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Prediction of fatigue damage progression in bonded composite repairs to aluminum aircraft structures Clark, Randal John 2000

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PREDICTION OF FATIGUE DAMAGE PROGRESSION IN BONDED COMPOSITE REPAIRS TO ALUMINUM AIRCRAFT STRUCTURES by RANDAL JOHN CLARK B. Eng., Carleton University, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering ) We accept this thesis as conforming to the required standard The University of British Columbia April 2000 © Randal John Clark, 2000 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of tkfJiAA^OoX £o.y*-ta,ri <j The University of British Columbia Vancouver, Canada Date 4pr/( Z7 . 2~Q&> DE-6 (2/88) Abstract The focus of this thesis is the development of a predictive model for fatigue damage progression in unidirectional bonded composite repairs of cracked isotropic plates. The principal use of this technology is in the design of repairs for aluminum aircraft structures. The ability to predict the rate of fatigue damage is critical to damage tolerance analysis of a repair. A damage tolerance analysis will allow designers to assess the design life, assign inspection intervals, and determine likely failure modes for a repair, and will be required for airworthiness certification for a long or indefinite period of operation. In this thesis, classical methods of bonded joint analysis are presented, and extended to the case of reversed plasticity of an internally pressurized lap joint. The crack bridging effect of the repair is examined using a boundary element method employing the Green's functions for a point load applied to a center-cracked plate. This results in a system of linear equations solvable by Gauss-Seidell iteration. The boundary element method allows calculation of the stress intensity and adhesive shear stresses under the bonded patch. These parameters govern the fatigue and static strength of the repair. The boundary element model combines engineering fracture mechanics and bonded joint analysis techniques in a very direct and straightforward manner. Results from the boundary-element model are compared to approximate analytical methods for a disbonding patch, and an improved analytical model employing correction factors is presented. Power law methods are then used to predict crack and disbond growth rates, which are compared to experimental results. The influence on patch life of various secondary effects, such as adhesive plasticity, process-induced thermal residual stresses, patch bending, shear deformation of the patch, and cracked-plate geometry are investigated. Based on this work, conclusions are drawn regarding patch behavior, limitations of modeling techniques, and experimental results still necessary to validate patch mechanics models. The techniques developed are also of interest in the study of cracking in fiber-reinforced metal laminates (FRML). i i Table of Contents A B S T R A C T » T A B L E OF C O N T E N T S i i i LIST OF T A B L E S vi LIST OF F I G U R E S vi i LIST OF S Y M B O L S x 1. I N T R O D U C T I O N 1 1.1 B A C K G R O U N D 1 1.2 DEFINITION OF A B O N D E D COMPOSITE R E P A I R 2 1.3 D A M A G E T O L E R A N C E A S S E S S M E N T OF A REPAIR 3 1.4 THESIS SCOPE A N D OBJECTIVES 4 1.5 O V E R V I E W OF THESIS 5 1.6 S U M M A R Y 7 2. L I T E R A T U R E R E V I E W 8 2.1 O V E R V I E W 8 2.2 B O N D E D JOINT A N A L Y S I S 8 2.3 C R A C K E D P L A T E A N A L Y S I S 13 2.4 P A T C H STRESS A N A L Y S I S IV 2.5 P A T C H F A T I G U E P E R F O R M A N C E 2 2 2.6 CRITICAL A N A L Y S I S 2 4 2.7 S U M M A R Y 2 7 i i i 3. B O N D E D JOINT A N A L Y S I S 28 3.1 OVERVIEW • 28 3.2 CLASSICAL MECHANICS OF BONDED JOINTS 28 3.3 FATIGUE AND FRACTURE OF BONDED JOINTS 3 0 3.4 SUMMARY 34 4. C R A C K B R I D G I N G A N A L Y S I S 36 4.1 OVERVIEW 36 4.2 T H E CRACK BRIDGING PROBLEM - INTERPOLATION SOLUTION 3 6 4.3 T H E CRACK BRIDGING PROBLEM - BOUNDARY ELEMENT SOLUTION 38 4.4 COMPARISON OF BOUNDARY ELEMENT AND INTERPOLATION MODELS 42 4.5 SUMMARY 53 5. P A T C H F A T I G U E A N A L Y S I S 55 5.1 OVERVIEW 55 5.2 CRACK-OPENING STRESS 55 5.3 FATIGUE D A M A G E PROGRESSION 63 5.4 LOAD INTERACTION AND OVERLOAD EFFECTS 64 5.5 SUMMARY 71 6. D A M A G E P R O G R E S S I O N : C O M P A R I S O N T O E X P E R I M E N T S 72 6.1 OVERVIEW 72 6.2 A M R L SPECIMEN 72 6.3 W P A F B SPECIMEN 74 iv 6.4 SUMMARY 77 7. DISCUSSION 78 7.1 OVERVIEW 78 7.2 RESULTS 78 7.3 EXPERIMENTAL VALIDATION 80 7.4 CERTIFICATION 85 7.5 D A M A G E TOLERANCE ASSESSMENT METHOD 87 7.6 SUMMARY 90 8. CONCLUSIONS .....92 9. RECOMMENDATIONS FOR FUTURE WORK 95 10. BIBLIOGRAPHY 97 APPENDIX A: CRACK BRIDGING MODEL WITHOUT A DISBOND 104 APPENDIX B: CRACK BRIDGING MODEL WITH A DISBOND 107 APPENDIX C: GREEN'S FUNCTION - POINT LOAD APPLIED TO THE CRACK FACE Il l APPENDIX D: GREEN'S FUNCTION - POINT LOAD APPLIED AT AN ARBITRARY LOCATION 114 APPENDIX E: THE FORTRAN PROGRAM COMPARE 119 APPENDIX F: THE FORTRAN PROGRAM DAMAGE 125 APPENDIX G: AMRL SPECIMEN DIMENSIONS 145 APPENDIX H: MATERIAL PROPERTIES 146 v List of Tables TABLE 4 - 1 : COMPARISON OF INTERPOLATION AND BOUNDARY ELEMENT MODELS BY RELATIVE ERROR 52 vi List of Figures FIGURE 1-1: PATCH SCHEMATIC DRAWING 2 FIGURE 2 -1 : SINGLE LAP JOINT 9 FIGURE 2-2: POLYMER CRACK TOUGHENING MECHANISMS 11 FIGURE 2 -3 : CRACK GROWTH PROCESS 14 FIGURE 3-1: LAP JOINT FREE BODY DIAGRAM 2 9 FIGURE 3-2: L A P JOINT SHOWING REVERSED PLASTIC ZONE 3 0 FIGURE 3-3: LAP JOINT WITH DISBOND '. 3 2 FIGURE 4 -1 : GREEN'S FUNCTION POINT LOAD CONFIGURATION 40 FIGURE 4 -2 : PATCH SCHEMATIC DRAWING SHOWING LOAD TRANSFER AROUND CRACK 41 FIGURE 4 -3 : STRESS INTENSITY FACTOR RESULTS COMPARED TO INTERPOLATION M O D E L 43 FIGURE 4-4: CRACK F A C E DISPLACEMENT RESULTS COMPARED WITH INTERPOLATION M O D E L ...44 FIGURE 4 -5 : PERCENT ERROR COMPARISON 44 FIGURE 4-6: ELLIPTIC DISBOND SHAPE 45 FIGURE 4 -7 : STRESS INTENSITY FACTOR RESULTS WITH DISBONDS 46 FIGURE 4-8: DISPLACEMENT RESULTS WITH DISBONDS 47 FIGURE 4-9: DISBOND-EDGE DISPLACEMENT 48 FIGURE 4-10: ADHESIVE SHEAR STRESS DISTRIBUTION WITH VARIOUS AMOUNTS OF DISBONDING49 FIGURE 4 -11 : ADHESIVE SHEAR STRESS DISTRIBUTION FOR A LONG CRACK 49 vii FIGURE 4 -12 : COMPARISON OF BOUNDARY ELEMENT AND INTERPOLATION CRACK BRIDGING MODELS 51 FIGURE 5-1: PATCH SCHEMATIC DRAWING 56 FIGURE 5-2: SUPERPOSITION AND THE TWO-STEP METHOD OF ANALYSIS. ONLY THE UPPER HALF OF THE PATCH IS SHOWN 56 FIGURE 5-3: COORDINATE SYSTEM FOR BIMETALLIC STRIP ANALYSIS 59 FIGURE 5-4: A BENDING FINITE REINFORCEMENT 61 FIGURE 5-5: CRACK TIP PLASTIC DEFORMATION 65 FIGURE 5-6: STRESSES NEAR THE CRACK TIP 68 FIGURE 5-7: CYCLIC STRESS/STRAIN CURVES FOR NEAR THE CRACK TIP 6 9 FIGURE 5-8: RELAXATION VS. CURE TIME 70 FIGURE 6-1: COMPARISON OF INTERPOLATION M O D E L AND EXPERIMENTAL RESULTS, WITH A SMALL DISBOND '. 73 FIGURE 6-2: COMPARISON OF INTERPOLATION M O D E L AND EXPERIMENTAL RESULTS, WITH A LARGE DISBOND 73 FIGURE 6-3: COMPARISON OF INTERPOLATION M O D E L AND EXPERIMENTAL RESULTS, W P A F B SPECIMEN 75 FIGURE 6-4: COMPARISON OF BOUNDARY ELEMENT M O D E L AND EXPERIMENTAL RESULTS, W P A F B SPECIMEN 76 FIGURE 7-1: CLIP GAUGE AND POTENTIAL DROP PROBE CONFIGURATION 82 FIGURE 7-2: VOLTAGE RATIO PREDICTED BY JOHNSON'S EQUATION 83 FIGURE 7-3: DAMAGE TOLERANCE ASSESSMENT FLOW CHART 88 viii FIGURE C - 1 : LOADING CONFIGURATION 111 FIGURE C - 2: DISPLACEMENT RESULT FOR UNIFORM OPENING PRESSURE 113 FIGURE C - 3: COMPARISON TO FINITE ELEMENT ANALYSIS 113 FIGURE D - 1 : LOADING CONFIGURATION 114 FIGURE D - 2: Y-DISPLACEMENT UNDER POINT-SYMMETRIC LOADING 117 FIGURE E - 1 : A M R L SPECIMEN DIMENSIONS 145 ix List of Symbols Symbols <J,T F = X + i-Y G,, Gn, Gm K,, K„, Km a,b,t Normal stress, shear stress A complex stress function, its derivative, and its indefinite integral A complex point load Crack driving force Stress intensity Geometries — Half crack length, length of disbond, thickness z — x + i • y, z — x — i • y A complex coordinate, and its conjugate u , or u + i • v Displacement Patch load transfer length Patch stiffness ratio or an engineering strength A G E v I(z) = K 3 - 4 v 3 - v A characteristic length representing the stiffness of a lap joint Shear modulus Elastic modulus Poisson's ratio r~2 2 yz - a Re(z) > 0 An elliptical coordinate. Conditional function fully accounts for - 4z2 - a2 > Re(z) < 0 multivaluedness of any complex solution in this report. I 1 + v plane strain , plane stress A parameter allowing results to apply to plane stress or strain Subscripts for Symbols co a u 0 r P b d An asymptotic limit value Remote applied load, or a property related to the adhesive Upper