for 8th-order plate bending formulation  107  Figure 4-8: Stress intensity difference plotted against normalized crack length  108  Figure 4-9: Crack face displacement plotted against normalized crack length  108  Figure 4-10: Finite element map showing boundary conditions  110  Figure 4-11: Through-thickness variation of the stress intensity  111  Figure 4-12: Theoretical and finite element bending stress intensities  112  Figure 4-13: Theoretical and finite element model crack face deflections  114  Figure 4-14: A bending crack reinforced by springs  115  Figure 4-15: Stress intensity from the line spring model  116  Figure 4-16: Normalized stress intensity plotted against crack length  117  Figure 4-17: Stress intensity predicted by line-spring and interpolation models  118  Figure 4-18: Crack face displacement for various spring stiffnesses  119  Figure 4-19: Crack face displacement from line-spring and interpolation models  120  xi  Figure 5-1: Repair configurations: balanced (top) and unbalanced (bottom)  125  Figure 5-2: Equivalency of forces and moments  132  Figure 5-3: Three-dimensional finite element model of a repair  142  Figure 5-4: Membrane stress intensity  143  Figure 5-5: Bending stress intensity  144  Figure 5-6: Membrane crack face displacement  145  Figure 5-7: Bending crack face displacement  145  Figure 5-8: F E mesh showing the (a) cracked, (b) tapered, and (c) grip regions  146  Figure 5-9: Nominal stresses  147  Figure 5-10: Spring compliances  148  Figure 6-1: A n unbalanced repair  157  Figure 6-2: H y b r i d bonded joint element  158  Figure 6-3: Assumed variation of transverse adherend stresses  159  Figure 6-4: Single sided lap joint specimen  170  Figure 6-5: Lap joint finite element model  171  Figure 6-6: Substrate membrane and bending stresses under remote applied tension.... 172 Figure 6-7: Substrate residual stresses at 20" C for a 120" C adhesive cure cycle  172  Figure 6-8: M o d e l configuration for spring compliances  173  Figure 6-9: Adhesive stresses developed by an applied membrane load  174  Figure 6-10: Adhesive stresses developed by an applied bending load  174  Figure 6-11: Adhesive thermal residual stresses  175  Figure 6-12: Repair compliance under membrane and bending loading  176  Figure 6-13: Comparison of membrane stress intensity predictions  178  Figure 6-14: Comparison of bending stress intensity predictions  178  Figure 6-15: Membrane deflection results  179  Figure 6-16: Bending deflection results  179  Figure 7-1: A bonded repair and a mechanically fastened repair  185  Figure 7-2: Lap joint specimens. From the top: S L J , D L J , and H S - D L J specimens  187  Figure 7-3: Strain gauge locations  187 xii  Figure 7-4: Disbonding of the D L J specimen  188  Figure 7-5: Failure surface of the S L J specimen  189  Figure 7-6: F E mesh showing the (a) cracked, (b) tapered, and (c) grip regions  190  Figure 7-7: Strains for the cracked S L J specimen  191  Figure 7-8: Strains for the uncracked S L J specimen  192  Figure 7-9: Edge position of cracked S L J  192  Figure 7-10: Edge position of uncracked S L J  •• 193  Figure 7-11: Cracked S L J lateral deflection  194  Figure 7-12: Uncracked S L J lateral deflections  195  Figure 7-13: Load vs. displacement curves  196  Figure 7-14: The energy method: A - A and B - B show cracked and uncracked joints.... 197 Figure 7-15: Change in strain energy  199  Figure 7-16: Finite element model (a) front view and (b) crack-tip region  201  Figure 7-17: Nonlinear variation in R M S stress intensity  202  Figure 7-18: Stress-stiffening beam model  203  Figure 7-19: Membrane stress intensity  205  Figure 7-20: Bending stress intensity  206  Figure 7-21: Membrane crack face displacement  206  Figure 7-22: Bending crack face displacement  207  Figure 7-23: Adhesive shear stress  209  Figure 7-24: Adhesive peel stress  209  Figure 7-25: Patch membrane stress  210  Figure 7-26: Patch bending stress  .-  211  Figure A - l : Single lap joint specimen  231  Figure A - 2 : Double lap joint specimen  231  Figure A - 3 : Honeycomb separated double lap joint specimen  232  Figure A - 4 : Cracked lap-shear specimen  232  Figure A - 5 : L J specimen composite panel assembly drawing  234  Figure A - 6 : D L J specimen composite panel assembly drawing  235  xiii  Figure A - 7 : H S - D L J specimen composite panel assembly drawing  236  Figure A - 8 : C L S specimen composite panel assembly drawing  237  Figure A - 9 : L J specimen face sheet drawing  238  Figure A - 1 0 : D L J and H S - D L J specimen face sheet drawing  239  Figure A - l l : C L S specimen face sheet drawing  240  Figure A - 1 2 : Guide plug drawing  241  Figure A - 1 3 : H S - D L J spacer block drawing  242  Figure A - 1 4 : H S - D L J honeycomb drawing  243  Figure A - 1 5 : E n d tab drawing  244  Figure A - 1 6 : Workspace for patch lay-up  245  Figure A - 1 7 : Application of peel ply  246  Figure A - 1 8 : Workspace for C L S panel lay-up  246  Figure A - 1 9 : Strain gauge placement (finished panel in background)  247  Figure A - 2 0 : Layout on the tool plate  247  Figure A - 2 1 : Vacuum bagging and debulking  248  Figure A - 2 2 : Grit blasting  249  Figure A - 2 3 : Silane application  250  Figure A - 2 4 : Workspace for construction of the H S - D L J panel  251  Figure A - 2 5 : Rolling the adhesive on to the back of the face plate  251  Figure A - 2 6 : Positioning the spacer blocks and honeycomb  251  Figure A - 2 7 : Ready to apply the composite patches  252  Figure A - 2 8 : Applying the adhesive  252  Figure A - 2 9 : Applying an instrumented patch  252  Figure A - 3 0 : Double lap joint specimens, ready for vacuum bagging  253  Figure B - l : A M R L specimen dimensions  259  Figure B - 2 : Instrumented A M R L specimen drawing  261  Figure B - 3 : Instrumented A R L specimen drawing, side A  262  Figure B - 4 : Instrumented A M R L specimen drawing, side B  263  xiv  List of symbols a  half crack length  p  pressure  r  radius  t  thickness or distance from centre of crack  u  crack face deflection  y  coordinate taken from centre of crack  z  coordinate taken from plate centre line  k  spring stiffness  c  spring compliance  D  plate bending constant, Et* l\2  G  strain energy release rate or shear modulus  K  stress intensity factor  E  Young's modulus  5  stiffness ratio, E t  M  line moment, or moment per unit length acting on a cut in a plate  N  line force, or force per unit length acting on a cut in a plate  Y  geometry correction factor  a  normalized crack length or coefficient of thermal expansion  6  rotation  ju  shear modulus  v  Poisson's ratio  77  effective Poisson's ratio for transversally isotropic plate, 77 =  o~  normal stress  T  shear stress  K  parameter arising from differential equations  r  r  IE t p  p  Subscripts p  plate  r  reinforcement  a  adhesive  xv  v 4EIE z  z  m  membrane  b  bending  w  denotes relation to transverse deflection  (j)  denotes relation to pressurization effects  y/  denotes relation to transverse shear effects  Superscripts r  reinforced result from interpolation model  n  numerical result from line-spring model  0  nominal result without including crack bringing springs  °°  long crack limit result including crack bringing springs  z  transverse property  pres.  correction for transverse pressurization  shear  correction for transverse shear  xvi  Acknowledgments I would like to thank my advisor, Dr. Douglas Romilly, for his guidance and patience with this large body of work. This is a long thesis covering a broad topic and the last year of proofreading, editing, and journal submission has required constant attention. I must also extend my gratitude to my committee, and particularly Dr. Gary Schajer, for their support and advice. I would also like to take the opportunity to mention M r . D o n Raizenne and the staff of the Canadian National Research Council Institute for Aerospace Research, who deserve much credit for their advice and for their help in the manufacture and instrumentation of the bonded composite repair and hybrid bonded joint specimens employed in this research. Second, I would like to thank my colleagues in the lab and office for the years of support and companionship during the (at times) destitution of student living and the inevitable low points of post-graduate work. Never again shall we eat 'fridge stew'! A n d finally, I dedicate this thesis to friends and family, the people who make life a joy. It has been a long road.  xvii  Co-authorship statement The six main chapters of this thesis (Chapters 2 through 7) have been published or w i l l be published as co-authored papers with authorship as indicated in the heading of each chapter.  The  authors  have  maintained  a  traditional supervisor/graduate  student  relationship through all stages of the work, with the student becoming more independent as the program progressed, as would be expected. Details are provided below for each stage of the work as they are defined by the University of British Columbia Faculty of Graduate Studies requirements for the co-authorship statement. The identification and design of the overall research project was accomplished by Dr. Romilly, with the stated goals of the University of British Columbia Bonded Repair Program being defined in research grant applications allowing provision of laboratory equipment and supplies, travel, and training. The identification and design of the specific steps in the research work for this thesis were carried out by R . J . Clark under the supervision of Dr. D . P . R o m i l l y and following from the goals of the greater research program. Both the research work and the data analysis were performed by R . J . Clark, with Dr. D.P. R o m i l l y providing supervision, guidance, and oversight. The A M R L specimen used to obtain the test results reported in Chapter 2 was manufactured by A . Albat, and the strain gauge instrumentation was installed by the technical staff of the Canadian National Research Council - Institute for Aerospace Research. Hybrid bonded joint specimen manufacture and instrumentation was carried out by R.J. Clark. Initial manuscript preparation (i.e. completed first drafts) was performed by R.J. Clark, with Dr. D . P . R o m i l l y reviewing the manuscripts, providing constructive comments and suggesting both technical and presentation-oriented additions to the text. R . J . Clark served as corresponding author on all submissions. When the examining committee signs the Dissertation/Thesis Approval Form, this shall be a confirmation of the accuracy of this co-authorship statement.  xviii  Chapter 1: Damage tolerance of bonded composite aircraft repairs R.J. Clark University of British Columbia Department of Mechanical Engineering  1.1 Introduction A typical bonded repair is a multi-layer composite patch bonded to a weak or damaged aircraft structure. The repair acts to reduce stresses in the damaged region and to prevent crack growth by restricting the opening of any cracks. In this opening chapter, the bonded composite repair technology applied for life extension of aluminum airframes is introduced. The chapter begins with a description of the technology, including an overview of its history, benefits and limitations, the materials used for patch construction, and application methods. Some terminology specific to bonded composite repairs is introduced, along with some basic concepts for assessing damage tolerance, durability, and assurance of structural integrity. The second section is a detailed literature review of subjects related to the damage tolerance analysis of repairs, and includes a critical review identifying predominant issues that need to be addressed in order to certify a repair. Finally, in light of the difficulties facing the technology, the objectives of this thesis are defined and the content of upcoming chapters is described.  1.2 Bonded composite repair technology A bonded composite repair is a filamentary composite and epoxy patch bonded to a cracked or damaged aircraft structure. The original use of this repair technology was for life extension of military aircraft and rapid repair of battle damage in the field. The most common methods of application involve either pre-cured composite repairs that are bonded to the damaged structure using an epoxy adhesive, or repairs built up from layers of composite 'pre-preg', which are co-cured with the epoxy onto the underlying structure. The result is a hybrid structure consisting of the original damaged structure, a layer of epoxy adhesive, and an outer layer of composite material. If accessible, repairs are applied to both sides of the structure to prevent induced bending and hence reduce the stresses in the adhesive. The bonded patch technology originates in bonded metal reinforcements used to strengthen fatigue-prone components in military and commercial aircraft. Metal patches have also been considered for the repair of cracked structures, but have primarily been  1  used for reinforcement, lowering stresses in fatigue-prone parts with simple surfaces. There are several reasons for metal patches being limited to a reinforcing role - first, it is difficult to inspect through a metallic repair to see i f a crack is growing, meaning metallic repairs shouldn't be used over riveted plates or to repair cracks. Second, it is difficult to apply enough reinforcement to prevent crack growth using a matching material as it requires at least a doubling of the thickness of the structure to return the strength, and perhaps an even thicker patch to prevent induced bending. A n d finally, it provides a second metallic surface that must be chemically treated for bonding. It w i l l  be  demonstrated that the durability (i.e. the long term strength in service) of adhesive/metal bonds is a formidable obstacle to bonded repairs. In practice, boned composite repairs are more commonly made from composite materials such as boron-epoxy, which is three times stiffer and much stronger than aluminum, and hence can be made very thin compared to the original structure.  Boron-epoxy also provides the advantage of  inspectability, as eddy-current probes can be used to detect crack growth through a patch. There is also significant interest in G L A R E (a fibreglass and aluminum laminate) as a low cost material that is easier to apply. G L A R E has lower modulus and strength than boron-epoxy but is more impact-resistant and is reasonably compatible with eddy-current inspection. A final advantage of a composite repair is the ability to tailor the stiffness of the patch to the stresses in the structure, allowing one to provide reinforcement only in the direction required. This minimizes changes in the load flow within the broader aircraft structure caused by the application of the repair, as an overly stiff patch may detrimentally attract load to the region in which it is applied. Carbon fibre reinforced polymers have also been considered as a repair material but are limited by concerns regarding larger thermally induced strains, corrosion, and inspectability. A repair attracts loads from the surrounding structure due to its high stiffness, which is the means by which it provides reinforcement. Generally, the larger and stiffer the repair, the more it attracts loading from the underlying structure. The load transfer from the structure to the repair occurs through adhesive shear stresses that build up near the edges of the repair, and about the crack. Accordingly, the repair acts both to reduce stresses in the vicinity of the crack, and prevent the crack from opening. These beneficial effects may be partially offset by thermal stresses arising both during patch application and during service due to mismatched thermal expansion coefficients in the patch and structure materials, and by attraction of loads from the surrounding structures due to the increase in stiffness caused by the patch. In this thesis the term repair w i l l refer to a bonded patch used to repair a cracked or damaged structure, and reinforcement w i l l apply to a patch used to strengthen an undamaged or sound structure.  2  Figure 1-1: Bonded (left) and mechanically fastened (right) repairs Figure 1-1 illustrates a bonded and a conventional repair. Bonding offers many advantages over mechanical fastening. The principal advantages are the smooth load transfer provided by bonding, the high stiffness and hence high load transfer capability of the composite materials, and the restraint that a bonded patch provides against crack opening. These advantages make possible a many-fold increase in fatigue life. Other advantages include improvements in manufacturability, cost, and inspectability. Despite this potential, the bonded composite repair technology has not seen widespread use outside of military applications. Examples of commercial application include a Boeing demonstration  program where small reinforcing doublers were applied to sound  structures to test the technology for the fast repair of engine cowling bird-strikes and for localized fuselage fatigue enhancement or repair of localized damage. Another example is the application of very large bonded composite reinforcements during the overhaul of Lockheed L-1011 passenger aircraft to repair cracking doorframes, enabling the return to service of a fleet of aircraft. Primarily, the issues of damage tolerance and bond durability prevent airworthiness certification and widespread use [1,2]. Damage tolerance refers to the ability of a structure to hold up under design loads in a damaged state without failure or excessive deformation before the damage can be detected and repaired. Damage can arise from a number of sources including fatigue, corrosion, manufacturing flaws, or accidental loads such as impacts. Durability refers to the ability of a structure to withstand  environmental  exposure  and  mechanical  loading  without  undue  or  unpredictable degradation. This is a significant issue for bonded repairs, as poorly applied adhesives are prone to failure under service loads and environmental exposure. It is presently very difficult and costly to certify a major bonded composite repair, as it requires all of the damage tolerance, materials and process controls that are necessary for the certification of any primary composite structure.  3  1.2.1 Assurance of structural integrity Regulatory authorities are rightly concerned about the structural integrity of bonded repairs over the life of the structure. Repairs are generally applied to highly loaded structures that have already experienced significant damage. By F A R 25.571, damage tolerance analysis is required for bonded composite repairs and is one of the three means of assuring the structural integrity, as summarized below [3]. (1) Safe-life: cracking will not occur during the life of the repair, (2) Fail-safe: multiple load paths prevent catastrophic failure before detection, or (3) Damage tolerance: inspections during service ensure adequate residual strength. It is a widely held view that a damage tolerance analysis is the only way to ensure the structural integrity of a bonded repair [3] for Principal Structural Elements (PSE) and Structurally Significant Items (SSI) on an aircraft. A PSE is an element of an aircraft whose failure could result in catastrophic failure of the aircraft. A n SSI is a structure that contributes significantly to the carrying of flight, ground, or pressurization loads. In this thesis, we will simply refer to these safety-critical components as primary structure. Safe life methods alone are not feasible for repairs applied to cracked structures, as both crack growth and disbonding are likely to occur. Stopping crack growth requires significant restraint against crack opening, and hence, large adhesive stresses near the crack, which will promote disbonding. The best that can be achieved is a compromise to optimize the life of the repair. Baker has proposed that a safe life approach to certification could be used about the edges of a patch [3] to ensure that adhesive disbonding will not occur under environmental and fatigue loading. Safe life methods may also be useful for bonded reinforcements that act to strengthen weak or weakened structures without cracks. This would preclude an analysis of disbonding from the edges of the repair or reinforcement, but would require stresses to be kept very low, which can be accomplished by tapering the edges of the patch. The remaining method, damage tolerance, is then the method of choice and will require a means for predicting the rate of damage accumulation and load-carrying capacity of the patch, such that failures are preventable through a program of inspections during service.  1.2.2 Durability Durability issues complicate the damage tolerance assessment of bonded joints [4-10]. Durable bonds predictably withstand wear and tear; they last, are stable, and will degrade in a consistent manner. Poorly formed bonds exhibit unpredictable and varying damage rates and are sensitive to environmental conditions, particularly intermittently hot and wet environments. This sensitivity is due to the nature of the adhesive/metal oxide interface, and oxide instability in the service environment [11-21]. As chemical bonds at the  4  adhesive/oxide interface tend to hydrate, surface treatments are designed to increase their strength in the presence of water and ensure a reversible reaction, allowing them to reform when the solvent (water) is removed [22]. Process variability, contamination, and surface treatment errors result in reduced durability. Variations in primer thickness, silane surface treatment concentration, surface abrasion, and etching treatments can impact durability [13]. Metal oxide layers may be initially weak or corrode in service [15]. Adhesive porosity weakens the joint and speed the transport of water [23-26], and results from gases or liquids entrapped during assembly, evaporation of water absorbed by the adhesive prior to curing, and by entrapment of gases formed by chemical reactions. Accordingly, bond degradation under the effects of stress, moisture and temperature depends greatly on the control exerted over shop cleanliness, material storage, surface preparation, and other aspects of design and fabrication. Bond durability is difficult to assess, as there is no non-destructive test for it. Instead, it is evaluated by destructive testing of 'travellers' - test coupons made with adhesives from the same lot and from aluminum or composite adherends that have received a surface treatment concurrently with the aircraft structure [16]. Physical tests in a hot and wet environment, performed on wedge specimens made from the traveller, ensure the durability of the adhesive/oxide interface. The assumption is that loss of durability due to bonding process errors or contaminated materials will be detected by early failure of the traveller. A poor traveller result must be seen as a reflection of the facilities' process control and quality assurance regime and not just the lack of fitness of an individual component. The goal is to reliably produce sound bonds with controlled processes, and to have a track record of good results - possibly a challenge for smaller repair facilities or in the field. Adhesive bonds are a common joining method for aerospace structures, and methods exist to ensure bond durability. As such, this issue alone should not impede certification. The required quality assurance controls, training, and extra steps in the manufacturing process are onerous but not undue given the risks. The main issue for bonded repairs when compared to other bonded structures is the existence of cracks in the underlying structure, and the presence of a highly stressed bond line that might severely test a poor bond. In this thesis, the issue of bond durability will only be examined indirectly, by providing a means for the assessment and hence reduction of the adhesive stresses (particularly peel stresses) in the region of the repaired crack.  1.3 Literature review In this section, the author isolates the elements that are required for the damage tolerance analysis of a repair, and reviews the literature for significant experimental and analytical work that has relevance to the subject. The section begins with a review of  5  mechanical testing that has been carried out to evaluate the repair technology, a review of the failure mechanisms that have been observed, and an assessment of the regulatory controls and requirements for damage tolerance analysis. The section continues with a review of techniques available for assessing the mechanical behaviour and hence the life and strength of a repair, and concludes with a critical assessment of the capabilities and shortcomings of the available methods for damage tolerance assessment of a repair.  1.3.1 Repair experiments and failure modes Many  experiments  have  been  performed to  investigate  failure  mechanisms,  to  demonstrate the predictive performance of models, and to evaluate repair materials and mechanical phenomena in a repair. A widely used repair specimen is the Australian Aeronautical and Maritime Research Laboratories ( A M R L ) specimen, described in detail in Appendix B . This test article consists of two repaired plates that are bonded to an aluminum honeycomb employed to restrict the bending induced by the neutral axis offset of the repair and to model the support of underlying structures that often exist in highly stressed sections of aircraft. Baker has published experimental results for the A M R L specimen that illustrate the effects of cure cycle parameters [26,27], disbonding [26,28], patch shear deformation [27,29], adhesive plasticity [29], and test temperature [26] on performance. In Canada, the A M R L specimen has been used by the National Research Council - Institute for Aerospace Research ( N R C - I A R ) to examine the effects of load spectrum, compressive loading, and adverse environments on patch performance [30,31], and by Albat and Romilly [32] to evaluate thermal strains. Without bending, the failure mode of the repair has consistently been continued cracking of the existing crack in the aluminum plate with accompanying adhesive disbonding. Testing of double lap-shear specimens  has been used to investigate the effects of patch shear deformation and  adhesive plasticity on the apparent stiffness of the patch in the region of the crack [29]. Patch shear deformation and adhesive plasticity have been shown to reduce patch life by reducing the restraint against crack opening imposed by the repair. It has also been shown that increasing the cure cycle temperature and time can impose two penalties on patch performance; 1) tensile thermal residual stresses from a mismatch in thermal expansion coefficients  can accelerate fatigue crack growth, and 2) crack growth retardation is  reduced due to annealing of crack tip plasticity-induced residual stresses. Sharp, Clayton, and Clark [37] have examined the effects of adhesive infiltration into the crack, which was found to increase the stress required to open the crack, and hence improve the fatigue life. Researchers have also examined bonded repair issues using other types of bonded repair specimens. Poole, Lock, and Young [33] have investigated fatigue damage propagation in thick aluminum plates repaired with graphite/epoxy patches. Similar to the  6  AMRL  specimen, repaired plates were bonded to a honeycomb core and tested in pairs to restrict bending. They examined the effect of the F A L S T A F F loading spectrum on repair performance, and compared the results to boundary element analyses that included the effects of adhesive plasticity. The results showed lifetime improvement by a factor of 3.2 for the F A L S T A F F loaded specimen, and up to 17 for a constant-amplitude cycled specimen when compared to an unrepaired sheet. Denney and M a l l  [34-36] have  investigated boron/epoxy repairs applied to a single side of a thin centre-cracked aluminum panel, where patch bending is a significant factor due to an offset of the neutral axis. They investigated the effect of variously sized disbonds at different locations within the repair, and concluded that disbonds over the crack reduce patch life whereas disbonds away from the crack are not detrimental. They suggested that disbonds away from the crack may increase patch life by reducing the apparent size and stiffness of the patch and hence the amount of load it attracts from the surrounding structure. Failure occurred due to crack growth in the aluminum and little disbonding was noted. K l u g et al. [38] performed a similar test using thick and thin unbalanced repairs, and noted significant disbonding near the crack and near the edges of the patch for a thick repair and only near the crack for a thin repair. For these specimens, the patch was not tapered, leading to increased stresses about the edge of the repair. Jones, Chiu, and Smith [39] have presented a thorough review of the failure modes and locations of damage experienced in laboratory testing and military use of bonded repairs. W h i l e most of the work reported in the literature examines the primary failure modes of the repair, i.e. substrate cracking and accompanying cohesive disbonding of the adhesive, they conclude that the assessment should also include composite failure modes such as fibre failure, adhesive failure, cohesion failure at the patch-adhesive interface, adhesive failure at the adhesive-substrate interface, and inter-laminar failure and delamination. In practice, unexpected failure due to crack growth and disbonding about the crack has not been a problem, as mitigation of this form of crack growth is the primary design goal of the repair and it is well accounted for in the analysis, testing and inspection regimes. Another factor is that the technology has mainly been limited to double-sided repair of flat plates, to structures with bending restraint, and to structures with residual strength. Unexpected composite failures have usually only occurred during laboratory testing for evaluation of a repair, rather than during service. F A A Advisory Circular 25.571-1A identifies the failure modes and locations of damage that must be considered, and requires the assessment of the effects of impact damage, inter-ply delamination, and disbonding on the composite, adhesive, and substrate. Composite and adhesive damage mechanisms are most likely to arise in areas of high stress, or due to impact damage or manufacturing flaws. Generally, the fatigue strength of a composite laminate is high unless it is damaged by environmental degradation or  7  impact, or it is loaded in a weak direction. One would not expect growth of inter-ply or impact damage unless it is accompanied by fibre damage or occurs in a region experiencing significant bending and through-thickness stresses, compressive loads, or very large transfers of load through shear, such as might exist near a repaired crack or about the edges of the patch. In practice, this has proven to be true, with experimental results suggesting that delamination growth from impact damage or manufacturing flaws away from load transfer regions is minimal [34]. A final concern is the possibility of undetected widespread or multi-site fatigue damage in the region of the repair, which must be considered when designing repairs for older structures. The Australian A i r Force has had some experience in crack patching with widespread fatigue damage [1], and Molent and Jones [43] have successfully applied bonded repairs to heavily damaged transport aircraft lap joint sections, demonstrating that the technology can be applied in areas of widespread or multi-site damage.  1.3.2 Certification and damage tolerance analysis Regulations require the damage tolerance analysis of all components of the repair system. According  to F A A Advisory  Circular 25.571-1 A ,  the  damage  tolerance  evaluation must "define the loading and the environmental conditions, extent of damage or degradation of the structure, adhesive and repair, conduct of the associated structural tests or analyses, or both, used to substantiate that the repair design objective has been achieved, and establish data for inspection programs necessary to ensure detection of damage". Damage tolerance assessment is required for impacts, manufacturing flaws corresponding to limits of inspection methods, and progressive failure near the crack in the underlying structure. The result is that the certification of a repair must meet most of the requirements for the design of a new composite structure. Baker [3] has postulated that the outer edge of the repair could be treated in a safe life manner, with the inner region over the repaired crack being subject to damage tolerance assessment. The somewhat unpredictable nature of the failure mechanism and strength of composite structures has lead to the general requirement for physical testing of representative specimens to catch unexpected failure modes. Requirements for certification include the documentation and control of the manufacturing process such that the test articles closely match the actual composite parts in service. The only possible exceptions to the rule are for structures that are very similar in loading, composition, and manufacture to ones that have already been tested and certified. Jones et al. [39] have suggested a policy for damage tolerance assessment based on F A A requirements, by incorporating a two-stage approval process whereby a repair may be certified for a short term based on static strength  analysis  and  testing.  Physical  testing  of representative  structures  in a  representative environment would be required for cases where sufficient test data or  8  operational experience from similar structures was not available. Continuing operation of the aircraft would require the registration of a detailed damage tolerance analysis including an inspection plan. Jones et al [39] stress the need to perform residual strength analysis based on the limits of the inspection criteria, rate of damage growth, and strength of the damaged structure, including the possibility of wide-spread fatigue damage (the possibility of many cracks smaller than the detection limit). F A A circular 25.571-1A effectively makes this a requirement. It is stated that, unless stresses are so low that serious damage growth is extremely improbable, the structure must be shown to always be capable of supporting the design limit loads, and that the rate of damage growth and the inspectability of the structure provide a practical basis for an inspection program. Experimental testing of similar articles in representative environments must support the analysis. Damage may be simulated by saw-cuts, manufactured disbonds or delaminations, and by low-energy impacts, with the caveat that the damage must be placed in regions of maximum stress, where the residual strength of the structure will be most affected, or in areas where the additional compliance caused by damage will cause increased loads in surrounding structural elements and elevated stresses in the region of the patch. It is also required to examine the consequences of failure - e.g. structural failure and redundancy or criticality of the component, engine intake of loose parts, excessive drag, etc. Jones et al [39] also outline the analysis and testing requirements for static strength analysis and damage tolerance of the composite patch itself. Design allowables for the laminate must be generated through testing in appropriate environments, unless such data already exists or service experience shows that the performance of the material is unaffected. In addition, the static strength and damage tolerance of the composite must be tested as an entire system for the given application. The testing must include load dwells, and the specimens must be produced according to the production specifications. They also emphasized the need to include several composite-specific failure modes in the damage tolerance analysis of the patch system. These include failure by fibre failure, adhesive failure, cohesion failure at the patch-adhesive interface, adhesive failure at the adhesive-substrate interface, and inter-laminar failure.  1.3.3 Mechanics of a repair Several methods are available for the stress analysis of a bonded repair, including both numerical and closed-form approaches. Many researchers have used finite element analysis to investigate the mechanical response of a repair. Early studies by Ratwani [44] used three-dimensional linear finite element analysis to develop correction factors to account for bending, and Jones [45] developed 'three layer' finite element models that required fewer computational resources by modeling the underlying structure and the  9  repair as plates, and by modelling the adhesive as shear and extensional springs. More recent work includes nonlinear analysis performed by K l u g et al. [38] and by Naboulsi and M a l l [46,47], where three-layer solutions have been used to assess the effects of nonlinear deformation and stress-stiffening in a single-sided repair. These techniques have the advantage of allowing evaluation of complex geometries and loading, but tend to be computationally demanding due to the large number of elements required to accurately model the crack in the substrate and the stresses in the load transfer regions of the patch. They also demand a large amount of operator time to accurately model a repair scenario. A n alternative choice is the boundary element method, whereby the patch and plate are modelled as cracked plates, with the deflection, plate loads, and stress intensity being characterized  by Green's  functions,  and the  adhesive  being modelled as  independent shear and extensional springs. Erdogan and A r i n [48] first used the boundary element techniques to analyze the problem of a disbonding patch bonded to a cracked aluminum plate without bending. Ratwani [44] has shown that this model provides accurate results for determining the stress intensity when compared with finite element analysis and experiments. Dowrick, Cartwright, and Rooke [49] and Young, Cartwright, and Rooke [50-52] have developed a boundary element model applicable to highly orthotropic patches. Here, the patch is divided into cells, each of which is 'riveted' at regular intervals to the underlying plate. The 'rivets' each have a shear spring stiffness determined from the adhesive thickness and shear modulus, and the area of the patch represented by the cell. This model also has the capability to include nonlinear adhesive response. Poole, Lock, and Young [53] have verified this model against experimental results for a graphite/epoxy patch applied to a thick cracked plate. M o r e recent efforts have applied the dual method, a means of simplifying the analysis of the region near the crack, to solve the problem of combined bending and extension of a repaired plate [54,55]. A n analytic method for analysis of a repair is to consider the load transfer region about the crack separately from the one about the edges of the repair, and is known as the twostep method [56]. The presence of the crack and its accompanying adhesive load transfer region  is the  distinguishing characteristic  for  a repair  versus  a  reinforcement,  differentiating between the safe life (outer) and damage tolerance (inner) regions, and allowing significant conveniences to the analyst. The implicit assumption is that the change in compliance introduced by the crack and disbond do not significantly affect load transfer about the edges of the repair. This is justified only for large and stiff repairs, where limited disbonding or crack growth do not greatly affect the interaction of the patch and the greater structure of the aircraft. Figure 1-2 shows these load transfer areas.  10  Load Transfer Regions Crack  Figure 1-2: Load transfer regions For a stiff repair the load transfer regions are small compared to the planar dimensions of the patch, effectively become independent, and may be analysed separately [56-60]. The problem then resolves into two steps: (1) finding the stresses in a reinforcement, and (2) determining the state of the repair by introducing a crack and solving the resulting crack bridging problem. This two-step process simplifies the analysis, making closedform solutions feasible. The state of the repair may then be determined by superposition of the reinforcement and fracture mechanics solutions. This process is shown in Figure 13, where (a) shows the stresses in a reinforced plate, (b) shows the stresses resulting from a crack-bridging analysis by applying the crack-plane stresses from (a) as crack-opening stresses, and (c) shows their superposition.  (a)  (b)  (c)  Figure 1-3: Two step method In the first step, the patch acts as a 'reinforcement' and may be considered as a stiffened inclusion. The inclusion attracts load - the larger and stiffer it is, the more load it attracts from the surrounding structure.  Analytical solutions allow the calculation of the  distribution of stresses within and about the stiffened inclusion, providing the stresses in the underlying plate. Due to the reinforcing effect of the patch, these stresses are reduced from the remote applied stresses. In addition to the stresses resulting from the inclusion  11  analogy, the reinforcement may experience detrimental bending stresses due to a neutral axis offset, and due to thermal residual stresses from curing of the adhesive. For the second step, a crack is introduced and the reinforcement, now referred to as a 'repair' is modelled as springs bridging the crack. Here, the stresses found in the first step are applied as crack opening tractions and the springs restrict the opening of the crack reduce the stress intensity. This crack-bridging problem occurs often in materials science, and its solution determines the stress intensity at the crack tip and the adhesive stresses near the crack, characterizing the effects of adhesive load transfer about the crack. Rose refined this inclusion analogy and applied it to the crack-patching problem [56], applying the analytical result for a stiffened elliptical inclusion developed by Eshelby [61]. In addition to the stresses resulting from the inclusion analogy, patch bending and process-induced thermal residual stresses may also occur. Patch bending results from the offset in neutral axis caused by the patch, and may be estimated using simple beam theory [44]. Residual stresses occur when the patch application process requires an elevated temperature cure cycle and the patch and plate have differing thermal expansion coefficients. Residual stresses are particularly large for laboratory specimens which are often cured without restraint against thermal expansion. The residual stress in the region of the crack may be calculated by analytical means, where solutions have been provided by Rose [60] for an isotropic repair, by Wang et al [62] for a circular orthotropic repair, and by Albat and R o m i l l y [63] for a tapered repair. These stresses depend upon the cure temperature of the adhesive and the degree of physical restraint available during curing, i.e. whether the cure was accomplished with local heating or immersion in an autoclave and on the support conditions of the cracked plate. Local heating and mechanical restraint can significantly reduce residual stresses. Without support, thermal strains can result in significant bending, or warpage. To calculate the thermal residual strains, it is necessary to know the effective stress free temperature, i.e. the temperature at which thermal residual strains vanish. Albat and R o m i l l y [32] tested an instrumented A M R L specimen, and found an effective stress free temperature of 81 °C, significantly lower than the oftenused cure temperature. For the second part of the analysis, a crack is introduced and the reinforcement, now referred to as a repair, may be modelled as springs bridging the crack. Here, the stress found in the first step is applied as a crack opening traction and the springs bridging the crack partially prevent the crack from opening, reducing the stress intensity. This crackbridging problem occurs often in materials science. Its solution determines the stress intensity at the crack tip and the adhesive stresses near the crack, and characterises the adhesive load transfer about the crack, allowing the determination of the rate of crack growth and near-crack disbonding. Crack-bridging calculations may be accomplished  12  using finite elements, boundary elements, or analytical solutions based on interpolation between limit states. Exact analytical solutions exist for the limiting cases of a patch over a very short crack or a very long crack. N o exact closed-form solution exists for an arbitrary length crack. For small cracks, the stress intensity closely approaches the basic fracture mechanics solution for a crack with an opening pressure equal to the edge traction found in the first step of the analysis. This solution becomes less valid as the crack length, and hence, the crack opening displacement, increase. In the limit state of a very long crack, the crack opening displacement well away from the crack tip can be considered to be constrained only by the stiffness of the patch bridging the crack, and the state of the crack tip becomes independent  of crack length. Rose [60] has examined this problem and  developed an interpolation model which asymptotically approaches these two limit cases, and provides accurate results for an arbitrary crack length. C o x and Rose [64] have extended this approximate solution to the cases of near-tip reinforcement and elasticperfectly plastic springs. One considerable limitation of this method is that the reinforcement is assumed to act at the faces of the cracks. Baker [3] has investigated the effects of disbonding and temperature on fatigue crack reinitiation and growth. In this study, he developed a few simple relations for estimating the effect of a measured disbond on the crack-tip stress intensity and patching efficiency. Here, the limit case of a long crack is modified to account for the additional compliance of a disbonded portion of patch. The disbonding geometry is limited to a disbond of a constant size extending across the entire patch. Building on this work, Albat and R o m i l l y [65] developed solutions that are applicable for any degree of patch disbonding and include the effect of thermal residual stresses, patch lay-up, and cracked plate geometry. Thermal residual stresses were shown to contribute significantly toward reducing patch efficiency, particularly for laboratory specimens, where little restraint exists during the curing cycle to reduce thermal residual stresses. Clark and R o m i l l y [66] tested this model against a boundary element model and found that the results were valid for a wide range of patch stiffness parameters, and extended the models to include adhesive plasticity and allow the calculation of adhesive stresses and the load carried by the patch. 