bound Nominal, or resulting from nominal loads Reinforced, or a property of the reinforcement Property of the cracked plate or substrate Including the effects of crack bridging Including the effects of disbonding xi 1. Introduction 1.1 Background Over the last few decades, economic pressures have been the impetus for a fundamental change in life management of airframes. Safe life methods have given way to damage tolerant design and engineered inspection and maintenance programs based on risk assessment and fracture analysis. These programs aim to ensure that cracks in primary structure will not grow undetected to a critical size. As a result, many airframes are lasting well beyond their original design life. In conjunction with this change in life management strategy, industry has developed new technologies for inspection and maintenance of aircraft. In the 1970's, the Australian Aeronautical and Maritime Research Laboratory (AMRL) pioneered the use of bonded boron/epoxy composite repairs. Here [1], composite doublers bonded over a weakened or cracked metal structure reduce stresses in the underlying structure and restrict crack opening. These effects retard crack growth and, upon re-initiation, reduce growth rates from those experienced before the repair. Bonded repairs have several advantages over riveted repairs. They eliminate the need to drill holes, are lighter weight, have a lower profile, are not susceptible to galvanic corrosion, and provide smoother load transfer. This technology offers the possibility of significant gains in both economy and safety. Several issues prevent widespread use of bonded repairs. These include stringent surface treatment and material controls to achieve bond durability, environmental and loading rate sensitivity of structural adhesives, possible high thermal residual stresses, and the difficulty of analyzing complex configurations. The issue of assurance of bond durability will require either strict quality assurance and material control requirements, or the development of an inspection method for bond durability. Certification will require an initial assessment of the repair static strength, allowing short term operation, followed up by a full damage tolerance assessment [2, 3]. Thus there is the need for a simple analytical method for damage tolerance assessment, suitable for adoption in industry codes and standards, which can account for disbonding, temperature and moisture effects, loading rate, and thermal residual stress effects in a conservative fashion. 1 1.2 Definition of a Bonded Composite Repair A bonded composite repair consists of a boron/epoxy or graphite/epoxy patch, bonded to a weakened or cracked structure. Patch fibers are generally oriented perpendicular to the loading direction, or crack, to provide maximum reinforcement. The patch attracts load from the surrounding structure, effectively reducing the stress in the underlying plate. Load transfer occurs predominantly at the edges of the patch and around the crack, and is transferred through shear stresses in the adhesive. Thus, crack growth is reduced in two ways; by reducing the stresses in the region of the crack, and by restricting crack opening. Figure 1-1: Patch Schematic Drawing The process of applying a patch involves several steps. The surface to be repaired must be carefully cleaned and chemically treated to achieve maximum bond strength and durability. The individual plies of the repair must be cut from a pre-impregnated composite fabric, and either formed and cured before being bonded to the structure, or cured concurrently with the bonding process. The first process results in a pre-cured repair, and the second in a co-cured repair. A pre-cured repair offers the possibility of reduced thermal residual stresses induced by the mismatch in thermal expansion coefficients between the patch and the underlying plate. These process-induced stresses can be a significant detriment to the efficiency of a patch. Bonding to the weakened structure may be achieved in-situ, using a vacuum bag and localized heating or by 2 placing the entire component in an autoclave. Mechanical restraint against thermal expansion can be used to reduce thermal residual stresses. Baker [1, 73] provides a good overview of method of application, advantages, and disadvantages of the crack patching technology. In this thesis, the terms reinforcement or doubler will refer to a bonded composite joint applied to strengthen an undamaged structure. A repair or patch will mean a bonded composite joint applied to a weakened or cracked structure. A n adherend may refer to any component which is bonded to any other component. 1.3 Damage Tolerance Assessment of a Repair A damage tolerance assessment is a procedure by which a structure may be evaluated regarding its ability to function in a weakened or degraded state. This process allows for rational control of the risks associated with operation of a degrading structure, both through optimized design and by selection of appropriate measures to control risk during operation. The requirements for a damage tolerance assessment depend on the purpose of the structure. Redundant, fail-safe, or non-safety critical structures may not require a stringent damage tolerance analysis or operational safety controls. Primary structures operated in an application with large consequences of failure may require all of the following measures to control risk; definition of a safety goal for the structure, identification of loads or stresses acting on the structure, an assessment of the likely initial state of the structure, a means of analysis for the rate of damage progression under the applied loads, and a means of determining the failure state of the structure. These can be combined to predict the useful life of the structure and plan operational safety measures such as scheduled inspections and repair or replacement of parts. A bonded composite repair may fail by disbonding, by failure of the patch itself, or by failure of the underlying plate. Disbonding may initiate due to a bond-line defect formed during patch application, or due to alternating adhesive stresses at the edges of the patch or near the underlying crack. Disbonding caused by alternating loads may occur in the bulk of the adhesive, at the interface of the composite matrix and the layer of fibers closest to the cracked plate, or the interface between the adhesive and plate. Failure of the patch itself is unlikely, because of the relatively high strength of the patch material, however inter-laminar fatigue cracking has, on occasion, been observed, usually emanating from the ply drop-offs at the edge of the patch [100, 3 101]. Failure of the underlying plate may occur due to growth of the repaired crack or due to initiation of a new crack near the edge of the patch. For a bonded composite repair applied to the primary structure of an aircraft, all of the requirements of a damage tolerance analysis will be necessary to allow long-term operation. A l l of the above noted failure mechanisms must be taken account of, as must the effects of loading conditions and environment. As will be demonstrated at a later point, one of the primary missing components is the ability to predict the rate of degradation of a patch, particularly given the complex geometry, and loading of a real-life application. A straightforward, conservative means of life prediction for composite patches is necessary to allow certification and wide-spread use of the technology. The next section describes the objectives of this thesis in light of the requirements for a damage tolerance analysis of a composite repair. 1.4 Thesis Scope and Objectives The ultimate objective of the University of British Columbia composite patch repair program is to move this beneficial technology toward airworthiness certification and wide-spread use. The objective of this thesis is to develop and validate methods for the prediction of fatigue degradation of bonded boron-epoxy repairs to cracked aluminum plates, and to implement this method using engineering bonded joint and fracture mechanics techniques. In the framework of a damage tolerance analysis for a bonded composite repair, this will allow assessment of patch failure modes and life, allowing the determination of suitable inspection intervals, and a rational assessment of the risks involved in the use of bonded repairs. The scope of the work will be limited to the prediction of the rate of crack propagation and near-crack disbonding, as these are the elements which define a patch repair as opposed to a reinforcement. Other failure modes of a patch repair may be analyzed using the more general methods applicable to a bonded doubler or reinforcement. The issue of assurance of bond durability during patch fabrication is also left outside the scope of this thesis, as it is primarily the concern of regulatory bodies and repair facilities, and outside the scope of academic study, except, perhaps, the development of a method for detecting low durability bonds. The issues of adhesive performance under environmental loading, and loading rate sensitivity are 4 enhancements which can be included within a damage progression model at a later stage, and are also not considered in this thesis. Thus, the scope of this thesis will be the prediction of bonded repair performance under nearly constant and moderate temperature, humidity, and loading rate conditions, with the bond assumed to be well-manufactured. The following sub-objectives must be met to enable prediction of the fatigue performance of a repair; (a) Development of a model to predict fatigue damage progression in a patch-repaired structure The model will be a synthesis of existing models for fatigue crack re-initiation and growth in aluminum plates, disbonding of adhesive joints, and traditional bonded patch analysis. Disbonds impair patch efficiency [5], and a model combining the effects of crack and disbond growth will improve design and life assessment of boron-epoxy patches. The focus is to develop means of analysis which are amenable to inclusion in codes and standards, but which are sufficiently rigorous and accurate to provide either accurate or appropriately conservative results when compared to experiments. Such a means of analysis will be a significant contribution toward airworthiness certification of the bonded composite repair technology. (b) Validation of the model based on experimental results To reduce duplication of effort, results from experiments in the open literature will be used to validate the model wherever possible. No new experimental results will be presented in this thesis, but a detailed testing program directed toward specific needs of the patching community will be developed. The test program will address any weaknesses in the models by direct measurement of relevant parameters and by specific tests directed toward shortcomings in the available pool of knowledge. The next section describes the layout of this thesis, from which it should be apparent that the objective, quantification of the progression of fatigue damage in a composite repair, will be met. 1.5 Overview of Thesis In subsequent chapters, a framework for analysis of damage progression in a bonded composite repair is developed, models are applied to predict the performance of the repair, and the results 5 are compared to existing experimental data. This framework and the underlying modeling of physical processes are based both on previous researcher's work and original work, and are fully explained in this thesis. The differences between predicted and observed results are noted, and the sources of discrepancies are discussed. Following is a breakdown of the thesis by chapters. Chapter 2 contains a literature review of existing analysis techniques and other relevant background information regarding fracture mechanics and bonded joint technology, and describes how other researcher's work relates to this thesis. The chapter concludes with a description of how the objectives of this thesis will fulfill a demonstrated need regarding airworthiness certification of bonded repairs. Chapter 3 describes the classical method of analysis for a bonded joint, which is extended to the case of reversing plastic fatigue loading of an internally pressurized lap joint. Methods of predicting the stiffness, static strength, and rate of fatigue degradation of a bonded joint are presented, and subsequently applied to the prediction of the bond performance of a composite repair. Chapter 4 describes fracture mechanics principles relevant to the analysis of a bonded repair to a cracked plate. The crack-bridging problem is introduced, and a boundary-element solution is developed which employs Green's functions. An approximate analytical solution, based on interpolation between limit states, is also explained, and it's use in the case of a disbonding patch is discussed. The results of the boundary-element and interpolation models are compared, and corrections to the interpolation solution are developed which retain the accuracy of the boundary-element solution. In Chapter 5, the remaining elements for the analysis of a bonded repair are assembled. Methods for evaluation of the stresses in the plane of the crack are presented, including consideration of thermal residual stresses and bending. The problem is expressed in terms of the crack-bridging problem solved in Chapter 4. Fatigue crack growth retardation is also discussed, including the effect of the cure cycle and localized annealing of crack-tip residual stresses. Chapter 6 compares the results of the models developed in Chapter 5 to experimental results from the open literature. The effect of different bonded repair parameters and secondary effects such as cracked plate geometry, bending, and thermal residual stresses on the performance of the patch are investigated. Chapter 7 contains a discussion of the results, and the possible sources of 6 discrepancies between predicted and observed results. It is demonstrated both that the thesis objectives have been met, and that future work is necessary to meet the requirements for airworthiness certification. In Chapter 8, conclusions are drawn regarding patch behavior and the capabilities and limitations of current analysis techniques. The appendices contain mathematical derivations, computer code listings, a sketch of the geometry of the A M R L and specimen, and a listing of relevant material properties. 1.6 Summary This chapter has explained the composition of a bonded composite repair and introduced some basic definitions and concepts regarding repairs and damage tolerance of structures. Based on the defined needs for a full damage tolerance assessment of a repair, the objectives of this thesis have been formed, and an outline of the thesis has been prepared to demonstrate how the objectives will be met in subsequent chapters. It has been established that prediction of the rate of damage progression in a repair is critical for a full damage tolerance assessment and hence airworthiness certification of a repair for long-term operation. 7 2. Literature Review 2.1 Overview In this chapter, a summary of literature relevant to prediction of fatigue damage progression in a bonded repair is presented. The review is limited to discussion of crack growth and near-crack disbond growth, as these are the modes of fatigue damage progression which are special to a repair, as opposed to a reinforcement. The first two sections discuss methods for predicting fatigue damage progression in bonded joints and cracked plates, establishing the standard engineering practice for these two mechanical systems. Following this, an overview of current modeling techniques for repairs is presented. Next, a review of available experimental data on damage progression in patches is presented. The chapter is concluded with a critical analysis of the methods available in the open literature, and a summary describing the limitations of the available techniques and how these limitations will be addressed in subsequent chapters. 2.2 Bonded Joint Analysis Mechanics of Bonded Joints In adhesive joints, the adhesive transfers a load between adherends. Adhesives have lower stiffness and strength than most engineering materials, thus requiring loads to be transferred through a larger area. This larger area of load transfer has the benefit of reduced stresses when compared to mechanical attachment methods such as bolting or riveting. Because strain in the adherends decreases as the load sheds from the adherend to the adhesive, there is a large mismatch in strains in the adherends, and a corresponding peak in adhesive shear stresses at the edges of the joint. Most load transfer occurs in this region. Methods to increase the load transfer length and reduce the peak stress in the adhesive include; tapering adherends, reducing adhesive shear modulus, and increasing adhesive thickness. Outside the load transfer region, the strain in the base plate and reinforcement are nearly equal, and the adhesive is not greatly stressed. This region is referred to as the 'elastic well'. It provides a buffer against adhesive creep under sustained loads and will increase the useful life of the bond under progressive disbonding. 