1.3.4 D a m a g e m e c h a n i c s  Here, we seek to define the best practice for predicting damage growth and failure of the various components of a repair: the plate, the adhesive joint connecting the plate and patch, and the composite patch itself. The damage mechanics of a metal plate, being a topic of undergraduate or graduate courses, are generally well understood. For a cracked plate, fracture and fatigue may be characterized by the stress intensity, but difficulties can arise due to load interaction and crack closure, load spectrum, adverse environments, and  13  plasticity or thickness effects. These topics often require the use of experiments for evaluation of structural integrity under realistic conditions; examples are the load spectrum testing of airframes and the use of empirical models for crack-closure. For bonded repairs, Duong and Wang [67] have noted that inadequate methods for characterizing fatigue of cracked plates experiencing bending loads are a significant impediment to life assessment. They have proposed the use of a modified stress intensity, developed by linear interpolation between two limit states where the crack growth rate can be accurately assessed: for a crack experiencing only a membrane stress intensity, and a crack with equal membrane and bending stress intensities. The method provided good results when applied to the analysis of unbalanced repairs. Mechanical analysis of bonded joints is a complex subject, as the adhesive stresses are related to the loading, the bending and extensional stiffness of the adherends, and the elastic/plastic properties and thickness of the adhesive. In general, both shear and peel (transverse normal) adhesive stresses arise. For a repair, it is necessary to assess both the adhesive stresses and the stiffness of the patch as it acts to restrict the opening of the crack. Closed-form elastic and elastic/perfectly plastic solutions for the shear and peel stresses are available for simple geometries, usually developed by modeling the adhesive as a row of shear and extensional springs connecting the adherends. The standard techniques are the shear-lag solution for lap joints developed by Volkerson [67] and the bending analysis of Goland and Reissner [68]. Hart-Smith [69] has used these to develop elastic-plastic failure criteria and analytical closed-form solutions for several practical geometries. Delale et al. [70] and Bigwood and Crocombe [71] have improved on these solutions by including mechanical coupling of the bending and extensional responses of the joint. Coupling occurs due to moments resulting from the adhesive shear stress acting at the adhesive/adherend interface, and from adhesive shear stresses induced by rotation of the adherends under bending. In practice, finite elements are often used to analyse the complex geometries that arise in the design of airframes, with the closed-form equations used to inform the design process. The methods of Hart-Smith have been prevalent in the analysis of bonded repairs. For life assessment, bonded joints are often treated in a safelife manner, where peak stresses are controlled by tapering the adherends, using a long overlap length, and choosing an appropriate adhesive thickness. This approach does not work for a repair, as it is usually not possible to reduce the adhesive stresses near the crack to the point where safe-life methods may be applied. Damage mechanics analysis of bonded joints poses similar challenges. Here, two modes of cracking may occur, cohesive and adhesive. Cohesive failures occur when the adhesive fails by yielding and cracking, while adhesive failures occur at the bond line. There is debate over whether pure interfacial failure can occur or i f it instead occurs by lowenergy fracture of the weaker material due to constraint from the stiffer and (usually)  14  stronger material. Adhesive failures should not occur unless a poorly prepared surface results in low durability, as discussed in section 1.2.2. Several different parameters have been used to characterize adhesive fatigue and fracture, including the plastic or elastic strain range [e.g. 59], strain energy release rate [e.g. 38], or stress intensity. The strength and stiffness of an adhesive results from molecular and inter-molecular forces, exhibiting strain-rate, moisture, and temperature sensitivity, while the stresses near the fracture zone are dependent on the joint geometry, including the location of the crack within the adhesive and type of loading. Illustrating the effects of the crack location on fracture, for a boron-epoxy repair, interfacial crack growth may require 230 Jim 3000 Jim  compared to over  for cohesive failure [3].  Fundamentally, the use of energy-based fracture parameters is complicated by the dependence of the stresses on the geometry and composition of the joint. This violates the similarity criterion, whereby in fracture  mechanics it is assumed that cracks have  similarly-shaped stress distributions regardless of crack length, position, or geometry. This explains why different geometries and loading conditions lead to different critical strain energy release rate and disbond growth rate parameters [66, 72], even though the same adhesive has been employed. Strain energy release rates are used to predict fracture and fatigue of materials for which the rate of plastic work with disbond or crack growth is either negligible, or the rate of plastic work is constant, for an increment in crack growth [73]. For bonded joints, this means that there should not be a discernible critical strain energy release rate as a material property - instead, the adhesive must be evaluated on a case-by-case basis where test data must be developed for a particular joint composition. Hart-Smith [69] has noted that strength under static loading depends mainly on the adhesive strain energy, and advocates an energy-density criterion for crack growth, which may necessitate a detailed assessment of the elastic/plastic strains in the adhesive at the tip of the crack. This method explicitly considers the strains near the crack tip, and has been used [2] to develop damage tolerance criteria for bonded joints. A n alternative approach is to use alternating plastic strains, or an approach based on the rate of plastic work with disbond growth. Either alternative directly accounts  for the effects of  geometry. A n additional advantage is that the adhesive shear strength provides a very simple and direct approach with which to include the effects of temperature, loading rate, and environment on disbond growth rates. It is noted that test data is readily available for F M 7 3 M , the adhesive most commonly used in bonded repairs. Strength and fatigue analysis for axial loading of unidirectional composites is relatively straight-forward. Accurate estimates of the ultimate strength of the laminate may be found by considering the strain in the fibres and matrix to be equal, a reasonable assumption for high stiffness laminates with long fibres [74]. For a polymer-reinforced composite such as boron-epoxy, composite failure occurs when the stress in the boron 15  fibres reaches its maximum, corresponding to a nominal tensile stress in the orthotropic solid. This is the principal strength of the composite. Fatigue of composites under such loading may be characterized by a stress-life (S-N) diagram, with small reductions in stiffness and strength occurring as the matrix cracks and delaminates near fibre breaks and stress concentrations. When bending is a significant component of the load on the plate, transverse stresses and interlaminar failure have a detrimental effect on fatigue life and strength. Tensile interlaminar stresses develop due to the buckling of fibres on the compression side of the plate, and transverse stresses may cause interlaminar failure. The mechanisms of failure under transverse loading are matrix failure by fracture and interfacial failure, and by fibre splitting i f they are weak in the transverse direction [74]. Stiff fibres lead to stress concentrations in the matrix, causing it to fail before the nominal transverse stress exceeds the strength of the matrix. Models for predicting such failures are often developed by using a periodic fibre distribution to characterize the peak stresses, whereby a stress concentration or strength reduction factor is applied to the matrix and used as a failure criteria for the composite. From mechanical testing and fractography, it has been shown that the exact geometrical distribution of the fibres has a significant role, and more recent attempts to predict failure strength have used finite element analysis to predict the matrix stresses given a randomly determined fibre spacing or by arranging the fibres in a configuration consistent with the manufacturing technique [75]. In all cases, the thrust of the analysis is the same, to predict the stresses in the matrix and fibre given a nominal loading, and then develop failure criteria.  1.3.5 Critical analysis Airworthiness certification of a bonded repair requires the assessment of the resulting structure's damage tolerance considering all plausible damage sources and failure modes, including an inspection plan to detect damage growth before catastrophic failure. Bond durability is a persistent problem, but one that can be ameliorated by adopting best practices in the manufacturing processes and reducing the adhesive stresses to whatever degree possible. Baker [3] has argued that the design of the outer, tapered portion of the patch could be governed by safe life principles, but the fitness of the design can also be assured through damage tolerance analysis, which would ensure adequate life in the instance of an impact or manufacturing flaw. In this region the repair does not vary significantly from other types of bonded joints, and an analysis suitable to certification should be possible by standard methods. It is possible, however, that the inner and outer regions of the repair could be assessed independently, as Denney and M a l l [34] have shown that damage in this outer region can actually improve the efficiency of the repair by reducing stresses in the region of the crack. For the inner part of the repair, applied over the original damaged aircraft structure, three types of damage must be assessed: 1)  16  cracking in the structure beneath the repair, 2) disbonding of the adhesive joining the patch to the structure, and 3) interply delamination or composite failure in the region of the crack. The issues that prevent widespread use of the technology are related to these final three forms of damage, with near-crack disbonding and composite failure being issues that have lead to unexpected rapid failure during proof testing and fatigue life assessment. T o address these issues, the damage tolerance assessment must consider all of the following elements, which need to be in place to assure structural integrity of a repair. (1) Loading conditions: assessment of the fatigue loading, service environment, and thermal stresses. The repair must be able to carry the design limit load. (2) Failure modes: the type of damage, location, and mechanisms for growth must be identified. Crack growth and accompanying disbonds are expected, but damage may also arise from process errors (e.g. adhesive voids) or accidents (e.g. impact damage). (3) Damage parameters: means of calculating parameters such as stress intensity that are used to predict damage growth rates with a given service loading and environment. (4) Test data: empirical relations correlating damage growth rates observed in testing with the damage parameter, and structural test data for representative repair specimens. (5) Inspection methods: inspection methods must be qualified to ensure that they can detect the damage and that the inspections will occur often enough to prevent failures. A l l of these elements must be combined to ensure that the structure w i l l meet a set of safety goals, which may be statistical and/or deterministic in nature. A n example of a very broad statistical goal would be a 10~  7  chance of catastrophic failure over the  operational life of the part, which is a rough industry guideline for primary structures. A n example of a deterministic goal would be the specification that the part must be capable of withstanding the design limit load throughout its working life. Widespread use of bonded composite repairs w i l l not occur until all of these elements and a set of acceptable safety goals can be presented in a form that is easy to understand and apply. A n assessment of the service loads in the structure is an element of the analysis. For aircraft manufacturers, load data is available in design and flight test documents. For overhaul and repair facilities, this data is often not available, and loads must be estimated by analysis of the existing structure. Here, the goal is to return the static and fatigue strength without disrupting the flow of load in surrounding structures. When compared to riveted repairs, the only unique issue is the assessment of thermal stresses arising from mismatched thermal expansion coefficients. There are three types of damage growth unique to a bonded repair: 1) substrate cracking and accompanying disbonding, 2) interply delamination, or 3) composite failure. The  17  damage tolerance analysis requires accepted methods by which damage parameters may be calculated and compared with test data in order to predict the life and strength of the repair. The two-step process and the reduction of the mechanics of a repair to a crackbridging problem are fundamental concepts for assessment of these types of damage growth, but several difficulties arise. These include: a) thickness effects in fracture mechanics energy analysis, b) the coupled and geometrically nonlinear membrane and bending responses of the repair, c) difficulties inherent in the stress analysis of hybrid bonded joints, and d) complications in predicting the fracture and fatigue of bending plates, adhesive joints, and interlaminar failure of composites. These problems are inherent to the mechanical analysis of a repair and w i l l be discussed in more detail below. The final element is identification of all damage to the structure, including initial damage, repair application flaws, and damage accrued during subsequent service. The existing damage may be assessed by the conventional means of visual, dye penetrant, and eddy current inspection, with macroscopic defects such as fatigue cracks and corrosion characterized by measured defect sizes and locations. Widespread fatigue damage and/or undetectable discrete cracks must be assumed to exist, and included in the worst possible locations. Guidelines for the design of new aluminum aircraft structures exist whereby small cracks of a specified size are assumed to exist at rivet holes and other stress-raisers, and certification requires life assessment under the assumption that these flaws are present. For widespread damage, undetectable flaws should be assumed to exist at locations adjacent to stress-raisers, and the possibility of crack interaction should be investigated. Several deficiencies may be noted in the application of the two-step analysis process and interpolation model. The first problem is in the analysis of geometrically nonlinear bending of a repair. The two-step method relies on superposition, and, strictly speaking, is only applicable to linear problems. Rose [60] has calculated the long-crack limit for the strain energy release in the presence of nonlinear bending by applying composite beam theory. This calculation provides the total available strain energy but does not separate it into bending and membrane components. A general method is required for the treatment of  geometrically nonlinear  bending.  Perhaps  worse,  energy  methods  have  been  effectively abandoned as Wang and Rose [76,77] have shown that for linear coupled bending and extension, they incorrectly predict the membrane  and bending stress  intensity. A n additional problem arises due to the somewhat arbitrary application of plane stress or plane strain conditions at the crack tip. For extension of generalized plane strain cracked plates, Wang and Rose [78] have shown that, when transverse stresses are considered, the crack tip is effectively in a state of plane strain. For bending of cracked plates, the actual state of the crack tip is less clear. Joseph and Erdogan [79] have noted that classical plate bending models employing the Kirchoff boundary condition lead to  18  conflicting definitions of the stress intensity depending on whether a stress- or strainbased definition is employed. Errors greater than 60% have been demonstrated when the classical model is compared to more sophisticated plate bending models that include shear deflections and allow the satisfaction of all three plate bending  boundary  conditions. Wang and Rose [80] have developed a numerical line-spring model for linear coupled bending and extension of repairs, but it employs a plane stress plate extension model and the Reissner plate bending model, both of which neglect the important role of transverse normal stresses. Hence, methods must be made available to properly account for plate thickness effects" i f energy methods are to be accurately applied and design equations for combined bending and extension to be developed. A s a final point, interpolation solutions such as those originally developed by Rose [60] hold great promise for regulatory purposes and stochastic risk analysis or optimization analyses, but have not been properly developed for the case of coupled extension and bending due to difficulties in developing appropriate long-crack limits. Putting aside issues specific to the analysis of bonded composite repairs, there are difficulties inherent in the stress analysis of hybrid bonded joints which must be accounted for. In particular, it is necessary to include all of the relevant deformation modes particular to orthotropic composite adherends when calculating the stiffness of the repair, the stresses in the adhesive, and the load carried by the composite repair as it acts to restrict the opening of an underlying crack. Complications also exist in the prediction of the fracture and fatigue of bending plates, adhesive joints, and interlaminar failure of composites. It appears that in the absence of a universally acceptable model, the best approach is to characterize the potential for composite failure by calculating the plate loads and hence stresses in the composite, and attempt to reduce them however possible by design. Guidelines for what is an acceptable level of peel stress w i l l develop over time as experience with the technology develops. Finally, there is a deficiency in the available experimental data. The A M R L specimen, which has proven invaluable in the assessment and development of the bonded composite repair technology, has not been tested under varying degrees of bending restraint. Such testing would provide data that may help in the avoidance of unexpected failures due to patch bending. It would also prove to be useful to carry out testing in the manner of Baker [29] whereby lap joint specimens would be tested under varying degrees of bending restraint to examine the response of the repair as it acts to bridge the crack. The best practice when safety is a critical issue is to perform mechanical testing to determine whether transverse failure/delamination is a problem. Analytical modeling and strength prediction remain useful in order to understand the problem and to minimize it. In design, it is always good practice to minimize transverse stresses wherever possible.  19  1.4 Thesis objectives The objective of this thesis is the development and validation of a basic damage tolerance analysis methodology to assure the structural integrity of a repair, with particular attention to the effects of bending and plate thickness on the crack tip stress intensity, adhesive stresses, and interlaminar composite failure. The desired product is a set of validated engineering equations that capture the relevant physical phenomena in a conservative manner, and are based upon a theoretical analysis of the mechanics of a repair. This necessitates experimental work for the investigation of bonded composite repair failure mechanisms and the validation of developed predictive models, and theoretical work on the fracture of bending plates with reinforced cracks and fracture mechanics of repairs under non-linear bending conditions. This w i l l enable the damage tolerance assessment and certification of a repair. This project falls under a larger research and development program at U B C directed toward certification and widespread use  of bonded repairs  in civilian aviation.  Advancements to this program have been made through the development of design and damage tolerance analysis tools and test methods for repairs. W i t h the assistance of the N R C - I A R , much effort has been applied toward the design, analysis, manufacture, and testing of repair specimens. Assistance and co-operation has also been obtained from the A M R L , who pioneered the use of this technology for military applications. W i t h the completion of a P h . D . dissertation by Andreas Albat and a M . A . S c . thesis by Randal Clark, this thesis marks the third graduate degree devoted to this work. Seven A M R L repair specimens have been constructed, including an instrumented specimen used to study thermal stresses. Procedures for the manufacture and application of repairs have been documented, and the two-step model for double-sided repairs has been extended to include the effects of patch-edge tapering, thermal strains, adhesive plasticity, crackedplate geometry correction factors, and disbonding. The result is a set of validated closedform models suitable for engineering analysis, optimization, or use in codes and standards. The goal of this thesis is to provide a similar analysis capability for singlesided, or unbalanced repairs. The bonded composite repair technology has great potential in the management and life extension of ageing transport airframes. It is now common practice to rebuild and re-fit aircraft to extend their life, often repeatedly. Ultimately, aircraft are retired when the cost of this refurbishment and the requirements for inspection become too onerous or the design of the airframe has become obsolete and is too inefficient to compete with newer models. Other financial factors are also important: e.g. an aircraft maintains its value i f new aircraft of the same type are still being built but may lose a large amount of value i f the line is discontinued. For aircraft in an advanced stage of usage, the consequential defects are well known and documented, and many of them could profitably be returned  20  to useful service with a few well-placed bonded repairs. There is also a significant opportunity for the use of repairs on newer aircraft, as when such a defect is initially discovered it is likely that it can be permanently repaired with a bonded patch, instead of being 'managed' using conventional riveted repairs. Obsolete or discontinued aircraft often see service in utility markets, such as cargo transport or forest firefighting and it can be profitable to refit an airframe that has virtually no value as a passenger transport aircraft into a new role. Given the advantages of the bonded repair technology in terms of cost, applications time, and effectiveness, there is a very large market for the application of the technology. A t any given time, there are literally thousands of aircraft that are approaching a refit or retirement, and even most aircraft that are receiving their first overhaul would benefit from bonded repairs for defects that may ultimately manifest themselves into a life-threatening problem i f a riveted patch is employed. This illustrates the importance of realizing an easy-to-use damage tolerance methodology that w i l l allow them to compete against conventional riveted repairs and be more widely used. Several barriers prevent the realization of a comprehensive damage tolerance analysis of a repair as required for certification. Significant limitations exist in our ability to include the effects of finite plate thickness and nonlinear coupled bending and extension on the performance of repairs. Significant issues also affect our ability to predict the strength and rate of fatigue of bending plates, hybrid bonded joints, and composites under transverse  loading. Engineering methods  for analysis of repairs  generally ignore  composite failures, and there is a lack of experimental data available for validation of mechanical models of geometrically non-linear bending of bonded composite repairs and fatigue of hybrid bonded joints. T o address these issues, a program of experimental and theoretical analysis has been carried out, involving the testing of a heavily instrumented repair with and without bending constraints, the construction and testing of representative lap joint specimens, the development of line-spring models for cracked plates that include the relevant through-thickness effects, and the reformulation of the energy methods for repair analysis to allow the calculation of the parameters governing the life of a repair. Chapter two describes the structural testing of an instrumented repair specimen, with and without bending restraint, which is used to validate a three-dimensional finite element model of a repair. The results are used as a basis for the development of a methodology and criteria for damage tolerance substantiation of a repair. The issue of plate thickness effects is addressed in Chapter three by adoption of a generalized plane strain plate model, allowing the analysis of transversally isotropic plates of arbitrary thickness. Engineering equations are developed that relate the stress intensity factor to the strain energy release rate, and the Rose model, a classical model for analysis of repaired plates is shown to hold for a generalized plane strain plate.  21  In Chapter four, a series of engineering plate models are developed to illustrate the effects of through-thickness stresses, shear forces, and shear deflection on the bending of cracked transversally isotropic plates. The result is a set of equations relating the bending stress intensity to the strain energy release rate, which are used to extend the Rose model to the analysis of bending in reinforced cracked plates. In Chapter five, the problems of coupled linear bending and extension of a cracked plate repaired with an isotropic patch are addressed. A line-spring model, developed from the results of Chapters three and four, is used to develop a new interpolation model applicable to repaired plates with arbitrary crack lengths. In Chapter six, this model is extended to the analysis of an orthotropic repair by the adoption of more sophisticated bonded joint models that take in to consideration several composite-specific deformation modes and more accurate analysis of the coupling of the extensional and bending responses of an unbalanced joint. In Chapter seven, nonlinear effects  caused by stress-stiffening of the repair are  accounted for by the use of the generalized Rose model, and by the extension of the new interpolation model to the nonlinear analysis of a repair, allowing the calculation of the extensional and bending stress intensity factors under a single-sided repair. In Chapter 8, the results of the work are reviewed in the light of the thesis objectives, topics for future work are discussed, and the contributions of this work are outlined.  1.5 Summary This chapter has introduced the bonded composite repair technology, demonstrated the need for damage tolerance assessment of a repair, identified shortcomings in the understanding of repair mechanics that make it difficult to certify a bonded composite repair, and described the objectives and content of this thesis. This goal of this work is to develop the elements required for the damage tolerance analysis of a repair. Subsequent chapters w i l l describe experimental and analytical work directed toward this result.  1.6 References [1] R. Jones. Assessing and maintaining continued airworthiness in the presence of widespread fatigue damage: an Australian perspective. Engineering Fracture Mechanics 60, 1998, 109-130. [2] R. Jones, W K C h i u , J. Paul. Designing for damage tolerant bonded joints. Composite Structures 25, 1993, 201-207. [3] A A Baker. O n the certification of bonded composite repairs to primary aircraft structure. Proceedings of I C C M - 1 1 , G o l d Coast, Australia, 1997,1-1 to 1-23. [4] A A Baker. Evaluation of adhesives for fibre composite reinforcement of fatiguecracked aluminum panels. S A M P E Journal, March/April, 1979, 10-17.  22  [5] J Cognard. Influence of water on the cleavage of adhesive joints. International Journal of.Adhesion and Adhesives,8/2, 1988, 93-99. [6] A D Crocombe. Durability modelling concepts and tools for cohesive environmental degradation of bonded structures. International Journal of Adhesion and Adhesives 17, 1997, 229-238. [7] W P deWilde, G VanVinckenroy, L Tirry, A H Cardon. Effects of the environment and curing on the strength of adhesive joints. Journal of Adhesion Science and Technology 9/2, 1995, 149-158. [8] M Fernando, W W Harjoprayitno, A J Kinloch. A fracture mechanics study of the influence of moisture on the fatigue behaviour of adhesively bonded joints. International Journal of Adhesion and Adhesives 16, 1996, 113-119. [9] W S Johnston, L M Butkus. Considering environmental conditions in the design of bonded structures: a fracture mechanics approach. Fatigue and Fracture of Engineering Materials 21, 1998, 465-478. [10] A N Rider, D R Arnott, A R Wilson, O Vargas. Influence of simple surface treatments on the durability of bonded aluminium alloy plates. Materials Science Forum 189-190, 1995, 235-240. [11] K B Armstrong. Long-Term Durability in Water of A l u m i n i u m A l l o y Adhesive Joints Bonded with Epoxy Adhesives', International Journal of Adhesion and Adhesives 17, 1997, 89-105. [12] M R Bowditch, SJ Shaw. Adhesive bonding for high-performance Advanced Performance Materials 3, 1996, 325-342.  materials.  [13] N C Cave, A J Kinloch. The effects of silane deposition conditions on the durability of aluminum joints pretreated using 3-aminopropyltrimethoxysilane. Journal of Adhesion 34, 1991, 175-18. [14] G D Davis, T S Sun, JS Ahearn, J D Venables. Application of surfgace behaviour diagrams to the study of hydration of phosphoric acid-anodized aluminum. Journal of Materials Science 17, 1982, 1807-1818. [15] R P Digby, D E Packham. Pretreatment of aluminium: topography, surface chemistry and bond durability. International Journal of Adhesion and Adhesives 15, 1995, 61-71. [16] L J Hart-Smith. A peel-type durability test coupon to assess interfaces in bonded, cobonded, and co-cured composite structures. International Journal of Adhesion and Adhesives 19, 1999, 181-191. [17] J D Minford. Durability evaluation of adhesive bonded Bonding. Plenum Press, N e w York, U S A , 1991, 239-290.  structures.  Adhesive  [18] P J Pearce, F C Tolan. Epoxy silane - a preferred surface treatment for bonded aluminium repairs. International Conference on Aircraft Damage Assessment and Repair, 1991,188-193. [19] A N Rider, D R Arnott. B o i l i n g water and silane pre-treatment of aluminum alloys for durable adhesive bonding. International Journal of Adhesion and Adhesives 20, 2000, 209-220. [20] W Thiedman, F C Tolan, P J Pearce, C E M Morris. Silane coupling agents as adhesion promoters for aerospace structural adhesives', Journal of Adhesion 22, 1987, 197-210.  23  [21] J R J Wingfield. Treatment of composite surfaces for adhesive bonding. International Journal of Adhesion and Adhesives 13/3, 1993, 151-156. [22] D R Arnott, P R Pearce, A R Wilson, R J Chester. The effect on mechanical properties of void formation during vacuum bag processing of epoxy film adhesives. Second Pacific International Conference on Aerospace Science and Technology, 1995, 811-816. [23] W D Bascom, R L Cottington. A i r entrapment in the use of structural adhesive films. Journal of Adhesion 4, 1972, 193-209. [24] R J Chester, J D Roberts. V o i d minimization in adhesive joints. International Journal of Adhesion and Adhesives 9/3, 1989, 129-138. [25] P R Pearce, D R Arnott, A Camilleri, M R Kindermann. Cause and effect of void formation during vacuum bag curing of epoxy film adhesives. Journal of Adhesion Science and Technology 12/6, 1998, 567-584. [26] A A Baker. Repair efficiency in fatigue-cracked aluminum components reinforced with boron/epoxy patches. Fatigue and Fracture of Engineering Structures 16/7, 1993, 753-765. [27] A A Baker. Fiber composite repair of cracked metallic aircraft components - practical and basic aspects. Composites 18/4, 1987, 293-308. [28] A A Baker. Bonded composite repair of metallic aircraft components - overview of Australian activities. A G A R D - 79th Meeting of the Structures and Materials Panel. Specialists Meeting: Composite Repairs of Military Aircraft Structures, 1994, 1-14. [29] A A Baker, J D Roberts, L R F Rose. Experimental study of overlap joint parameters relevant to k eduction due to crack patching. 28th National S A M P E Symposium, 1983, 627-639. [30] M D Raizenne, T J Benak, J B R Heath, D L Simpson, A A Baker. Bonded composite repair of thin metallic materials: variable load amplitude and temperature cycling effects. A G A R D - 79th Meeting of the Structures and Materials Panel. Specialists Meeting: Composite Repairs of Military Aircraft Structures, 1994, 5-1. [31] P Hucalek, M D Raizenne, R F Scott. Influences of environment and stress history on the composite patch repair of cracked metallic structures. Canadian Aeronautics and Space Institute Journal. 34/2, 1988, 85-91. [32] A M Albat, D P Romilly, M D Raizenne. Thermal residual strain measurement in a composite repair on a cracked aluminum structure. I C C M - 1 1 , G o l d Coast, Australia, 14th to 18th July 1997, p. VI-279-VI-288. [33] P Poole, D Lock, A Young. Composite patch repair of thick aluminum sections. International Conference on Aircraft Damage Assessment and Repair, 1991, 85-91. [34] J Denney, S M a l l . Characterization of disbond effects on fatigue crack growth behavior in an aluminum plate with a bonded composite patch. Engineering Fracture Mechanics 57, 1997, 507-525. [35] J Denney, S M a l l . Fatigue behaviour in thick aluminum panels with a composite repair. Collection of Technical Papers, 39th Annual A I A A - A S M E - A S C E - A H S Structures, Structural Dynamics, and Materials Conference 3, 1998, 2434-2443.  24  [36] J Denney, S M a l l . Effect of disbond on fatigue behavior of cracked aluminum panel with bonded composite patch. American Institute of Aeronautics and Astronautics A I A A 96-1322-CP, A I A A , 1996, 14-21. [37] P K Sharp, J Q Clayton, G Clark. Retardation and repair of fatigue cracks by adhesive infiltration. Fatigue and Fracture of Engineering Materials and Structures 20/4, 1997, 605-614. [38] J K l u g , S Malley, C T Sun. Characterization of fatigue behavior of bonded composite repairs. Journal of Aircraft 36/6, 1999, 1017-1022. [39] R Jones, W K Chiu, R Smith. Airworthiness of composite repairs: mechanisms. Engineering Failure Analsysis 2, 1995, 117-128  failure  [40] F . A . A . Advisory Circular 25.1529-1. Instructions for continued airworthiness of structural repairs on transport airplanes. U . S . Department of Transportation, Federal Aviation Administration, 1991. [41] F . A . A . Advisory Circular 20.107A,. Composite aircraft structure. U . S . Department of Transportation, Federal Aviation Administration, 1984. [42] F . A . A . Advisory Circular 25.571-1C. Damage tolerance and fatigue evaluation of structures. U . S . Department of Transportation, Federal Aviation Administration, 1998. [43] L Molent, R Jones. Crack growth and repair of muli-site damage of fuselage lap joints. Engineering Fracture Mechanics 44/4, 1993, 627-637. [44] Ratwani M . M . , 1979. Cracked, adhesively bonded laminated structures. Journal 17(9), 988-994.  AIAA  [45] Jones R , Calinan R J , Aggarwal K C , 1983. Analysis of bonded repairs to damaged fibre composite structures. Engineering Fracture Mechanics 17(1), 37-46. [46] S Naboulsi, S M a l l . Modeling of cracked metallic structure with bonded composite patch using three layer technique. Composite Structures 35, 1996, 295-308. [47] S Naboulsi, S M a l l . Nonlinear analysis of bonded composite patch repair of cracked aluminum panels. Composite Structures 41, 1998, 303-313. [48] F Erdogan, K A r i n . A sandwich plate with a part-through and disbonding crack. Engineering Fracture Mechanics 4, 1972, 449-458. [49] G Dowrick, D J Cartwright, D P Rooke. Boundary effects for a reinforced cracked sheet using the boundary element method. Theoretical and Applied Fracture Mechanics 12,1990,251-260. [50] A Young, D J Cartwright, D P Rooke. Influence of tapering on the stresses in repair patches. Proceedings of the Third International Conference on Numerical Methods in Fracture Mechanics, 1984, 721-737. [51] A Young, D P Rooke, D J Cartwright. A numerical study of balanced patch repairs to cracked sheets. Aeronautical Journal, 1989, 327-334. [52] A Young, D P Rooke, D J Cartwright. Boundary element method for analyzing repair patches on cracked finite sheets. Aeronautical Journal 92, 1988, 416-421. [53] P Poole, D Lock, A Young. Composite repair of thick aluminum alloy sections. International Conference on Aircraft Damage Assessment and Repair, 1991, 85-91.  25  [54] N K Salgado, M H Aliabadi. The boundary element analysis of cracked stiffened sheets, reinforced by adhesively bonded patches. International Journal for Numerical methods in Engineering 42, 1998, 195-217. [55] P H W e n , M H Aliabadi, A Young. Boundary element analysis of flat cracked panels with adhesively bonded patches. Engineering Fracture Mechanics 69, 2002, 2129-2146. [56] L R F Rose. A n application of the inclusion analogy for bonded reinforcements. International Journal of Solids and Structures 17, 1981, 827-838. [57] L R F Rose. A cracked plate repaired by bonded reinforcements. International Journal of Fracture 18/2, 1982, 135-144. [58] L R F Rose. Crack reinforcement by distributed springs. Journal of the Mechanics and Physics of Solids 35/4, 1987, 383-405. [59] L R F Rose. Basic concepts in assessing the efficiency of bonded repairs. International Conference on Aircraft Damage Assessment and Repair, Melbourne, Australia, 1991, 202-208. [60] L R F Rose. Theoretical aspects of crack-patching. In: A A Baker, R . Jones. Bonded repair of aircraft structures. Martinus Nijhoff, Dordrecht, The Netherlands, 1988. [61] J D Eshelby. The determination of the elastic field of an elliptical inclusion, and related problems. Proceedings of the Royal Society of London A 2 4 1 , 1957, 376-391. [62] C H Wang, L R F Rose, R Callinan, A A Baker. Thermal stresses in a plate with a circular reinforcement. International Journal of Solids and Structures 37, 2000, 45774599. [63] A M Albat, D P Romilly. Direct linear-elastic analysis of double symmetric bonded joints and reinforcements. Composites Science and Technology 59/7, 1999, 1127-1137. [64] B N C o x , L R F Rose. A self-consistent approximation for crack bridging by elastic/perfectly plastic ligaments. Mechanics of Materials 22, 1996, 249-263. [65] A M Albat. Thermal residual stresses in bonded composite repairs on cracked metal structures. P h . D . thesis, The University of British Columbia, 1998. [66] R J Clark, D P Romilly. Fatigue damage prediction for bonded composite repairs applied to metal aircraft structures. 2001 S A E Transactions - Journal of Aerospace, 10 p. [67] Volkersen O. Rivet strength distribution in tensile-stresses rivet joints with constant cross-section. Luftfahrforschung 1938;15(l):41-47. [68] Goland M and Reissner E . The stresses in cemented lap joints. Journal of Applied Mechanics, Transactions of the A S M E 1944;66:A17-A27. [69] Hart-Smith L J . Stress analysis: a continuum mechanics approach. In: Developments in Adhesives, volume 2, applied science publishers, London, 1973. [70] Delale F , Erdogan F , Aydinoglu M N . Stresses in adhesively bonded joints: a closedform solution. Journal of Composite Materials, 1981;15:249-271. [71] B i g w o o d D A and Crocombe A D . Elastic analysis and engineering design formulae for bonded joints. International Journal of Adhesion and Adhesives 1989;9(4):229-242. [72] C L i n , K M Lietchi. Similarity concepts in the fatigue fracture of adhesively bonded joints. Journal of Adhesion 21, 1987, 1-24.  26  [73] D Broek, Elementary engineering fracture mechanics, revised 4 academic publishers, Dordrecht, The Netherlands, 1997.  edition, Kluwer  [74] B D Agarwal, L J Broutman. Analysis and performance of fibre Composites. 2 Edition, John W i l e y and Sons, Inc., N e w York, 1990.  n d  [75] V N Bulsaraa, R Talrejab, J Qua. Damage initiation under transverse loading of unidirectional composites with arbitrarily distributed fibres. Composites Science and Technology, 59, 1999, 673-682. [76] C H Wang, L R F Rose. O n the design of bonded patches for one-sided repair. Proceedings of the 11th International Conference on Composite materials, G o l d Coast, Australia, 5, 1997, 347-356. [77] C H Wang, R J Callinan, L R F Rose. Analysis of out-of-plane bending of a one-sided repair. International Journal of Solids and Structures 35/14, 1998, 1653-1675. [78] C H Wang, L R F Rose. Bonded repair of cracks under mixed-mode loading. International Journal of Solids and Structures 35/21, 1998, 2749-2773. [79] P F Joseph and F Erdogan. Surface crack problems in plates. International Journal of Fracture 41, 1989, 105-131. [80] C H Wang, L R F Rose. A crack bridging model for bonded plates subjected to tension and bending. International Journal of Solids and Structures 36, 1991, 1985-2014.  27  Chapter 2: Bending of bonded composite repairs: an experimental and finite element design study R. J. Clark and D . P. R o m i l l y University of British Columbia Department of Mechanical Engineering  2.1 Introduction This chapter describes an experimental and finite element study of a bonded composite repair applied to a metallic aircraft structure. The experimental study involves the fatigue testing and fractographic examination of 2024-T3 aluminum plates repaired  with  boron/epoxy composite patches. The two plates are tested simultaneously in a sandwich panel configuration with and without an aluminum honeycomb spacer to provide bending restraint. The specimen was heavily instrumented, with strains being measured on the surface of the patch at many locations, thus allowing the validation of a threedimensional finite element model and an assessment of the effects of geometrically nonlinear bending, buckling, and crack closure on the mechanical response and strength of the repair. The experimental and finite element results illustrate the large influence of bending and composite failure modes on the life and strength of the repair. The authors use the experimental results and the validated finite element analysis to evaluate the effectiveness of the methods and data currently available in the open literature for damage tolerance substantiation, and then, by applying probabilistic methods and the principle of compounded conservatism to the findings, propose a new process for classifying bonded repairs and for assigning testing and inspection requirements. A bonded composite patch is a fibre-reinforced epoxy doubler used to reinforce weak structure or repair cracked aircraft structure. The patch acts both to reduce stresses in the underlying structure, and to restrict the opening of cracks that may lie beneath it. These effects and the smooth load transfer inherent to bonding offer the potential for a significant increase in the fatigue life when compared to mechanically fastened repairs. The technology also offers improvements in cost, weight, and inspectability. One of the main barriers to the widespread use of this technology is the lack of a comprehensive process for damage tolerance analysis, particularly regarding the mechanical assessment of the structure in the presence of significant bending loads, which can lead to early failure. The other significant impediment is the assurance of bond line durability for repairs applied in a maintenance and repair facility.  * A version of this chapter will be submitted for publication. R.J. Clark, D.P. Romilly, "Bending of bonded composite aircraft repairs: an experimental and finite element design study", A I A A Journal.  28  Figure 2-1: A bonded repair and a mechanically fastened repair  Many  experiments  have been  performed  to investigate  failure mechanisms,  to  demonstrate the predictive performance of models, and to evaluate repair materials and mechanical phenomena in a repair. A widely used repair specimen is the Australian Aeronautical and Maritime Research Laboratories ( A M R L ) specimen, which is the type of specimen used for this study. The A M R L test article consists of two repaired plates that are bonded to an aluminum honeycomb support to restrict the bending induced by the neutral axis offset of the repair and to model the support of underlying structures that often exist in the highly stressed regions of an aircraft. Engineering drawings of the A M R L specimen are provided in Appendix B . Baker has published experimental results for  the A M R L specimen that illustrate the effects of cure cycle parameters [1,2],  disbonding [1,3], patch shear deformation  [2,4], adhesive plasticity [4], and test  temperature [1] on performance. In Canada, the A M R L specimen has been used by the National Research Council - Institute for Aerospace Research ( N R C - I A R ) to examine the effects of load spectrum, compressive loading, and adverse environments on patch performance [5,6], and by Albat and R o m i l l y [7] to evaluate thermal strains. W i t h the honeycomb bending restraint, the failure mode of the repair has consistently been continued cracking of the aluminum with accompanying adhesive disbonding. Double lap-shear hybrid bonded joint specimens matching the composition and geometry of the A M R L specimen geometry have been used to investigate the effects of patch shear deformation and adhesive plasticity on the apparent stiffness of the patch in the region of the crack [4], to determine the rate of disbonding in the region of the crack [8], to examine disbonding at the edges of the repair. Patch shear deformation and adhesive plasticity have been shown to reduce patch life by reducing the restraint against crack opening imposed by the repair [4]. It has also been shown that increasing the cure cycle temperature and time can impose two penalties on patch performance; 1) tensile thermal residual stresses from a mismatch in thermal expansion coefficients can accelerate fatigue crack growth, and 2) crack growth retardation is reduced due to annealing of crack tip  29  plasticity-induced residual stresses. Sharp, Clayton, and Clark [9] have also examined the effects of adhesive infiltration into the crack, which was found to increase the stress required to open the crack and hence improve fatigue life. Other researchers  have examined both similar and distinctly different types of  specimens to assess the effects of bending on the life of a repair. Poole, Lock, and Young [10] have investigated fatigue damage propagation in thick aluminum plates repaired with graphite/epoxy patches to investigate structural failure behaviour and assess fatigue life. Similar to the A M R L specimen, repaired plates were bonded to a honeycomb core and tested in pairs to restrict bending. They examined the effect of the F A L S T A F F loading spectrum on repair performance, and compared the results to boundary element analyses  that  included the  effects  of  adhesive  plasticity. The  results  showed  improvements on life of a factor of 3.2 for the F A L S T A F F spectrum-loaded specimen, and up to a factor of 17 for a constant-amplitude cyclically loaded specimen when compared  to  an  unrepaired  sheet. Denney and M a l l  [11-13]  have  investigated  boron/epoxy repairs applied to a single side of a thin center-cracked aluminum panel (i.e. an unbalanced repair), where patch bending is a significant factor due to an offset of the neutral axis. They investigated the effect of variously sized disbonds at different locations within the repair, and concluded that disbonds over the crack reduce life whereas disbonds away from the crack are not detrimental. They suggested that disbonds away from the crack may actually increase patch life by reducing the apparent size and stiffness of the patch and hence the amount of load attracted from the surrounding structure. During this testing, failure occurred due to crack growth in the aluminum and little disbonding was noted. K l u g et al. [14] performed a similar test using both thick and thin unbalanced repairs, and noted significant disbonding near the crack and near the edges of the patch for a thick repair and only near the crack for a thin repair. For these specimens, the patch was not tapered, leading to increased stresses about the edge of the repair. In more recent work, Sabelkin et al. [15] performed tests to investigate the effects of a supporting stringer on the life of a single-sided repair applied to a thin cracked aluminum plate. Their results show that the stiffeners have an important role in reducing the stress intensity, and that the thermal residual stresses must be accurately evaluated. A s previously observed for repair of thin plates, they found that disbonding extended only a few millimeters from the edge of the crack. Jones, et al. [16] have presented a thorough review of the failure modes and locations of damage experienced in laboratory and military use of bonded repairs. W h i l e most of the work reported in the literature examines the primary failure modes of the repair, i.e. substrate cracking and accompanying cohesive disbonding of the adhesive, Jones et al. state that any damage tolerance assessment must also include composite failure modes such as fiber failure, adhesive failure, cohesion failure at the patch-adhesive interface,  30  adhesive  failure at the  adhesive-substrate interface,  and inter-laminar failure and  delamination. In practice, unexpected failure due to crack growth and disbonding about the crack has not been a problem, as mitigation of this form of crack growth is the primary design goal of the repair and it is well accounted for in the analysis, testing and inspection regimes. Another important factor is that the technology has mainly been limited to double-sided repairs of flat plates, to structures with bending restraint, and to structures with residual strength. Accordingly, unexpected composite failures  have  usually only occurred during laboratory testing for evaluation of a repair, rather than during service. F A A Advisory Circular 25.571-1A identifies the failure modes and locations of damage that must be considered, and requires the assessment of the effects of impact damage, inter-ply delamination, and disbonding on the composite, adhesive, and substrate. This paper describes a unique experimental study investigating the bending mechanics of a bonded composite structural repair tested with and without bending restraint. The specimen is relatively thick and is repaired with a patch of sub-optimal  planar  dimensions, an extreme case that allows the authors to assess the potential effects of induced bending on early failure of the structure. The specimen was comprised of two edge-cracked aluminum panels, each repaired with a boron-epoxy patch arid joined together as an aluminum honeycomb sandwich. Strain results are reported for the fatiguedamaged specimen under both tensile and compressive applied loading, and before and after removal of the honeycomb separating the aluminum panels. Strain data from gauges fitted to the surface of the repair in the region of the crack are used to validate a threedimensional finite element model, allowing the calculation of the stress intensity and the stresses in the adhesive and the patch as it acts to reinforce the cracked plate. Fractographic examination of the failure surfaces of the cracked plate and the composite patch illustrate the failure mechanisms active in a bending repair. It is shown that geometrically nonlinear bending and stress stiffening have a significant role in the type of failure and rate of damage, thus decreasing the time to failure of the structure. In addition, a technique is described for three-dimensional finite element linear elastic  fracture  analysis of a structure experiencing large deflections. The method employs the stressbased definition of the stress intensity factor, meaning that the analyst does not need to assume a state of plane stress or plane strain. Finally, the authors attempt to use the finite element stress intensity factor, adhesive stresses, and composite stresses near the repaired crack to calculate the residual strength of the repair and rate of damage progression in light of the failure modes observed during the fatigue test. It is shown that significant problems remain for the damage tolerance assessment of repairs, particularly in the characterization of the observed composite and adhesive failure modes.  31  2.2 Experimental study The experimental study involves the fatigue testing of a symmetrical honeycomb sandwich structure, comprised of two edge-cracked aluminum panels, each repaired with a boron-epoxy patch, as shown in Figures 2-2 through 2-4. This specimen was originally developed by the A M R L , and has also been used extensively by both the N R C - I A R . A s described in the introduction, the A M R L specimen has seen extensive use in the evaluation of the bonded composite repair technology for balanced or symmetric doublesided repairs. Without bending, the failure mode of the repair has consistently been found to be continued cracking of the aluminum with accompanying adhesive disbonding. In this study, the authors test the specimen with and without the bending restraint provided by the aluminum honeycomb spacer in order to: 1) investigate the failure mechanics of a repair under bending and 2) to collect strain data to validate models of bending repairs. It should be noted that the A M R L specimen was designed to model a repair applied to cracked plates that are supported by underlying structures (e.g. ribs, stringers, etc.). If designed as a one-sided, or unbalanced repair, a much larger patch would typically be used to reduce the bending loads in the cracked region. A s such, the A M R L specimen without bending restraint provides an extreme test of the technology. Figure 2-2 shows a picture of the instrumented patch and the testing apparatus. A photoelastic coating was applied to the upper half of one of the patches to reveal the distribution of strains and estimate the extent of disbonding. A n A C potential drop probe was used to measure the crack length. Unfortunately, the utility of the potential drop probe was greatly affected by electrical contact between the honeycomb core and aluminum face-sheets, which had been electrically insulated in specimens built by the A M R L . A traveling microscope was used to estimate the degree of disbonding at the edge of the patch. T o investigate the stress distribution within the repair, strain gauges were applied to the specimen by technicians at the N R C - I A R .  32  Figure 2-2: Experimental apparatus The specimen contains 100 strain gauges, most in 10-gauge strips, located as shown in Figure 2-3. Engineering drawings illustrating the type, orientation, and location of the gauges are included in Appendix B . This chapter will mainly report the strain in gauges located above the repaired crack (location A on the figure), as this region has the greatest influence on cracking and disbonding. The gauges at location B allow the examination of the stresses about the edges of the repair and the gauges at locations C and D measure the strains in the patch as a disbond forms during fatigue testing. The gauges at location E allow the measurement of the strains in the tapered region of the repair and the stresses at the apex of the patch, where a stress concentration in the aluminum plate is known to exist. The remaining gauges allow the assessment of the distribution of the strains in the structure, and allow the analyst to evaluate the grip boundary conditions. Note that the gauges w i l l , in general, over-report any strains caused by bending as due to the thickness of the gauge and the adhesive used to bond it to the structure, particularly when the gauge is applied to the surface of a disbonded portion of the repair, which is fairly thin (-1 mm).  This effect  was taken into consideration during subsequent analysis of the  measured strain data.  33  Figure 2-3: Strain gauge locations The fatigue testing was carried out with a load (expressed as a remote applied stress) of 138 M P a at an applied stress ratio R = 0.1 and test frequency of 3 H z . Strain measurements were taken at several stages, including at the start of the test and after 175,000 cycles, at which point the honeycomb support joining the specimens was removed to evaluate the bending of a restrained and an unrestrained patch. The initial fatigue pre-crack length was 19.9 mm, and the crack length when the honeycomb was removed was 45 mm, as measured by fractographic examination. A l s o at 175,000 cycles, isopropyl alcohol and ink were injected into the adhesive disbond, revealing a disbond of up to 6.4 m m extending from each side of the crack. Strains were measured during static testing using a Schlumberger Solartron Orion datalogger, 10 gauges at a time, using a 3wire strain guage connection. During each measurement, the strain readings at each location were taken for remote applied stresses ranging from 138 M P a to - 5 5 M P a with the honeycomb support, and from 138 to -28 M P a without the honeycomb support, allowing the evaluation of both non-linear bending and buckling. A complete record of the measured strain gauge data is tabulated in Appendix C . Figure 2-4 shows the deflections of the structure  under maximum tensile and  compressive loads with the honeycomb removed. Under tension the neutral axis of the patch moves to the load line, causing large moments and accompanying adhesive peel stresses. Under compressive loads the structure bows outward, approaching collapse. This clearly demonstrates the large influence of bending on the mechanics of the structure.  34  Figure 2-4: Bending under tension (left) and compression (right)  Figure 2-5 shows the extent of the disbonding at 175,000 cycles and at failure after 185,000 cycles, revealing that the  specimen  failed  rapidly after removal of the  honeycomb spacer that provided resistance to bending. This demonstrates the need to account for induced bending during damage tolerance assessment of a single-sided repair.  6.4 mm in 175,00 cycles  4.1 mm in 10,000 cycles  Figure 2-5: Adhesive failure surface  The disbond was initiated by the failure of the adhesive that infiltrated the crack during curing, followed by cohesive disbond growth towards the adhesive/composite  interface.  The disbond then grew along the interface with the scrim carrier and boron fibres. Final failure occurred by simultaneous failure of the composite patch and fracture of the  35  cracked plate. Figure 2-6 shows the interfacial failure surface. Fibres that failed after removal of the bending restraint remained bonded to the underlying plate and pieces of adhesive remain attached to the boron fibres, indicating high-energy failure and good interfacial bond strength.  11 mm  Figure 2-6: Interfacial failure surface Figure 2-7 shows the failure surface of the aluminum plate. The left picture shows the fatigue pre-crack and the right figure shows the transition from high to low cycle fatigue after removal of the honeycomb at 175,000 cycles. U p to this point the crack front grew at an angle, indicating that bending affected crack growth even before removal of the honeycomb. After removal of the honeycomb, the surface attains rougher appearance and begins to show signs of shear failure.  3.175 mm  crack front  Figure 2-7: The crack face after pre-cracking (left) and after 175,000 cycles (right) Selected strain measurements from this study are used in subsequent sections to validate a finite element model of the tested repair. A complete record of the strain measurements is provided in Appendix C . Strain results w i l l be compared to finite element data in subsequent sections.  36  2.3 Finite element modeling The authors have constructed a finite element model to simulate the experiments described above. The model was developed using A N S Y S [17], and consists of ' S O L I D 95' 20-node brick elements with 15 node wedges at the crack tip. Table 2-1 shows the material properties used in the model. The Textron 5521 boron-epoxy prepreg composite patch is modeled as an orthotropic solid. The Cytec F M 7 3 M adhesive and 2024-T3 aluminum plate are modelled as isotropic solids, and the aluminum honeycomb spacer is modelled as an orthotropic solid with a young's modulus of 1.86 G P a in the throughthickness direction and very low modulii in all other directions. G , GPa  V  a, jus fc  210.0  7.24  0.21  4.61  —  Patch, transverse  2.5  1.0  0.019  25.87  0.924  Adhesive  2.14  0.805  0.33  50  0.25  Plate  72.4 1.86 0.001  27.2 0.001  0.33 0  23.45 23.45  3.125 6.35  Material  £,GPa  Patch, longitudinal  Honeycomb  t, mm  Table 2-1: Material properties and dimensions Thermal residual strains imparted by the cure process and operating temperature are a very important consideration for bonded composite repairs, and can be characterized by an effective stress free temperature that accounts for creep of the adhesive and polymer matrix as the specimen is cooled during the cure cycle. For the materials systems and specimen geometry used for this experiment, using a cure temperature of 1 2 1 ° C , it has been shown that the effective stress-free temperature for this specimen is 81° C [7]. The experiments  were  conducted  at  room  temperature  (approximately  20° C ) ,  and  accordingly a temperature difference of 61° C was applied in the finite element model. Figures 2-8 is a planar view of the model. Inset (a) shows the crack tip elements and (b) shows the aluminum substrate symmetry and pressure boundary conditions for a crack of length a and patch of radius r (-75 mm). T o model the constraint of the grips, the nodes at the loaded edge are forced to move together in the direction of the load. A s in the static testing, the applied stress was varied from  138 to - 5 5 M P a for the double-sided  configuration and 138 to - 2 8 M P a for the single-sided configuration.  37  Figure 2-8: Finite element mesh, (a) the crack tip and (b) the entire model  The model was solved using a Newton-Raphson method with an iterative solver, large deflections, and stress stiffening. Pivot checking was disabled to allow the use of singularity elements. A user-defined macro generated  singular elements  that were  properly oriented with respect to the crack tip and with mid-side nodes placed at the quarter-point position. A stress-based  method was employed to calculate the stress  intensity through the thickness of the plate, as described in reference 18. The membrane and bending stress intensities K  m  and K  h  were calculated as the average stress intensity  and first moment of the stress intensity through the thickness of the plate, and were obtained by a linear least-squares fit to the stress intensities calculated at the integration point planes near the crack tip. Figure 2-9 shows a three-dimensional view of the finite element model in the region of the repaired crack, illustrating the singular crack tip elements and the layering of the plate, adhesive, and composite patch elements.  38  Figure 2-9: Three-dimensional view of the cracked region The experimental and finite element strain data are compared in the next section to demonstrate the mechanics of a repair and validate the finite element model.  2.4 Validation For validation, the finite element model was solved in an edge-cracked tension configuration, as shown in Figure 2-8(b). Figure 2-10 compares the finite element model strain results to the strains measured on the outside of the patch, as recorded before the specimen was subjected to fatigue loading. In this section, the distance x from the free edge of the repair w i l l be referred to repeatedly, and is shown on Figure 2-8(b). The finite element model strains have been adjusted to include the effect of gauge thickness, as the strain gauge sensing element is offset from the surface of the repair by a small but not insignificant distance that tends to amplify bending effects. The experimental strains are reported as the average strain for gauges mounted on the patches on either side of the specimen and results are shown for an applied tensile stress of 138 M P a and a compressive stress of -55 M P a . In this case, the crack was 19.9 m m long.  39  1400  -600 -800 Position x [mm]  Figure 2-10: Strain distribution in undamaged repair The strains determined for the outer surface of the patch are lowest in the region of the crack in both the model and experiments. This is contrary to what might be expected, as load should be diverted to the patch as it bridges the crack, increasing the strain. The cause is induced bending due to the neutral axis offset caused by the crack and the overlying patch, which reduces the surface strain in the patch. Even with the honeycomb support, under tensile loads the structure experienced bending sufficient to noticeably reduce the strains on the outer surface of the patch. After 175,000 cycles, the crack was nearly 45 mm long, and the interface between the patch and underlying cracked plate had a disbond of approximately 5 m m in length. Figure 2-11 shows the adhesive layer mesh, where the grey elements have been removed to impose the disbond condition observed in the fatigue experiment after 175,000 cycles.  Figure 2-11: Adhesive elements in the region of the crack, showing the disbond  40  Similar to Figure 2-10, Figure 2-12 shows the surface strains in the patch as it reinforces the cracked plate. It is unfortunate that the strain gauges did not extend far enough to measure the strains at the crack tip, but much can still be made of the results. For both the finite element model and the actual specimen, the strains reach a peak at the free edge of the patch. A traditional centre-cracked repair model would predict a maximum strain at this location, but it should be reached gradually. There are two reasons for this abrupt peak: 1) the edge-cracked geometry of the plate has as significantly lower stiffness than a centre-cracked plate at this location and hence it w i l l shed more load to the patch, and 2) that the disbonded part of the patch has a low bending stiffness and w i l l carry mainly axial loads, an effect that increases when the honeycomb core is removed. The increase in load at the edge of the patch and adhesive edge effects also lead to larger adhesive stresses and a faster rate of disbonding near the free edge. 2000  1500  1000  E  s  500  CA  -500  -1000  Position [mm]  Figure 2-12: Strain distribution, with honeycomb Figure 2-13 shows the surface strains measured by selected gauges plotted against the remote applied stress. The gauges are shown at selected distance from the free edge of the repair. The strains are linear until the applied stress is well into the compressive region, when tensile residual stresses in the substrate are overcome and crack closure reduces the proportion of the load carried by the repair. The distances in the legend are the position of the strain gauges, (i.e. x on Figure 2-8(b)). The finite element results are from a linear model, and do not include crack closure, which is seen to reduce the compressive strains observed in the experimental results when compared to the strains measured under tensile loading.  41  20GG-  1500  • Experimental, 1.3 mm A Experimental 19.5 mm x Experimental, 40.3 mm -o-Finite Element, 1.3 mm -A-Finite Element, 19.5 mm -x-Finite Element, 40.3 mm  E  1  c  "5 to  -1000  Remote Applied Stress [MPa]  Figure 2-13: Variation of strains with applied stress, with honeycomb The next stage of testing occurred after the removal of the honeycomb core separating the aluminum plates. The specimen experienced no further fatigue cycling at this point, so the crack and disbond sizes are the same as for the previous case. Figure 2-14 again plots the strain distribution on the surface of the repair. Note that applied tension results in compressive strains on the surface of the patch. The honeycomb provided a large degree of constraint against bending, and now that it has been removed, bending stresses dominate the response of the repair. This bending and the resulting adhesive peel stresses must be accounted for in order to predict the rate at which a repair w i l l fail. After removal of the honeycomb, the maximum applied compressive stress was only -28 M P a , and the patch appeared to be approaching collapse. Here, the effect of the edge-cracked plate geometry combined with the reduced bending stiffness of the disbonded section of the patch is again evident in the dramatically increasing strains near the edge of the patch. It can be seen that crack closure plays a significant role for compressive loading for both the finite element model results and the measured strains, and hence crack-face contact should be included to obtain accurate results under compressive loading.  42  5000 4000  • Experimental, 138 MPa A Experimental,-28 MPa -o-Finite element, 138 MPa - a - F i n i t e element, -28 MPa -*-..with contact, -28 MPa  3000 -\ 2000  -4000 Position [mm]  Figure 2-14: Strains, without honeycomb In Figure 2-15, selected strain results are plotted against applied load. The strains are clearly non-linear without the honeycomb support, and show good agreement except at 40.3 m m , the strain gauge nearest to the crack tip. This discrepancy is likely due to a small amount disbonding near the crack tip or adhesive plasticity, neither of which was included in the finite element model. Both of these effects would act to relieve the severity o f bending of the composite patch as it bridges the crack. The compression results again show the importance of crack closure, which acts to unload the patch near the crack.  43  -2000-  •4000Remote Applied Stress [MPa]  Figure 2-15: Variation of strain with applied stress, without honeycomb The finite element model and experimental results capture the geometrically nonlinear mechanical response of the bending repair and demonstrate the importance of bending on the stresses within the patch itself. The importance of modeling the finite geometry of the edge-cracked plate has also been demonstrated, as a significant peak in the patch strains was observed at the edge of the repair. Crack closure was also shown to have a large role under compressive loading, acting to greatly relieve the load carried by the patch. This beneficial effect, however, is offset by the potential for elastic buckling, which appeared to be imminent for the unsupported repair under an applied compressive stress of only 28 M P a . In the next section, this validated finite element model w i l l be used to calculate the residual strength of the repair and to examine factors affecting its fracture strength and fatigue life.  2.5 Residual strength and damage tolerance assessment The fractographic examination illustrates the failure mechanisms at work in the repair. One important observation is that crack growth and disbond growth appear to occur together, which should be expected, as crack growth in the aluminum plate w i l l generally shed load to the patch and promote disbonding. Similarly, disbond or delamination growth w i l l reduce the effective stiffness of the patch as it bridges the crack and promote further cracking in the aluminum. The final failure of the structure occurred by simultaneous fracture of the composite patch and the repaired plate. Another important observation is that composite delamination and composite fracture are significant failure modes when a repair is free to bend, neither of which are often observed in double-sided  44  repairs or repairs with bending restraint. It is clear that in order to assess the strength of a one-sided repair, full account must be made of the potential for failure in three areas: a) cracking in the aluminum substrate, b) failure of the adhesive and c) failure of the composite. The following sections address these three areas of concern and place into perspective the relative merits of both existing and proposed methods of evaluation (resulting from the current investigations) for the purposes of damage tolerant design of bonded composite repairs on metallic structures.  2.5.1 Aluminum substrate fracture Figure 2-16 shows ^ a n d K as m  h  predicted by the finite element model for two cases:  i.e. with and without the honeycomb bending restraint. The predicted results reflect the previously experimentally observed nonlinear deformations, with stress-stiffening acting to reduce the rate of increase in stress intensity under larger applied loads. This happens when the structure moves toward the load line and sheds load to regions remote from the crack.  Remote Applied Stress [MPaVm]  Figure 2-16: Membrane and bending stress intensity factors B y L E F M (Linear Elastic Fracture Mechanics) criteria, fracture is considered imminent when the sum of K  m  and K  h  exceeds the plane strain fracture toughness, K . lc  aircraft skins, the actual fracture toughness (denoted as K ) c  is often used. K  c  obtained from testing and as it is often significantly greater than K , Xc  For thin is usually  it w i l l result in a  longer critical crack length thus can greatly easing inspection requirements.  The  increasing acceptance of the N A S G R O [19] damage tolerance software package allows  45  the use of a calculated K  c  in most circumstances, but caution is required for short cracks  where the non-singular stresses near the crack-tip are significant. For the 2024-T3 plate used in this investigation K = 34MPaVm ]c  2024-T3 plate N A S G R O provides a K  c  [20], whereas for a 3.175 m m thickness  value that is over twice as large. For the A M R L  specimen with a crack size of 45 mm, K  Xc  is exceeded by K  m  + K with an applied b  remote stress of nearly 90 M P a for unrestrained bending, but requires well over 140 M P a to exceed K with the honeycomb bending restraint present. u  Fatigue is also well characterized by the stress intensity, with crack growth properties being catalogued in well-accepted software tools such as N A S G R O or A F C R A C K . Difficulties can arise due to load interaction and crack closure, load spectrum, adverse environments, and plasticity or thickness effects, often requiring the use of experiments to evaluate crack growth under realistic conditions. Doung and W a n g [21, E q . 5] have characterized crack growth under combined bending and extension using a modified stress intensity, ^ f  m o d  , which, when compared to test data for A L 7 0 7 5 specimens,  demonstrates good results.  is developed by interpolation between two states for  which crack growth can be accurately assessed: 1) a crack experiencing only a membrane stress intensity, and 2) a crack with equal membrane and bending stress intensities. From Figure 2-16 it can be seem that there is a significant increase in  after removal of the  honeycomb. Given that as little as a 15% increase in AK can halve fatigue life, it is apparent that bending has a large role.  2.5.2 Adhesive failure Adhesive failure is dependent on the adhesive stress or strain state that develops as the repair acts to bridge the crack. Figure 2-17 shows the peak shear and peel stresses at the edge of the crack for location A on Figure 2-11. The results exhibit nonlinear behaviour, with the  adhesive  stresses for the unconstrained case nearly vanishing under  a  compressive remote applied stress, where the crack w i l l close as the thermal residual strains are overcome. These results include thermal residual strains in the adhesive using an effective stress-free temperature of 81°C [7]. It must be noted that in the finite element model, the peel and pressurization stresses vary significantly through the thickness of the adhesive, an observation that other researchers have also observed in bonded joint models. It was not possible to examine this effect in detail with the developed finite element model because of size restrictions on the university version of  ANSYS.  Similarly, finite element results for the adhesive compression in the unrestrained case are likely to be over-reported as no attempt was made to model crack closure between the  46  surfaces of the adhesive disbond. Closure of the adhesive crack faces would be expected to act to relieve a portion of the compressive adhesive stresses in the restrained specimen.  Q. S o> in  o> k _  +->  tn  > '55  CD £ T3 <  -40  Remote Applied Stress [MPa] Figure 2-17: Adhesive stresses  Adhesive strength depends on whether yielding or fracture is prevalent. In hot and wet conditions, high temperatures and the plasticizing effect of absorbed water act to make yielding dominant. A t low temperatures associated with high altitude flight, fracture becomes prevalent. The main design criterion is the adhesive stress, which should be kept low to ensure that fatigue failures w i l l not initiate and gross yielding w i l l not occur under limit loads. Shear and peel stresses are controlled by using an appropriate adhesive thickness, providing long joint overlap lengths to minimize bending loads and peel stresses, and by tapering the edges of the joint. The strength of the Cytec F M 7 3 M adhesive under various degrees of constraint and for laboratory or hot and wet environmental conditions is well characterized by the work of Chalkley and van den Berg [22] and by Ignjatovic, Chalkley and Wang [23,24]. The failure surface for this adhesive is pressure-dependant,  and is best characterized by the Drucker-Prager cap plasticity  model [23], particularly in the presence of compressive pressurization. It can be seen in Figure 2-17 that compression of the adhesive is not a concern for unbalanced repairs (see the curves for the no constraint case under compressive remote loading), as the crack faces w i l l close and the aluminum w i l l carry the stresses, unloading the adhesive. The authors use a simpler model, based on [22] and [23], whereby the adhesive w i l l obey the Tresca or von Mises criterion under compression, and the modified Tresca or modified von Mises criterion under tensile pressures, thus providing the limit shear yield strength as defined below. 47  _ f T y  p < OMPa  38.6MP«  ~ 13S.6MPa -\A3p  p>  (1)  OMPa  This best fit criterion is plotted in Figure 2-18 along with a second criterion that has been developed to account for environmental effects, and employs a penalty factor of two against both the shear and pressurization stresses, as suggested by Chalkley and Baker [26] for the shear stress allowable. It is noted that R A A F design guidelines allow adhesive loading to 80% of the allowable shear strain [25]. For comparison, allowable shear stress results from A S T M D5656 testing (pure shear) by Chalkley and van den Berg [22] are shown in Figure 2-18 for various environmental conditions. —  -60-  O Tresca [23] • Von Mises [23] - - Best fit (damage tolerant allowable) Penalty factor (safe life allowable) • Chalkley and van den Berg [22] - Unrestrained 'load line' - Restrained 'load line'  - 4 0 ° C , dry Restrained bending c a s e allowable remote applied  50  stress using the damage tolerant criterion is 84 M P a  a. (0 in a> Tn  Unrestrained bending  a> >.  c a s e allowable remote applied stress using the  na>  damage tolerant criterion is 66 M P a  .c  (A 4)  U a> UJ  -15  -10  -5  0  5  10  15  Pressure (MPa) (tensile positive)  Figure 2-18: Adhesive yield criteria andfiniteelement model adhesive stresses The trace or 'load line' of the finite element model adhesive pressurization and shear stresses is plotted in Figure 2-18 for both the restrained bending and unrestrained bending cases. For restrained and unrestrained bending the adhesive should begin to yield under a remote applied stresses of 84 and 66 M P a , respectively. It is apparent that the adhesive was repeatedly loaded into the plastic range during the fatigue testing, and that without bending restraint, significant peel stresses and pressurization developed in the adhesive near the crack, explaining the rapid failure of the repair. Applying the safe life design criterion that includes factors to account for environmental effects, neither of the two cases would result in a useful load carrying capability. It is clear that the safe life allowable stress is very conservative and would be difficult to meet in the presence of a 48  long crack under a repair. It is an appropriate criterion, however, in the tapered region where good design can reduce the adhesive stresses. In the region near the crack, environmental concerns are less likely to be critical, and the damage tolerant allowable may be more suitable, perhaps with a reduction such as the 80% of allowable shear strain policy used by the R A A F . One must also assess adhesive fracture and fatigue. Since adhesive strength and stiffness result from molecular and inter-molecular forces which exhibit strain-rate, moisture, and temperature sensitivity, damage tolerance substantiation must include both the detrimental effect of cold temperature on fracture toughness [27] and the effect of high temperatures that induce plasticity and accelerate fatigue crack growth. The use of energy-based  fracture parameters  is complicated by the dependence  of the stress  distribution on the joint loading type, geometry, and material composition. This violates the similarity criterion, whereby it is assumed that cracks have similarly-shaped stress distributions regardless of crack length, position, or geometry and thus explains why inconsistent results can arise when they are used to predict fracture and fatigue [8, 28]. A final complication is that either of two modes of cracking may occur, cohesive and adhesive. Cohesive failures occur when the adhesive fails by yielding and cracking, while adhesive failures occur at the bond line. There is debate over whether pure interfacial failure can occur or i f it occurs by low-energy fracture of the weaker material due to constraint from the stiffer and (usually) stronger material present. But, it is generally agreed that for a well-designed joint, adhesive failures should not occur unless a bond surface has been poorly prepared, i.e. the principle forming the basis for quality assurance testing of bonded joints. These are significant challenges, and in practice test data from specimens with a similar composition, geometry, and loading is required. Many researchers in the bonded composite repair field have been critical of the strain-energy release rate approach to adhesive fatigue and fracture, and other options have been suggested including use of the Hart-Smith plastic strain energy density [29] or plastic strain range [8] approaches. In spite of this criticism, the use of strain energy release rates for specific geometries and loading conditions has provided some excellent results [3037], and a body of literature exists for the Textron 2251 boron prepreg and Cytec F M 7 3 M patch  system  [e.g. 38-40] which  includes information on fatigue  threshold and  environmental effects. Here, the authors test the use of elastic adhesive stresses as a conservative criterion for fracture and fatigue assessment of the adhesive. In a previous round-robin test of bonded joint analysis methodologies for the cracked lap-shear specimen [41], Joseph and Erdogan demonstrated that the opening and shear mode strain energy release rates are related to the peak peel and shear adhesive stresses. This approach builds on the strain energy release rate arguments of Rice [42], an approach which is considered to provide  49  exact results for many fracture problems, and has been applied by other researchers [e.g. 8, 43]. The round-robin testing showed that this method of strain energy release rate calculation provides reasonable results when compared to other techniques such as modified crack-closure. Accordingly, the opening and sliding mode strain energy release rates may be calculated from the peel and shear adhesive stresses as follows:  6 , = - ^ ,  For  611=77*-  (  a linear-elastic homogeneous  ' ) B  material, the strain energy release rate may be  converted to a stress intensity using the plane strain relationship, K  2  Thus the adhesive stresses, G,  2 A  and K  = GE / ( l —v^). a  should be more or less equally capable of  predicting fatigue and fracture in the elastic regime for joints of similar composition and loading. In the plastic regime, the elastic-plastic strain energy release rate, J, can be calculated by defining contours outside of the crack tip process zone and using energy arguments to determine the rate of work required for crack extension. J is effectively independent of the state of the process zone unless the plastic strains are sufficient to cause significant changes in the energy state of the elastic region of the specimen. In this case, the method of Erdogan and Joseph w i l l still provide accurate results and the adhesive stresses, G (or / ) , and K based on an elastic analysis are still more or less equally capable of predicting fatigue and fracture. Similarly by the method of Erdogan and Joseph, the strain energy density and adhesive stresses are also simply related for a linear elastic material. Accordingly, while each of these various measures of the state of the adhesive has advantages concerning how it is calculated or assessed from a numerical model or experimental results, and some provide a better representation for an adhesive experiencing creep or gross plasticity, there is in many ways little to choose between them, particularly when one considers that the design criteria w i l l essentially restrict the thickness-averaged  adhesive  stresses to the elastic range. The main difference  is  convention, whereby an analyst w i l l understand a critical stress intensity to imply a material property whereas a strain energy release rate should be understood as a measure of the energy absorbed during the failure of a particular specimen under particular conditions. For repair and overhaul there is not the time, information, or expertise to perform  sophisticated  analysis,  and even  with  conventional  repairs,  simple  and  conservative means are often preferred, even at the expense of increased operational burdens such as higher inspection frequency or early scrapping or rework. Accordingly, the authors w i l l use the available experimental data to test the stress-based adhesive failure criteria. A s a comparison of adhesive failure criteria, Figure 2-19 plots experimental data from several sources against the yield criteria by assuming an adhesive thickness of 0.17 mm,  50  which is the value used by Butkus and Johnson [39]. The data has been converted from adhesive stresses and strain energy release rates to opening mode ( K , ) and sliding mode (K„)  stress intensities as described above. Given the complex interactions between the  boron fibres, matrix, and adhesive, this conversion does not imply that the actual stress intensity at the tip of a crack has been calculated. The plotted results indicate that the stress-based criteria are conservative in all cases, even when including environmental effects and matrix or interface cracking. It is also seen that, for this adhesive thickness, the yield criterion and room temperature fatigue threshold (including environmental effects) are similar and a criterion based upon the adhesive stresses should be adequate for both fracture and fatigue. Note that Butkus and Johnson used a threshold of 10~ mlcycle , whereas Chalkley and Baker used 6 • 10~ mlcycle. 