8 Figure 2-1: Single Lap Joint Analytical results for the adhesive shear stress distribution are available for simple joint configurations. The original solution to the problem was developed by Volkerson [6]. Hart-Smith [7,8] provides analytical elastic and elastic-plastic solutions for adhesive shear stresses and strains, and adherend stresses for several joint configurations. Hart-Smith advocates the use of adhesive strain energies to determine joint strength. Similar calculations are applicable to the stress analysis of the load transfer in bonded composiie patches. Albat and Romilly [9] have developed a technique to find adhesive stresses in composite doublers in the region where ply drop-offs reduce the stiffness of the reinforcement in discrete steps. These solutions are based on the one-dimensional theory of joints, and yield approximate solutions which are suitable for simple joint configurations and loading conditions. The standard methods of lap joint analysis mentioned above include assumptions that are difficult to justify from a solid mechanics viewpoint [10,12], but are satisfactory for estimation of shear stresses in the adhesive. These solutions ignore the boundary condition which necessitates zero shear stress at the ends of the joint. Goland and Reissner [11] have developed solutions which include these boundary conditions. One additional limitation is that the standard relations are not useful for estimating peel stresses. More recently [12], Wang and Rose have developed approximate closed-form solutions that do not make any such assumptions and are useful for estimating peel stresses and examining the three dimensional stress state of the adhesive. In any case, engineering methods of bonded joint analysis will be approximate because of the visco-elastic or visco-plastic nature of adhesives, and the dependency on environmental loading such as temperature and moisture. Visco-elastic or visco-plastic analyses are difficult to apply outside a research setting and are even more difficult to employ in a regulatory setting. The consideration of time-, temperature-, and moisture-varying adhesive properties requires a thorough understanding of the adhesive properties and specialized solution techniques. The analysis of elastic-plastic adhesive behavior provided by Hart-Smith is the most accessible manner by which to include a first-order assessment of nonlinear adhesive behavior. For the adhesive most commonly used for repairs, i.e. FM-73, approximate values of the adhesive stiffness and shear yield strength are available for a wide range of operating conditions [13]. Results of methods which do not consider the varying properties of adhesives will only be valid for a limited range of conditions. Fatigue Mechanisms When compared to metals, mechanical models for fracture and fatigue of adhesives are not well defined. Two methods of crack growth exist for the bond ~ cohesive and adhesive. Cohesive failure means that the adhesive fails before the bonds between the adhesive and the adherend. Adhesive failure means a failure in the bond-line. When composite materials are bonded, the situation is complicated by the additional interface between the composite matrix and fibres. In terms of Mode I fracture energy, failure along this interface is favored [5]: crack growth along the interface may require 230 compared to over 3000 for cohesive or adhesive failure. m m In practice, it is found that the adhesive failure mode occurs principally due to a poorly manufactured bond. A n interfacial failure [14] is thought to occur when stresses are highly concentrated at the interface due to the use of a very thin or high modulus adhesive. Fracture mechanics can describe cohesive failure. Instead of inter-atomic bonds in metals, in an adhesive the strength of the material is due to molecular and inter-molecular forces. As a result, the adhesive often behaves visco-elastically or visco-plastically, exhibiting strain-rate and temperature sensitive behavior. Elastic solutions may be adequate at low temperature and low strains. For adhesive failure, chemical, thermodynamic, electrostatic, and fluid considerations are as important as solid mechanics. Bond-line strength comes from: chemical bonds (van der Waals and Lewis acid-base); "mechanical hooks" into the adherend surface roughness; electrostatic forces; and diffusion. With the proper design of the adhesive joint and bonding procedure, adhesive failures should not occur unless manufacturing errors, such as improperly prepared surfaces, result in a low bond-line strength. For specimens developed under this research program, great care was taken in the preparation of the specimens to be tested in this project, and adhesive failure is unlikely. 10 Interfacial failure may be described by fracture mechanics, but is complicated by both the large dependence of adhesive properties on testing conditions, and the presence of two materials with different properties. Analytical linear-elastic methods of analysis exist for the case of a bi-material strip with a crack on the interface, and could likely be applied to this problem. Similar to the analysis of crack growth in metals, adhesives have an energy associated with an increment of crack growth. For rubber-toughened polymeric adhesives, several mechanisms exist to absorb energy during crack growth [15, 19, 76]. Shown in Figure 2-2, they are; (a) stretching, disbonding, and fracture of rubber particles and other inclusions, (b) formation of shear bands and crazing in the adhesive, (c) diffuse shear yielding, and (d) plastic deformation at the crack tip. At high temperature, creep deformation is another energy absorbing mechanism that must be considered. So, in contrast to metals, elastic methods of analysis cannot be relied upon except when dealing with high cycle fatigue or very glassy adhesives, where the effect of these energy absorbing mechanisms is reduced. More generalized methods [16, 17] are required to account for non-recoverable losses associated with creep, plasticity, hygrodynamic effects, etc. Figure 2-2: Polymer Crack Toughening Mechanisms This research deals with high cycle fatigue of adhesives in a lap joint configuration. Adhesive stresses are low except in the load transfer region near the crack faces. Test conditions will be such that thermal and creep effects are negligible. Prediction of crack growth should not require generalized fracture-mechanics approaches. The effects of mean stress will be significant, due to 11 thermal residual stresses from curing the patch, and the limitations of the specimen geometry, and as such should be included in any analysis. Similar to metals, fatigue in adhesives may be characterized by the S-N curve, a curve plotting stress against cycles to failure [18]. Reports suggest a fatigue endurance strength as low as 30% of the static strength for some adhesives, with a typical toughened adhesive having a fatigue endurance strength of approximately 50% of the ultimate shear strength. Some researchers suggest that no endurance limit exists for adhesives. This technique may be applicable to prediction of disbond initiation. The S-N relation is; log 5 = 1 . - 3 l 0 g | r0.9S, ^ log N + log (0.9SJ 2 (2-1) At this point in the research program, Se will be assumed to be simply related to the ultimate strength of the adhesive. It may prove that coupon testing will be necessary at a future point in this research program to better estimate this parameter. A study of the effects of environment and loading rate on Se may also be necessary to predict disbond initiation under arbitrary conditions. For constant amplitude high cycle fatigue crack growth, an equation similar to the Paris equation for metals may be used. In this equation, n is between 4 and 4.5 for most adhesives. The following illustrates the form of this equation [19]; da dN = C(AG)" (2-2) Where A G = A G , + AG,, + AG m • For adhesive lap joints, the strain energy release rates, A G , and A G / 7 relate to peel and shear stresses respectively. This relationship is applicable to disbond growth and can be incorporated into an engineering lap-joint analysis. The strain energy release rate is the rate at which the stored elastic energy in the structure changes with an increment in crack growth. The strain energy release rate has been criticized [21-23] as a parameter for fracture and fatigue of bonded joints. Typically, it provides good results for fatigue when applied to the geometry from which the empirical fit was derived. Fundamentally, the use of a fracture parameter for an adhesive is complicated by the effect of the thickness of the adhesive, which invalidates the similarity criterion for analysis of cracks. One of the principal assumptions behind fracture 12 mechanics is that cracks will have a similar stress distribution regardless of crack length, which is not the case with an adhesive joint. This may necessitate a detailed assessment of the elastic/plastic strain state of the adhesive at the tip of the crack. Hart-Smith [24] advocated an energy density criterion for static strength of adhesive joints which has been extended to provide design guidelines for the case of rate-dependent response of joints. This appears to be the best method of analysis for an adhesive joint because it explicitly considers the strain state of the crack tip, and has been used [4] to develop damage tolerance criteria for bonded joints. 