9  9  1.6  1.4  •  Butkus and Johnson [39] fracture data  •  Cheulk et al [44] fracture data  A  Chalkley and Baker [26] fatigue threshold  - - Damage tolerant allowable (fig. 2-18) 1.2  1  Safe life allowable (fig 2-18)  cold (-40 C), dry  1.0 -  O  Ignjatovic et al [23] yield data  A  Chalkley and van den Berg [22] yield data  r—i  E  as-received thermal  •g  |  n  cycling  R T ( 2 0 C ) , dry  exposed and  0.8  dessicated cold, -54 C  exposed  0  0.6  O  v  »  fatigue threshold (AG =80 Jim ) 2  T  fatigue threshold (AG =100 T  0.0  0.2  0.4  Jim)  0.6  0.8  1.0  1.2  K, [MPaVm]  Figure 2-19: Comparison of adhesive fracture data and yield criteria For fatigue assessment, it has been proposed by many [e.g. 8, 30, 36] that the adhesive stresses be low enough that disbond growth will not occur. Hart-Smith [45] has also emphasized that well designed bonded joints should designed for an infinite life, and that the locus of failure or weak link in the structure should not be the bond. This proposition arises because adhesive joints exhibit high disbond growth rates that increase very quickly with an increase in the applied fatigue load, and because through good design it is usually possible to reduce the adhesive stresses to the point where fatigue cracking is extremely unlikely. It is suggested that the unmodified adhesive failure criteria be  51  employed in the damage-tolerant region about the repaired crack, and that the safe life criteria (to account for environmental effects) be used for the safe-life region about the edges of the repair. This is in accordance with the definition of the safe life and damage tolerant regions of a repair as defined by Baker [46], but to some degree this runs counter to the advice of Chalkley et al [8], who have suggested that a stress-based approach for fatigue assessment may not be appropriate because in their experiments, the specimens with the highest stresses were not always the ones that failed first. However, given the experimental threshold data shown in Figure 2-19, which by nature accounts for the effects  of local stress raisers (e.g. due to fibre geometry and lay-out, adhesive  imperfection, etc.) that are not included in most repair models, it should be adequate to simply show that limit loads do not result in adhesive stresses that exceed these allowables. This is particularly true i f one considers that the effective once-per-flight fatigue loads (the load that would be applied once per flight cycle to represent all the fatigue damage accrued by the actual fatigue spectrum) are typically well below limit load (usually below 60% of limit load) and that the fatigue threshold data applied is typically for room temperature testing while flight loads w i l l tend to occur at high altitude where the lower temperatures enhance fatigue strength. Accordingly, the authors suggest that a stress-based approach, while imperfect, would still be applicable by the principle of compounded conservatism [47, sec 9.5]. Compounded conservatism is the practice of applying many 'worst-case' conservative assumptions together to ensure safety, and whije it may lead to an inefficient design it does ensure safety in the face of a number of factors that are difficult to quantify (e.g. temperature and environmental effects, fatigue that may be dominated by very local structural details, or stochastic variations in properties or material processes).  2.5.3 Composite fracture and delamination The textbook approach to composite failure analysis is to use interactive failure criteria such as the Tsai-Wu theory [e.g. 48]. These theories extend the energy-based failure criteria that have been successful for homogeneous anisotropic ductile solids to the analysis of heterogeneous  fibre-polymer composites that exhibit fracture-dominated  failure modes. Pioneering data on the application of the Tsai-Wu failure theory is available for boron fibres with an older matrix system [49, 50]. The other general class of failure criteria arise from mechanistic modelling, in which individual failure modes are analysed to form a set of equations that form a failure envelope based upon stresses or strains. The W W F E (World W i d e Failure Exercise) [52] tested several composite failure criteria against a set of tests that are representative of design problems arising in different industries. Interactive theories were found to give good results in most cases, but caution was urged in their application to unidirectional lamina, where unconservative results could be obtained. Most criteria provided reasonable results for the types of problems for  52  which they were designed, including maximum stress or strain criteria that are often used for specific applications in codes and standards.  It was emphasised that careful  interpretation of the results is required, and that for large deformations or final failure strength, non-linearity and progressive damage are important considerations. Given that for a repair patch, weight optimization is not critical, simple criteria are preferable and adequate when supported by experimental data, and a mechanistic approach expressed in terms of simple stress or strain limits may be preferable. Hart-Smith [51] has made a convincing case that the various constituents and failure modes of the composite should be treated separately, and that interaction should be handled by superposition of loads, allowing the strength of the composite to be assessed through separate analysis of the constituents (fibres, matrix, and interface) under the combined loads (including cure-induced thermal residual stresses in the matrix), leading to a failure envelope comprised of several individual criteria for each constituent. M i c r o mechanical failure models are often only required for the development of engineering design allowables, and resulting individual failure criteria are often quite simple. HartSmith has catalogued the typical failure mechanisms, which after elimination of those that are not relevant to crack patching with a unidirectional patch, are summarized in Table 2-2. Fracture of fibres at flaws and defects, and fibre failure under tensile loads Fibre micro-instability and fibre shear failure under compressive loads Ductile matrix failure under in-plane loads Matrix cracking under transverse-tension (geometry dependent) Interfacial failure between fibre and matrix Interlaminar failure at edges and discontinuities Delamination under impact or transverse shear loads  Table 2-2: Possible composite failure modes for unidirectional patch Within F A A guidance documents and the crack-patching literature, it has been stressed that composite and adhesive failure modes must be addressed. A s the technology has been used primarily in structures with significant bending restraint and primarily tensile loads, composite failure mechanisms other than progressive adhesive disbonding or delamination have rarely been observed. The requirement for fatigue test validation has also provided a definitive control on composite failure modes. W i t h increasing interest in the use of single-sided repairs and wider application of generic repairs without specific tests, it w i l l be necessary to specifically address each composite failure mode, especially considering that single-sided repairs are much more susceptible to composite failures and instability under compressive loads, as demonstrated by the testing carried out by the authors as described above.  53  Given that: a) for most boron-epoxy repairs the primary load path is in the primary direction of the patch, b) the aluminum substrate w i l l support out-of-plane loads, c) crack closure w i l l shield the composite patch from the worst of the compressive stresses, and d) the tensile strength of the boron patch is high compared to the underlying aluminum plate, it should generally not be necessary to perform a sophisticated analysis of the many possible failure modes to assess the strength of the composite patch. The margins against both matrix and fibre failure should generally be high, except in the presence of significant bending and peel stresses which may promote matrix cracking (due to transverse stresses) and the failure of fibers on the inner or outer surface of the repair. Transverse loading can result in matrix or interfacial failure as stiff fibers lead to stress concentrations in the matrix, causing it to fail before the nominal transverse stress exceeds the strength of the matrix. From testing and fractography, it has been shown that the exact geometrical distribution of the fibres has a significant role, and a strength reduction factor must be used to develop a failure criterion for the composite [48]. It is suggested that the adhesive stress allowables already cover this assessment of the composite matrix, including the temperature and environmental effects and the stress concentrations in the matrix due to the constraint of the fibres. The adhesive stresses are representative of the critical matrix transverse stresses and the Butkus and Johnson data [39] provides assurance that they w i l l be held to an appropriate level. Under axial loads, for a unidirectional polymer-reinforced composite such as boronepoxy, failure occurs when the stress in the boron fibres reaches its maximum, corresponding to a nominal tensile stress in the orthotropic solid. This is the principal strength of the composite, and Textron product data [53] specifies a tensile strength of 1520 M P a at room temperature and 1450 M P a at elevated temperatures. Fatigue under such loading may be characterized by a stress-life (S-N) diagram, with small reductions in stiffness and strength occurring as the matrix cracks and delaminates near fibre breaks and stress concentrations. When compressive stresses occur (possibly due to bending), transverse stresses and interlaminar failure have a detrimental effect. Tensile interlaminar stresses develop due to the buckling of fibres, and transverse  stresses may cause  interlaminar failure. This form of fibre-buckling is controlled by the adhesive shear yield stress [48]. Textron product data provides a compressive strength of 2930 M P a at room temperature and a much lower 1250 M P a at elevated temperatures. A s the compressive strength is controlled by the matrix shear stress, it is suggested that the lower value be used as a basis for a conservative design allowable. In the experimental work described above, the composite repair was observed to have failed via partial delamination, and ultimately by fracture under combined bending and axial loads at the edge of the disbonded region. The composite stresses at point A of the finite element model are presented in Figure 2-20, where it can be seen that the sum of  54  the membrane and bending stresses is high but does not exceed the principal strength of the composite (as quoted in the paragraph above) for either restrained or unrestrained bending. Some fibre failure was observed in the experiments, and could be explained by either fatigue failure under these relatively high stresses, or due to the adhesive/matrix peel stresses. These adhesive/matrix peel stresses were shown by the authors to be very large after removal of the bending restraint and could act to break individual fibres that come free of the matrix due to delamination. -•{400  (0  Q. 5  10 (0 0)  k_ *-*  </> 0)  "55 o Q.  E o o  -200  Remote Applied Stress [MPa]  Figure 2-20: Composite stresses Although comparison to the results of this single test is not definitive, it is likely that the principal strength design criteria are adequate for the composite patch itself, so long as the adhesive stress design criteria are adequate to rule out early failure due to transverse stresses. This can be confidently stated because the strength of the boron/epoxy patch is very high relative to the substrate. It may prove to be necessary, however, to develop a knock-down factor to account for fatigue (i.e. to keep the stresses low enough to prevent composite fatigue failures during the life of a patch), but there should generally be a sufficient margin to cover such a strength reduction. Given that the adhesive design criteria adequately covers the possibility of early failure due to high transverse normal or shear stresses in the matrix, and the composite strength accounts for failure from tensile or compressive stresses, early patch failure should only be possible due to impact damage or manufacturing flaws. Currently, low-velocity impact tests are required to determine the susceptibility of a repair to such damage. A s "best practices" and the proposed adhesive and composite design criteria w i l l keep the stresses  55  low in the bulk of the repair through good design, only low rates of sub-critical damage growth should occur in the event of accidental flaws. One would not expect significant growth of inter-ply or impact damage unless it is accompanied by fiber damage or occurs in a region experiencing significant bending and through-thickness stresses, compressive loads, or very large transfers of load through shear (such as might exist near a repaired crack or about the edges of the patch). In practice, this has proven to be true, with experimental results  suggesting that delamination growth from impact damage or  manufacturing flaws away from load transfer regions is minimal [11]. Accordingly, damage growth should only occur in highly loaded regions such as about the crack and near the edges of the repair, where the stresses have already been controlled through the proposed design criteria.  2.5.4 Synergy of aluminum and composite/adhesive fracture criteria The final failure of the patch as observed in the experimental investigation was via simultaneous fracture of the repair and plate due to the overload of the composite by local bending and axial loads at the edge of the disbond. For symmetric repairs or repairs with significant bending restraint, failure has been observed to occur by fracture of the repaired plate and simultaneous growth of an adhesive disbond through to the edge of the repair, typically with the composite patch itself otherwise being undamaged. It is possible to model either scenario using a detailed finite element analysis, using nodal release and element removal to model the growing crack and disbond, or by generating a new model for every increment in crack growth or disbond growth. The actual residual strength can then be determined by whatever ratio of cracking and disbonding led to a natural balance between the failure of the patch and the repair/adhesive [e.g. 54]. Simplified  finite  element techniques [e.g. 12, 13], boundary element models [e.g. 10], and crack-bridging models [55] have all been used to evaluate the life of a repair with a growing disbond and crack. The authors have recently developed a new crack-bridging model for coupled bending and extension [56] and geometrically non-linear coupled bending and extension [57], which should allow for rapid assessment of unbalanced repairs. The concept of synergy between crack and disbond growth and a natural disbond size, when combined with the allowable stress criteria described above and effective models for combined crack and disbond growth, leads to a new opportunity to simplify the damage tolerance substantiation of a repair. Experience suggests that limited disbonding w i l l often occur near a repaired crack in a highly loaded structure, and that additional analysis w i l l be required to fulfill substantiation requirements and ensure that the patch is sufficiently  large to provide adequate damage tolerance  against  disbonding. It is  suggested that, in the absence of a fatigue test that demonstrates otherwise, one must assume that the adhesive has debonded about the repaired crack to the point where the  56  adhesive stresses drop to a threshold value (i.e. the damage tolerant allowable). This w i l l , in effect, result in disbond growth that is controlled by growth of the repaired crack under fatigue loading, where both the crack and disbond size are determined by applying standard L E F M techniques to the crack alone using the as-reinforced stress intensity (i.e. A"  mod  as demonstrated  above). A s crack growth in the metal substrate is well  characterized, the issue of fatigue crack growth rate prediction for the adhesive is avoided, allowing a rational determination of the inspection criteria.  2.6 Discussion The experiments and the validated finite element analysis demonstrate the potential for early failure of a single-sided bonded composite repair due to induced bending. The tested geometry was initially designed to include bending restraint and did not have a sufficient overlap length to minimize bending loads and adhesive peel stresses. Without bending restraint, failure occurred very rapidly, illustrating the need for adherence to bonded joint design guidelines, particularly the use of sufficient overlap lengths. The observed failure modes included rapid disbonding of the adhesive at the adhesivecomposite interface, fibre breaking, and ultimately simultaneous failure of both the repair and the underlying plate. It is shown that nonlinear analysis techniques are required for the accurate analysis of a repair, and that crack face contact must be modeled to evaluate a repair under compressive loads. These results show that composite and adhesive failure modes must be treated very seriously for single-sided repairs, and the specimen proves to be an extreme test for patch design methods. In the second half of the paper, finite element results for the stress intensity in the underlying plate, the stresses in the adhesive layer, and the composite patch stresses were compared to failure criteria for fatigue, yielding, and fracture. It was shown that the underlying plate can be treated using L E F M , and that adhesive and composite stresses can  be controlled to ensure integrity of the patch. Linear elastic analysis is used to  determine the adhesive stresses and a failure criterion and a method for fatigue assessment is proposed that accounts for adhesive pressurization and places a large penalty on peel stresses. Most of the complications of fatigue and fracture assessment of bonded joints are encapsulated within the proposed allowable stresses, allowing for a relatively simple assessment of the joint. W i t h more experimental work to justify the design allowables, and with appropriate closed-form tools for repair analysis, most repairs should be able to be certified using a simple analysis of this type. The principal strength of the composite was also shown to be an adequate criterion for longitudinal tensile loads in the unidirectional patch. A s the compressive strength of the composite is determined primarily by the shear strength of the matrix, the authors suggest that the high temperature compressive strength be used as the allowable stress in  57  compression. This corresponds to the approach developed by the authors for evaluation of the adhesive and of the composite matrix under transverse loads, whereby it was shown that a simple stress-based criterion is generally adequate for damage tolerance substantiation of a repair. The stress-based adhesive failure criteria were achieved by directly comparing the adhesive yield criteria to fracture criteria via Joseph and Erdogan's definition of the strain energy release rate for a bonded joint. The technique is promising but needs further investigation to validate. Specifically, the adhesive yield criteria under combined peel and shear stresses need to be established under hot and wet conditions, and more fracture strength  and  fatigue  threshold  data  needs  to  be  generated  for  the  aluminum/FM73M/boron-epoxy system considering the effects of adhesive thickness and environment. This would entail a detailed testing program, but would establish an authoritative stress-based design allowable for bonded repairs. A s the practical working range of an adhesive in this application is only 0.125 to 0.250 m m [58], the scope of this additional testing should not be overly expensive or onerous. A n additional problem that needs to be addressed is that, to ensure consistency in the comparison of the test results, Joseph and Erdogan's method should be used throughout. It is noted that the strain energy release rates reported by Butkus and Johnson were calculated using the virtual crack closure technique. Finally, the criteria should be tested against the working stresses in the many repairs that have been accepted for certification through a comprehensive damage tolerance substantiation process, to ensure that the proposed stress-based design criteria is sufficiently  conservative when compared to established practice. The  observation that the composite patch and the repaired plate fail in a synergistic manner, when combined with this simple stress-based design criteria that encapsulates the many complex damage mechanisms that can occur within a repair, leads to a new opportunity to evaluate the design life of a repair and the ability to assign an inspection interval without a detailed analysis of the fatigue response of the adhesive. In this proposed approach it should be assumed that an adhesive disbond w i l l exist that is large enough to reduce the adhesive stresses to the threshold level. The fatigue assessment and damage tolerance substantiation can then be can carried out based on the crack length alone, with the ultrasonic or thermographic inspection of the repair being carried out at the same time as the eddy current inspection of the underlying crack. Based upon the adhesive stress design criteria and the residual strength and fatigue life of the unrepaired crack, it is possible to divide repairs into four categories, as shown in Figure 2-21. B y applying the principle of compounded conservatism and by applying a simple risk assessment to ensure that the safety of the repaired structure is higher than that established during the design of the original damage tolerant structure, it is possible  58  to develop acceptance criteria for bonded repairs applied to weakened or damaged structures. Ouadrant B  Ouadrant A adhesive and composite qualifiable by compounded conservatism, using allowable stresses  adhesive and composite not qualifiable by compounded conservatism (eg allowable stresses exceeded near a crack)  and adequate residual strength in unrepaired original siruciuic  adequate residual sirenglh in unrepaired structure  Ouadrant C  Ouadrant I)  adhesive and composite qualifiable by compounded conservatism  adhesive and composite not qualifiable by compounded conservatism  inadequate residual strength in unrepaired original structure (e.g. there is a long crack or a significant corrosion grind-out)  inadequate residual strength in original structure  Figure 2-21: Proposed classification of repairs by DT substantiation requirements Repair designs in Quadrant A have very low working stresses in the composite/adhesive and the underlying structure has adequate residual strength even without the repair. In practice, the working stresses in many transport airframe components are l o w and the structure can tolerate very long (e.g. bay-to-bay) cracks and still survive limit load due to other reinforcing structures such as stringers and frames. Under this scenario, for the repair of a relatively short crack, the authors suggest that the repair may be substantiated without much further work with inspections based on the crack growth predicted in the equivalent unrepaired structure. The only additional work required would be to ensure that load attraction to the patch does not result in early failure in regions about the edge of the repair or in surrounding structures. In Quadrant B , the adhesive stresses are locally high, but the patch is not really required to establish inspection criteria for the underlying structure, as it has sufficient residual strength. This could occur in the event of a small flaw or corrosion grind-out in an area where due to geometric constraints it is not possible to reduce the stresses in the composite/adhesive sufficiently to meet the very stringent design criteria, or for a reinforcement used to strengthen an under-designed but undamaged structure with high working stresses. Here, the adhesive/composite interface is likely to fail in a progressive  59  manner, which could be addressed by re-examining the limit loads to see i f they are due to flight or landing loads (i.e. a fatigue case such as a vertical gust), or whether hot and wet conditions are likely to exist during the limit load maneuver. In the event that such a re-evaluation of the loads is insufficient, then a damage-mechanics analysis of the patch or a static test may be required to demonstrate that the patch itself w i l l have sufficient strength even in a highly damaged state. L i k e in Quadrant A , repairs in Quadrant C have low working stresses, but the underlying structure has significant damage and the reinforcing effect of the patch is required. A n example might be a large fatigue crack emanating from  a stress  concentration in an otherwise lightly stressed fuselage panel. Here, the primary concern is the possibility of an unpredictable early failure of the repair due to poor surface treatment of the aluminum surface for bonding. Quality assurance and bonding procedures minimize the probability of this type of failure, but cannot eliminate it, particularly for one-off repairs applied in the field. In this case, the authors propose the use of 'phenomenological risk assessment' [59] to assign a very conservative probability of failure to the patch and use this to redefine the limit load for the unrepaired structure. B y this method, the repaired structure with an intact patch must be capable of carrying the full limit load, but in the unlikely event of a poorly bonded repair, the underlying structure must be shown to be capable of withstanding the reduced limit load without a patch. The resulting level of safety would still be much higher than the original design because a very conservative probability was used to generate the reduced limit load. B y the tenets of phenomenological risk assessment, a conservative probability is assigned to the likelihood of failure based on expert opinion and regulatory acceptance. If it can be agreed upon that the likelihood of a bond lacking durability is much less than 1 in 100, then that is the likelihood that w i l l be used. Whatever probability is chosen, service experience should be used as a basis for verification where possible. This probability can then be used to redefine the limit load for the unrepaired structure to give credit to the patch. If a gust load case were critical, this would lead to the critical case being defined as the gust that w i l l occur 100 times during the life of the structure. This is consistent with providing a reasonable 'get home' ability to the unrepaired structure, and would work well with the requirement for once-per-flight visual check of the patch. This process of re-evaluation of the limit loads w i l l be called Probabilistic Load Assessment ( P L A ) in the remainder of the chapter. Quadrant D requires a more detailed assessment of a repair design according to the process shown in Figure 2-22.  60  Start:  Adhesive or matrix issues resolved using representative test data for the service environment or by reassessing the fatigue loads?  yes  Residual strength issues resolved by P L A and/or increased inspections?  Residual strength issues resolved if credit is taken for the repair?  no  Adhesive or matrix issues resolved using actual fatigue spectra and damage mechanics analysis?  yes  Residual strength issues resolved by P L A and/or increased inspections?  yes H  Acceptable, static strength testing and smart patch or initial once-per flight inspection required.  yes  yes  / Acceptable, no testing required.  / Acceptable, fatigue ~H testing required.  no Repair not Acceptable  1*"  Residual strength issues resolved if credit is taken for the repair?  yes  Acceptable, static strength and fatigue testing and smart patch or initial onceper-flight inspection required.  Figure 2-22: Process for 'Quadrant D' repairs In Quadrant D , the adhesive/composite stresses exceed the stress-based design criteria and the damage in the underlying structure is sufficient to require that the patch be credited in some manner to establish a useful life for the repair. Here, the methods used for damage tolerance substantiation in Quadrants A , B , and C are extended and we find imposed additional requirements for static or fatigue testing, comprehensive damage tolerance analysis, and increased frequency and rigor in inspections, each depending on the particulars of the repair scenario. The use of the methodology and its rationale w i l l be demonstrated by examining two scenarios. The  simplest scenario is one in which both the adhesive stresses and the residual  strength of the structure may be addressed by the methods described for Quadrants B and C . Here, we allow the removal of only one of the many factors leading to the certification of the adhesive and composite by compounded conservatism (e.g. re-assessment of the fatigue loads or the environmental knock-down factor) with the understanding that the likelihood of a patch lacking durability is also very conservative and accordingly the resulting structure w i l l still be much safer than the original design. If P L A is not sufficient and it is required to directly credit the patch to obtain an acceptable life, then additional testing and inspection requirements arise as shown.  61  A more complex scenario is one in which an accurate damage tolerance analysis of the repair is required to demonstrate adequate patch life. Here, we can no longer rely on simple stress-based criteria for the composite and adhesive, and a detailed fracture mechanics and fatigue analysis under the actual fatigue spectrum is required. Depending upon the residual strength of the structure, we find that a patch can be deemed to be outright unacceptable,  or acceptable  with possibly onerous  inspection and testing  requirements. In this case, we have essentially reverted to the full requirements for certification of composite structures, and detailed testing and analysis are required. Such a repair is likely to be uneconomical unless it is a fleet-wide repair, or is to be applied to an integral component of the aircraft that cannot be removed or replaced.  2.7 Concluding remarks The experimental and finite element results clearly demonstrate the importance of both bending and cracked plate geometry on the mechanics of a bonded composite repair. A s bending plays a critical role in the peel loading of the adhesive and failure of the fibres in the patch, it is critical to include bending effects in the damage tolerance and durability analysis of a repair. The authors have proposed an approach to damage  tolerance  assessment based on a conservative adhesive allowable stress that envelopes information regarding adhesive and composite matrix fatigue and fracture, including environmental effects. It is demonstrated that the hot and wet adhesive static strength represents the critical case for both the adhesive and the composite matrix, and accordingly provides adequate protection against fracture, fatigue, and other environmental effects. A s the method also places a large penalty upon peel stresses, the new criteria should be adequate to protect against poor designs. The authors have also proposed a quadrant classification scheme for repairs, based upon the residual strength of the unrepaired structure and the calculated adhesive and composite stresses. Different damage tolerance analysis methods and requirements for substantiation are outlined for each class of repair. W h i l e more adhesive test data is required (ideally using a consistent method to determine the strain energy release rates), and the effects of adhesive thickness on fracture and fatigue thresholds needs to be assessed with rigor, a process has been established that is applicable to most day-to-day repairs of scratches, dents, oversized holes, and minor corrosion grind-outs which should fall within quadrant A , allowing for rapid design and certification without much more engineering effort than is required for conventional riveted repairs. For more significant repairs, requirements for in-service inspection and testing of representative specimens are assigned by applying probabilistic risk assessment and the principle of compounded conservatism.  62  2.8 References [I] Baker, A . A . , "Repair efficiency in fatigue-cracked aluminum components reinforced with boron/epoxy patches", Fatigue and Fracture of Engineering Structures, V o l . 16, Num. 7, 1993, pp. 753-765 [2] Baker, A . A . , "Fiber composite repair of cracked metallic aircraft components practical and basic aspects", Composites V o l . 18, N u m . 4, 1987, pp. 293-308 [3] Baker, A . A . , "Bonded composite repair of metallic aircraft components - overview of Australian activities", A G A R D - 7 9 Structures and Materials Panel Meeting. Specialists Meeting: Composite Repairs of Military Aircraft Structures, 1994, pp. 1-14 th  [4] Baker, A . A . , Roberts, J.D., Rose, L . R . F . , "Experimental study of overlap joint parameters relevant to k reduction due to crack patching". 28th National S A M P E Symposium, 1983, pp. 627-639 [5] Raizenne, M . D . , Benak, T.J., Heath, J.B.R., Simpson, D . L . , Baker, A . A . "Bonded composite repair of thin metallic materials: variable load amplitude and temperature cycling effects", A G A R D - 79th Structures and Materials Panel meeting. Specialists Meeting: Composite Repairs of Military Aircraft Structures, 1994, pp. 5-1 [6] Hucalek, P., Raizenne, M . D . , Scott, R . F . , "Influences of environment and stress history on the composite patch repair of cracked metallic structures", Canadian Aeronautics and Space Institute Journal. V o l . 34, N u m . 2, 1988, pp. 85-91 [7] Albat A . M . , Romilly, D . P . , Raizenne, M . D . , "Thermal residual strain measurement in a composite repair on a cracked aluminum structure", I C C M - 1 1 , G o l d Coast, Australia, 14th to 18th July 1997, pp. VI-279-VI-288 [8] Chalkley,P.D., Wang, C . H . , Baker, A.A.,"Fatigue testing of generic bonded joints". In: "Advances in the Bonded Composite Repair of Metallic Aircraft Structure", Volume 1, Baker A A , Rose L R F , Jones R, Eds. Elesvier, 2002, pp. 103-126. [9] Sharp, P . K . , Clayton, J.Q., Clark, G . , "Retardation and repair of fatigue cracks by adhesive infiltration", Fatigue and Fracture of Engineering Materials and Structures, V o l . 20, N u m . 4, 1997, pp. 605-614. [10] Poole, P., Lock, D . , Young, A . , "Composite patch repair of thick aluminum sections", International Conference on Aircraft Damage Assessment and Repair, 1991, pp. 85-91. [II] Denney, J., M a l l , S., "Characterization of disbond effects on fatigue crack growth behavior in aluminum plate with bonded composite patch", Engineering Fracture Mechanics, V o l . 57, N u m . 5, 1997, pp. 507-525. [12] Denney, J., M a l l , S., "Fatigue behaviour in thick aluminum panels with a composite repair", Collection of Technical Papers, 39th Annual A I A A - A S M E - A S C E - A H S Structures, Structural Dynamics, and Materials Conference V o l . 3, 1998, pp. 2434-2443. [13] Denney, J., M a l l , S.,"Effect of disbond on fatigue behavior of cracked aluminum panel with bonded composite patch", Collection of Technical Papers AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conferences, V o l . 1, 1996, pp. 14-21. [14] K l u g , J., Malley, S., Sun. C.T.,"Characterization of fatigue behavior of bonded composite repairs", Journal of Aircraft V o l . 36, N u m . 6, 1999, pp. 1017-1022.  63  [15] Sabelkin, V . , M a l l , S., Avram, J.B., "fatigue crack growth analysis of stiffened cracked panel repaired with bonded composite patch", Engineerin Fracture Mechanics, 73, 2006, 1553-1567. [16] Jones, R., Chiu, W . K . , Smith, R.,"Airworthiness of composite repairs: failure mechanisms", Engineering Failure Analysis V o l . 2, 1995, pp. 117-128 [17] A N S Y S 9.0 University Intermediate, Release 9.0, A N S Y S , I n c , Canonsburg, P A , U S A , 2004 [18] Clark, R . J . and Romilly, D . P . , "Coupled bending and extension of bonded composite repairs", C A N C O M 2005, August 2005, Vancouver, British Columbia, Canada, Paper No. 41. [19] N A S G R O 4.0, Southwest Research Institute, San Antonio, Texas, U S A , 2002. [20] Anon, "Metallic Materials Properties Development and Standardization Handbook" (formerly M D L - H D B K - 5 ) , Federal Aviation Administration Technical Report. [21] Duong, C . N . , Wang, C . H . , " O n the characterization of fatigue crack growth in a plate, with a single-sided repair", Journal of Engineering Material and Technology, V o l . 126, A p r i l 2004, pp. 192-198. [22] Chalkley P., van den Berg, J., " O n obtaining design allowables for adhesives used in the bonded composite repair of aircraft", Australian Department of Defense, Defense Science and Technology Organization, D S T O - T R - 0 6 0 8 , 1998. [23] Ignjatovic, M . , Chalkley, P., Wang, C , "The yield behavior of a structural adhesive under complex loading", Australian Department of Defense, Defense Science and Technology Organization, D S T O - T R - 0 7 2 8 , 1998. [24] Wang, C . H . , and Chalkley, P., "Plastic yielding of a film adhesive under multiaxial stresses", International Journal of Adhesion and Adhesives", V o l . 20, 2000, pp. 155-164. [25] Chester, R.J., Walker, K . F., Chalkley, P . D . , "Adhesively bonded repairs to primary aircraft structure", International journal of Adhesion and Adhesives, V o l . 19, 1999, pp. 18. [26] Chalkley, P., and Baker A . A . , "Adhesives characterization and database". Chapter 4 in: Advances in the Bonded Composite Repair of Metallic Aircraft Structure", Volume 1, A . A . Baker, Rose, L . R . F , Jones, R., Eds. Elesvier, 2002, pp. 87-102. [27] W i l k i n s , D . J . , Eisenmann, J.R., Camin, R . A . , Margolis, W . S . , Benson, R . A . , "Characterizing delamination growth in graphite-epoxy", Damage in Composite Materials, A S T M S T P 775, K . L . Reifsnider, E d . , American Society for Testing and Materials, 1982, pp. 168-183 [28] L i n , C , Liechti, K . M . , "Similarity concepts in the fatigue fracture of adhesively bonded joints", Journal of adhesion, V o l . 21, N u m . 1, 1987, pp. 1-24. [29] Hart-Smith, L . J . , "Adhesively bonded double lap joints", N A S A Langley Research Report, N A S A CR-112235, January 1973. [30] M a l l , S., Johnson, W . S . , Everett, R . A . , " C y c l i c debonding of adhesively bonded composites". Adhesive joints: formation, characterization and testing: proceedings of an international symposium. Plenum press, New York, 1982, pp. 639-658. [31] Kinlock, A . J . , Osiyemi, S.O., "Predicting the fatigue life of adhesively bonded joints", Journal of adhesion, V o l . 43, N u m . 1, 1993, pp.79-90.  64  [32] Fernlund, G . , Papini, M . , M c C a m m o n d , D . , Spelt, J . K . , "Fracture load predictions for adhesive joints", Composites Science and Technology, V o l . 51, 1994, 587-600. [33] Williams, J . G . , " O n the calculation of energy release rates for cracked laminates", International Journal of Fracture, V o l . 36, 1988, pp. 101-119. [34] M a l l , S., Ramamurthy, G . , Rezaizdeh, M . A . , "Stress ratio effect on cyclic debonding in adhesively bonded composite joints", Composite Structures, V o l . 8, 1987, 31-45. [35] M a l l , S., Y u n , K . T . , "Effect of adhesive ductility on cyclic debond mechanism in composite-to-composite bonded joints", Journal of Adhesion, V o l . 23, 1987, pp. 215-231. [36] M a l l , S., Johnson, W . S . , "Characterization of mode 1 and mixed-mode failure of adhesive bonds between composite adherends". Composite Materials: Testing and Design (Seventh Conference), A S T M S T P 893, J . M . Whitney, E d . , American Society for Testing and Materials, Philadelpia, 1986, pp. 322-334. [37] H w u , C , K a o , C . J . , Chang, L . E . , "Delamination failure criteria for composite laminates", Journal of Composite Materials, V o l . 29, N o . 15, 1995, pp. 1962-1985 [38] Valentin, R . V . , Butkus, L . M . , Johnson, W . S . , " A finite element and experimental evaluation of boron-epoxy doublers bonded to an aluminum substrate", Journal of Composites Technology and Research, V o l . 20, N o . 2, 1998, pp. 108-119. [39] Butkus, L . M . , Johnson, W . S . , "Environmental effects on the opening mode fracture behavior of bonded boron-epoxy/aluminum joints", Proceedings of the American society for composites, 1 1 technical conference, October 7-9, 1996, Atlanta, Georgia, pp. 10391048. th  [40] Lubke, K . A . , Butkus, L . M . , Johnson, W . S . , "Effects of environment on fracture toughness and debond growth of aluminum/FM73/boron-epoxy adhesively bonded joints", Journal of Composites Technology and Research, V o l . 23, N o . 1, 200, pp. 42-49. [41] Joseph and Erdogan in: Johnson W . S . , "Stress Analysis of the cracked-lap shear specimen: an A S T M round-robin", Journal of Testing and Evaluation, V o l . 6., 1987, pp. 303-324. [42] Rice, J . C . "Stresses in an infinite strip containing a semi-infinite crack", Journal of Applied Mechanics - Transactions of the A S M E , 1967, pp. 248-249 [43] Krenk, S., "Energy release rate of symmetric adhesive joints", Engineering Fracture Mechanics, V o l . 43, N o . 4, 1992, pp. 549-559. [44] Cheulk, P.T., Tong, L . , Wang, C . H . , Baker, A . , Chalkley, P., "Fatigue crack growth in adhesively bonded composite-metal double-lap joints", Composite Structures, V o l . 57, 2002, pp. 109-115 [45] Hart-Smith, L J , "Design and analysis of bonded repairs for metal aircraft structures". In: "Bonded repair of aircraft structures", Baker A A and Jones R, eds. Martinus Nijhoff Publishers, 1988, pp.31-47. [46] Baker, A . A . " O n the certification of bonded composite repairs to primary aircraft structure". Proceedings of I C C M - 1 1 , G o l d Coast, Australia, 1997,1-1 to 1-23. [47] Anon., "Composite materials handbook volume 3. Polymer matrix composites materials usage design and analysis", M I L - H D B K - 1 7 - 3 F , U . S . A . Department of Defense, 2002.  65  [48] Hull, D., Cliyne, T.W., " A n Introduction to Composite Materials", 2 Cambridge University Press, 1996.  