2.3 Cracked Plate Analysis Mechanics of Cracked Plates In examining the mechanics of a cracked plate, one is primarily concerned with the stress distribution at the tip of the crack. The cracked plate is a particular instance of a planar boundary-value problem, which in the case of plane elasticity requires the solution of the bipotential equation in terms of stresses and strains. Several means exist to solve this problem. The first accurate stress analysis technique for a cracked plate was developed by Inglis [25], who obtained the exact solution for the stresses around an ellipse, and then examined the limit state of an infinitely thin ellipse. Muskhelishvili [26] popularized a very general complex variable solution of the cracked plate problem, using the Goursat-Kolosov formulation of the bipotential equation in terms of normal and shear stresses. Westergaard [27] simplified the Goursat-Kolosov formulation by considering a limited subset of crack problems. Here, only one stress function satisfying the boundary conditions is required. This technique has greatly increased the accessibility of fracture mechanics solutions. Other researchers have used Fourier transform techniques, for example Sneddon [28]. The principal result of all analyses of cracks is the severe singularity that occurs at the crack tip. Irwin [29-31] showed that the rate of increase of the stresses near the crack tip could be characterized by a single constant, the stress intensity factor. Based on the work of Griffith [32], in which an energy balance criterion was developed for fracture of brittle materials, the stress intensity factor was shown to be a good predictor of fracture strength of cracked plates. The principal limitations on this technique are that the strain state of the crack tip must be mainly elastic, and that the similarity criterion are satisfied. The similarity criterion are that the crack 13 sees the same loading, characterized by the stress intensity factor, and is in the same material and environment. Paris [33] showed that the stress intensity factor is also a good parameter for predicting the rate of fatigue crack growth in metals. Fatigue Mechanisms The fatigue process has three stages; initiation, growth, and final failure. Initiation is the stage before the service loading and environment create a crack of a finite length. For composite repairs, fatigue cracks already exist in the structure, so this review will not deal with initiation. Fatigue crack growth is the stage from first appearance of a finite length crack to final failure. The following is a brief review of standard engineering practice for predicting fatigue crack growth in metals. A n excellent comprehensive review [34] is available which explains the history and standard engineering practice for analysis of fatigue analysis. In one widely accepted qualitative model [35] for fatigue crack propagation, crack opening loads cause slip along a preferential plane in the direction with the highest shear stress, increasing the crack opening and length. Figure 1 (a), (b) and (c) illustrates this concept. The preferential plane is the one requiring the least energy to slip, and will change due to work hardening and increased stresses. Figure 1(d) shows the new slip bands, while (e) and (f) show a subsequent loading cycle and increment in crack length. This implies that fatigue crack growth is related to the alternating plastic strains at the crack tip. For long cracks, bulk plasticity also becomes important. For crack growth near threshold, environment and mean stress play an increasing role. Figure 2-3: Crack Growth Process 14 For constant amplitude loads, engineers often use the empirical power-law relation proposed by Paris [33] to predict fatigue crack growth. This works well for moderate loads, and modified forms exist [36, 37] to account for mean stress and the presence of a threshold load at which fatigue cracks will not grow. Predictions based on this power law equation break down at high loads due to loss of crack self-similitude and large strains. For low loads, environment, oxidation, and debris effects come into play. The general form is: % ) - C W (2-1) A repair patch reduces the loading on the crack. An understanding of load interaction effects is necessary to predict crack growth in the repaired structure. Plastic deformation from the load applied before patching may leave compressive residual stress at the crack tip. This stress acts to close the crack and reduce damage caused by subsequent loading. Experimental results show a period of reduced crack growth, often followed by a period of accelerated growth, and then a gradual return to the nominal rate. If loads subsequent to the overload are small enough, the crack will not open, and growth will be retarded or delayed. Wheeler [38] developed an empirical relation for crack growth following an overload. After an overload, an enlarged plastic zone exists ahead of the crack. Wheeler stated that crack growth rates are reduced by an amount determined by a "retardation factor" while the crack grows through this plastic zone. The following relations apply; I 1 =A 1 (2-2) RETARDED CONSTANT where <f> •• f V r pi for (a, + rpi) < (a0 + rp0), otherwise (j>=\. Willemborg [39] developed a model based on the overload plastic zone size. He assumes that loads applied subsequent to an overload will not cause crack growth unless its plastic zone extends beyond that caused by the overload. As the crack grows into the overload plastic zone, the stress intensity required to cancel the residual stresses, Krequjred, decreases. This leads to the following effective stress intensities; K = 2 K — K (2-3^ max.ejfeclive max required V / 15 K = K + K — K ('2-4) mm.effective min max required V ' While in good agreement with tests, this model is limited by the assumption that the remaining overload plastic zone fully characterizes the compressive stresses. Elber [40] has suggested that for intermediate growth rates, da I dN depends on crack closure. He defines the following effective stress intensity, where Kop is the stress intensity required to open the crack. Keff = ^applied ~ Kop (2_5) Over the last two decades, these ideas have been extended to include more accurate predictions of the plastic zone size, constraint, and the effects of strain hardening. Researchers have developed relations for the adjustable parameters ~ n, Kretluired, and Kop — allowing crack growth prediction for a particular material and application using the Paris power law equation. McCartney [79] was the first researcher to use the Dugdale strip yield approach to study closure. Four years later, Newman [80] developed a strip yield model to find Kop. Programmed in the code FAST-2 (now FASTRAN), this model uses an effective stress intensity range based on the Elber formulation to predict retardation. The model assumes an elastic-perfectly plastic material, and uses a constraint factor and a flow stress to include thickness effects. It has been compared to experimental results [81], and needs additional development. Particularly, the model does not include mean stress effects, and the model is sensitive to small changes in the estimate of plastic constraint. Clayton [82] developed a model similar to Newman's, but improved the estimation of the effect of thickness on plastic constraint. This allowed the model to predict crack growth for an arbitrary thickness and improved results for high overload ratios. Concurrently, Russell [83] used a complex variable method originally developed by Budiansky and Hutchison [84] and a Dugdale strip yield model to include variations in the plastic zone size as the crack proceeds through the overload plastic zone. Ibrahim [85] has found new empirical relations for mean stress effects in 2024-T3 aluminum. When used with the crack closure model, intermediate and near-threshold da I dN data are compressed onto one curve. Here, S and Smax are applied stresses. 