nd  Edition,  [49] Pipes, R.B.,"On the off-axis strength of anisotropic materials", Journal of Composite Materials, Vol. 7, 1973, pp. 248-256. [50] Hashin, Z., "Failure criteria for unidirectional fiber composites", Journal of Applied Mechanics, Vol. 47, 1980, pp. 329-334. [51] Hart-Smith, L.J., "What textbooks won't teach you about interactive composite failure criteria", Composite Structures: Theory and Practice, A S T M STP 1383. P. Grant and C.Q. Rousseau, Eds. American Society for Testing and Materials, West Conshohocken, P.A. 2000. pp. 413-436. [52] Sodena P.D., Kaddourb, A.S., Hintonc, M.J., "Recommendations for designers and researchers resulting from the world-wide failure exercise". Composites Science and Technology, Vol. 64, 2004, pp. 589-604. [53] Textron Specialty Materials, Boron Epoxy Prepreg Tape-5521 Product Data Sheet, 2 Industrial Avenue, Lowell, M A . [54] Papikanos, P., Tserpes, K.I., Labeas, G., Pantelakis, Sp., "Progressive damage modeling of bonded composite repairs", Theoretical and Applied Fracture Mechanics, Vol. 43, 2005, pp. 189-198. [55] Clark, R.J., Romilly, D.P., "Linear coupled bending and extension of an unbalanced bonded composite repair", International Journal of Solids and Structures. Vol. 44, 2007, pp. 3156-3176. [56] Clark, R.J., Romilly, D.P., "Nonlinear mechanics of hybrid bonded joints and bonded composite repairs", International Journal of Nonlinear Mechanics. Submitted November 2006. [57] Clark, R.J., Romilly, D.P., "Fatigue Damage Prediction for Bonded Composite Repairs Applied to Metallic Aircraft Structures", S A E Transactions - Journal of Aerospace, 2001. [58] Okafor, A . , Singh, N . , Enemuoh, U.E., Rao, S.V., "Design, analysis and performance of adhesively bonded composite patch repair of cracked aluminum aircraft panels", Composite Structures, Vol. 71, 2005, pp. 258-270 [59] Theofanous, T.G., "On the proper formulation of safety goals and assessment of safety margins for rare and high-consequence hazards", Reliability Engineering & System Safety, Vol. 54, 1996, pp. 243-257  66  Chapter 3: Extension of generalized plane strain plates with reinforced cracks* Randal J. Clark, Douglas P. Romilly University of British Columbia, Department of Mechanical Engineering  3.1 Introduction In this chapter, a line-spring model for a transversally isotropic generalized plane strain plate is developed and applied to the analysis of a reinforced crack under extensional loads. The generalized plane strain plate model of Mindlin and Kane, as refined by Kotousov and Wang, is applied to the analysis of a through crack in plates of arbitrary thickness and the governing equations are developed into a hyper-singular line-spring model by the application of Fourier transforms. This line-spring model is first used to develop standard linear-elastic fracture mechanics geometry correction factors to account for the effects of changes in the transverse modulus, Poisson's ratio, and plate thickness on the stress intensity and crack opening displacement. Next, the line-spring model is used to calculate the reinforcing effect of springs bridging the crack faces. In accordance with the Rose model for the analysis of reinforced cracks, the stress intensity and crack opening displacement are shown to become independent of crack length for long cracks, and an interpolation for the stress intensity and crack opening displacement is developed which employs the geometry correction factors developed in the first part of the chapter. This technique allows the inclusion of plate thickness effects in the fracture and fatigue analysis of reinforced cracks. Reinforced cracks occur in a wide range of engineered structures. Examples in the aerospace field include cracking of filamentary composite structures, repair of damaged aluminum panels by the application of adhesively bonded composite repairs, and cracking in hybrid materials such as A R A L L or G L A R E . Crack-bridging is also a common occurrence in other fields. Examples include failure of natural fibrous materials such as wood and engineered wood laminates or bonded composite repair of concrete structures. Crack bridging analysis and line-spring models are also used to evaluate the stress intensity in part-through cracks and in the assessment of crack-tip plasticity. Past efforts to develop line-spring models have employed plane stress or plane strain plate  * A version of this chapter has been submitted for publication. R.J. Clark, D.P. Romilly, "Extension of generalized plane strain plates with reinforced cracks", Engineering Fracture Mechanics.  67  models, in which either the transverse stress or the transverse strain are assumed to vanish. This requires the analyst to assume that the plate is in one of these two limiting states, a problem which is particularly significant during the analysis of a crack bridged by springs, as the portion of the load carried by the reinforcing springs is determined by the ratio of the stiffness of the plate compared to the stiffness of the springs. A plane stress plate has a larger displacement and hence the springs will carry a larger portion of the stresses than in a plane strain plate. In reality, the plate exists in a state between the limiting cases of plane stress and plane strain. T o obtain an accurate assessment of the fatigue and fracture resistance of a reinforced plate, this thickness effect must be included in the analysis. In this chapter, the governing equations for a transversally isotropic generalized plane strain plate are presented and are developed into a line-spring model. The line-spring model is then used to develop linear-elastic fracture mechanics geometry correction factors to account for plate thickness effects and to determine the stress intensity and crack opening displacement for a reinforced crack. These results are used to extend the very successful Rose model for the extension of reinforced cracks to include platethickness effects. The result is a relatively simple set of closed-form equations that can be used to assess the fracture and fatigue performance of reinforced structures and materials without requiring the analyst to make assumptions regarding the transverse stresses in the cracked plate.  3.2 The generalized plane strain plate Kane and Mindlin [1] introduced the generalized plane strain plate model for the analysis of extensional vibrations of plates, where plane stress and plane strain models respectively under and over predict the natural frequencies observed in experimental studies. More recently, Kotousov and Wang [2] have reformulated the model, decoupling the governing partial differential equations and developing fundamental solutions for inplane loading, greatly simplifying its use. They also developed closed-form solutions for circular holes, approximate solutions for notches with circular tips, and a dislocationbased solution for a through-crack [3]. Generalised plane strain is one example out of the many approximate plate theories that exist in the engineering literature. Plate models are usually based on the selection of functions to represent the through-thickness variation of either stresses or strains. If the chosen functions are sufficiently general, the exact solution may result, but this is only practical for the simplest problems. Accordingly, the selection of a plate theory entails the selection of an appropriate form for the transverse strains or stresses. A s an example, the plane stress model assumes vanishing throughthickness stresses and provides the exact result only for the simple case of a plate acting under a uniform or linearly varying stress field. Within the generalized plane strain plate  68  theory, it is assumed that the transverse strain is constant through the thickness o f the plate, and the effects of transverse stresses and strains may be explicitly included in the analysis. The development of the differential equation governing the displacement field for a generalized plane strain plate, as per Kotousov and W a n g [2], is brief and is provided here to introduce the symbols and concepts used in subsequent sections. For a plate of thickness 2h, the displacement field in the plate may be approximated as follows. u (x,y),  u  u =u (x,y)  x  r  y  u  y  =- (x,y) h  (1)  w  This displacement field is illustrated in Figure 3-1, along with a depiction o f typical stress distributions through the thickness of the plate.  40  1  u ( z )  +h  ( )  V A  -h  x,y  Figure 3-1: Distribution of plate displacements and stresses In displacement-based plate theories, the stresses are expressed i n terms of stress resultants or line-forces in the plate. These are obtained by integration through the thickness of the plate.  xx = \<r <k,  N  a  N  yy  =  \cr dz, -h yy  N  xy  (2) —h  The transverse normal and shear stresses may also be expressed as stress resultants. The transverse shear stress resultants are defined as the first moments o f the shear stresses taken about the plate centre line, and represent a 'pinching' force. Curvatures in the i n plane stress fields lead to transverse shear stresses and the accompanying normal stresses. u  *  N  =  n  fczz z, d  R =  n  fazdz,  x  R = y  (3)  \r zdz yz  -h Observing that the plate surfaces are stress free, the equilibrium equations may also be expressed in terms of stress resultants by integration through the thickness. N„,  y  +N  VtX  =0 ,  69  N,  (4)  The assumed displacement field satisfies all but one o f the compatibility equations. ~  xx,yy  +e  (5)  =y  yy,xx  I xy,xy  The constitutive equations may also be expressed in terms of the stress resultants. For a transversally isotropic material E = E = E and v = v = v, and the normal strains x  y  x  y  may be expressed as follows: £ = rr  /V„  .y—HL-y  (6a)  ^ a .  2h  'z  J  '/V.. yy  2h E  E  y  (6b)  E j z  1  (6c)  The shear constitutive equations may be expressed as follows. R  3 xy  x  w  2hG  •  (7)  x  T  2h G, 2  2h G, 2  The equilibrium equations i n the plane of the plate have the same form as for plane stress and plane strain, and one may use an A i r y stress function, <p, to satisfy equilibrium. 0  ',yy  Using  =N  <t> =N yy , 6 =-N >xy  ,  7  xx '  T ,xx  (8)  T  vv '  r ,xv  the average extensional  .  stress i n the plate,  2N = N +N XX  =V <p, the 2  yy  constitutive law for transverse strain (6c) yields an expression for pressurization o f the plate. N =2v N a  + 2E w  t  (9)  l  This result combined with equilibrium in the transverse direction (4) and the constitutive laws for transverse shear (7) lead to the following relationship between w andN. V w—^—w = ——N h G h G 2  z  Defining77 = v ^JE/E z  (10)  2  z  z  , a second condition for w and  arises from compatibility (5). (11)  70  The governing differential equation arises by de-coupling equations (10) and (11).  VV-fc VV = 0  (12)  2  Here, it is a parameter characterizing the thickness and properties of the plate. 3 E, k =-^^ — h G (l-/7 )  (13)  22  2  2  z  The generalized plane strain plate requires three boundary conditions. These are the inplane normal and shear stresses o f the classical plane stress and plane strain plate models, and the transverse shear 'pinching' stress resultant. In the next section this work w i l l be expanded and the problem of a crack in a plate w i l l be solved using a Fourier integral transform, thus allowing the development of a line-spring model for a cracked plate. 3.2.1 A cracked plate Given the governing differential equation, it is possible to develop a solution to the problem o f a cracked plate. T o achieve this, the authors use an integral transform technique in the manner o f Erdogan and Joseph [4,5] in the development of a line-spring model for a Reissner plate [6]. The Fourier transform o f a function and the inverse transform may be defined as follows. f = f(x,a)= \f(x,y)e dy, ,a>  f = f(x,y) = -±- \f{x^e^da  (14)  Applying this transform to the governing equation, it can be expressed as an ordinary differential equation. ( ^ - 3 « V , _  +3 a ^ - a V ) - ^ , _ - 2 a ^ - r - a V ) = 0  (15)  2  4  Keeping only roots that vanish for x —> ° o , the solution is ^ =^j(A (a)e" 1  To  W x  +A (a)xe 2  > k  +A («)e- )e-'^Jar, f o  3  R =a +k 2  2  2  (16)  simplify the notation a is assumed to be positive in the remainder o f the  development. T o satisfy symmetry about the crack plane, the shear and pinching stress resultants must vanish. A^(0,y) = -^,(0,y) = 0  (17)  #,(0,y) = 0  (18)  71  From (7), the pinching shear condition requires w (0,y)  = 0. Using equations (10) and  x  (11), w may be expressed in terms of the average in-plane stress. 1  w•  2  (19)  V JV-A^ 2  7] k  2  'J  This leads to an additional symmetry condition. (20)  V>, (0,y)-^ VV, (0,y) =0 2  ; c  ; c  Using the two symmetry conditions (17, 20), two constants may be eliminated from (16), and the stress function takes the following form.  1 _ ^2+X [ A-+—e-^A  0=  p j +  a  Rfi  +  26  l  J  (a) '  2 { a )  P  0 ~  2 k  - k rj  2 2  ^  (21)  To form the line-spring model, the remaining constant must be expressed in terms of the displacement of the crack faces. From compatibility (5) and the constitutive equations (6a,b,d). 1  '  N  N  N  .1  AT  r  2h  £xx yy  (22)  E 'J z  A n expression for  may be found from equations (9) and (19). This relationship  shows that the pressurization of the plate arises due to curvatures in the in-plane stress field. "«=2  VN 2,2 V  (23)  2  Equation (22) may now be expressed as follows. 2hEe^  yy  = 2(1 + v)N„' xy,xy - i*"yy,xx V _ +' vN "'xx.xx + ' ^ Jt  v  ajac  V  2  ^  (24)  Next we express the in-plane stresses in terms o f 0, and integrate to find the displacement. A s the displacement must vanish as [x,y]—>°°, the integration constant must also vanish.  2hEu <yy x  •  w  k'  (25)  Symmetry requires 0 (0, y) = 0 , and the first and third terms must vanish. Substituting xy  for 0, the crack face displacement may be expressed as follows.  72  hEu  (0, y) = — \a A 2K  (26)  (a)e' da  2  x  iay  2  J  Inverting the Fourier transform.  ~h~E  A (a) = \u „(t) 2  (27)  e da im  x  Integrating by parts twice, noting that again displacements must vanish as [x, y] —> °°, and that the crack opening displacement exists only over the interval of the crack, L. (28)  A (a) = hEJu (t)e dt ica  2  x  A n integral relating the crack opening displacement to  NjO,y)  =  l7t  J£\u (t)l \a\-2 x  may now be formed.  /3 R)  2  (29)  e '- dadt ai  /3  2  y)  Note the following integral identities, where K is the modified Bessel function o f the 2  second kind. [ae '- da ia(  y)  o  \a e - da 3  ia(,  y)  0  o  = -— (t-y)  (30a)  =(t-y)  (30b)  1  (30c)  R  Defining a stress boundary condition o~(y) = -N^ I2h, equation (29) may be written as follows. ~~I  \  E  r  J.  -1  \[(t-y)  2k IK  _ 2 1+k {t-yT\ P  12  A  3  ^M-yl] it (31)  2  fib-y)  Separating equation (31) into singular and regular parts in order to perform numerical integration, one finds the final form of the hyper-singular integral equation.  <r{y) =  k ( * ) \ r - ^ r + V k L ( k \ t - \)]dt {{t-y) 'J 2  2  y  27t\-ri [ 2  2  73  (32)  Here,  L(z)  includes the regular part of the integrand and a weak logarithmic  singularity. The logarithmic singularity is embedded in the Bessel function. L(z) = 2  i+^ikto- !-^1 z  03)  z  4  •u  2  Equation (32) may be compared to equations (34) and (35) for plane stress and plane strain respectively provided below.  lK  l(t-y)  (35) Note that the first term of the generalized plane strain solution given by equation (32) matches the plane strain solution given by equation (35). A s the contribution of  L(z)  vanishes (i.e. the crack interval reduces as in the case of a short crack), the plane strain solution is seen to dominate. The thickness effects causing deviations from the plane strain solution evolve from the contributions of L(z).  Failure to account for these effects  can lead to inaccuracies in the cracked plate analysis.  3.2.2 Numerical solution In this section, a collocation method and Chebyshev polynomial expansion are used to develop a line-spring model that will be used to generate geometry correction factors and a crack-bridging model for a reinforced cracked plate. W h i l e applied to a new application, this is similar to the technique used by Erdogan and Joseph to solve the Reissner plate bending problem [4,5], and proceeds as follows. B y expressing the crack opening displacement as a series of Chebyshev polynomials of the second kind, equation (32) may be decomposed into a linear system, solvable for the displacements, and hence the stresses in the plane of the crack. The first step is to normalise the bounds and parameters in the integral equation. Normalising all dimensions with respect to the halfcrack length, a , we define the following parameters. u(r) = u (t) I a, x  r = t/a,  s = yla,  K = ka  (36)  The hyper-singular integral equation then simplifies. — —1 (j{s) = —  E  1  [u(r)\  f  1  1 ^ + v, K L(rc\r - s\) \dr 2  2  74  (37)  The crack face deflection may be expanded using Chebyshev polynomials o f the second kind. u{r) =  W(r)Y f U (r) J  i  (38)  i  Here, W(r) = ^l-r  is the weighting function, U^r) = sin(0' + l)cos  2  (r))/'J\-r ~,  1  2  and the solution w i l l be expressed in terms of the displacement coefficients, f  r  The  hyper-singular part o f the integral may now be evaluated analytically. ,  [  ^ VTT7^ it *~  (s-r)  x  -(n + l)U (s)  \s\<l  n  ( =  \(n + l)  ( 3 g )  N>1  I  A  V* -i  {  2  The integral equation may now be written in terms of the Chebyshev polynomials. The logarithmic term in L(z) may be calculated analytically, but the regular part requires numerical integration. r 1  n  2,^2  N  U, (s) -  I  L  1  — \L(x\r - j|)W(r)l/,. (rydr  (40)  1=0  B y selecting a set o f collocation points, s , a linear system o f equations results, solvable t  for the coefficients, / . . The linear system may be expressed in the following form.  o- = Af,  Here, L  {j  A  :  \-lf  (41)  ••J  has the following definition, where V . includes the logarithmic singularity, i  and may be solved analytically, as demonstrated by Erdogan and Joseph [4,5] and by Wang and Rose [7]. L , . = J L(4;. - s | ) + |ln(|r,. - ^W(s)\J  (s)d --V .  s  -iL  4  1  J  S  (42)  i  4  K  ,i  2  ~2 - r , + - + ln2|j V. . = f i n k - s\W(s)U . (s)ds = \ ' T (r )-\ _* I 7t T (0 2  }  J+2  t  j + 2 'J  j =0 (43) j>0  The regular part requires numerical integration. A n y method is applicable, but for this problem, the authors chose to apply Gauss-Chebychev quadrature, as suggested by  75  Joseph and Erdogan. This method concentrates the integration points in the region of the crack tip, where the crack opening displacement has the greatest variability. Equation (37) is valid anywhere along the line x = 0, and hence may be used to examine the stresses near the crack tip and calculate the stress intensity. The only terms that contain a  singularity arise from the hyper-singular part of the equation. These  terms may be found by examining equation (39) for the region outside of the domain of the crack. The result follows from the conventional definition of the stress intensity. K = l i m J 2 a a ( r - 1 ) < 7 „ (0, r) =  „v  ; **v  l+  f™  E  Y f.U. (1) =  2(77 -l)to 2  '  f™  Y (i +1)/,  E  (44)  207 -l)to 2  It is now possible to calculate the crack face deflection and the stress intensity for a crack in a generalised plane strain plate using a displacement-based solution.  3.2.3 Geometry correction factors for a cracked plate The displacement-based solution of the cracked plate problem generated above is now available to calculate the stress intensity factor and crack face displacement for a transversally isotropic plate, such that the effects of plate thickness and the material properties through the thickness may be calculated. Table 3-1 shows the properties for some selected transversely isotropic materials that w i l l be used to test the model and develop a set of geometry correction factors. Material  E  (GPa)  V  E  (GPa)  z  (1) Isotropic  71.0  0.5  -  0.5  (2) Isotropic  71.0  0.33  -  0.33  (3) Isotropic  71.0  0.1  -  0.1  (4) Quasi-isotropic boron/epoxy  106  0.12  . 25  (5) Transversely loaded unidirectional boron/epoxy  25  0.019  0.028 0.21  210  Table 3-1: Material properties used to test the GPS model Figure 3-2 shows the generalized plane strain stress intensity results normalized to the classical result, K  = o^jm , and plotted against the crack length. The results match  0  those found by Kotousov and Wang [3], who used a dislocation density method. In the limit,  as  the  K/K =l/^j\-T]  2  0  crack  length  increases,  all  .  76  of  the  materials  obey  the  limit  1.2  -isotropic 0.1 - isotropic 0.3 - isotropic 0.5 1.15  - quasi-iso boron/epoxy - trans boron/epoxy  (A C  0> (0 (0 Q>  (A 1.05  -f  100  Crack Length (a/h) Figure 3-2: Stress intensity for a through crack The  stress  intensity  may also  be plotted  against  a  normalised  crack-length,  c = JC^I — TJ as shown in Figure 3 - 3 . 2  — o isotropic 0.1 A isotropic 0.3 o isotropic 0.5 0.75 -f  j5  • quasi-iso boron/epoxy  o u c  o  A trans boron/epoxy /  — model  '  I  i  1  '  1  t  '  '  i  I I  Q)  3=  T  Q  >.  0.5  V  '35  c a) c (0  (/> a> 5)  0.25  l_— 0.01  0.1  ,— 10  Normalized Crack Length (c)  Figure 3-3: Stress intensity difference plotted against normalized crack length 77  1 100  Here, c = 1 corresponds to a crack length of approximately one-third of the thickness of the plate. It may be observed that with this normalization, and by taking into account the asymptotic behaviour for long cracks, the difference in stress intensity between the classical solution, K ,  and generalized plane strain plate, K ,  0  d  collapses onto one curve  for all of the tested materials. K  d  {K/K )-l  (45)  0  =  This curve is well approximated by the simple interpolation of equation (46). A s shown in Figure 3-3, this approximate model fits the data very well. J 5  K  d  =  (46)  c +1.5 1 5  This relationship can be developed into an expression for a geometry correction factor. K  .1.5  = 1+  ,1.5  -1  + 1.5  (47)  It is also useful to examine the variation in strain energy release rates. Within the classical plate theories, the strain energy release rate may be expressed in terms of the stress intensity. EL E  plane stress (48) E  plane strain  Kotousov and W a n g [2] have made the observation that all cracks are in a state of plane strain near the crack tip. This occurs because the large stress gradients near the tip of the crack w i l l always pressurize the plate, regardless of the plate thickness. Thus, the plane stress assumption is not valid and only the plane strain relationship is required. A s the crack length becomes very large, Y — > \ l ^ \ - i j k  2  , and the strain energy release rate may  be expressed in terms of the nominal stress intensity as G-Kl  IE.  Similar results may  be developed for the crack face deflection. Figure 3-4 shows the deflection at the centre of the crack normalized to the plane strain result, u = 2a(l - TJ )cT / E . 2  0  78  0  o isotropic 0.1 A isotropic 0.3 O isotropic 0.5  • quasi-iso boron/epoxy 3  •  0)  o c  model  0)  I  trans boron/epoxy  0.5 4-  c 0)  E  0>  o ra a.  (0 0.25  0.01  10  0.1  100  Normalized Crack Length (c)  Figure 3-5: Difference in crack face displacement plotted against crack length. In the long-crack limit, u/u  = 1/(1 -TJ ) , and the following normalization collapses the 2  0  results for all of the materials onto one curve, as shown in Figure 3-5.  79  _ M r f =  (u I u ) - 1 0  (49)  (l/(l-^))-l  Once again, it is possible to develop a simple model that fits the data remarkably well. „1 5  U,  (50)  c +2.5 1 5  This relationship can be developed into a geometry correction factor, Y  d  ,15  Y =d  =l+  c  1 5  Y  +2.5 A X-rf  -1  (51)  The geometry correction factors derived in equations (47) and (51) provide a significant step forward towards the accurate closed-form engineering analysis of cracks in finite thickness plates.  3.3 Crack bridging: a line-spring model This section describes the development of a numerical line-spring model, which is used to test closed-form design equations that utilize the new correction factors for stress intensity, strain energy release rate, and crack face displacement. For a reinforced crack, the springs bridging the crack faces w i l l develop a force countering the applied stresses, acting to partially close the crack faces, as shown in Figure 3-6.  .ttttttt  1 I I 1 I I I I I I  r  G  Figure 3-6: A reinforced crack in a plate. Erdogan and Joseph [4,5] and Wang and Rose [7] have used the same technique to examine this phenomenon for the plane stress case. Including the effect of the springs,  80  equation (37) may be rewritten as follows, where k  s  is the line-spring stiffness, or the  stress generated by the springs per unit deflection of a crack face. —  - I E  cr(s)-ak u(s) s  1  = ——  r  Ji,  1  1  •+  \u{r)\-  Tj lc L(fc\r-s\)\dr 2  (52)  2  B y expanding w(r) and choosing a set of collocation points, one may develop a linear system of equations, solvable for the crack face displacement. <T =  Af,  A  :  -• I.J l-n  + ak W(r,)U (r )  :  s  j  (53)  i  2  The stress intensity may be calculated using equation (44). Figure 3-7 shows the results for an isotropic plate with <j = 1 0 M P a , v = 0.5 and £ = 71.0 GPa.  The results are  shown for a wide range of spring stiffnesses, which are reported in the legend.  0.01  0.1  1  10  100  Normalized Crack Length (a/h)  Figure 3-7: Stress intensity results from the line-spring model It is apparent that stiffer springs lead to lower stress intensities, reaching a limit for long cracks. Applying the method of Rose [8,9], one may analytically determine the limit strain energy release rate for a very long crack. Rose observed that stresses near the crack tip become independent of crack length. Hence, for linear problems, the strain energy released with an increment of crack growth can be found from the difference in stored elastic energy between two strips of material far removed from the crack tip. One strip is located well away from the crack, is undisturbed by the singular stress field near the  81  crack, and has no stored strain energy. The other is over the middle o f the long crack, has energy stored in the springs bridging the crack. The difference in stored energy is given solely by the energy stored in the springs bridging the crack, and the strain energy release rate may be expressed as follows: G  x  =  2JCT  dx-2^k x s  dx = k u  1  s  (54)  =—  A s the crack tip is in a plane strain state, the limit stress intensity ( K ) and an effective x  crack length a  (for a correction factor of Y ) for a long crack may now be defined.  eff  k  GE 1-/7  eff  U  ~  Yin  2  = < T  0  (55)  1  (56)  k (l-rj ) 2  s  Figure 3-8 shows the generalized plane strain results normalized with respect to this limit stress intensity and effective crack length. A s crack length increases, the theoretical limit is observed. Unfortunately, the polynomial expansion used in this study is limited in its ability to resolve this limit state in the case of a very long crack with very compliant springs. In this study, the authors used the first 128 even Chebyshev polynomials.  0.0001  0.001  0.01  0.1  1  10  1000  Normalized Crack Length (a/a ) et(  Figure 3-8: Normalized stress intensity plotted against crack length.  82  In  addition to an expression for the limit stress intensity, Rose developed an  interpolation for the stress intensity for an arbitrary length of crack. For the limiting state of a very short crack, the crack opening displacement is small, and the springs bridging the crack cannot develop a large stress to resist the applied load. The stress intensity w i l l be that calculated by considering an un-reinforced crack, given by equation (45). In the long crack limit, the stress intensity w i l l approach the limit given by equation (55). The stress intensity for an arbitrary crack length may then be approximated by a simple interpolation between these limiting states, expressed as follows. K aa eff a+a eff  K.  (57)  Figure 3-9 shows the stress intensity values predicted by the interpolation model normalized to values obtained by numerical analysis. Equation (57) provides results that are low by up to 4%, an error that has also been observed when it is applied to plane stress or plane strain plates. 1.02  0.0001  0.001  0.01  0.1  1  10  100  1000  Normalized Crack Length (a/a ) ef(  Figure 3-9: Comparison of stress intensity predictions. The concepts used to develop expressions for the stress intensity may also be used to estimate  the  crack face  displacements  [10]. A s the crack face  displacement  is  proportional to the stretch and hence the maximum stress in the springs, the maximum crack face displacement is a valuable parameter for estimating the strength of the structure and its resilience to fatigue loading, where performance of the springs may degrade over time. Figure 3-10 shows the crack-face displacement results from the line-  83  spring model. Once again, for very weak springs, the results approach the nominal solution for an un-reinforced crack. The limiting behaviour for long cracks is evident in the results for cracks bridged by stiff springs. 1.5 no springs -0.1 G Pa/mm -1.0 GPa/mm -10GPa/mm  E 3.  -100 GPa/mm -1000 GPa/mm  0) E <u o  ta O.  .2 0.5 Q  0.1  0.01  1  10  100  Normalized Crack Length (c)  Figure 3-10: Crack face displacement for various spring stiffnesses. The long-crack limit crack face displacement  is given by  —c / k , 0  s  and the  behaviour for a short crack may be determined from equation (51). One may then develop a simple interpolation between these two limit states: (2YAl-TJ )a/El\7k ) 2  u. =  s  "o+"~~  °(2Y (\-Tj )a/E)+{\/k )~  _  0  (  k + El(lY (1  2  d  _  s  s  5  g  )  -T] )a) 2  d  Figure 3-11 shows the ratio of the interpolation and line-spring model results for crack displacement. Once again, the interpolation model results in a reasonable error, in this case not exceeding 6%. The line-force in the springs is given by N  a  - 2hk u s  r  and may  be used to predict potential rupture or fatigue degradation o f the springs bridging the crack faces.  84  0.001  0.1  1  10  1000  Normalized Crack Length (a/a ) ef)  Figure 3-11: Comparison of crack face displacement predictions.  3.4 Discussion The generalized plane strain plate model allows one to include the effects of finite plate thickness and transverse stresses caused by curvatures in the stress field in the plate. This is particularly important in the analysis of cracks, which arguably contain the largest stress gradients possible in a real structure. Accordingly, the first important result of the generalized plane strain model is that, at least in terms of the stress resultants, the base of a crack in a plate can always be assumed to be in a state of plane strain. This result is of particularly great importance for the analysis of bridged cracks, where strain energy release rate arguments are used to determine the stress intensity at the reinforced crack tip. This result is also consistent with the definition of the opening mode strain energy release rate as it is used in the analysis of cracks in three-dimensional bodies. The asymptotic behaviour observed for un-reinforced long cracks can be explained as follows: away from the crack tip the stress gradients become small, and the plate is essentially in a state of plane stress. However, the crack tip is still in a state of plane strain, and the plate is effectively stiffened by the transverse pressures around the crack tip. For a long crack, subsequent increases in crack length lead to a dispersion of stresses that can be well-approximated by plane stress analysis, except in the region of the crack tip, where plane strain conditions exist and additional load transfer results due to the 'stiffened' region around the crack tip. The geometry correction factors developed in this paper are a simple extension of the geometry correction factors traditionally used in fracture mechanics, and are a very effective way to deal with this complication.  85  The long-crack limit stress intensity factor and the interpolation model developed by Rose offer a very effective method for the analysis of bridged cracks. The manner in which geometry correction factors are included in the model was first introduced by Rose [9]. This form of the interpolation model was shown to be accurate within 5%. This order of error is in line with what has been reported previously by the authors [10], and can be attributed to the approximate nature of the interpolation model, and not to the correction factors introduced in this paper. It is unfortunate that the Chebyshev expansion method is not more effective in dealing with weak springs or very long cracks. Most forms of expansion w i l l result in the same difficulties, as the crack face displacement gradients w i l l always be concentrated at the very tip of the crack under these conditions. This may be alleviated by considering the integral over an open domain (i.e. a semi-infinite crack) and using an appropriate orthogonal expansion, but as the existence of the longrcrack limit is well accepted, there is little practical reason to expend effort in this direction.  3.5 Concluding remarks The authors have developed the generalized plane strain plate model into a hypersingular integral equation relating crack-face deflections to stresses along the plane of the crack. This numerical model has been solved using a collocation method, and the results have been used to develop simple and accurate geometry correction factors to account for the effects of plate thickness and Poisson's ratio on the fracture of through-cracked plates. These geometry correction factors were then used to improve the classical models for  cracked plates reinforced by springs. The resulting closed-form equations  were  validated against the new line-spring model, and shown to be accurate to within 6%, an error which is mostly due to the approximate nature of the original interpolation model used for bridged cracks. The generalized plane strain line-spring model is an improvement upon existing linespring models, and may be used to refine the results for the calculation of stress intensity factors for surface cracks [e.g. 3,4], to include the effects of transverse stresses in Dugdale strip-yield models, and in the analysis of cracks partially reinforced by ligaments bridging the crack faces. The model is based on the assumption of a linearelastic material and is not applicable to problems involving gross plasticity near the crack tip unless the plastic zone is explicitly included as a set of plastic springs or a superposed crack-closing stress applied in the region of the crack tip. The improvements to the Rose model for fracture and fatigue of reinforced cracks are easy to apply, and are useful in many fields including fracture analysis of layered structures, composites, and bonded aircraft repairs.  86  3.6 References [1] Kane T.R., M i n d l i n R . D . High-frequency extensional vibrations of plates. Journal of Applied Mechanics, v. 55, p. 277-283, 1955. [2] Kotousov A . , Wang C . H . , A generalized plane strain theory for transversally isotropic plates. Acta Mechanica, v. 161, p. 53-64, 2003. [3] Kotousov A . , Wang C H . Fundamental solutions for the generalised plane strain theory. International Journal of Engineering Science, v. 40, p. 1775-1790, 2002. [4] Joseph P.F., Erdogan F. Surface crack problems in plates. International Journal of Fracture, v. 41, pp. 105-131, 1989. [5] Joseph, P.F., Erdogan, F . Plates and shells containing a surface crack under general loading conditions. N A S A C R 178232, N A S A Langley Research Centre, 1987. [6] Reissner E . The effect of transverse shear deformation on the bending of elastic plates. A S M E Journal of Applied Mechanics v. 12, pp. A69-77, 1945. [7] Wang, C . H . , Rose, L . R . F . A crack bridging model for bonded plates subjected to tension and bending. International Journal of Solids and Structures, v. 36, pp. 1985-2014, 1999. [8] Rose, L . R . F . A cracked plate repaired by bonded reinforcements. Journal of Fracture, v. 18, n.2, pp. 135-144, 1982.  International  [9] Rose, L . R . F . Theoretical Analysis of crack patching. In: Baker A . A . , Jones R. Bonded Repair of Aircraft Structures. Martinus Nijhoff Publishers, pp. 77-105, 1988. [10] Clark, R . J . , Romilly, D . P . Fatigue damage prediction for bonded composite repairs applied to metallic aircraft structures. S A E Trans. - Journal of Aerospace, 2001, 10 p.  87  Chapter 4: Transverse shear and pressurization in the bending of reinforced cracks Randal J. Clark and Douglas P. R o m i l l y University of British Columbia Department of Mechanical Engineering  4.1 Introduction The authors develop an 8 -order model for bending of transversally isotropic plates and th  use integral transforms and a collocation method to form a line-spring model for a cracked plate. The 8 -order model allows satisfaction of the three standard plate bending th  boundary conditions; the normal moment, twisting moment, and transverse shear force, and an additional shear stress resultant that allows analysis of transverse normal stresses near the crack tip. The line-spring model is used to develop geometry correction factors for bending of finite-thickness plates, accounting for transverse shear deformation and pressurization of the plate near the crack tip. The line-spring model is then applied to the problem of a plate with a reinforced crack, and the results are used to validate an interpolation solution based on an energy method. W h i l e not explicitly analysed, the models are applicable to many problems, including bending of bonded repairs, fracture and fatigue of composite and layered materials, surface cracks, crack tip plasticity and crack closure or crack face interaction. Plates are important structural elements in applications ranging from aircraft and naval structures to building construction and infrastructure. When compared to the analysis of in-plane loading, plate bending is complicated by transverse shear deformations and transverse loads, leading to higher-order models that are more difficult to solve. For this reason, the construction and analysis of plate bending models has been an ongoing topic, with researchers seeking models that are easy to apply and yet capture the essential characteristics of the three-dimensional behaviour of the plate. Accordingly, a range of models exists for engineering analysis of plates. Selection of the correct model for an application depends upon many factors including the geometry and composition of the plate, whether large stress gradients are present, and on the accuracy required of the analysis.  A version of this chapter has been accepted for publication. R.J. Clark, D.P. Romilly, "Transverse shear and pressurization effects in the bending of plates with reinforced cracks", International Journal of Fracture, v 142, 2006, p 81-102.  88  In the simplest models, transverse shear deformations are neglected and the Kirchoff boundary condition is used to combine the twisting moment and shear force boundary conditions. The result is a 4 -order governing equation expressed in terms of the th  transverse deflection of the plate. This biharmonic problem is relatively easy to solve, and exact solutions exist for many geometries. More sophisticated models exist that explicitly include all three standard plate bending boundary conditions and account for shear deflections, the most accepted being the 6th-order Reissner [1] and M i n d l i n [2] models. The Reissner model follows from the assumption that the transverse stresses in a plate w i l l assume the through-thickness distribution predicted by simple beam bending, and is an example of a stress-based solution, whereas the M i n d l i n model follows from the assumption of a plane strain form for the displacement field. More sophisticated models may be developed using higher-order functions to approximate the through-thickness variation of the stresses or strains, and require the adoption of extra boundary conditions to solve the arising differential equations. These models usually employ an infinite series expansion of the stresses or strains, and are of interest in the field of fracture mechanics as they allow an accurate assessment of the stresses near a crack tip [e.g. 3]. For most engineering problems, stress gradients are small, and the Reissner and M i n d l i n models are considered adequate. Other examples of applications requiring more advanced models are those involving large stress gradients (e.g. higher frequency vibrations), when failure is governed by through-thickness stresses, or i f there is otherwise the need for very accurate solutions. For plate extension, Kotousov and Wang [4,5] developed a dislocation-density solution for a crack in a generalized plane strain plate and found that, in terms of the stress resultant or average of the through-thickness stresses, the crack tip in plate extension is in a state of plane strain. This is contrary to the common belief that for very thin plates, the crack tip is effectively in a state of plane stress. This observation has consequences for the use of energy methods in fracture mechanics. The pertinent example is the analysis of reinforced cracks, where the strain energy release rate may be calculated analytically, but the conversion to stress intensity requires the assumption of either plane stress or plane strain conditions. Clark and Romilly [6] have used the generalized plane strain plate model to develop a line-spring model and geometry correction factors for extension of cracked plates. The line-spring model was then used to investigate the effects of reinforcing springs bridging the crack faces, and it was found that the  geometry  correction factors could be used to include the effects of plate thickness and transverse material properties on the stress intensity and crack-face deflection, leading to improved accuracy in the analysis of a reinforced crack. The problem of transverse stresses building up near the crack tip also exists for plate bending, but is not often recognized due to the already difficult problem of modelling the shear forces and shear deflections and  89  assessing the effects of crack face contact. Accordingly, the fracture mechanics of bending plates is less well understood than for extension. In this chapter, the authors develop an 8 -order plate bending model, which is th  effectively a generalized plane strain model for bending. This is the simplest model possible that includes the effects of both transverse normal stresses and shear deflections. The motivation for this work is to develop a practical line-spring model and corresponding engineering analysis methods that include these two plate thickness effects and allow the calculation of the thickness-averaged bending stress intensity factor. The goal is to expedite the design of bonded composite repairs where the repaired structure consists of plates containing reinforced cracks subjected to bending. The authors achieve this by using the integral transform method of Joseph and Erdorgan [7,8] whereby the new plate bending model is reduced to a hyper-singular integral expressed in terms of the bending stresses along the plane of the crack and the displacement of the crack faces. The line spring model is then used to develop equations suitable for the assessment of plates containing through-cracks and reinforced cracks subjected to bending. The resulting equations are accurate, easy to apply, and include both transverse shear deflection and transverse normal stresses effects.  4.2 Basic plate bending models and fracture In this section, the authors describe four basic engineering plate bending models. These models are similar to the classical plate bending models available in the literature, and form the basis for the 8 -order model developed in subsequent sections. The first two th  employ the Kirchoff or 'effective shear stress' boundary condition, and w i l l be referred to as the plane stress and plane strain plate bending models. In their development, it is assumed that either the transverse stresses or the transverse strains vanish. The Kirchoff boundary condition allows the twisting moment and shear force acting along a cut to be combined, i.e. an artificial condition that satisfies equilibrium but does not allow for the explicit inclusion of the transverse shear forces. For crack problems, these models cannot be trusted. Errors in the range of 60% to 70% may exist, depending on the crack length and the thickness and elastic properties of the plate. Another problem is that while the governing equations are equivalent to those arising in the plane stress and plane strain plate extension models, the Kirchoff boundary condition causes inconsistencies between the stress and displacement based definitions of the stress intensity factor, as observed by Joseph and Erdogan [8]. This occurs because the crack tip is a region o f large stress gradients, and hence it is important to properly account for the zero shear force boundary condition on the edge of the crack. For plane stress bending, the deflection of the crack face and the bending stress at the outer fibre are derived in Appendix D and may be expressed as follows.  