16 ( Y ^ ) « , o = ( | ^ ) ^ o - 0 - ^ ) 2 + ^ (2-6) max "max ("^  )fl<0 = ) « = 0 ' 0 _ ( ) ) max max y Ibrahim proposes these as a replacement to the following equation, developed by Schijve [86], and commonly used in fatigue crack growth models. ~ - = 0.45 + 0.2R + 0.25R2 + 0.1R3 (2-7) max Pantelakis [87] developed a fatigue crack re-initiation model based on a strip plastic zone and a material hardening effect. Following an overload, a maximum plastic strain exists near the crack tip, and decreases to zero at the edge of the plastic zone. Materials that soften or harden under plastic strain will have a yield stress that varies as the crack grows through the overload region. He used this effect to predict variation in the Wheeler retardation factor with crack growth. Crack closure and residual stresses due to crack tip plasticity are widely considered to be the phenomenon which cause retardation of fatigue cracks. The analysis is complicated by thickness, plastic constraint, debris, loading rate, and environmental effects. In Chapter 5, these concepts will be used to form the basis of a relationship which may be used to determine the effect of the cure cycle on crack growth retardation following the application of a patch. 2.4 Patch Stress Analysis Patch Mechanics In a patch repair, load transfer through the adhesive occurs primarily around the edges of the patch, and near the crack faces. Where the patch is sufficiently large that these two load transfer regions are effectively independent, it is possible to simplify the analysis of a repair by considering each region separately. The analysis of a patch will then involve, first, consideration of an equivalent reinforcement or doubler, and second, the solution of the problem of a 17 reinforced crack. The adhesive away from the load transfer regions experience only low stresses, and so disbonding in this region should not grow or significantly alter the stress distribution in the load-transfer regions. For the first part of the analysis, the patch may be considered as a stiffened inclusion. This inclusion analogy was refined by Rose and applied to the crack-patching problem [41]. The stiffened inclusion or patch attracts load - the larger and stiffer the patch, the more load it attracts. The analogy provides the force intensity distribution in the patch. This force intensity can then be used to calculate the stress in the crack plane of the underlying plate. Due to the reinforcing effect of the patch, the underlying crack experiences an edge traction which is reduced from the remote applied load. 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 by an effective bending moment intensity and simple beam theory [42]. Residual stresses occur when patch application 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 usually cured without restraint against thermal expansion. The residual stress in the region of the crack can be approximated by simple analytical means [1]. For the second part of the analysis, the patch may be modeled as springs bridging the crack. Here, the stress found in the first step is applied as a crack opening traction. The springs bridging the crack partially prevent the crack from opening, reducing the stress intensity from that which would occur without the reinforcing springs. This part of the analysis is critical to determination of the rate of crack growth and the rate of near-crack disbond growth. Several means exist by which an assessment of the structural mechanics of a patch repair may be evaluated. The available methods may be divided into three categories; finite elements, boundary elements, and analytical solutions based on interpolation between limit states. The boundary element and interpolation techniques have the advantage of being capable of easily implementing the inclusion analogy. 18 Finite Element Analysis The finite element analysis method is a powerful tool for investigating the stress distribution in structures with complex loading or geometry, and has been used for some time [42] in the analysis of bonded repairs. For the analysis of a patch, however, finite element solutions tend to be computationally demanding due to the large number of elements required to accurately model the stress distribution around the crack tip and in the load transfer regions of the patch. Specialized techniques have been developed to model the adhesively bonded patch [43,44]. These models employ a three-layer model in which the patch and underlying plate are modeled with Mindlin plate elements. These elements may employ non-isotropic material properties. Typically, the adhesive is modeled as a series of shear springs. The finite element alternating method has also been developed to reduce the computational requirements of modeling the stress distribution near the crack tip [45]. Here, a coarse-mesh finite element analysis of an uncracked plate is used to obtain the stresses in the crack plane. These stresses are then fed into an analytic model of the crack, and the resulting analytical stresses are used to update the finite-element analysis of the uncracked plate. The resulting stresses are then fed again into the analytical analysis. Over a number of iterations, the error shrinks until the solution converges. This method is capable of efficiently analyzing a body containing multiple cracks. Critical parameters, such as the crack tip stress intensity, are available through post-processing techniques. To evaluate the stress intensity, the modified crack-closure technique is often employed [46, 47]. Here, the nodal force and moment at the crack tip and the translational and rotational displacement of the nodes along the crack face are used to determine the energy required to close the crack. Alternatively, the stress distribution at the crack tip [48] and J-integral [49] methods have been used to estimate the stress intensity. Many commercially available finite element packages include crack tip elements for which the program will automatically estimate the stress intensity. To the knowledge of the author, no finite element analysis in the literature has included a calculation of a critical parameter for prediction of disbonding, such as the strain energy release rate. Instead, the adhesive shear strain at the edge of the disbond is reported, and could be used to predict the rate of disbonding. It is not clear whether finite element results based on shear springs will provide more accurate results for adhesive response than the approximate analytical methods for analysis of bonded joints. In the 19 general bonded joint literature, the strain energy release rate for a cracking adhesive layer have been calculated by a method similar to the modified crack-closure technique [50], but such an approach has not been applied to bonded repairs. These specialized techniques reduce the computational power needed to solve patch problems, and have been used with some success to arrive at solutions to particular patch problems. These methods have recently been used [44] to consider the effects of non-linear adhesive behavior and patch bending, both of which were shown to be significant. Most of the techniques mentioned above are not available in commercial codes, and so finite element methods are most often employed in the analysis of doublers or reinforcements where the added complexity of modeling a crack is not an issue, and special techniques are not required. Boundary Element Analysis Boundary element methods for cracked plates are typically based on the complex variable analysis of a cracked plate formulated by Muskhelishvili [26]. The simplest form of boundary element analysis uses superposition of point loads to approximate tractions applied to the cracked plate. The point forces are assumed to act at nodal coordinates. Other methods available outside of the patch literature [e.g. 51] use smoother interpolating functions, which reduce the computational effort required to solve the problem and avoid the stress and displacement singularity caused by the point load. The functions which relate the applied point force to the stresses, strains, and stress intensity factor are termed Green's functions. Several solutions are available in the literature [28, 52-55], and derivations of the Green's functions are appended to this thesis. Unfortunately, the Green's functions reported in the literature have errors, likely accidentally introduced during publication, and are nowhere presented in a single, clear derivation. Erdogan and Arin [53] used the boundary element techniques to analyze the problem of a disbonding composite patch bonded to a cracked aluminum plate. The model characterizes the response of the cracked plate and