90  o( 1 + v ^ u(y) = 2 \ja E 3+v  -y  2  ,  2  Jf~p,  o-(y) = CT  y  \y\<a  (1)  \y\>a  (2)  For plane strain plate bending of transversally isotropic plates, where J] = v ^JEIE z  z  is  an effective Poisson's ratio, the stresses are given by (2), and the crack face deflection is given by: 1  f  u(y) = 2o\  + 1/  Y l - ^  3+i / - 2 ^ [  2  E  2  ^ v V - y  2  |y|<a  ,  (3)  The two remaining basic models w i l l be referred to as the plane stress shear  deformable plate and the plane strain shear deformable plate. These are formed by including the transverse shear deflection in the analysis. These 6 order bending models th  satisfy all three engineering stress resultants; the normal moment, twisting moment, and transverse shear force. W h i l e the Reissner model assumes that the transverse stresses take the form given by simple beam bending, the plane stress shear deformable plate is somewhat simplified, as the transverse normal stresses are assumed to vanish. The plane strain shear deformable plate is identical to the M i n d l i n plate, where it is assumed that the transverse strains vanish. The Reissner plate is usually considered to be more accurate than the M i n d l i n plate, as the assumed transverse stresses more closely model those found in an actual plate. In the presence of large stress gradients, both the Reissner and M i n d l i n models are far more accurate than the Kirchoff model, but their use in fracture mechanics is limited by assumptions regarding plane stress or strain, similar to the problem that exists for extension of cracked plates. For the limiting case o f a very short crack in the plane stress shear deformable plate, the deflection of the crack face at the outer fibre has the same form as may be found for plane stress plate extension, as derived in Appendix D . 2<T I u(y)=—y]a-y , E  T  T~  2  I  I  \y\<a  (4)  Similarly, for a very short crack in the plane strain shear deformable plate, the crack face deflection takes the form for plane strain plate extension. «(y) = —  -  v  a  - y  »  \y\  ()  <a  5  E For both types of shear deformable plate, the stresses at the base of the crack assume the form given by equation (2). It w i l l be shown in subsequent sections that these basic  91  models describe limits on the behaviour o f the 8th-order, or generalized plane strain bending model.  4.3 An 8 -order plate bending model th  Here, we develop an 8 -order plate bending model for a transversally isotropic plate, th  allowing the evaluation of both the transverse shear deflections and transverse normal stresses. F o r a plate o f thickness 2h, the displacement field may be approximated by equations (6a,b,c) below. The series expansion of the transverse displacement is truncated to produce an engineering model that requires four boundary conditions and allows the evaluation of the effects of transverse pressurization at the crack tip. x =T *( 'y^>  u  u  u  x  h  =^u  (x,y),  h  u =w(x,y)  +  (6a,b,c)  Lr-u(x,y)  h  The displacement field and the stresses in the plate are illustrated in Figure 4-1. The depicted stresses are a schematic representation intended to illustrate odd or even behaviour and the zero-stress boundary conditions at the edges of the plate.  ° (Z) Z Z  v<) z  x,y  Figure 4-1: Displacements and stresses for the 8th-order plate bending model For displacement-based theories, stresses are expressed as stress resultants or lineforces. Integrating through the thickness, we find expressions for the moments and shear forces. n M  x x =  n  l^^dz ,  M = \a zdz yy  yy  -h  ,  M = \t zdz xy  -h  V = \r dz, x  n  xz  V = jr dz y  (7a,b,c)  xy  -h  (7d)  yz  -h  These are the loads evaluated in the basic plate models. Pressurization o f the plate may be modelled by integration of the transverse normal stress as shown below.  92  i  (7e) -h  A final set of 'pinching shear' stress resultants arise from the transverse shear stresses (resulting from curvatures in the in-plane stress fields) and act to pressurize the plate. K = \^^ dz , R = 2  y  (7f,g).  \r z dz 2  yz  —h  The equilibrium equations may similarly be formulated in terms of stress resultants. M,  +M^  a x  =V ,  y  M„, + M ,  X  y  xy x  =V ,  V , +V ,,=-q  y  x x  y  (8a,b,c)  A final equilibrium condition arises from the second moment o f the transverse equilibrium equations, relating the pressurization o f the plate to transverse loads and 'pinching shear stress resultants'. (8d) The constitutive laws for a transversally isotropic material may also be integrated over the thickness, and the stress resultants may be expressed in terms o f the plate displacements.  v (l + v)2  V + TJ  2  M =D  7  r  i-n  M =D\  .. +  yy  zz  =D  (9a)  v + rj —u +— l-fi'  — u, +—  1-TJ  (9b)  z  rr  2  1-v  2  P  1  h  2  \-TJ  2  r 2  (9c)  « „ +u V  *•*  *"|J  2h G 2  (9d)  V, = 2G,  (9e)  '  u  h  1  (9f)  The second moment of the transverse shear stress constitutive equations results in expressions for the pinching shears.  93  3h 1 u +hw +—u *  2Gh  2  x  3  x  (9g)  zx  3h 1 U +hW +— z,y'  2Gh  2  (9h)  U  y  y  The plate bending constant, D, takes the form for plane strain. \h  D -  ,2  2 (  "\  (10)  \-v-2if  Substitution o f the constitutive (9) into the equilibrium equations (8) w i l l yield the governing equations for the plate. It proves useful to define the functions 0 and y/. 0 = ,x+ y,y> U  ¥ = U ~U  U  X  xy  (11)  yx  Eliminating q from the transverse equilibrium equations (8c,d) yields the following. 0 + hV w + -V u 2  2  z  =0  (12) 2 \ 77  \-V-2T] 2 V <t>= — U +-. r h {l-v)v 2  <*vv  G  (13)  j KG 15  ?  z  z  Adding the derivatives of equations (8a) and (8b), equilibrium in the plane o f the plate, and using equation (8c) to cancel the shear stress resultants leads to the following: (l + v)v 2  ^-VV z  l-Tj  2  h  (14)  =0  Subtracting the derivatives of equations (8a) and (8b) leads to the following additional condition: V -k ys 2  = 0,  2  ¥  k  2  3G^  (15)  hG 2  Decoupling equations (12) through (14), we find a fourth order governing equation for 0 along with the second order equation for y/ given by equation (15).  vv-*, vV = o,  i+i/  2  l-77  30v G 2  z  Jh 7] G 2  (16)  2  The governing equation w i l l be expressed in terms o f the average deflection o f the plate. 1 w=— 2h  z  J  f w + —u  -hL  h  1  (17)  dz = w + —u  1  94  B y equation (12), we note that <p = -hV w, and we find the following expression for w. 2  V w-k V w 6  2  = 0, where k  4  2  =kl  (18)  Equations (18) and (15) form an 8 -order system, requiring four boundary conditions. th  The first three are the normal and twisting moments and the shear force. The fourth is the 'pinching shear'. For £ acting along a cut a n d f along the outward normal, these are: M =M , ??  Mf =Mz ,  g?  ?  V =V ,  ?  (  R =R  C  ?  (19)  ?  The functions 0 and ^describe the normal and shear deformations of the plate, and lead to a simple derivation o f the governing equations, but are not convenient for calculating the moments or shear forces. F o r this purpose, one may define a stress function. From equilibrium, we note that a function % must exist such that: V =Z. > x  V =- , .  y  y  (20)  Z x  From the definition of y/, and constitutive equations (9e,f), we find the following. V  2 Z  = 2G y  (21)  zV  A differential equation for % may now be found from equation (16). V Z-kp Z 4  =0  2  (22)  Equations (18), (15), and (22) allow the analysis o f transverse shear stresses and pressurization in plate bending.  4.4 Hyper-singular integral equations In this section, the governing equations are reduced to a hyper-singular integral relating the crack face deflection to the bending stresses along the crack plane. The authors apply the integral transform method used by Erdogan and Joseph [7,8] to analyse a Reissner plate. The Fourier transform of a function and its inverse are defined as follows. oo  -  oo  / = f(x,y) = — [f\x,a)e^da  / = f(x,a) = \f(x,y)e >dy , ia  —oo  (23)  —oo  Applying this transform, the governing equations may be written as ordinary differential equations. (w,xax«  - 3 « r w , « « +3a w, 2  4  -a w)-kl(w, -2a w, +a*w)=0 6  xx  2  xxxx  (^-«V)-^> = 0  xx  (24)  (25) 95  Keeping only the roots that vanish as x —> ° ° , one finds the following expressions for w and \j/.  w = — [(A, (a)e~  + A {d)xe~  Hx  IK  = ±-  ¥  w  2  + A (a)e~ ) e da, Kx  iay  3  R  2 w  = a +k 2  (26)  2 w  J  J(B, ( o r ) * "  v  e-"»da,  )  R  = a +k  2  2  ¥  (27)  2 y  Symmetry requires vanishing shear stresses along the line of the crack. Accordingly: M (0,y)  = 0,  xy  V (0,y) = 0,  R (0,y) = 0  x  (28a,b,c)  x  From the constitutive equations, the deflections along the line of the crack are limited as follows: u + hw.x = 0 ,  x,y + y,x = 0 ,  U  U  u  x  zx  =0  (29a,b,c)  One must express these in terms of w and y/. The following arise from (29a): 1  ]_  (30)  2  The first symmetry condition arises from (29a) by substitution from (29b) and (30b). y/+2hw,  xy  =0  (31)  T o find the second symmetry condition, we require an expression for M (0,y) xxx  under  the restrictions (29). The following may be obtained from constitutive equation (9a). 2  1-V-2TJ  r,x  \  (32)  2  The second symmetry condition w i l l then arise from equilibrium equation (8a). r  \-v-2r, ^ 2  W, =0  (33)  y  A general equation for the transverse displacement arises from equations 13 and 14. 2  z  /i  z  ,2 V * 1-v  1/ i  1  k  2  v  rs„2 A l - ^  2  J  After taking the derivative, the final symmetry condition may be found from (29c).  96  (34)  1+ v  f  l-v-2/7  ^  (35)  2  Substitution o f (24) and (25) into symmetry conditions (31), (33), and (35) leads to the following restrictions on the integration constants. fl, + 2h^A a  2  3  i  w  l  3  2 (  a  w  2 A  ia \-v-2r] — 2h  2A a -KR A + 2  (36a)  - A a +A R a\ = 0  2  x  V[ 1, +, v , ^  „2 r\  L  R..A, =  (36b)  IB, = 0  I-77  2  (36c)  kl-v  After some manipulation, we find that A, and A, may be expressed as follows. 1 + v^ A -R A„ l-v  oA, =-  2  - L 2h  W  B  = - ^ A l - v  i  (37a,b)  2  The symmetry conditions eliminate all but one integration constant, and the crack face rotation,  = —w, (0, y), may now be written in terms of A (a) . x  2n h - v  2  (38)  2  The last integration constant arises from the boundary condition M^(0,y) but one must first find an expression for either u (0,y) xx  or u (0,y). yy  =-M(y),  Transforming  equation (22) and applying the symmetry conditions, the stress function x is given by: 1  -r-iihG.  ~2x±kl(l-v)  Z  (39)  .(e^'-e-^e-^da  From constitutive equations (9e,f) we find expressions for u  xx  -  X xy  =  U x,x  2  *• —-flW,xx,  H, y,y  G  -=  ' -hw  Z  2  (  (0, y) and u  y  (0, y).  (40)  xy  yy  J  From (9a), we now find an expression for the boundary condition M ( y ) = -M(0, y) 4h G 2  M(y) =  u  x x  +—0  ^  V  2  J  97  (41)  A n integral equation arises by substituting for u  x x  and w.  1+v ^ - ^ \f(.a)A e- *da 7>7T 11 - V J  M(y) = -  (42)  ia  2  J  Here, f(a) 3  f(cc) =  is given as follows.  +  \  v  Aa {a-R )  \  2  v  +-  1+v  •+•  k a + v) 2  2n  ,  £  1-77  a 2  -  a4 A  (43)  Rw J  W e may now form a hyper-singular equation for the crack face rotation. Inverting equation (38) we find an expression for integration constant A . 2  (44)  \ m e da ,ca  -±= 1-v  2  3  B y substitution, an integral arises relating the moment to the crack face rotation. . h?E M(y)=—jP(t)  TT,  \f(a)e °- da ia  (45)  dt  y)  Note the following integral identities, where K  0  and K  2  are modified Bessel functions  of the second kind. -1  \ a e ^ d a =  (46a)  {t-y)  o  ^\Ra e' ^da 2  1  (46b)  (t-y)  o  rflr  a  = ±(K (k\t-  y\)-K {k\t-y\))+  2  *  0  K (k\t-  (46c)  y\)  2  4  JaO-y)  da  1+k (t-y) 2  (46d)  K (k\t-y\) 2  2  Applying these identities, equation (45) may be written as a hyper-singular integral. The regular and weakly singular parts have been separated into the functions L  M(y):  -tiE 3^(1  -rf)  [Pit)  1  (t-y)  2  ,*>l-7?  2  2 1+V  98  L (k^-y\) ¥  +  and  1  kl7i UkJ\t-y\i dt 2  .  (47)  The function L  describes the transverse shear deflections, and function  ¥  is the same  expression found by the authors [6] for generalized plane strain plate extension, and accounts for the effects of pressurization near the crack tip. L (z) = ~ ¥  L(z) = 2 11 + —  4  +  z 3  z  + 4 ( * (z) - K (z)) + ^ z 2  V /\  12  1  | | ^ 2 U ) - —  -  z 'J  (48)  K (z)  0  2  (49)  —  z  z  Equation (47) may be written in terms of the bending stress in the outer fibre as follows. -hE  <x(y) =  i7t(\-v )i  2  (t-yf  l  \+v  L (k \t-y\) ¥  kl7i L{k \t-y\) dt  +  ¥  2  w  (50)  The equivalent equations for the four basic bending models are derived in Appendix D .  4.5 Bending of a cracked plate Here, the shear deformable plate models are solved by expanding the crack face deflection into a series of Chebychev polynomials. W i t h this expansion, a set of collocation points may be chosen to form a linear system, solvable for the expansion coefficients. First one must normalize the bounds  and parameters in the integral  equations. W e define the normalized quantities: h  u(r) = —fi(t), n  r = tla,  a  s = yla,  fc = k a,  K =k a w  w  (51)  Equations (52) through (54) show the normalized form of the hyper-singular integrals for the plane stress shear deformable plate, the plane strain shear deformable plate, and the eighth order plate, respectively 1 2n  (r-sY  \u(r)  2n(\-J] ) 2  +  2(1 +  -fj ) 2  1/)  1  w  2  \  1-77 L (K \r-s\) l +v j w  (r-s)—  +  (r-s)  f  v  2 +  (52)  ir  w  r  2  2/r(l  —L {K r-s\)  Hr  1 + 1/ K Jj L(/c \r-s\) 2  2  w  w  99  ir  w  (53)  (54)  The crack face deflection may be expanded into Chebyshev polynomials o f the second  (r)) / VTT ", with the weight function W(r) = VTT ". 2  kind, £/,(/-) = sin((/ + l)cos u(r) =  2  (55)  W(rj£g U (r) l  i  (=0  The hyper-singular part may be evaluated analytically. (n + l)U (s)  \s\<\  n  -Vi+i  ~ L — i — \ 2 —  d  r  =  i ,  As-Js -i  (56)  2  s>\  s -\ 2  Equations (52), (53), and (54) may now be written in terms of the expansion. CT (s) = b  g. -±±U (s)^ 4x(l + v) to [ 2 4^(1+ i  X-lf  to  { 2  E  A  2  11  7 7  V  (57)-  [L (K \r-s\W(r)U (rydr  l  ¥  r  t  (58)  4x1 1 + v I_J  '  2  1=0  V  4/r 2 2 27t  I(^(^^-^(^..(ryr v  (59)  1  JL^r-slWirW^dr  Analytical solutions exist for the logarithmic parts o f Z y ( z ) and L ( z ) , but the regular w  parts require numerical integration. The term V , . includes the logarithmic singularity and l  may be solved analytically, as demonstrated by Joseph and Erdogan [7] and by Wang and Rose [9]. T o evaluate the regular part, the authors used Gauss-Chebychev quadrature, as suggested by Joseph and Erdogan. i  L'j = jlL^fC^-s^-M^-s^isWjisyJs  + V.j  (60)  -i i  (61)  100  - r , ' ± ln2|j 2 ' T (r l it TM) +  V,, = flnlr. - s\W(s)U . (s)ds = \ .  I  1  7=0  +  ]+2  j  (62)  t)  j  >  q  7 + 2 'j  Selecting collocation points s,, a linear system arises, solvable for the coefficients,  gj.  V =Ag  (63)  b  For the three bending models, the coefficient matrix may be expressed as follows. i +l 4tf(l + v)  (64)  i J  u. 11+" J  (65) J  2 \  E  1-/7 l+v  jiff  _  2 ^wH  2 jw  (66)  Equations (57), (58), and (59) are valid along the entire line x = 0, and allow the evaluation of the stresses near the crack tip. A s for plate extension, the only terms containing a V r singularity arise from the hyper-singular part of the equation. These terms may be found by examining integral identity equation (56) for the region outside of the crack domain. The results for plane stress and plane strain bending (equations (67) and (68) respectively) follow from the stress-based definition of the stress intensity. The result reported is for the stress intensity in the outer fibre of the plate. K = lim  fixaJr-Y)^  (0, r) =  2  r_>  U, (1) =  (i  8l  ,  Z  = 0  +  l)g,  (67)  i=0  , =j™V^^(o, )^I, /,a,=^Io- o, r  (  +  (68)  For the 8 -order model, in the evaluation of the part of the stress field containing the th  *Jr singularity, the stresses and crack face displacements are related by the equations that arise in the analysis of plane strain plate extension, meaning that the crack tip singular stress field exists in a state of plane strain. This is analogous to the result of Kotousov and Wang [5] for extension of cracked plates. Equations (67) and (68) allow the evaluation of the stress intensity for the various plate bending models. Table 4-1 shows properties for selected transversely isotropic materials that will be used to evaluate them.  101  E  Material  V  (GPa)  (GPa)  (1) Isotropic  71.0  0.5  -  0.5  (2) Isotropic  71.0  0.33  -  0.33  (3) Isotropic  71.0  0.1  -  0.1  (4) Quasi-isotropic boron/epoxy  106  0.12  25  0.028  (5) Transversely loaded unidirectional boron/epoxy  25  0.019  210  0.21  Table 4-1: Material properties used to test the GPS model Figure 4-2 shows stress intensity results for the plane stress shear deformable bending model, which are normalized to the nominal result K  0  = a *Jm h  and plotted against the  crack length.  0.9  $2  0.8  c  Q>  (0 (0 0)  0.7 + •isotropic 0.1  +->  CO  isotropic 0.3  0.6  isotropic 0.5 • quasi-iso boron/epoxy - transverse boron/epoxy  0.5 0.01  0.1  1  10  100  Crack Length (a/h)  Figure 4-2: Stress intensities for shear deformable plane stress bending For a very long crack, the stress intensity reaches a limit that can be calculated according to the following argument. Large stress gradients and hence large shear forces only exist near the crack tip. Thus, for a long crack, shear deflections have a vanishing contribution to the deflection well away from the crack tip, and the plane stress plate model gives the correct result. The strain energy release rate for an increment in crack growth is proportional to the work done by the applied stress on the crack face, which may be calculated considering the deflection given by equation (1). For the plane stress  102  shear-deformable plate, the crack tip is in plane stress, and the limit stress intensity follows. (69) The stress intensity result may be plotted against a normalized crack-length, defined as: 3+v  (70)  Applying the normalized crack length and the long crack limit stress intensity, the difference between the plane stress and the plane stress shear deformable plate solutions for stress intensity (i.e. denoted  K) d  collapses onto one curve for all of the tested  materials. K =  {K/K )-\ 0  (71)  d  \3 + v  g 0.75  -  U C  a> a> 3= Q  'iii c 0) c  -  0.5 •  -  (0 (0  a> 0.25 CO  0.01  0.1  1  10  Normalized Crack Length (c)  Figure 4-3: Stress intensity difference plotted against normalized crack length  103  100  Using a statistical analysis package, equation (72) was fit to K . Figure 4-3 shows that d  this approximate model fits the data reasonably well, with an error not exceeding 2.5% . J.2  K  d  =  (72)  c^+3.5  This relationship can be developed into an expression for a geometry correction factor. c  f  Y= —  = 1-  k  1 2  1+ v  v  (73)  3+ v  < +3.5 2  Similar results may be found for the crack face displacement at the outer fibre of the plate and the centre of the crack. Figure 4-4 shows this displacement normalized to the result for plane stress bending, u = 2a al 0  1  h  Eh.  T  0.9  3 0.8 4| 0.7 + JS Q.  5 0.6 + <D O fl  isotropic 0.1  3n 0.5  isotropic 0.3  +  isotropic 0.5 - • - quasi-iso boron/epoxy  0.4  - A - transverse boron/epoxy  0.3  I  0.01  I  1—l_L  0.1  10  100  Crack length (a/h)  Figure 4-4: Crack face displacements for shear-deformable plane stress bending The following normalization collapses these results onto one curve (Figure 4-5). _ u = d  (u/u )-\ 0  l+v 3+v  (74)  -1  104  1  — i  T-t  o isotropic 0.1 A isotropic 0.3 o isotropic 0.5  ^0.75  • quasi-isotropic boron/epoxy • transverse boron/epoxy  u c  V  — model  <D i—  0)  !t  2 0.5 c  -  Of  V  0)  E  -  o «  I  0.25  •  •'  •  M&W&M ytUt^Mv^^  0.01  i  1, . i . . i i i i  1  0.1  !  i  10  100  Normalized crack length (c)  Figure 4-5: Difference in crack face displacements plotted against crack length Once again, one may develop a simple model that fits the data remarkably well. ¥  (75)  This relationship can be developed into a geometry correction factor, Y  d  Y  d  =  —  1-  = \  1+ v  (76)  3+ v  Very similar results are found for the plane strain shear deformable plate. The limit stress intensity factor, crack length normalization,  and geometry correction factors  follow. cr a(\-ri ) 2  K  0  = c r  h  s [ n n  u =2  ,  h  1+v \-rf  (77a,b)  0  \3 + v-2r]  1+v -K >  3 + V-2/7  0  105  2  Ur.  (78a,b)  3 + V-27]  2  (79)  1+ v V  ,12  Y = — = lK  ,1.2  k  0  ,1.3  Y =-  = l-  d  ,1.3  + 3.5  1+ v 2  A  Y  + 5X  (80)  3 + v - 2rj  1-  1+ v  (81)  3 + V-2/7'  It was found above that the crack tip is effectively in a state of plane strain, and that the crack face displacement away from the crack tip w i l l approach the displacement given by the plane stress model as the crack length gets very large. The developed 8 -order th  bending formulation allows one to investigate this varying effect of shear and transverse deformations on bending of a cracked plate. Figure 4-6 shows the stress intensity results obtained for the five trial materials analysed. The solid lines are the values obtained from the approximate model that will be developed over the next few pages. The points are data generated by solution of the hyper-singular equation for generalized plane strain plate bending. The nominal solutions follow: <V*(l-?7 ) 2  K = o 4~na , 0  (82)  h  1.1  0.9 (0  c CO  (0  (0  0.8 o A o 0.7 • A — 0.6 + — — — — 0.5 0.01  isotropic 0.1 isotropic 0.3 isotropic 0.5 quasi-iso b/epoxy trans b/epoxy isotropic 0.1 model isotropic 0.3 model isotropic 0.5 model quasi-iso b/epoxy model trans b/epoxy model  A A  * o  0.1  10  C r a c k length, a/h  Figure 4-6: Stress intensity for 8th-order plate bending formulation  106  100  Transverse deformations (the difference between plane stress and plane strain) result in larger stress intensities, whereas shear deflections decrease the stress intensity and ultimately have a larger effect as crack length increases. The true stress intensity arises as a result of the combination of these effects. For an isotropic plate with a large Poisson's ratio, the increase in stress intensity is large enough and occurs at a short enough crack length that it results in a net increase in stress intensity for cracks somewhat smaller than the thickness of the plate. Figure 4-7 shows similar results for the crack face displacement. 1.2 oo o 3  "3 c 0)  E 0.8 a> o ro o. (A  "•5 o (B  o a  o A 0.6 4- o • A — — 0.4 — — —  isotropic 0.1 isotropic 0.3 isotropic 0.5 quasi-iso boron/epoxy transverse boron/epoxy) isotropic 0.1 model isotropic 0.3 model isotropic 0.5 model quasi-iso boron/epoxy transverse boron/epoxy  0.2 0.01  0.1  1 C r a c k length, a/h  10  100  Figure 4-7: Crack face displacement for 8th-order plate bending formulation N o convenient normalization exists to collapse these results, but they do approach a limit as crack length increases. The stress intensity and crack face displacement for a long crack follow.  K.  \\ + v K V3 + vJ —f  1+ v  n  \-rf  ,  A  3+ v  l-V  (83a,b)  Figures 4-8 and 4-9 show the generalized plane strain model results normalized to these limit values. The results are plotted against a normalized crack length. |3 + v r  5  (84)  ° ~^vT+7 v  107  o isotropic 0.1 A isotropic 0.3 0.75  o*  o isotropic 0.5  of  • quasi-iso boron/epoxy  CD  o*°*° ° °  °  A trans boron/epoxy  u c d)  N o r m a l i z e d c r a c k length, c  Figure 4-8: Stress intensity difference plotted against normalized crack length  o4 o ^ o i  +o* o  o isotropic 0.1 0.75  of o  oA o 1  A isotropic 0.3 o isotropic 0.5 • quasi-isotropic boron/epoxy  V  o c tt)  A transverse boron/epoxy  B  0.5 OA o  N o r m a l i z e d c r a c k length, c  Figure 4-9: Crack face displacement plotted against normalized crack length  108  Correction factors for the generalized plane strain plate arise by multiplication of the correction factors for shear and transverse deformation. This is indicated by the form of equations (83a,b). For stress intensity, the correction factor is given by the following: f  4  17=1V  Y  1.5  Y  w  k  (85)  V  ( Y =l +  2  u  5  +1.5 , A.  (86)  1-  (87)  K = Y 'Y "K k  k  0  A similar result may be found for the crack face displacement as shown below. 1.3  y;=i-  ,1.3  J 5  y ; = i+  v1-  1+ v  (88)  3+ v  Y (89)  V^ +2.5 A 5  (90)  u = Y*Y u w  d  0  Equations (85) and (88) only account for transverse shear deformations, and hence one should use the normalized crack length given by c  w v  = KA  \3 + v  W  . Note that this should  ^Vl+v  not be confused with the normalization given by equation (84) which includes a term to account for pressurization. Figures 4-8 and 4-9 show the results obtained by this method. Equations (87) and (90) allow for a simple engineering analysis for bending of cracked finite thickness plates. A n additional result is that the crack tip is observed to be in a state of plane strain, regardless of plate thickness. In the next section, we w i l l develop a model for a crack bridged by springs, and use these newly developed correction factors for stress intensity, strain energy release rate, and crack face displacement to develop design equations for bridged cracks in plates of arbitrary thickness.  4.6 Comparison to three dimensional finite element results In this section, the thickness-averaged stress intensity and the crack face displacement are compared to the results of a three-dimensional finite element model. The A N S Y S finite element software package was used to analyse bending of a 20 m m crack in an isotropic plate with a Poisson's ratio of 0.495. The plate thickness was varied from 1 m m  109  to 64 mm, and both the width and height of a quarter-section of the plate were set at 600 mm, sufficient to minimize the effects finite plate dimensions on the through-thickness averaged bending stress intensity. A coarse model of the plate employs 4 elements through the thickness of the plate, with a finer sub-model employing 10 elements being used in the region of the crack tip. Figure 4-10 shows the planar element distribution for both the coarse model and fine sub-model.  (a) sub model  (b) full model  Figure 4-10: Finite element map showing boundary conditions The crack tip elements are 20-noded solid brick elements collapsed into quarter-point singular wedges. These elements have an edge length of 1.0 mm, which by numerical experimentation was determined to be the smallest element size that could be reasonably used for this crack length. In the numerical experiments, the finite element stress intensity factor results with different sized crack tip elements were compared to the exact plane strain results. The 1 m m crack tip elements lead to an underestimate of the stress intensity of 2.8%. The results presented below were corrected to account for this error. During this verification process, the nodes of the three-dimensional finite element model were constrained in the thickness direction, and a membrane stress was applied to the edge of the model. The stress intensity factor was calculated by the displacement-interpolation method using the displacement of the crack face nodes nearest the crack tip [10, equation 2]. A stress-based  method was not used because of problems with oscillations occurring  through the thickness of the plate, a problem that has been observed by others. Note that in the displacement-interpolation method, as is common in the three-dimensional analysis of cracks, the crack tip is assumed to be in a state of plane strain and the stress intensity is calculated based upon the theoretical plane-strain solution of the displacement of the crack faces. The plate-averaged bending stress intensity factor was calculated by least-  110  squares regression to results for each plane of nodes through the thickness of the plate at the crack tip. In this calculation, the nodes at the surface of the plate were given one half the weighting of interior nodes. Figure 4-11 shows the through-thickness variation of the stress-intensity factor for a selected set of the finite element results. 1.0 j  y*>-+  0.9 -  / /  0.8 -  •y? /  /  1 /  /  i^y^x  o.6 %  0.5-j - » - 2 h = 2 mm ^>-2h = 8 mm ^ 2 h = 32 mm  - a - 2 h = 128 mm 0.0  0  .  1  1  1  1  20  40  60  80  100  k [MPaVmm]  Figure 4-11: Through-thickness variation of the stress intensity Note that the effect of the free edge is apparent at the edge of the plate, resulting in a somewhat reduced stress intensity factor at this location. This effect has been noted by many others [e.g. 3], and its characterization requires a higher order analysis than that presented in this chapter. It is also well known that at the intersection of the crack and the free edge of the plate, the singular stress field does not observe the classical l / V r singularity except for the special case of v = 0 . Figure 4-12 shows the bending stress intensity factor at the edge of the plate as determined by a least-squared fit to the slope of the through-thickness finite element results. The result is compared to the bending stress intensity factor determined by the geometry correction factors for the sixth- and eighthorder models described above. For comparison, additional displacement-interpolation method finite element results are shown in which the crack tip is assumed to exist in a state of plane stress.  Ill  85  50  -I  ,  ,  ,  0.01  0.1  1  10  1 100  a/h Figure 4-12: Theoretical and finite element bending stress intensities It is readily apparent that the  new eighth-order  model provides a significant  improvement for short cracks in thick plates. For long cracks in thin plates, the results are more difficult to explain and the accuracy of the proposed model is difficult to verify. When comparing the plane stress based displacement-interpolation finite element method to the 6 -order model results, the finite element model predicts a higher stress intensity. th  This demonstrates that some degree of local stiffening and load attraction is occurring that is not predicted by the 6-th order models. Comparing the plane strain based finite element results to the S^-order model, the finite element results are also seen to be higher. The reason for this is that the displacement field predicted using a plane strain model is not identical to that predicted using the 8 -order model except for very short cracks in th  thick plates, particularly as regards the regular and weakly-singular terms in the generalized plane strain formulation. Here, the finite element results must be seen to be in error, as the 8 -order results represent an upper-bound on the actual stress intensity. The th  true solution should lie between the plane-stress based finite element results and the results predicted by the 8 -order model. th  It is possible to show that, as regards the generalized plane strain plate bending singular stress field, the crack tip is in a state of plane strain regardless of the thickness of the plate. Beginning with either of equations (8d) or (9c), it is possible to show that:  112  B y the method used to determine M (y) in section 4.4, it can also be shown that: Ah Gv  i'  Pit)  • + weakly singular + reg. terms W  (92)  (t-y)  :  Comparing with (47) and noting that M^(y)  =-M(y),  the singular stresses at the  crack tip are found to obey the following relationship: = 2v,  (93)  A t the crack tip the singular parts of o'^  and a  are equal in magnitude, and  accordingly the generalized plane strain model predicts that, in terms of the plateaveraged quantities  and M^,  the singular stresses at the crack tip exist in a state of  plane strain. This is only true as regards the singular stresses in a small region close to the crack tip, and is in accord with the exact eigenfunction expansion results of Hartranft and Sin [11]. Significant differences between the finite element and the generalized plane strain plate bending model results were noted above which would not be expected i f the crack tip were in a state of pure plane strain. It is possible that the finite element method used is sensitive to the lower-order terms. It is also possible that for very thin plates, the boundary layer at the plate edges begins to become large when compared to the dimensions of the singular region at the crack tip and the region dominated by purely plane strain conditions decreases in size. For the purposes of this chapter, it is unfortunate that the standard finite element approaches to stress intensity factor determination are built upon the assumption of the validity of the plane stress or plane strain states (i.e. classical models) and that a significant amount of engineering judgement is required in order to interpret the results. Figure 4-13 shows a similar improvement in the prediction of the crack  face  displacement at the centre of the crack. Note that the results are very good for both short cracks in thick plates and long cracks in thin plates. Here, the results for the deflection of the crack face at the outer fibre of the plate are also fitted to the slope of the finite element results. Unlike the determination of the stress intensity factor, these results are not subject to interpretation.  113  6E-06  OE+00 H 0.01  .  •  1  1  0.1  1  10  100  a/h Figure 4-13: Theoretical andfiniteelement model crack face deflections  4.7 Bending of reinforced cracked plates Here we consider the action of springs bridging the crack, developing a line-spring model for a generalized plane strain plate under bending. Figure 4-14 shows a cracked plate in bending reinforced by springs. The goal here is to determine the bending stress intensity for the reinforced crack and to calculate the moment carried by the springs bridging the crack, such that the strength of the plate can be determined.  114  M or a  b  U U 0 <J (j 0 U U (j M or a  k  Figure 4-14: A bending crack reinforced by springs To account for crack bridging, equation (59) may be modified to include a stress due to the reinforcing springs, which acts to restrict the motion of the crack faces. Here, k  s  is  the line-spring stiffness, or stress generated by the springs on the outer fibre of the plate per unit deflection of the crack surfaces. This definition of the bending spring stiffness leads to line-spring equations with a similar form to that found by Clark and R o m i l l y for plate extension [6].  c (s)—ak u(s) b  =  s  Expanding u(r)  -E  \u(r)  2n(\-r] ): L  -,)  2  2  +  1+v  M  ' V  A  (94)  JC Tj L(/c \r-s\) 2  2  w  w  and choosing a set of collocation points, equation (94) w i l l reduce to  the linear system cr = Ag . Here, A is expressed as follows, and includes a term to h  account for the springs. r  1  i i +  2  ( 1-/7 2 N 1+ v  2 2 In  , J  +  ak W( )Uj( ) 5  Si  Si  (95)  The generalized plane strain plate bending problem may now be solved for the case of a cracked plate bridged by springs. The stress intensity may be calculated from the expansion coefficients using equation (68). Figure 4-15 shows results for a cracked isotropic plate with cr =10 MPa, h  v = 0.5 and £ = 72.4 GPa. The results are plotted  for a wide range of spring stiffnesses, as shown in the legend.  115  2 - - - no springs —•—0.1 GPa/mm 1 GPa/mm — A — 10 GPa/mm —A—100  GPa/mm  —x—1000 GPa/mm  P3P5—x—x-!-x—x—x—x—x—x—x—x^-x—x—x—x—x—x—x—»i—x—x—x—x—x 0 0.01  0.1  10  100  Normalized Crack Length (a/h)  Figure 4-15: Stress intensity from the line spring model A s was the case for plate extension, it is apparent that stiffer springs lead to lower stress intensities, reaching a limit for long cracks. Applying the method of Rose [12,13], one may analytically determine the limit strain energy release rate for a very long crack. In this case, we examine the strain energy release rate at the outer fibre of the plate. In the limit of a very long crack reinforced by springs, the energy available for crack growth is given solely by considering the energy balance in the springs bridging the crack under the applied (nominal) bending stresses. The strain energy release rate may be expressed as follows: (96) The equations governing bending are identical to those found for extension. A s the crack tip has been shown to exist in a state of plane strain, the limit stress intensity may now be found. (97) A n effective crack length may also be defined, including factors to correct for plate geometry.  116  (98)  eff  l  Figure 16 shows the generalized plane strain results normalised with respect to these two theoretical limits. Note that as the crack becomes longer, the stress intensity limit is obeyed. Based on equation 97, the use of the 6 -order Riessner formulation would lead to th  an error in the determination of the long-crack limit stress intensity analogous to the difference between plane stress and plane strain, i.e. on the order of a 5% to 10% increase for most metals.  0.0001  0.001  0.01  0.1  1  10  100  1000  Normalized Crack Length (a/a ) etf  Figure 4-16: Normalized stress intensity plotted against crack length It is also possible to define an interpolation for stress intensity between the limit states of a very short and a very long crack. For a short crack one may use equation (87), developed for an un-reinforced crack. In the long crack limit, the stress intensity w i l l approach that determined by equation (97). The stress intensity for an arbitrary crack length follows.  JC, = Y~Y 'a, k  K aaeff  (99)  1  a + aeff  From equations (98) and (99), the transverse shear correction factor (based on the 6 th  order models) w i l l have a large role for moderately short cracks reinforced by relatively weak springs. In this case, based upon the previous analysis of an unreinforced crack, the  117  inclusion of the correction factor may improve the accuracy of the interpolation model by as much as 60%. The effectiveness of the interpolation model for bending may be tested against the line spring model. Figure 4-17 compares the stress intensity predicted by equation (99) normalized with respect to values determined by numerical analysis. The interpolation model is observed to lead- to results that are low by up to 5%, a difference that has also been found for extension of reinforced plates. This over-riding trend in the error is principally due to the form of the interpolation model, but other trends may be noted that result from the interaction of the geometry correction factors and the numerical linespring model. For long cracks, the numerical model cannot accurately model the crack tip and tends to over-predict the stress intensity. A t the same time, shear effects (modelled by the shear correction factor) act to reduce the stress intensity at the crack tip. The observed oscillations are due to the interaction between these effects. 1.02  (0  oc >. "c5 Q)  C  10 CO  a>  0.0001  0.001  0.01  0.1  1  10  100  1000  Normalized Crack Length (a/a ) eff  Figure 4-17: Stress intensity predicted by line-spring and interpolation models One may apply the same methods to the crack face displacements. Figure 4-18 shows crack-face displacements from the line-spring model. For weak springs, they approach the nominal solution for an un-reinforced crack. The long crack limit is evident for cracks bridged by stiff springs.  118  0.1  1  10  Normalized crack length (a/h)  Figure 4-18: Crack face displacement for various spring stiffnesses The long-crack limit crack face deflection is u = cr lk . x  b  s  The displacement for short  cracks is given by equation (90). A simple interpolation between these limits is given by: uu  u +u„ 0  (2Y Yj(\-7 )alE\\lk ) w  0  _  2  d  1  s  (2Y;Yf(l-rj )a/E)+{\/k )  CJ  b  (  1  0  0  )  k +E/(2Y;Yf(\-rj )a)  2  2  s  s  Figure 4-19 shows the ratio of the interpolation model and line-spring model results for crack displacement. This simple interpolation results in an error not exceeding 7%.  119  The maximum moment carried by the springs is given by equation (101). (101) Knowledge of this moment can be used to predict rupture or fatigue degradation of the spring materials (e.g. patch repair materials) bridging the crack faces.  4.8 Discussion Similar to the case for extension of cracked plates, for plates in bending the crack tip is subject to very large stress gradients that induce pressurization of the plate near the crack. A s for extension, this pressurization stiffens the plate locally in the region of the crack tip, effectively increasing the local stresses and reducing the overall crack  face  deflections from what would be predicted by models that neglect the build-up of transverse stresses. Different from the case of extension of cracked plates, in the solution of plate bending problems, the transverse shear stresses enter into the equilibrium equations and are accompanied by shear deflections. This additional set of loads and deflections requires the adoption of a higher-order plate model to obtain an accurate solution, particularly for thick plates and plates experiencing large stress gradients. The 8 -order plate model developed in this chapter is the simplest possible model that can th  120  include both of these effects, and has been used to develop a line-spring model which may now be used anywhere that line-spring models based on classical plate models have been applied, for example in the analysis of bonded composite repairs, surface cracks, crack tip plasticity, and crack closure. The stresses at the base of a crack in bending, expressed in terms of the moment stress resultant, have been shown to exist in generalized plane strain conditions. This study has developed a set of standard engineering geometry correction factors to describe the effects of plate thickness and transverse material properties on the stress intensity and crack face deflection, allowing for a simple and accurate analysis of through-cracks in finite thickness plates. The correction factors account for transverse shear deflections and the effects of transverse stresses caused by curvatures in the stress field in the plane of the plate. This is an important phenomenon in the, analysis of cracks, which contain very large stress gradients. These results are of particular significance for the analysis of bridged cracks, where strain energy release rate arguments are used to determine the stress intensity at the reinforced crack tip, and an appropriate conversion factor is required to determine the stress intensity. These results are in accord with the exact eigenfunction expansion results of Hartranft and Sih [11] and Hartranft [14], who found that the through-thickness stresses are singular and that plane strain conditions should exist at the crack tip, a property which is not possessed by many other approximate theories. Lower-order plate bending theories are also not capable of modelling the local 'stiffening' effect caused by this region of generalized plane strain, and the authors see the 8 -order model as an excellent engineering approach due to its relative ease of th  application and the ability to capture the relevant phenomena in a conservative but reasonably accurate manner. A s for extension, the long-crack limit stress intensity factor and the interpolation model developed by Rose offer a very effective method for the analysis of bending of bridged cracks. The interpolation model for the stress intensity in a repaired crack has been shown to be accurate within 5%, a similar result to that found for extension of reinforced cracks [15,16]. This error is principally attributed to the approximate nature of the interpolation model, and not to the correction factors introduced in this chapter. A t its worst, the interpolation model result for the crack face deflection is low by approximately 7%, a difference which should be accounted for when using the model to predict rupture or fatigue degradation of the spring materials bridging the crack. It is important to note that this study has not addressed the subject of crack closure and crack face interaction, a limitation that is general to any basic cracked-plate plate bending theory. It is widely known that such an interaction will act to reduce the bending stress intensity factor at the tension side of the plate [e.g. 17]. The study of this effect requires the superposition of extensional and bending models for cracked plates, and nonlinear  121  analysis is required to determine the area of contact of the crack surfaces. Several approaches exist to address this problem, including the use of line-spring models'[8]. The authors have applied the 8 -order plate bending model to the analysis of linear coupled th  bending and extension of reinforced cracks in a recently accepted paper [15] that could easily be extended to such a nonlinear analysis.  