orthotropic patch by using Green's functions for a point load applied at an arbitrary location. The adhesive is modeled as linear-elastic shear springs. The solution of the problem involved dividing the patch into square elements, and solving the resulting series of linear equations which arise through superposition. Ratwani [42] has shown 20 that this model provides accurate results for determining the stress intensity when compared with finite element analysis and experiments. Dowrick, Cartwright, and Rooke [56] and Young, Cartwright, and Rooke [57-59] have developed a simplified 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 underyling 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 has the capability to model nonlinear adhesive response. Poole, Lock, and Young [60] have verified this model against experimental results for a graphite/epoxy patch applied to a thick cracked plate. Recently, Guo and Wu [61] have applied the boundary element method to analysis of cracked Fiber Reinforced Metal Laminates (FRML), a problem very similar to the analysis of a repair patch. This model used approximate Green's functions based on work by Suzuki and Sakai [62], and the resulting stress distribution demonstrates a good correlation with experiments. Suzuki and Sakai have additionally proposed power-law curve fits for the stress distribution in F R M L , which Cain and Tan [74] have validated against boundary-element and finite element models. Interpolation Method Exact analytical solutions exist for the limiting cases of a patch over a very short crack or a very long crack. No exact closed-form solution exists for an arbitrary-lengthed 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 [41, 63, 64] 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. Cox and Rose [65] have extended this approximate solution to the cases of near-tip reinforcement and elastic-perfectly plastic springs. One considerable limitation of this method is that the reinforcement is assumed to act at the faces of the cracks. 21 Baker [5] has investigated the influence of disbonding and temperature on fatigue crack re-initiation 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 Romilly [48] 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. 2.5 Patch Fatigue Performance Patch Fatigue Experiments Many experiments have been performed both to demonstrate the predictive performance of models, and to investigate specific phenomenon which occur in a patch. These experiments have greatly shaped the understanding of patch behavior and emphasized the importance of many subtle effects including shear deformation of the patch, thermal residual stresses, disbonding, and adhesive plasticity. Baker has published experimental results illustrating the effects of cure cycle parameters [5, 67], disbonding [5, 66], patch shear deformation [67, 68], adhesive plasticity [68], and test temperature [5]. The A M R L patch specimen has been developed, which minimizes the bending in the patch, allowing for a simpler correlation between modeled and experimental results. This specimen uses two edge-cracked plates repaired with single-sided boron/epoxy patches. American Cyanamid FM-73 film adhesive was used to bond the patches to the plates, and the plates to a honeycomb core. Additionally, testing of lap-shear specimens has been used to further investigate the effects of patch shear deformation and adhesive plasticity on the apparent stiffness of the patch in the region of the crack. It has been shown that increasing the cure cycle temperature and time can impose two penalties on patch performance; tensile thermal residual stresses from a mismatch in thermal expansion coefficients can accelerate fatigue crack growth, and crack growth retardation is reduced due to annealing of crack tip plasticity-induced residual 22 stresses. Disbonding has been shown to result in up to a factor of four decrease in patch life. Patch shear deformation and adhesive plasticity have been shown to reduce patch life by weakening the amount of restraint against crack-opening imposed by the repair. Test temperature was found not to have a significant effect on crack growth rates for temperatures ranging from ambient temperature to 100 degrees centigrade. Other researchers have performed fatigue experiments on patched panels. Poole, Lock, and Young [60] have investigated fatigue damage propagation in 12.5 mm thick aluminum plates repaired with graphite/epoxy patches and bonded with Redux 312/5 film adhesive. Similar to the A M R L specimen, the patched specimens were bonded to a honeycomb core and tested in pairs to restrict bending. Here, the authors examined the effect of the F A L S T A F F loading spectrum on performance of a repair, and compared the results to boundary-element analysis techniques which 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 specimen and 17 for a constant-amplitude specimen when compared to an unrepaired sheet. These results cannot be compared because of different applied loading levels. Denney and Mall [69-71] have investigated repairs to thin center-cracked panels with patches on a single side of the panel. Here, patch bending may be a significant factor due to an offset of the neutral axis. The specimens consisted of 0.381 mm boron/epoxy patches bonded to a 1 mm thick aluminum sheet using AF-163-2 film adhesive. The authors investigated the effect of variously sized disbonds at different locations within the repair. It was concluded that disbonds over the crack reduce patch life, whereas disbonds away from the crack are generally not detrimental, and in fact may increase patch life by reducing the apparent size and stiffness of the patch and hence the amount of load it attracts from the rest of the structure. Additionally, Sharp, Clayton, and Clark [72] have examined the effects of adhesive infiltration on the efficiency of a repair. Here, it was found that infiltration of adhesive into the crack increases the stress required to open the crack, and hence improves fatigue life. Patch Fatigue Analysis Rose [21] has set out the framework for an analysis of composite patches including the effect of disbond growth on patch efficiency. This paper contains solutions from lap joint and bonded patch technology reduced to a form useful to a disbonding and crack growth model. The model chosen for disbonding is based on alternating shear stress range. Also presented are equations for the cyclic energy release rate more commonly used in disbonding analyses. The emphasis is on a 23 closed-form analytical approach where distributed springs acting between the crack faces model the crack bridging effect. This could allow a closed form solution, which is highly desirable, but the integration and results are difficult to apply. The effect of disbonding is included by including the compliance of the disbonded portion of the joint in the springs used to model the patch. Baker [5] has analyzed the effects of crack-patching on fatigue- and disbond-growth rates in patched plates by using the long-crack limit solution for the stress intensity under the patch. Disbond growth rates were calculated using the average rate determined in experiments. 2.6 Critical Analysis In this section, a critical review of the information presented in the previous sections will reveal the essential elements determining the life of a repair, and the simplest means of accurately including them in an analysis. This approach is driven by the objective and desired end-use of the work in this thesis. The objective is to develop methods for prediction of fatigue degradation of bonded boron-epoxy repairs to cracked aluminum plates, focusing on the rate of crack propagation and near-crack disbonding. The desired use of these models is inclusion in design codes and standards, which means that standard engineering techniques for bonded joint and fracture mechanics will be applied wherever possible. Ideally, simple relationships will be adapted which are sufficiently rigorous and accurate to provide conservative results when compared to experiments. The two-step analysis approach cleanly divides the problem into two separate components; (1) assessment of a reinforcement or doubler, and (2) assessment of a patched crack. The requirements of this method are not overly restrictive as the load transfer areas of a practical patch application are typically much smaller than the patch. This method also cleanly separates those analysis elements that are special to patch analysis from elements which are common to both a patch and a reinforcement. It is thus desirable to use this approach. Engineering approaches to bonded joint analysis allow a