4.9 Concluding remarks The authors have developed a high-order plate model and by the method of integral transforms have developed a hyper-singular integral equation useful for the bending analysis of plates containing cracks. The model allows for the inclusion of both shear i,  deformations of the plate and pressurization of the crack tip under the extreme stress gradients induced by the presence of a crack. The resulting line-spring model and engineering analysis equations for through-cracks and reinforced cracks are applicable to a wide range of plate fracture problems and provide a significant improvement in accuracy over models based on the classical plate models, and thus should be should be widely adopted.  4.10 References [1] Reissner E . The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 12, 1945, A69-77 [2] M i n d l i n R . D . Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 18, 1951, 31-38 [3] S i h G . C . A review of the three-dimensional stress problem for a cracked plate, International Journal of Fracture 7, 1971, 39-61 [4] Kotousov A . and Wang C H . Fundamental solutions for the generalised plane strain theory. International Journal of Engineering Science 40, 2002, 1775-1790 [5] Kotousov A . and Wang C H . A generalized plane strain theory for transversally isotropic plates. Acta Mechanica 161, 2003, 53-64 [6] Clark R . J . and R o m i l l y D . P . Reinforced cracks in transversally isotropic generalized pane strain plates. T o be submitted to: Engineering Fracture Mechanics. [7] Joseph P.F. and Erdogan F. Plates and shells containing a surface crack under general loading conditions. N A S A contractor report 178232, N A S A Langley Research Centre, 1987. [8] Joseph P.F. and Erdogan F. Surface crack problems in plates. International Journal of Fracture 41, 1989, 105-131 [9] Wang C H . and Rose L . R . F . A crack bridging model for bonded plates subjected to tension and bending. International Journal of Solids and Structures 36, 1999, 1985-201 [10] He W . J . , L i n Y . , and D i n g H . J . A three-dimensional formula for determining stress intensity factors in finite element analysis of cracked bodies. Engineering Fracture Mechanics 57, 1997, 409-415  122  [11] Hartranft R . J . and S i h G . C . The use of eigenfunction expansions in the general solution of three-dimensional crack problems. Journal of mathematics and mechanics. 19, 1969,123-138. [12] Rose L . R . F . A cracked plate repaired by bonded reinforcements. Journal of Fracture 18, 1982, 135-144  International  [13] Rose L . R . F . Theoretical Analysis of crack patching. In: A . A . Baker and R. Jones (eds.) Bonded Repair of Aircraft Structures. Martinus Nijhoff, 1988, 77-105 [14] Hartranft R . J . Improved approximate theories of the bending and extension of flat plates. In: G . C . S i h (ed.), Mechanics of Facture 3 - Plates and Shells with Cracks, Noordhoff, 1977, 45-83. [15] Clark R.J. and R o m i l l y D . P . Linear coupled bending and extension of an unbalanced bonded repair. Accepted by: International Journal of Solids and Structures, 2006. [16] Clark R . J . and R o m i l l y D . P . Fatigue damage prediction for bonded composite repairs applied to metallic aircraft structures. S A E Transactions - Journal of Aerospace, paper #2001-01-26, 2001 [17] L i u R., Zhang T., W u X . J . , Wang C H . Determination of stress intensity factors for a cracked shell under bending with improved shell theories. Journal of Aerospace Engineering, A S C E , 19(1), 2006, 21-28  123  Chapter 5: Linear coupled bending and extension of a bonded repair R.J. Clark and D . P . R o m i l l y University of British Columbia Department of Mechanical Engineering  5.1 Introduction A n unbalanced repair is a composite patch bonded to one side of a cracked structure for the purpose of preventing or reducing damage growth in the substrate. A single-sided repair offsets the load path within the structure, inducing out-of-plane bending. This bending increases the stress intensity of the underlying crack and causes adhesive peel stresses and bending of the repair which can, relative to a repair that is restrained against bending, lead to early failure. In this chapter the authors correct the analysis of Wang and Rose [1], developed by using an energy analysis of a single-sided or unbalanced repair applied to a very long crack, to comply with M a x w e l l ' s reciprocal theorem and to account for transverse normal and shear stresses at the crack tip and the accompanying shear deflections. The authors then develop closed-form equations useful for bonded composite repair design and damage tolerance assessment of cracks of arbitrary length by developing a new method for interpolation between this long-crack limit and a shortcrack limit based on the stress intensity and crack face displacements for an unreinforced crack. The interpolation method is then tested against an advanced line-spring model that has been created by using a 6 extension and a new 8  th  th  order generalized plane strain plate formulation in  order formulation in bending, thus allowing for the inclusion of  transverse shear and normal stresses. The closed-form equations are found to be accurate when compared to the line-spring model, and to provide reasonable results when compared to a three-dimensional finite element model of a bonded repair. Inaccuracies are shown to exist principally in the determination of the nominal stresses in the vicinity of the crack. A bonded composite repair consists of a filamentary composite reinforcement that is bonded to a damaged or weakened structure to reduce stresses and prevent or reduce continuing damage growth. Such repairs are an effective method by which to extend the  A version of this chapter has been published. R.J. Clark, D.P. Romilly, "Linear coupled bending and extension of an unbalanced bonded repair", International Journal of Solids and Structures, v 44, 2007, p 3156-3176.  124  life of aircraft structures, offering advantages in effectiveness, weight, profile, application time, cost, and inspectability when compared to mechanically fastened repairs. Bonded composite repairs have seen significant use in military applications but limited use in civil aviation, primarily due to certification issues. Repairs to primary or structurally significant aircraft structures are required to meet Damage Tolerance (DT) requirements, which  are outlined in F A A Circular 25.571-1C (1998), effectively requiring an  assessment of the residual life of the part and the development of an inspection schedule suitable for the detection of damage before the failure of the structure. Accordingly, models are required for the prediction of the fatigue life and residual strength of repaired structures. In this chapter, the authors develop closed-form equations for the analysis of an unbalanced repair, where the patch is bonded to only one side of a cracked structure, and induces out-of-plane bending by offsetting the load path. Bending increases the stress intensity in the underlying crack and causes adhesive peel stresses and bending of the repair that, i f not properly accounted for in the analysis, could lead to premature failure.  Figure 5-1: Repair configurations: balanced (top) and unbalanced (bottom) A significant body of work exists for the analysis of bonded composite repairs, which can be separated into two broad classes, balanced or double-sided repairs and unbalanced or single-sided repairs (Figure 5-1). Closed-form equations useful for composite repair design and damage tolerance analysis have existed for some time for balanced repairs, in which patches are applied to both sides of a cracked plate or in which the repaired plate exhibits a large degree of bending restraint due to underlying structure. The most significant contributions have been from Rose [2-4], who established an upper limit on  125  the stress intensity for a very long reinforced crack by means of an energy analysis. B y using this long-crack solution as an upper limit and the nominal, or unreinforced solution as a short-crack limit, Rose developed a very effective interpolation for cracks of arbitrary length. Wang and Rose [5] have demonstrated that for extension of reinforced plates the long-crack limit needs to be modified to account for the plane strain conditions present at the crack tip. Clark and R o m i l l y [6] have developed a generalized plane strain line-spring model for plate extension and developed a set of geometry correction factors for finite thickness plates which allows for accurate determination of both the stress intensity and crack opening displacement in balanced repairs. The closed-form equations of Rose have been adapted to the analysis of repairs with finite planar dimensions v i a the inclusion analogy [2], with significant shear lag in the composite patch [7,8], for partially disbonded repairs [9,10], and for tapered repairs [11]. These closed-form methods allow for simplified damage tolerance analysis, reliability assessment, and design optimization of repairs. There is much interest in the application of unbalanced repairs, as often only one face of a structure is exposed. Often the interior structure of the aircraft does not provide restraint against bending, and application of the repair w i l l result in significant out-of plane loading due to a shift in the neutral axis. For balanced repairs, failure typically occurs by progressive cracking and adhesive disbonding. Composite fibre and matrix failures are rare except in the presence of compressive loading or bending induced by geometric details or lack of restraint. For unbalanced repairs, composite failure mechanisms such as interply delamination and fracture are more likely to occur in conjunction with cracking and disbonding. Jones, Chiu, and Smith [12] have reviewed the failure modes and damage locations observed in both laboratory and in-service military use of bonded repairs. They conclude that, besides substrate cracking and cohesive disbonding, damage tolerance assessment should include fibre failure, adhesive failure, cohesive failure at the patch-adhesive interface, adhesive failure at the adhesive-substrate interface, plus interlaminar failure and delamination. These aspects are reflected in the F A A circular, which requires the assessment of the repaired structure, adhesive, and composite repair, including impact damage, inter-ply delamination, and adhesive disbonding. T o streamline the design and certification of repairs, particularly single-sided repairs, closed-form methods are needed to assess the effects of bending on the rate of failure of the repaired structure, but perhaps more importantly, methods are required to assess the life and strength of the repair itself under bending and transverse normal (peel) stresses. Many studies have been carried out in an effort to characterize the effect of bending. The first result is due to Ratwani [13] who developed a bending correction factor by adding the bending stress expected in the outer fibre of the plate to the nominal stresses in the crack plane, allowing the results for a symmetric repair to be applied to a one-sided  126  repair. Early finite element studies by Jones [e.g. 7,8] set out the framework by which two-dimensional finite element models of plates can be extended to the analysis of adhesively bonded layered plates. Several studies have focussed on producing numerical methods such as three-layer finite element analyses [14,15], boundary element methods [16], or automated damage growth in fully three-dimensional models [e.g. 17] in order to predict the life of a repaired structure. The focus of this chapter is the development of closed-form methods that are easily adaptable to damage tolerance or reliability analysis and design optimization. A s for the analysis of balanced repairs, Australian researchers have led the field in the development of closed-form methods. Rose [4] developed a bending correction factor for the long-crack case that includes geometrically nonlinear (stress-stiffening) effects by considering the adhesive as providing a rigid bond and applying an energy analysis to determine the strain energy release rate for a very long crack. More recently, Wang and Rose [18] have developed an expression for the total strain energy release rate for a long crack bridged by coupled bending and extensional springs, and a method for separation of this strain energy release rate into the membrane and bending stress intensity components. Unfortunately, by comparison to a threedimensional  finite  element  analysis,  they  concluded  that  their  energy  method  significantly over-predicts the bending stress intensity and thus was not valid or useful for the design of a repair. Wang, Rose, and Calinan [19] went on to develop an analytical approach for separation of the strain energy release rate (expressed as the root-meansquare stress intensity) by assuming that the bending and membrane components act in the same ratio regardless of the length of the crack. B y considering the short-crack limit, where the springs bridging the crack faces do not provide any constraint to the motion of the crack faces, the membrane and bending stress intensity components are linearly related to the membrane and bending stresses in the substrate in the plane of the crack, and this ratio could easily be determined. They continued by developing an interpolation between the long-and short-crack limit solutions for the root-mean-square stress intensity, allowing the membrane and bending stress intensity to be calculated for a crack of arbitrary length. Unfortunately, as this approach assumes that the membrane and bending stress intensities act in the same proportion for any crack length, the validity of the approach must be assessed on a case-by-case basis, particularly i f it is to be extended to non-linear analysis of a repair or to new repair geometries. This approach also provides no information regarding the stresses in the adhesive or the composite patch near the crack. Through the work of Australian researchers (primarily Rose and Wang), closed-form equations exist that establish a conservative upper limit on the root-mean-square stress intensity at the crack tip. Unfortunately, the energy methods that have been very successful for the analysis of tensile loading of double-sided repairs, have otherwise been  127  abandoned in favour of line-spring models for analyzing the case of coupled bending and extension [20]. In a recent conference paper, Romilly and Clark [21] stated that a corrected form of the energy method of Wang and Rose [18], which establishes longcrack limits for the membrane and bending stress intensity in a repair, is in fact very accurate when compared to the predictions of generalized plane strain line-spring models which capture the three-dimensional behaviour of the plate near the crack tip. In this chapter, the authors continue by fully describing the line-spring model and using it to validate an interpolation approach that is valid for cracks of arbitrary length. This methodology and the resulting analysis are then tested against a three-dimensional finite element model, demonstrating very accurate results for both the membrane and bending stress intensity and the crack face displacements, which may be related to the stresses in the adhesive and in the composite patch.  5.2 A methodology for damage tolerance assessment In keeping with the two-step approach introduced by Rose [2,3], several elements are required to complete the analysis of a bonded composite repair. The first step is to determine the stresses in the plane of the crack for the prospective repair. Due to load transfer from the plate to the patch, this stress will normally be reduced from the stress in the plate before application of the repair. The second step is to perform an analysis to determine the stress intensity and the stresses in the repair in the region of the crack. To achieve this, it is first necessary to characterize the stiffness of the repair as it acts to restrain the opening of the crack. Once these two steps are completed, a crack-bridging model is used to predict the stress intensity in the plate and the stresses in the adhesive and the composite repair. This is usually achieved by interpolation between limit states developed from an energy analysis of a very long crack bridged by springs and a very short crack without reinforcement. The resulting interpolation solution is applicable to a crack of arbitrary length.  5.2.1 Stress in the crack plane The work of Wang and Rose [1] provides a set of useful expressions for the plate stresses in the vicinity of the crack. By their analysis one can consider that the shear and peel stresses in the adhesive bond joining the repair patch to the plate will vanish in areas remote from the edges of the repair, the plate and repair may be considered to be rigidly bonded, and that the membrane and bending stresses in the plate may be calculated by considering the structure to be a simple composite beam. The neutral axis of the combined structure is offset from the centreline of the repaired plate by a distance z, calculated as follows.  128  -_S(t +t ) p  r  (1)  2(1 + S)  Z  The moment of inertia of the reinforced plate is then calculated from:  t*  I, =-^ + t z '  12  B  "  V.l  f  + — ^- + t £ „ 12 \2  2  (2)  With these two quantities defined, the membrane and bending stress under a remote applied extensional stress, a", are given as follows: 1  2~2  "\  -+  (3)  l+S  (4)  " 21,  For nonlinear bending of a repair of large planar extent with low bending restraint, the neutral axis will move to the load line upon the first instance of loading and the moment carried by the patched structure at the crack plane will vanish. In this case, the stresses on the crack plane simplify to CT° = a" /(l + S) and al = 0.  5.2.2 Characterization of crack-bridging springs T o perform the crack-bridging analysis of a repair, it is necessary to model the patch as a set of springs bridging the crack faces. The desired result is a set of spring constants relating the crack-opening stresses to the displacement of the crack face:  (5)  B y inversion of equation (5), the springs may also be characterized using a compliance matrix, as shown below. ' mb  \ b u  j  "bin  'bb  (6) K^b  J  The spring compliance coefficients are determined by the analysis of a bonded lap joint that is equivalent in geometry and composition to the cross-section of the repair taken in the region of the crack. The governing equations for a bonded joint under tension and bending are originally from Volkerson [22] and from Goland and Reissner [23], and may be expressed in terms of the adhesive stresses as:  129  T (x)-K T (x) m  (7)  =0  2  a  a  CT™'(x) + 4K* (J (x) h  a  =0  (8)  The two characteristic parameters K and K describe the shear and bending response M  H  of the joint, and are given as follows:  K  =  2  4G„  1 V  f  (9)  P P  i  V  1  j _ P  D  +  D  (10)  r j  B y applying the boundary conditions at the crack face, the compliance o f the springs bridging the crack faces may be calculated as follows. First, the general solutions to equations (7) and (8), considering only those parts that vanish as x —> » , are given by:  T (x) = Ae- " K  (11)  x  a  a (x) = [Bcos{K x) + Csin{K x)]e- " K  a  b  (12)  x  h  The derivation then follows the method of Wang and Rose [18] with the exception o f the treatment o f the adhesive shear stress boundary condition. The integration constants are determined by the boundary conditions of the joint. O f interest are the displacements under the action of an applied force and moment acting on the plate.  N (0) = -a° p  (13)  Jp  M (0) = -<7° t /6  (14)  2  p  h  p  The force and moment acting on the reinforcement  are determined considering  equilibrium. N (0) = cr°J  (15)  p  r  M (0) = CT° t /6 + <T j (t +t )/2 2  r  h  (16)  0  p  p  p  r  The final condition is symmetry, which requires the transverse shear forces to vanish. V (0) = V (0) = 0 p  (17)  r  The stresses may be expressed in terms o f the boundary conditions using the adhesive constitutive equations. The process is identical to that of Wang and Rose [1], with the  130  exception that here the shear forces are made to vanish according to (17). W a n g and Rose included an extra term to balance the shear stress at the edge of the adhesive layer, T (0), a  which is of minor consequence and will requires C = -B,  vanish in a real joint. Equation  (17)  and the remaining constants are found to be: 1  ,  1  Et  3(f +f,).l  Et  r r  EX  p p  r ~ p  l  m p  1  .1  1  1  r  E t)  E/  r  1+- Dr D  V  (18)  p  (19)  p J  The displacements of the plate are then given by equations (20) and (21).  (0) = % K (0) - vi (0)) = - c r r  "  m  (0) =  2E„  K £-B  IE.  b  B  (0) = K (u (0) - u (0))+ u (0) = ±- A + K p  r  h  h  (20)  (21)  2E..  G,.  A t this point, the spring compliance coefficients may easily be found, and are given by equations (22) through (25). 1  1  3(t +t )  Et  Et'  (  K^r  r  p p  .3  mb ~ '  p  J f  (t  r  + 2K • D H  E rt ^r  E pt J ^p l  J _  J  K  J _  12xv  D.  D p  (23) J  J _  (24)  12IC  K  D  '  +  D  (22)  R  C  l  +t„  , J  t. +t.  = 3c  2DK  U  (25)  mb  A s found by Wang and Rose, c is asymmetric. This asymmetry is required to satisfy M a x w e l l ' s reciprocal theorem, whereby it can be shown that the compliance matrix must obey the relationship c  bm  = 3c . mh  For a linear system, a set of forces is related to a set of  displacements at the points of application of the forces by a symmetric compliance matrix, denoted here as C. u  = [C]F  (26)  131  For a moment acting on a plate, the moment may be equivalently expressed as a pair of forces as shown in Figure 5-2. m  'm  Figure 5-2: Equivalency of forces and moments  For the bending and membrane stress distributions shown in Figure 5-2, the equivalent forces are given as follows: (27)  (28) The compliance matrix may now be expressed in terms of the bending and membrane stresses and displacements.  m \b U  J  c  mm  = t.  c„  n  c,bm  B y M a x w e l l ' s reciprocal theorem, for the spring compliances, c  bm  bh \ b c, a  Comparison with equation (6) shows that  hm  mh  (29)  3 'J  - C .  =3c .  j  Small variations from this relationship w i l l also  result from approximations made during the derivation of the bonded joint model. For example, the simple analytical models described by equations (7) and (8) do not include bending moments in the adherends caused by adhesive shear stresses acting at the adherend/adhesive  interface. Coupling of extension and bending results only from  equilibrium and compatibility requirements  employed during the application of the  boundary conditions, and small errors w i l l result because the bonded joint model and boundary conditions are inconsistent. Additional minor errors w i l l arise i f the adhesive thickness is ignored during derivation of the bonded joint model but included when one determines the boundary conditions.  132  5.2.3 Long-crack limit behaviour Within the crack-bridging analysis, the long-crack limit behaviour o f a repair may be determined by energy analysis [3,18], providing a limiting value for the stress intensity in the plate. T o analyze linear coupled bending and extension of a repair, W a n g and Rose [18] have developed an energy method for the separation of the membrane and bending components of the stress intensity in the repaired structure. In this paper, the authors have corrected their method to account for asymmetry of the compliance matrix and the presence of plane strain conditions at the crack tip, as found by Kotousov and Wang [24] for extension of cracked plates, and by Clark and Romilly [25] for plate bending. For consistency with line-spring models developed by Clark and R o m i l l y [6,25], the lineforces acting on the crack faces are expressed in terms of the membrane and bending stresses in the plane of the crack. Rose [2-4] observed that for a very long reinforced crack, the stresses near the crack tip become independent of crack length. The strain energy released with an increment of crack growth may then be found from the difference in stored elastic energy between strips of material far removed from the crack tip. One strip is located over the middle of the long crack, and its displacement is governed by the stretch of the springs bridging the crack. The other is well away from the crack and is undisturbed by the singular stress field near the crack. The difference in stored energy between the two strips is given by the energy stored in the springs bridging the crack. For a very long reinforced crack, the energy available to drive crack growth w i l l then be equal to the strain energy stored in the springs, and is evaluated as follows [18].  G.  = — \cr(z)u(z)dz p -t i2 P  2z -tJ2 V  > =  +  mm l  C  (J  "m+Uby-  t  > A  2  C  m b  (  T m  ^ b  + ^  \k  ( °) 3  "P J C  b b ^ h  Based on the results o f the generalized plane strain models for extension [24] and bending [25] o f plates, the crack tip must be in a state of plane strain. The strain energy release rate at the outer fibre of the plate w i l l have two components, one for bending and one for extension. Here, K and G are the stress intensity and strain energy release rate b  h  at the outer fibre.  ^ ±. lzR  G  Gm+  Gh=  3  2  (  A  i  (31)  K1+-K 3 E„ V  2  133  The applied stresses are linearly related to the membrane and bending stress intensities.  K"  (32)  m  bm  A s for the spring compliance matrix, by M a x w e l l ' s reciprocal theorem, the authors w i l l demonstrate that d  = 3d . This is a significant correction to the method o f W a n g and  hm  mh  Rose [18]. Since 1997, it has also been shown that plane strain conditions always exist at the crack tip, and the stress intensity at any point along the crack front can be related to the crack face displacement as follows, where r is the distance from the crack tip.  27t  lim  K  r->0  4 ( l - 77  2  (33)  u(r)  r  )  In the region of the crack tip, equation (33) can be rewritten as follows: lim  E  CO  Fix  p  4(l-/7 ) V r  r->0  (34)  m  _ bm  2  d  B y the same argument used for the spring compliances (comparison with equation 29) it can be stated that d  = 3d . Substitution o f (32) into (31) provides a second expression  hm  forG  M  , which is a corrected version of that given by Rose and W a n g [18]. (d  2  E  mb  G  lh)el  +3d mm  " ~ • + 2d {d +d )cr CX 1-/7' + {dlb+dl l3\rl mh  mm  M  hh  (35)  H  b  Repeated differentiation of equations (30) and (35) with respect to the applied stresses w i l l lead to a set of equations solvable for the coefficients of matrix d , as follows:  mh  I-77 d mb  { + bb) d mm  = C>  d  lb + h b 3  d  d  2  /  =  1  2  I-77  (37)  2  E - bbT- T c  3  (36)  2  i  (38)  L  1 - 7]  A unique solution can be found by noting that d , m  leading to a system o f equations  d  h  must be positive and non-zero,  solvable for the coefficients o f matrix  convenient to define three intermediate variables.  134  d.  It is  A = c„  B =c  I-/7'  r =c  -  p  ""X-rf  '""l-Tj ' 1  (39)  Equations (36) through (38) may then be decoupled to form a single equation for d . mm  (d  + JC-A + d \lA-d 2  mm  2  mm  mm  = SB  (40)  Equation (40) has one positive real root, bounded by upper and lower limits given by-s/A and V A — C . The correct value of d  mm  may be found in this interval using a  simple root finding procedure. The remaining coefficients of the d matrix follow:  d =^C-A bb  + d" mm  (41)  B mb  =•  d  (42)  d +d mm  mm  hh  on  The resulting long-crack limit membrane and bending stress intensities follow from equation (32), and the limit spring displacements are given by equation (6). These corrected quantities define the effectiveness  o f a repair and are essential for the  development o f an interpolation applicable to repaired cracks o f arbitrary length.  5.2.4 Interpolation for arbitrary crack lengths This section describes the development of a new engineering model for the stress intensity and crack opening displacement for bridged cracks o f arbitrary length. These quantities may be estimated by developing an interpolation between the long-crack limit solutions given by equations (32) and (6), and the nominal solutions for an unreinforced crack. In both cases, the stresses given by equations (3) and (4) are assumed to act to open the crack. The first step is to establish the short-crack limit, or nominal solutions for a through-thickness crack without reinforcing springs. For finite thickness plates, Clark and R o m i l l y [6,25] have developed a set of geometry correction factors by using generalized plane strain models for extension and bending of plates. These correction factors account for the through-thickness material properties and the thickness of the plate. For the stress intensity, they take the following form:  Y  pres k  •(«) = ! + v  a 1.5 a  , 5  +1.5 . a .1.2  rshear  a - + 3.5 K 1  2  1  -1  (43)  .4  1 + v.  (44)  3+v  p J  135  Using these correction factors, the nominal membrane and bending stress intensityfollow: (45) (46) Equations (47) and (48) provide the necessary geometry correction factors for the maximum displacement of the crack faces, which occurs at the centre of the crack.  V  1.5  Y  (a) = \ +  d  a +2.5 15  a  f  Y  (a)  13  =h  shear d  1  pres  A  Y  A  1-/7 1  1  (47)  -1 p  J  \  +  (48)  3 + vp  p J  Without any crack-bridging springs, the extensional and bending displacements o f the crack faces under the action of the crack-opening stresses, t r ° and cr°, w i l l then be:  u°=2Yr(aD  ul =  2Y  -  pres d  a = 2Y,  (a  - )YJ (al —— )  pres h  hear  hear  (49)  a  dm  '^-a = 2Y.  db  a  (50)  Each of the correction factors is a function of a, a normalized crack length that accounts for the effects  o f the transverse properties and thickness o f the plate. The three  normalized crack lengths characterize the effects  o f transverse normal stresses i n  extension, transverse normal stresses in bending, and transverse shear i n bending, respectively. (51)  2a v „ z  pres.  \5E  p  (52)  =  <P  shear 2a _  n 6G;(3 + v ) p  (53)  These nominal solutions for the stress intensities and displacements provide the shortcrack limit solution. The long-crack limit stress intensities may be calculated using  136  equation (32), and the corresponding long-crack limit displacements are given by equation (6). A t this point the authors propose a new method of interpolation between these limits to establish the state of a reinforced crack of arbitrary length. For the case of coupled extension and bending, a very effective interpolation may be achieved by considering the stiffness of both the cracked plate and the crack-bridging springs. Equations (49) and (50) may be inverted to develop expressions for the stiffness of the cracked plate.  k" m  \ 1-/7  pJ  k  (54)  2Y a dm  (55)  p  1-/7  pJ  2i>  A stiffness matrix for the as-reinforced crack may now be developed by considering the crack and the patch as a set of springs acting in parallel. The result is an as-reinforced stiffness matrix applicable to repaired cracks of arbitrary length.  k  m  \b a  j  ~T~  k^  mb  K  mm m  ^bb  +  1f..T  \  J\ b  J  U  (56)  B y inverting the combined stiffness matrix of equation (56), a new compliance matrix may be generated for any particular crack length that accounts for the stiffness of the cracked pate:  .To- > 0  ^mbi  \b U  j  "bm  _0  (57)  J  The as-reinforced crack face displacements can now be calculated directly, and the asreinforced membrane and bending stress intensity may be calculated by the applying the new compliance matrix to the energy method described in Section 5.2.3. This is a new application of the energy method, which has in the past only been used to calculate longcrack limit solutions. This method of interpolation does not require any assumptions regarding the ratio of the membrane and bending components, overcoming a significant problem with the method of Wang, Rose and Calinan [19]. This new approach to the interpolation allows for the simplified analysis of linear coupled bending and extension of a repair and includes the effects of transverse normal and shear stresses and accompanying shear deflections. It is a valuable tool for patch design and damage tolerance. In subsequent sections, this new interpolation model will be validated against a numerical line-spring model for generalized plane strain plates and against a three-dimensional finite element model of a bonded composite repair. 137  5.3 A line-spring model for crack-bridging In this section we combine the generalized plane strain crack-bridging models for extension and bending [6,25] to address the problem o f a crack bridged by coupled springs. The results w i l l be used to validate the closed-form equations developed i n Section 5.2 above. The springs bridging the crack are characterized by equation (5), i.e. the stiffness  matrix relating the membrane and bending stresses to the crack face  displacements. Using an approach similar to that of Wang and Rose [1], the line-spring models for extension and bending may be combined to include the effect o f the coupling springs. The significant differences in the current approach are the use o f the generalized plane strain plate models and the choice to express the bending moment and displacement in terms of the crack face displacement and the bending stress in the outer fibre of the plate. The generalized plane strain plate models lead to additional terms that account for pressurization and transverse shear about the crack tip. This leads to the following set o f coupled Fredholm integral equations with hyper-singular integrands that are solvable for the crack face displacements. The left side of the equation is the net crack opening stress along the crack face, and the right side describes the reaction of the plate.  -1  mm Jy) U  k  -k  2K  rE u„(t)  a  p  J  -a  (58)  (t-y)  2  1-JJ  2  +T] (Kr) L (Kr\t-y\) 2  '  2  pres  (t-y)  2  <T {y)-k»">(y)_-i'r > 0  E  b  Ub(t)  2x1 l - ? 7  -k ujy) bm  2  Y ^ L* (rf \t-y\)  +  2  +  +  ) 7] L 2  2  pres  ear  ear  (K \t pres  \it  (59)  - y\)  In equations (58) and (59), the following parameters characterize the through-thickness properties of the plate. pre,  K  3£ =  J  (60)  L  G (\-rj )  l  2  z  2v  15E  z  „pres.  _  p  n  P  (61)  WF (i-/?;) z  -shear  2 3G,  (62)  138  The functions L  pres  (z) and L (z)  describe the effects of pressurization and transverse  shear  shear stresses near the crack tip. If the L (z) pres  functions are neglected, the line-spring  model w i l l take the forms for classical plane strain extension and plane strain plate bending with shear deflections (i.e. plate bending according to the M i n d l i n formulation). If the term L  {z) is also removed, then the effects of transverse shear stresses and the  shear  accompanying shear deflections w i l l be removed. The formulation of W a n g and Rose [1] can be recovered by neglecting the pressurization terms, removing the plane-strain modification o f the Young's modulus, and using isotropic rather than isotropic material properties. The functions L {z) pres  and Lf (z) hear  transversely  have the following  form, where K are various orders of modified Bessel functions of the second kind. n  L (z) = 2 l + ^ . K z 'J pres  AO  { z ) ~ - \ z z  2  (63)  A  L^ (z) = -™+^ + 4{K (z)-K (z)) z z ar  2  0  94  + ^K (z) z  (64)  2  The domain o f the integration in equations (58) and (59) may be normalized with respect to the half-crack length, a, and the displacements expanded using Chebyshev polynomials of the second kind: J7,.(r) = sin((i + l)cos-- (r))/A/T^r I  (65)  r  For this set of orthogonal polynomials, the weight function is given by: W{r) = j \ - r  (66)  2  Given this expansion, the crack face displacements may now be decomposed as follows:  I  " « =?(r)(/,.(r) t  (68)  g i  i  Choosing a set o f collocation points, a coupled linear system may now be defined from equations (58) and (59), which may be solved for the expansion coefficients.  Uric JJU  .  In this system, the matrix coefficients have the following definitions.  139  (69)  IJ  +  2n  ak W(r )U (r ) mm  i  j  (70)  i  B =ak W(r )U (r ) iJ  mb  i  j  (71)  i  i + K, . .  {aKt )\{\-rf}IT shear ear  4K +  J pres.  1+v  (72)  2K  ak W(s )U (s ) hh  i  j  i  D =ak W(r yU ir ) iJ  km  i  J  (73)  l  Analytical solutions exist for the logarithmic parts o f L  (z) and L  sHear  pres  (z), but the  regular parts require numerical integration. The term V, j arises from the logarithmic singularity, and may be solved analytically, as demonstrated by Joseph and Erdogan [26,27] and by W a n g and Rose [1]. T o evaluate the regular part, the authors used GaussChebyshev quadrature, as suggested by Joseph and Erdogan. Z0"  =  YL^iaK^lr.-ty-lnfa-tytyWjisyis  (74)  + Vu  -i  tfi* = j f L  p r e s  ( a K ™ \ - s \ ) + j\n(\ p  r i  -$W{syUj{s)ds-jV  n  u  K  u  {  j =0  a  ~2  V = Jln|r,-s|VvW,(^M ^  - r , + i + ln2|  (75)  (76)  TM)  j  J  J +  2  >  Q  'j  For the generalized plane strain plate models, the stress intensities in extension and bending may be found from the expansion coefficients in the following manner. (77)  2(l-TJ )tt z  (78)  2(1-^)7=^  These stress intensity results w i l l be compared to the interpolation model results and finite element results in Section 5.4.1. Equation (5) relates the displacement of the crack faces to the line-forces carried by the springs. Expressing the membrane and bending stresses as their line-force equivalents: 140  N (y) x  x(y)  M  (79)  = 2h{k u (y) + k u am  m  = -h { bn " (y) i  k  l  m  ab  +  b  (80)  kbb u  b  Given the crack face displacements, these expressions for loading may be used to predict fatigue cracking and/or rupture of the springs (i.e. the patch materials) bridging the crack faces as required for patch design and/or damage tolerance assessment.  5.4 Comparison to finite element models In this section, finite element models are used to determine the capability of the closedform equations. The stress intensities and crack face displacement results from the closed-form equations and the line-spring model are compared to a three-dimensional bonded repair model, and the nominal stresses in the substrate and compliance o f the repair as it bridges the crack are investigated using two-dimensional bonded joint models.  5.4.1 Stress intensity and crack face displacements The authors have developed three-dimensional solid finite element models representing the A M R L (Australian Maritime Research Laboratory) specimen, a specimen type that has been widely adopted for the investigation of bonded repairs [e.g. 9]. The models have been validated against mechanical testing of an instrumented specimen, during which strains were measured at nearly 100 positions on the patch and repaired plate, including 40 strain gauges applied to the face of the repair as it bridges the crack [28]. Figure 5-3(a) shows the model in a large-plate configuration for a crack length of 160 mm. W i t h the appropriate boundary conditions, symmetry allows the use of a one-quarter plate model. The large-plate model is designed to test the closed-form equations and line-spring model for the idealized case from which they where derived. Figure 5-3 (b) shows a close-up of the elements near the crack tip, with elements removed to show the repair patch, epoxy, and plate layers. The crack face is identified by the cross-hatched region.  141  (a)  (b)  Figure 5-3: Three-dimensionalfiniteelement model of a repair The finite element models were developed in A N S Y S , utilizing three-dimensional 20node brick elements and 15-node wedges. The wedge elements use quarter-point midside nodes to model the singularity at the crack tip, from which the stress intensity was calculated using the stresses at three integration points in the wedges ahead o f the crack tip. This stress-based approach allowed an accurate evaluation o f the stress intensity as it varied through the thickness of the plate. Validation models demonstrated  stress  intensities accurate to within 2% compared to theoretical models corrected for throughthickness effects. Table 5-1 shows the material properties and dimensions used in the finite element models. F o r the purposes of this chapter, the patch was modelled as an isotropic solid to remove the complicating effects  of transverse  shear  and peel  deformations as would otherwise exist in an orthotropic composite. Boron/Epoxy Patch  Property  FM-73M Adhesive  2024-T3 Aluminum Plate  Elastic Modulus, E  210 GPa  2.14 GPa  72.4 GPa  Shear Modulus, G  7.24 GPa  0.805 GPa  27.2 GPa  Poisson's Ratio, V  0.21  0.33  0.33  Thermal Expansion, OC  4.61  Thickness, t  0.924 mm  fie /'C  50.0  jue r°c  0.25 mm  23.45 /IE /'C 3.125 mm  Table 5-1: Material properties and dimensions The membrane and bending stress intensities are the average stress intensity and the stress intensity at the outer fibre of the plate, respectively, as found from the average and the first moment o f the stress intensity taken through the thickness o f the plate. These  142  quantities were calculated using a linear least-squares fit to the stress intensities found at the integration points through the thickness of the plate. T o determine the actual transverse variation of the stress intensity through the thickness of the plate would require the use of many elements, as a thin boundary layer exists at the top and bottom surfaces. However, convergence studies have shown that few elements are required to obtain accurate values for the plate-averaged quantities, K  M  and K . H  The three-dimensional  finite element models employ two elements through the thickness of the plate, enough to ensure that errors due to through-thickness mesh refinement are much smaller than the 2% error resulting from stress intensity determination. Figures 5-4 and 5-5 show the membrane and bending stress intensities predicted by the interpolation model and by the line-spring model when compared to the large plate finite element model, under a remote applied stress of 1 MPa. The nominal results are the membrane and bending stress intensities that would arise i f the patch only acted to reduce the stresses in the underlying plate but did not act to restrict the opening of the crack. For both the interpolation model and the line-spring model, the nominal stresses were calculated using plane strain results from a two-dimensional finite element model (described in Section 5.4.2) and the spring stiffnesses were calculated using the plane strain analytical results from Section 5.2.2. 0 .4  o A  A  A  finite element model interpolation model long-crack limit nominal result  O  line-spring  0 0  50  100 Crack length [mm] Figure 5-4: Membrane stress intensity  143  150  200  0.25  100  150  200  C r a c k length [mm]  Figure 5-5: Bending stress intensity The interpolation and line-spring models provide very similar results for short cracks but diverge when the crack length becomes large. For long cracks, the line-spring model develops errors due to the increasing importance of the higher order Chebychev polynomials and the errors arising from numerical integration of the regular parts of the hyper-singular integrals. For relatively short cracks with a < 50 mm, the interpolation model membrane stress intensity is lower by less than 2% and the interpolation model bending stress intensity is higher by as much as 10% when compared to the F E M results. This 10% error can be attributed in roughly equal parts to three sources. First, equation (31) slightly over-predicts the nominal stress intensity for this plate, next, very small variations from M a x w e l l ' s reciprocal theorem that arise due to simplifications made during the development of the spring compliance matrix lead to disproportionately large errors for this crack length and last, the interpolation method is approximate and errors of up to 5% should be expected. The finite element results are generally lower than those calculated using either analytical model, a difference that could arise from a number of sources. Using the limited .academic version of A N S Y S , it is difficult to include enough elements in the adhesive layer near the crack, and at the same time model a repaired plate of large planar extent. A n insufficiently dense mesh would cause the repair to appear to be overly stiff and artificially reduce the stress intensity and crack face displacement. Figures 5-6 and 5-7 compare the membrane and bending crack face  displacements  calculated using the interpolation model and the line-spring model to the results of the large-plate finite element model.  144  200  100 C r a c k length [mm] Figure 5-6: Membrane crack face displacement 1.25  100 C r a c k length [mm]  150  200  Figure 5-7: Bending crack face displacement  The crack displacement results follow a similar trend to that observed for the stress intensities. For short cracks with a < 50 mm, both the interpolation model membrane and bending displacements are lower by less than 4% when compared to the F E M results. For long cracks the interpolation and line-spring model results diverge for the same reasons as stated above. In both cases, interpolation provides reasonable results, and the long145  crack limits established by the energy analysis are obeyed. This contradicts the findings of Wang and Rose [18], who, by comparison to a three-dimensional finite element model, observed that the energy method significantly over-predicts the bending stress intensity and is not a useful approach. This supports the validity o f the modifications described in Section 5.2, i.e. that the crack tip must be considered to be in a state o f plane strain and that M a x w e l l ' s reciprocal theorem must be satisfied in the manner shown. In the next section, a two-dimensional finite element model w i l l be used to examine the effectiveness of the means used to calculate the crack opening pressures and the definition o f the spring constants.  5.4.2 Spring stiffness and nominal stress determination The composite beam model used to calculate the nominal stresses under a repair is, strictly speaking, only applicable to a very large patch without boundary conditions that lead to restraint against bending. The effect of a finite patch applied to a plate with rotational constraints at the grips can be estimated using finite element analysis. Using the same materials and geometry as were used for the large plate finite element model, a two-dimensional finite element model has been developed for this purpose and to calculate the compliance of the repair as it bridges the crack. This model has been validated against load-deflection curves, strain gauge data, and lateral deflections from experiments [29]. Figure 5-8 shows the finite element mesh and boundary conditions for two configurations of the model, one with a crack under the patch (top), and an uncracked version used to calculate the nominal stresses under the repair (bottom). patch  \ \  \ \ \ \ \ \\  \  T/  / / / /  J  ii  \ 1 1  \  \\  dob  (c)  (b)  Figure 5-8: FE mesh showing the (a) cracked, (b) tapered, and (c) grip regions In this particular analysis, the bonded repair is assumed to be isotropic to eliminate composite deformations that cannot be predicted using the bonded joint equations presented in this chapter. Figure 5-9 shows the nominal stresses predicted by the closedform equations for a remote applied stress of 1 MPa. Both the finite element and closed-  146  form equation results are shown for two cases: plane stress and plane strain. It is likely that the correct case is the intermediate one of generalized plane strain, where the structure is allowed to deform uniformly in the transverse direction. For the A M R L geometry, it is seen that the closed-form equations provide reasonable results, but that there are sufficient differences to account for the errors observed in the comparison of the line-spring and interpolation model results to the three-dimensional finite element results. The closed-form equations are seen to generally over-predict the nominal stresses, which would lead to an over-estimate of the stress intensities and crack face displacements.  0.9 • analytic plane stress EH analytic plane strain E3 finite element plane stress • finite element plane strain  0.8 0.7  to  w OS  0.6  W  «" 0.5  C  S  E  o  o.4 0.3 0.2 0.1 0.0  Figure 5-9: Nominal stresses The other factor that would affect the interpolation results is the calculation of the spring compliances. Figure 5-10 shows a comparison of the spring compliances predicted by the closed-form equations and the finite element model. These displacements were derived by separately applying a uniform membrane stress and a linearly varying bending stress to the face of the crack in the two-dimensional finite element model shown above. Again, the results are shown for two cases, plane stress and plane strain.  147  cmb  cmm  cbm  ebb  Figure 5-10: Spring compliances  The closed-form equations are shown to provide very reasonable results, generally slightly under-estimating the stiffness of the patch when compared to finite elements. In the generation of Figures 5-4 to 5-7, the plane stress closed-form equations were used, which when compared to the plane strain results, may demonstrate a slight over-estimate of the compliance of the repair. A s in the determination of the nominal stresses, the actual state of the repair in the idealized large plate case is likely to be one of generalized plane strain, and intermediate to the plane stress and plane strain results. Accordingly, it is difficult to precisely determine what role the spring stiffness equations might contribute to the differences observed between the three-dimensional finite element models and the interpolation and line-spring models.  5.5 Results and discussion Through comparison with three-dimensional finite element results, it has been shown that the line-spring and new interpolation models capture the crack-bridging behaviour of a repair. A l s o through comparison with finite element analysis, it has been shown that the composite beam model may lead to some inaccuracies in the determination of the nominal stresses acting under the repair. Factors affecting its applicability include the length of the repair, the support conditions, and the planar stress state assumed to exist in the repair (i.e. plane stress or plane strain conditions). Some additional complicating effects that have not been investigated for the case of linear coupled bending and extension are the shape of the patch (whereby patches of finite planar geometry are known to generally increase the stresses in the region of the repair) and the effects of a highly anisotropic composite patch. Another pressing issue is that, while the common  148  bonded joint model presented in this paper proved adequate in the case of an isotropic repair, the composite nature of a real repair is a complicating factor in the determination of the compliance of the repair as it bridges the crack. The modelling of such a hybrid bonded joint is complicated by many secondary effects such as shear and peel (transverse normal) deformations that act to increase the compliance of the repair, pressurization of the adhesive due to peel stresses, the plane state of the adherends and the adhesive, and the complementary shear deformations resulting from the rapidly changing bending stresses in the patch and plate. These effects can be particularly important in the prediction of failure of the repair by disbonding, where the peel and shear stresses in the adhesive must be known. The alternative to accurately determining the adhesive stresses is to calculate the strain energy release rate of the bonded joint system, which may be safely compared to experimental results for a similar specimen under similar loading conditions to predict fracture and fatigue, but is not directly comparable to any adhesive material property. This would necessitate a significant amount of testing for each new combination of materials or for new loading combinations, which is currently the norm in the development of composite airframe structures. A n additional issue that has not been addressed is the crack closure phenomenon that occurs on the compressive side of the bending plate when the applied tensile loads are not sufficient to prevent closure. This has been identified by Duong and Wang [30] as one of the principal impediments to the prediction of the rate of failure of a cracked plate under tensile and bending loads. They have proposed an effective or modified stress intensity to account for combined bending and tensile loads, but admit that their technique requires further work before it can be used generally. The authors hope that the development of the generalized plane strain plate line-spring models w i l l prove useful in the analysis of this phenomenon. Crack closure would pose a nonlinear contact problem, which could be examined by using a one-dimensional finite element mesh along the crack face instead of using Chebychev polynomials to obtain a numerical solution. Without a method to address this issue, experimental verification or very conservative means should be employed in the prediction of failure, such as simply adding the membrane and bending stress intensities and comparing them to the critical stress intensity for the material or by using them together in the Paris equation. This approach should prove to be conservative, as it w i l l effectively over-predict the energy available for fracture.  5.6 Concluding remarks In this chapter, the authors have, based on a corrected form of the energy method of Wang and Rose, developed a new interpolation model applicable to the analysis of bridged cracks under combined tension and bending. The interpolation model has been tested against a line-spring model of for a generalized plane strain cracked plate and a full  149  three-dimensional analysis of a repaired plate, and proven to be very accurate for the A M R L specimen geometry. Some deficiencies have been noted in the methods used to determine the nominal stresses acting in the regions of the repair and in the extension of linear crack-bridging models to the analysis of unbalanced repairs experiencing stress stiffening, and accordingly the authors urge bonded repair designers to verify their results against finite element models or experiments when thick plates or repairs of limited planar dimensions are analysed. It has also been noted that for an anisotropic composite repair, standard bonded joint models may not be adequate for the determination of the spring compliances or the prediction of disbonding and fracture. Prediction of the rate of failure or ultimate load of the structure is also complicated by the lack of an adequate and widely accepted means of translating the stress intensities or adhesive stresses into crack or disbond fracture or growth rates, which w i l l continue to require the structural testing of repairs (and composite structures in general) to determine their suitability for service.  5.7 References [I] Wang C . H . , Rose L . R . F . A crack-bridging model for bonded plates subjected to tension and bending. International Journal of Solids and Structures 36, 1999, 1985-2014. [2] Rose L . R . F . A n application of the inclusion analogy for bonded reinforcements. International Journal of Solids and Structures 17, 1981, 827-838. [3] Rose L . R . F . A cracked plate repaired by bonded reinforcements. International Journal of Fracture 18(2), 1982, 135-144. [4] Rose L . R . F . Theoretical Analysis of crack patching. In: A . A . Baker and R. Jones (eds.) Bonded Repair of Aircraft Structures. Martinus Nijhoff, 1988, 77-105. [5] Wang C . H . , Rose L . R . F . Bonded repair of cracks under mixed-mode loading. International Journal of Solids and Structures 35(21), 1998, 2749-2773. [6] Clark R.J., R o m i l l y D.P. Reinforced cracks in transversally isotropic generalized pane strain plates. T o be submitted to Engineering Fracture Mechanics. [7] Jones R. Neutral axis offset effects due to crack patching. Composite Structures 1, 1983, 163-174. [8] Jones R., Callinan R . J . , Aggarwal K . C . Analysis of bonded repairs to damaged fibre composite structures. Engineering Fracture Mechanics 17(1), 1983, 37-46. [9] Baker A . A . Repair efficiency in fatigue-cracked aluminum components reinforced with boron/epoxy patches. Fatigue and Fracture of Engineering Materials and Structures 16(7), 1993, 753-765. [10] Clark R . J . , R o m i l l y D . P . Fatigue damage prediction for bonded composite repairs applied to metallic aircraft structures. S A E Journal of Aerospace, paper # 2001-01-26, 2001. [II] Albat A . M . , R o m i l l y D . P . A direct linear-elastic analysis of double-symmetric bonded joints and reinforcements. Composites Science and Technology 59, 1999, 11271137.  150  [12] Jones R., C h i u W . K . , Smith R. Airworthiness of Composite Repairs: Failure Mechanisms. Engineering Failure Analysis 2(2), 1995, 117-128. [13] Ratwani M . M . Cracked, adhesively bonded laminated structures. A I A A Journal 17(9), 1979, 988-994. [14] Arendt C , Sun C . T . Bending effects of asymmetric adhesively bonded composite repairs on cracked aluminum panels. Proceedings of the F A A / N A S A International Symposium on Advanced Structural integrity for Airframe Durability and Damage Tolerance, N A S A conference publication 3274, P a r t i , 1994, 33-48. [15] Schubbe, J.J, M a l l , S.S. Modelling of cracked thick metallic structure with bonded composite patch repair using three-layer technique. Composite Structures 45(3), 1999, 185-193. [16] W e n P . H . , Aliabadi, M . H . , Young, A . Boundary element analysis of flat cracked panels with adhesively bonded patches. Engineering Fracture Mechanics 69, 2002, 21292147 [17] Papanikos P., Tserpes K.I., Labeas G . , Pantelakis Sp. Progressive damage modelling of bonded composite repairs. Theoretical and Applied Fracture Mechanics 43, 2005, 189198. [18] Wang C . H . , Rose L . R . F . O n the design of bonded patches for one-sided repair. Proceedings of the 1 1 Int Conference on Composite materials, G o l d Coast, Australia, 5, 1997, 347-356. th  [19] Wang C . H . , Callinan R . J . , Rose L . R . F . Analysis of out-of-plane bending of a onesided repair. International Journal of Solids and Structures 35(14), 1998, 1653-1675. [20] Rose L . R . F . and Wang C H . Analytical methods for designing composite repairs. Chapter 7 in Advances in the bonded composite repair of metallic aircraft structure. Baker A . A . , Rose L . R . F , Jones R., eds. Elsevier, 2002. [21] R o m i l l y D . P . , Clark R . J . Damage tolerance analysis and testing of unbalanced bonded composite repairs for aluminum airframes. S A E 0 5 - A M T - 4 1 , 2005. [22] Volkersen O. Rivet strength distribution in tensile-stressed rivet joints with constant cross-section. Luftfahrforschung 15(1), 1938, 41-47 [23] Goland M . , Reissner E . The stresses in cemented joints. Journal of Applied Mechanics - Transactions of the A S M E 66, 1949, A 1 7 - A 2 7 . [24] Kotousov A . , W a n g C H . Fundamental solutions for the generalized plane strain theory. International Journal of Engineering Science 40, 2002, 1775-1790. [25] Clark R.J., R o m i l l y D . P . Bending of transversally isotropic generalized plane strain plates with reinforced cracks. Accepted by The International Journal of Fracture, 2006. [26] Joseph P.F., Erdogan F . Surface crack problems in plates. International Journal of Fracture 41, 1989, 105-131. [27] Joseph P.F., Erdogan F . Plates and shells containing a surface crack under general loading conditions. N A S A contractor report 178232, N A S A Langley Research Center, 1987. [28] Clark R . J . , R o m i l l y D . P . Experimental study of a bonded composite repair. 2002 C S M E Forum, M a y 21-24, Kingston, Ontario, Canada, 2002.  151  [29] Clark R . J . , R o m i l l y D . P . Non-linear bending study of hybrid aluminum and boronepoxy bonded joints. C A N C O M 2003, Ottawa, Ontario, Canada, 2003. [30] Duong C . N . , Wang C H . O n the characterization of fatigue crack growth in a plate with a single-sided repair. Transactions of the A S M E 126, 2004, 192-198.  152  Chapter 6: Elastic analysis of hybrid bonded joints and bonded composite repairs* D . P . R o m i l l y and R . J . Clark University of British Columbia Department of Mechanical Engineering  6.1 Introduction In this chapter, the authors extend the closed-form bonded joint linear elastic analysis method of Delale et al [15] and Bigwood and Crocrombie [7] to include the composite deformation mechanisms and the thermal residual strains that arise in hybrid metalcomposite joints such as those presented by bonded composite repairs applied to metallic aircraft structures. The analytical predictions for the adhesive stresses and the compliance are compared to the results of a linear elastic finite element model that has itself been validated by comparison with experimental results. The results are applied to the problem of coupled linear extension and bending of a bonded composite repair applied to a cracked aluminum substrate. The resulting stress intensity factor and crack opening displacement in the repaired plate are compared to the results of a three-dimensional finite element  analysis, and also exhibit excellent results.  Throughout  the  text,  observations are made regarding the practical application of the results to failure prediction in hybrid joints, whereby the authors demonstrate the need for consistency in the analytical methods used to determine the fatigue and failure of composites from the coupon level to the analysis of the final structural details. A bonded composite aircraft repair consists of a composite patch that is adhesively bonded to an underlying aluminum plate. The result is a hybrid structure  with  significantly mismatched adherends that w i l l exhibit significant thermal residual strains and nonlinear deformation (stress stiffening) under applied tensile loads. In the damage tolerance analysis of such a repair, the stiffness of the structure and the stresses in the adhesive and the composite patch all require accurate evaluation. Accordingly, improved methods are required to analyse hybrid bonded joints with mismatched adherends and account for secondary effects such as coupled extensional and bending response, shear deformation, and axial adhesive stresses. In this paper, the authors extend and enhance the elastic analysis of Delale et al. [15] and Bigwood and Crocombe [7] to include the  * A version of this chapter has been accepted for publication. D.P. Romilly, R.J. Clark, "Elastic analysis of hybrid bonded joints and bonded composite repairs", Composite Structures.  153  effects of orthotropic adherends, adherend lateral shear deflection, transverse (throughthickness) adherend shear deformation caused by adhesive shear stresses (often termed adherend shear lag), transverse deformation of the adherends caused by adhesive peel stresses (peel lag), adhesive longitudinal stresses, and thermal stresses resulting from coefficient of thermal expansion mismatches, all of which may be included within their analysis method without raising the order of the governing equations. The enhanced hybrid bonded joint analysis method uses a linear elastic analysis to calculate the strength and compliance of the repair in the region of the underlying crack. The strain energy release rate and stress intensity values in the adhesive are then calculated from the stresses in the adhesive using the method of Joseph and Erdogan [ 9 ] , and the results are verified against finite element model results for a single-lap joint geometry. The results are then applied to the linear analysis of a bonded repair to show the effectiveness of the resulting design equations. The elastic analysis of bonded joints is an important subject, particularly in the design of aircraft structures. The accurate evaluation of adhesive stresses has been critical to the application of safe-life and damage  tolerance methodologies to the  airworthiness  certification of such structures. In practice, four general methods are available to reduce the adhesive stresses in a joint: adherend tapering, increasing the adhesive thickness, decreasing the adhesive modulus, and the choice of an adequate overlap length. A l s o , due to the relatively high rate of crack growth in an adhesive joint, it is generally desired to keep the working stresses below a fatigue threshold.  In the bonding of metal sheets,  failure criteria for the sheets are usually easy to obtain and the elastic response of the isotropic adherends is easily modelled. In addition, the plates are often matched in stiffness, reducing the complexity of the analysis. For composite-composite bonded joints, the failure criteria and analysis requirements of the joint become complex as lowenergy composite failure modes (such as interply delamination and  fibre-buckling)  become factors in the life of the structure, and the composite sheets require modeling as orthotropic materials to reflect the composition of the plates. For a composite repair applied to a metal plate, the panels being joined are often well matched in terms of extensional stiffness, but poorly matched in terms of bending stiffness. In addition, composite failure modes are a major concern, and physical testing or reference to physical tests is required to rule them out. For most applications of bonded joining during original manufacture,  whether  metal-metal or composite-composite, designers  take  significant measures to reduce peel stresses (e.g. adherend tapering), thus reducing or eliminating the risk of peel-induced failures. The result is that the original structure can generally be designed according to fail-safe principles. In the application of bonded composite repairs to metal structures, in the region around the repaired crack, the joint cannot be tapered, and it is typically not desirable to increase the thickness or compliance  154  of the adhesive to reduce the stresses as this w i l l reduce the crack-stopping ability of the repair. Accordingly, composite delamination or adhesive disbonding may result near the repaired crack. Damage tolerance analysis is therefore necessary, and a method is required to predict the rate of damage progression and the load at failure. Finally, since the joint is typically a hybrid of aluminum and composite materials, the thermal expansion properties of the adherends are mismatched and the joint experiences large thermal residual strains from both patch application and temperature changes in service. In this paper, the authors develop a methodology to allow the accurate evaluation of stresses in hybrid bonded joints. T o demonstrate the utility of the method, the results are applied to the analysis of a bonded composite repair. The methods developed should also be of use in the analysis of other composite-metal joints such as metal inserts for fastening of composite structures.  6.2 Background Bonded joints are widely used in many industries, and accordingly much effort has been expended on characterizing their mechanics. The first effective analyses of the stresses in bonded joints are due to Volkersen [1] and to Goland and Reissner [2], by which the tensile and bending response are assumed to be linear and independent, and second and fourth order ordinary differential equations may be formed and solved for the shear and peel (transverse normal) stresses in the adhesive. In these analyses, the adhesive is assumed to act as a series of shear and tension springs, and the adherends are modelled as simple beams in tension and bending. These models have proven effective in the analysis and understanding of the mechanics of bonded joints. Models based on these equations have been adapted to many applications, usually by examining the boundary conditions specific to the particular application. For composite-composite joints, Renton and Vinson [5] have modified the equations to account for the lay-up of the adherends, and Jones et al. [16] have developed modifications to account for the relatively low in-plane shear stiffness and transverse or peel stiffness of composite adherends. For aluminum aircraft structures, Hart-Smith [e.g. 3] has used the basic bonded joint models to develop widely used design equations and failure criteria considering both elastic and perfectly plastic adhesive response.  In the field of bonded composite repair of aluminum aircraft  structures, Australian researchers have used the methods of Hart-Smith to characterize the strength and stiffness of the repair as it bridges the underlying crack [e.g. 14, 11]. Improved models for bonded joints have been developed to account for secondary effects that lead to coupling of the tensile and bending responses of the structure, to explicitly consider the equilibrium and stress-free boundary condition of the adhesive, and to include variations in the distribution of adhesive stresses through the thickness of the adhesive. Equilibrium requires the shear and bending response of the joint to be  155  coupled, as the adhesive shear stresses act on the face of the adherends and therefore must induce bending. Renton and Vinson [4] have developed a higher-order analytical model to examine the effects of adhesive equilibrium and the stress-free boundary condition, and predicted that the actual peak adhesive shear stress was shifted inboard from the edge of the joint and reduced when compared to the value found by traditional analysis methods. It should be noted that their method did not include coupling of the shear and bending responses of the joint, which has a significant effect on the stress distribution. Using an alternate approach, A l l m a n [6] has expressed the stresses in the joint as a set of stress functions, and, by minimizing the strain energy in the joint, developed solutions that satisfied the adhesive stress free boundary condition and allowed the satisfaction of the full equilibrium equations for the adherends. Adams and M a l l i c k [13] used this approach to develop a simple one-dimensional finite element solution that is easier to apply and to extend to nonlinear or viscoelastic adhesive response than closed-form bonded joint formulations. They found that, while the energy minimization method predicts the vanishing stresses at the edge and the stresses in the bulk of the joint, it generally under-predicts the peak stresses. In other work, Delale et al [15] extended the classical analysis of a bonded joint to include coupling between bending and extension, which resulted in modified constitutive equations for the adhesive that include the longitudinal stresses in the adhesive, derived from the average extensional strains in the adherends. Their results were shown to be accurate when compared to detailed finite element results for the adhesive stresses at the edge of a bonded joint. This allowed them to make the significant observation that it is not important to impose a stress-free boundary condition at the edge of the adhesive, since it is well known that the peak stresses occur at the singularity at the interface of the adherend and adhesive at the extreme edge of the joint. Bigwood and Crocombe [7] subsequently used an equivalent approach to test and develop design equations for bonded joints, but did not include the effects of mismatched adherend thermal expansion coefficients or the presence of adhesive axial stresses in their analysis. In this paper, the authors modify the analysis techniques of Delale et. al., and Bigwood and Crocrombe to characterize the performance of bonded composite repairs. A bonded composite repair consists of a filamentary composite reinforcement that is bonded to a damaged or weakened structure to reduce stresses and prevent or reduce continuing damage growth, as shown in Figure 6-1. When the repair patch is bonded to only one side of a cracked structure, its presence induces significant out-of-plane bending by offsetting the load path. Bending increases the stress intensity in the underlying crack and causes adhesive peel stresses and bending of the repair that, i f not accounted for in the design of the bonded structural repair, could lead to rapid failure.  156  Figure 6-1: An unbalanced repair There is much interest in the application of unbalanced repairs, as often only one face of a structure is exposed. In many cases the interior structure of the aircraft does not provide significant  restraint  against  bending, potentially leading to rapid disbonding and  composite and/or composite matrix failures. T o design and certify single-sided repairs, closed-form methods are needed to assess the effects of bending on the rate of failure of the repaired structure, but perhaps more importantly, methods are required to assess the life and strength of the repair itself under bending and transverse normal (peel) stresses. Accordingly, the authors seek to develop closed-form solutions for the compliance of the bonded joint (characterizing its ability to stop the growth of the underlying crack) and the stress distribution in the adhesive and the composite repair.  6.3 Linear bonded joint formulation In this section, the elastic analysis of Delale et. al. [15] and B i g w o o d and Crocombe [7] is extended to include phenomena that are critical to the understanding of the mechanics of hybrid composite-metal joints. These phenomena include: shear and peel deformation of the adherends, complementary shear of the adherends caused by the large bending stress gradients near the ends of the joint, pressurization of the adhesive due to out-ofplane and longitudinal constraint, and thermal residual strains. W e begin with the definition of an element of the adherend/adhesive sandwich, the external loads that act upon it, and the internal (adhesive) stresses that hold it together. Figure 6-2 illustrates such an element.  157  Ti + dx-T^x V,+dx-V  Adherend 1  1>x  Mi + dx-M^x  z,w  41 x,u  4T +dxT , 2  2  V +dxV  Adherend 2  2  x  2>x  M +dxM 2  2 l X  Figure 6-2: Hybrid bonded joint element In the analysis that follows, the joint can be analysed in either plane stress or plane strain by choosing 'effective' elastic properties for the adherends and adhesive. The derivation w i l l be carried out assuming that plane stress conditions exist. A simple substitution as follows w i l l produce a plane strain analysis for an orthotropic material. For the plane strain case, it is assumed that the adherends are specially orthotropic, with vanishing through-thickness stresses. plane stress  plane stress  E,.  plane strain'  plane strain  (1 a,b)  Similarly, the shear and transverse modulus for the adherends are expressed as follows. G, =G LT  (2 a,b)  '  For adherends of thickness h, the plate bending stiffness may now be defined. Eyh[ 12  (3 a,b)  12  The strain at the interfaces between adhesive and adherends is given by the following. The third term i n each of the equations accounts for 'shear lag', the deformation o f the adherend caused by the adhesive shear stresses acting on the inner surface. The fourth term accounts for the lateral deformations caused by the shear force in each adherend. 3/z, r , E\ x  ~2~b~  V,\,x  (4 a)  8 G, 2G,  158  u 2,x  •  Eh 2  Af,  h  3/i, 2 T  + ——- + • 2  2 £>,  2  8  2,x  G,  (4 b)  2G-,  The curvature at the interfaces between adhesive and adherends has been separated into two parts to include bending and shear deflections. A term has been included to account for deformation caused by peel stresses, or peel lag. These peel deformations should be accompanied by lateral deformations due to the Poisson effect, leading to some additional degree of coupling between the shear and peel responses of the joint. This has been neglected since the inclusion of the primary 'peel lag' term is already very much a secondary effect. t W h x x  _M  =  (5 a,b)  ±  D, 3ft, &a,x  6V, 5G,h  8  t  6V,  3h  2  w.2,x  E,  5G h  2  2  2  8  C7  ax  (6 a,b)  £,  The shear and peel lag terms are found by assuming a linear variation of stresses in the adherends, as shown in Figure 6-3, and the shear strain w i l l be: u (x,z) =  T(X,Z)  G  1  (V)  < J  <v  The shear contribution to (4 a,b) follows by integration through the thickness. The peel lag terms in (6) follow by assuming a linear variation of transverse stresses. This method is due to Jones et. al. [16] who originally applied it to finite element analysis of repairs.  T  a  4_  (x,D  ^  T (x,z)+dx.r (x,z) a  f—  a  iX  O (x,z) + dx.cr (x z) ay  ay  >  iX  Figure 6-3: Assumed variation of transverse adherend stresses Equations (4-6) describe the behaviour of the adherends. The adhesive is often modelled as independent shear and extensional springs, but one may instead model the adhesive by  159  employing the full form of the constitutive laws to relate the adhesive stresses to the relative displacements o f the adherends.  The last set of equations required to perform the analysis arise from  adherend  equilibrium. T =T , hx  T =-t  a  1<x  (9 c,d)  = -cr,  V,.x =  <T.,  M  = V , - ^ ,  U  (9a,b)  a  M  2  ,  x  =V  2  -^T  (9e,f)  a  To develop the governing equations, it proves convenient to introduce two functions.  y/ = ||CTUdxdx  0 — jV <£t, a  (10 a,b)  The governing equations may be found as follows. First, eliminate cr^ from (8 b,c) and then substitute 0,y/ for the peel and shear stresses in (4), (6), (8), and (9). Eliminate the displacements from (8) using (4), (6), and (8), and then eliminate the forces and moments using the equilibrium equations (9). The following ordinary differential equations arise:  K 0 -K 0 o  xx  = -K y,-K,  x  2  K.W,^ ~ K  = -K 0  +K  i¥xx  (11a)  ¥xx  7  eW  :XXXX  + K 0^ + K 0 %  9  (lib)  The constants for the shear equation are given as follows, and account for shear lag of the adhesive and adherends, the extensional stiffness of the adherends, shear stresses caused by rotation of the adherends, and shear deformation of the adherends.  K0  =  i + -^ + A_ G  2G,  K =J X  Eh x  x  +  12a)  2  J _ Eh 2  (  2G  a  2  +  M ^  +  4D  X  M ^ ± 0  ( 1 2 b )  4D  2  K =-^— + -^— EX E hl  (12c)  2  2  160  2G, 2 G  (12 d) 2  The constants for the peel equation follow and account for peel lag of the adherends and adhesive, adherend shear deflections, bending deflection, transverse adhesive stresses, and moments resulting from adhesive shear acting on the edge of the adherend. A l l terms that include v arise from the inclusion of the longitudinal adhesive stress, c r ^ . a  *1  * 4  =  2E.  Gh j  J  D,1  D ~ 2 f  =  2  J _  \  h,  1  yG,/i,  s  (12 e)  2E.T2 1  K  ' 2  0  (12 f)  2  (12 g)  h  /i, G  V  2  h  — + — D, D j  (12 h)  2  \  2  G  J  2vj.  (12 i) K  A  E  E  \[h +t x  2 2 J h  h +t  a  2  D  a  (12 j) J  2  These defined constants are now available for use in quantifying the parameters of the characteristic equation for the system as developed in the following section.  6.3.1 Solution and boundary conditions Introducing the trial functions <f> = Ae* and y/ = Be* , the characteristic equation for x  x  the system may be found from (11a) and (11 b).  Z -btf +cA -d = 0 6  (13)  2  The coefficients follow. ^_K K X  A  + KK 0  5  +KK 2  KK 0  KQK^  1  3  2  X  A  +  K^Kg  3  S  (14 a)  7  + K K% + K K 0  KK  -K K  4  KK  -  - K  5  3  K  (14b)  7  161  D  =  K  K  I  6  - K  K  2  KK0  9  (  L  4  C  )  KK  4  3  7  Setting A = & + b/3,the equation (13) may be expressed as a 'depressed' cubic. 2  /i3  b  , be n=a  2  ^  6 +m6 — n,  m—c  ,  3  3  2b  ....  1  (15)  3  27  The roots of the depressed cubic may be found as follows. Note that q is complex.  ^ + ^V4m +27n 2 18 3  (16)  2  m  —+ iV3| q+— 3q 3?  -  (17)  V  m q + 3tf V  / 3?  The roots of (13) are given by X = ±^0 + b/3 . The solutions for <> / and  follow. (18 a)  i  (18 b) The constants in (18 b) arise from the governing equations (11 a,b). KK 4  a  —  Q  KK 7  3  (19 a)  K K +^(K K +K K ) 3  6  3  KK 3  5  2  4  -K K  S  ]  - —  4  b=  (K K 3  +K K )  5  2  4  (19 b)  ± 3  K K +^(K K +K K ) 3  KT,K  9  6  3  +—  5  2  (K K  4  3  5  +  KK) 2  4  (19 c)  c= K K +^(K K +K K ) 3  6  3  5  2  4  162  The adhesive stresses follow easily from the definitions (10 a,b).  The axial adhesive stress may be calculated from (8 b).  E  E  +-d k + h r + T r  (21)  L  vA  *„=-r[*8*-*7*»]--r  A,  It remains to calculate the constants A, given the forces applied to each end o f each adherend. Here, we w i l l examine the case o f a very long joint, by keeping only those roots that vanish as x —> ° o . T o obtain the solution for a finite length of joint, one must retain all o f the eigenvalues and consider the forces at the far end o f the joint. F o r the very long joint, the boundary conditions may be expressed as follows. 7/,(0) = r,,  T (0) = T ,  7/(oo) = 7 / 7 /  V,(0) = V , ,  V (0) = V ,  V(oo) = V V  2  2  2  Af,(0) = A f , ,  1+  2  i+  M (0) = M , 2  (22a)  2  (22b)  2  M ( ° o ) = M , +M + S{T -T )+x{V +V ) (22c)  2  2  ]  2  x  2  The method of B i g w o o d and Crocombe w i l l be used to find the integration constants, thus developing the expressions relating the adhesive stresses to the applied loads. From the derivative o f the adhesive shear constitutive equation (8 a), with substitution for the adherend deflections from equations (4 a,b), the first equation is obtained as  K  o 0 x x  +  K  3 V x x = —  —  1  E\  3  x  Eh 2  2  —  2D, 2 D  •  ( ) 23  2  From the second derivative o f the adhesive peel constitutive equation (8 c), with substitution o f the adherend shear and bending deflections from equations (5,6) the second equation is obtained as  ~K  s¥ja  -  +  M M = -_ L + - 1 .  A  (24)  2  D  From the third derivative o f the adhesive peel constitutive equation (8 c), with substitution o f the adherend deflections from equations (5) and (6), and equilibrium equation (9), the final equation is obtained. F o r a very long joint with boundary conditions that satisfy equilibrium at x = 0 , equations (23)-(25) allow the definition o f a set of three linear equations that may be solved for the integration constants. For a finitelength joint, equations (23)-(25) must also be defined for the far end, increasing the number of equations from three to six.  163  V KAI/  -KAI/  41r ,xxxx  -K <f>-K d> Q  5r  +K d)  s  ,xx  9r  =—  %r,xx  V  (25)  + -^-  L  n  IT.xxxx  For loading systems that require the existence of forces or moments acting at a remote part of the joint (i.e. as x — > °°), a complimentary solution is required to satisfy the farfield boundary conditions. For the unbalanced shear condition (V, - V * 0 ) , the limit 2  solution has little physical meaning. Accordingly the complementary solution w i l l be examined in the absence of the shear forces. A s x —> °°, the loads carried by the adherends do not vary and adherend equilibrium (8) requires the shear and peel stresses to vanish. The adherend deformations simplify as follows.  h  x  Eh x  i.xx  2  M  2 D,  x  2,,  =  +  M  2  (26 a,b)  2  2  Eh 2  .Ml A'1  -  w  r h  M,  2  (27 a,b) A^2  Vanishing stresses force the adhesive constitutive equations to take the following form. f  u. — u = 0,  u  U  +  2 , x  U  ^  "2+VJa  2  = 0  (28 a,b)  These may be expressed in terms of the forces by substitution from (26) and (27).  h M_ h x  Eh 2  l  +  2  2 D  2D,  2  M  2  2  M  x  M  D  x  D  2 = Q  (29 a,b)  2  Equations (29 a,b) and the third parts of equations (22 a,b) form a set of equations from which the force and moment in the adherends may be determined. The results follow.  A±^V  ( O O ) +  M(OO)) ZM +  D D/  Z =•  l+  Eh  2  1  2  -+  1  h +h  f  +  x  2  ^  2  T =TM-T M,  (30 b)  X  A  =  M, =  iMM-S(T -T )) x  D +D  K  X  D,  A  (30 a)  yD +D ; x  2  2  (30 c)  2  2  (30 d)  - M. L  164  h,  h  7  Where S = — + t„ + — . The adhesive axial stress is given as follows:  hM  hM  2D,  2D 2 J  x  Eh 2  2  x  2  2  (31)  6.3.2 Thermal strains Thermal strains often arise in hybrid composite joints due to the elevated temperature required to cure the adhesive, and due to thermal loading during service. During the cooling phase of the cure cycle for thermally activated adhesives, the temperature w i l l drop below the adhesive glass-transition temperature, T,  and residual strains w i l l  develop. A s creep occurs near T , it is common practice to use an effective stress free g  temperature, T , to assist in the analysis o f a joint. The effective stress free temperature sf  T may be determined from experiments in which a specimen is heated until the warping sf  due to thermal strains is eliminated. Thermal strains may be calculated by adding an additional complementary solution to the homogeneous solution from previous sections. The thermal load, AT, is the difference between T  sf  and the service temperature. One  may calculate the interfacial adherend strains as follows.  h M x  x  3fy T !  Eh x  2,x  l  x 1  x  2  (32 b)  +- —2- + —^-*- + a AT 2 D2 8 G L  2  Eh 2  (32 a)  x  — —— + a AT 8 G,  2  2  Using this new definition of the extensional strains, the governing equations follow:  ~KJ  = -K y/- K 2  Ktf,^ ~ K  i¥xx  i¥xx  + (or, - a )AT  (33 a)  2  + K y/ = - K ^  + K0  6  s  <xx  +K0 9  (33 b)  The complementary solution requires consideration of polynomials of sufficient order to satisfy the forcing function. A s the temperature is uniform, it is sufficient for <f>, y/ to be constant, and the adhesive stresses w i l l vanish. The method used in the previous section for the remote loading case is applicable, with the adherend deformations described by (32 a,b). The forces are required to obey the following relationships.  M  M  D,  D  x  Eh j 2  2  h,—- + K  2  + {a -a )AT = 0 x  2  J  165  2  (34 a)  M,  M  2  =0  (34 b)  The remaining consideration is equilibrium. For unrestrained thermal expansion, the net force and moment acting on a cut through the joint must both vanish. T +T =O, X  M +M  2  L  + (T -T )S  2  L  (35 a,b)  = 0  2  A s in the previous section, equations (34) and (35) may be solved for the forces and moments in the adherends. a 1  T = -T = —  x  -a  (36 a)  -AT  2  _ h, + h  7  + s  1  2  D,  +D  2  -2D, 1  D, + D  (36 b,c)  D,  1 2  The axial adhesive stress may be determined from the axial strains.  T  1  .+.  T  hM hM  2  Eh  1  2  2  ±+  2D,  2  +  2  2D,  (a +a -2a )AT x  2  a  The thermal strains w i l l result in adhesive stresses and accompanying  (37) adherend  deflections at the ends of a finite length joint, where the thermal forces and moments on the adherends are required to vanish. These stresses and deflections can be calculated by applying the thermal loads in equations (36 a-c) as boundary conditions in equations (2325). The complete solution w i l l then arise by superposition of the homogeneous and complementary (thermal) solution.  6.3.3 Joint failure Given the techniques presented above, it is possible to calculate the strain energy release rate and stress intensity in the adhesive layer near the end of the joint. The total strain energy release rate is defined as follows, where dA is an increment in the area of the disbond or crack front.  G =^-(W-U) dA  (38)  T  Here, W is the potential of the externally applied loads, and U is the change in stored elastic energy. The standard calculation of G  T  for a very long bonded joint is easily  accomplished. Noting that the state of the joint is self-similar for an increment in disbond length, the energy released by crack extension may be determined by comparing a part of  166  the joint remote from the loaded end to the newly disbonded part. The segment far removed from the loaded end typically has no internal energy or loading, except in the presence of thermal residual strains, and the strain energy stored in the adhesive in the disbonded part is normally very small. G  T  is then given by the energy balance in the  disbonded part o f the adherends. l  T  T  2\  A  E  2  1  ,  1  G ——  Calculation of G  T  iK  E  1  j +— 2  yG,ft,  JJ )  +  Gh 2  2  1 {Ml ——  1A +  M\ \ —D  2  (39)  2  allows prediction of fracture or fatigue given test data for a joint with  a very similar composition and loading. In practice, bonded joints are sensitive to loading mode, i.e. the relative proportions of energy associated with adhesive shear or peel. Consideration of mixed mode loading allows extrapolation of test data to different load cases, but not between joints of differing geometry or composition, even i f the same adhesive is used. The reasoning is that the actual stresses and hence the stress intensity in the disbonding process zone are dependent upon the composition and geometry of the joint, including the adhesive thickness. Joint geometry plays a strong role in determining the actual stresses present in the part given a particular strain energy release rate. Accordingly, it is useful to partition the total strain energy release rate into opening and shear components.  G =G,+ T  G„  (40)  The peel and shear strain energy release rates may be calculated using the method of Joseph and Erdogan [9], which is very convenient for this type of closed-form analysis. In round-robin testing for the Cracked Lap Shear ( C L S ) specimen, this method has been shown to provide results comparable to those obtained by other means, such as finite element analysis employing crack-closure techniques. The C L S specimen exhibits mixedmode loading and non-linearity, and as such is a good test of the validity of the method. A description of the