sufficiently accurate prediction of the stress distribution and stiffness of a bonded joint to provide good correlation between as-modeled and experimental results. The relatively simple nature of these relationships, and the ability to include complex secondary effects such as adhesive plasticity and shear deformation of the patch 24 make them valuable for patch analysis. In addition, models retaining these approximate analytic techniques will retain the duality between simple bonded joints and a repair patch, which allows use of experimental data from bonded joints to be easily applied to analysis of a patch repair. General numerical analysis techniques such as finite elements are, by nature, not amenable to the development of sound analytical, closed-form expressions required for inclusion in codes and standards. They are, however, invaluable for the detailed analysis of a particular configuration, particularly in an application with a complex geometry or load. Additionally, use of such numerical techniques negates the advantage of the two-step analysis process, which allows us to focus on the elements of the analysis special to a repair, and prevents the use of the engineering approaches to bonded joints. For these reasons, finite element analysis and advanced boundary element techniques are not employed in this thesis. The aspect which is unique to a repair patch, as opposed to a reinforcement or even a riveted repair, is the load transfer through the adhesive near the crack faces. This load transfer acts to restrict crack opening, and is the ultimate determinant of the crack tip stress intensity and adhesive stresses which cause crack growth and disbonding, respectively. Crack-bridging analyses address this situation, and may be most easily performed using either the interpolation model or a rudimentary boundary element model. The power-law methods used to approximate crack-bridging stresses in FRML's do not have a sound analytical basis. They could, however, prove useful in characterizing the state of a particular repair geometry with complex adhesive behavior if experimental data or a more advanced analysis were available for validation. The interpolation model provides a simple closed-form solution in which the synthesis of bonded joint mechanics and fracture mechanics is carried out in an elegant and straightforward manner. It has been extended to include the effects of disbonding and plate geometry, but the extended interpolation model has not been validated against numerical studies or experiments. It is also assumed in the formulation of the model that the disbond is of constant width. Additionally, no method exists for the determination of the adhesive shear strains, which must be known to predict the rate of debonding. The boundary element methods presented in the literature do not take advantage of the engineering relations for lap joints. The boundary element technique has the further disadvantage of being somewhat complex and specialized. The Green's functions solutions for a point load applied to a cracked plate are not easily accessible, and use of this numerical technique requires a 25 significant amount of programming. For the case of no disbond, a good correlation has been shown between the interpolation model and boundary element methods. These models have the advantage of being able to analyze an arbitrarily-shaped disbond, with nonlinear adhesive properties, without a large computational effort. Unfortunately, the relatively simple boundary element models employed in the crack-patching literature are not capable of analyzing a complex cracked plate geometry. To overcome some of these disadvantages, a new, simpler boundary element model will be employed in this thesis. Here, the patch will be modeled by springs which have constitutive properties determined from the standard engineering analysis of a lap joint, allowing one to easily include the effects of patch shear deformation and adhesive plasticity. The springs will terminate at the edges of the disbonded region, allowing consideration of an arbitrary disbond shape. The main advantages of the new boundary element model are; a) parameters needed to estimate disbond growth are readily available, b) modeling of non-linear adhesive response is simplified, c) results are easily compared to the results of lap joint tests, and d) it is computationally more efficient. Results from the boundary element model will be used to validate the interpolation model for the case of disbonding, and extend it to allow prediction of the rate of disbonding. The boundary element model will also allow the interpolation model to be 'calibrated' to a particular shape of disbond. In addition to disbonding, adhesive plasticity, plate geometry, and patch shear deformation, patch bending and thermal residual stresses can greatly influence patch life. Here, the crack-opening stresses employed in the boundary element and interpolation models must be adjusted. The loads applied to the structure may also have a significant mean stress component, which will affect the crack and disbond growth rates predicted by the model. Fatigue crack growth and disbond growth laws and parameters will correspond to those used in common engineering practice. Crack growth retardation is an important effect which may largely determine the fatigue life of a repair, however, to ensure a conservative analysis, it is typically not included in codes and standards for aircraft. Thus, while not likely to be accepted for airworthiness certification of a repair, fatigue crack growth models should include a mechanism whereby crack growth retardation can be assessed, such that the true life, and hence, the economic feasibility of a repair may be accurately assessed. The effect of the cure cycle and 26 annealing of the crack tip plastic zone must be included to accurately assess fatigue crack growth re-initiation. An opportunity exists to include fatigue, disbonding, and methods for composite patch analysis in one model, thereby improving the design and damage tolerance analysis of bonded composite repairs. Crack and disbond growth rates are linked: a change in crack length influences disbond growth and vice-versa. The engineering equations for bonded joint analysis and the interpolation model provide good engineering descriptions of fracture mechanics and stress fields for the composite repair, and in this thesis will be extended to the case of a disbonding patch and validated against numerical models and experimental results from the literature. 2.7 Summary Literature relevant to predicting fatigue damage progression in a bonded repair has been presented. This review covered crack growth and near-crack disbond growth, as these are the modes of fatigue damage progression which are special to a repair, when compared to a reinforcement. It has been shown that the objectives of this thesis may be obtained through the consideration of both a numerical boundary element model of a patch, and a model based on interpolation between limit states. In presenting the analytical techniques and experimental studies relevant to the patching problem, it has been shown that the effects of disbonding, crack-growth retardation, thermal residual stresses, bending, adhesive plasticity, patch shear deformation, and cracked plate geometry may all be important considerations. The complicating effects of loading rate and service environment have not been addressed, as they are outside the scope of the present work. In subsequent chapters, the essential elements of bonded joint analysis and fracture mechanics required for prediction of patch fatigue performance will be refined and applied to the modeling of a patch using both the boundary element and interpolation model techniques. 27 3. BondedJoint Analysis 3.1 Overview In the analysis of a patch repair, one approach is to model the patch as a series of springs bridging the crack. The stiffness of these springs is based on the one dimensional theory of joints. Thus, an understanding of the mechanics of adhesive joints becomes critical to understanding the behavior of a repair patch and the development of a model to predict the fatigue life of the repaired structure. Particularly important is the stiffness of the bonded joint, as this is what determines how well the repair will restrict the opening of the underlying crack. Additionally, many of the difficulties facing this repair technology are related to the behavior of the bonded joint. In this chapter, classical methods of bonded joint analysis are applied to the case of an internally pressurized lap joint. Equations illustrating the effects of fatigue loading on the adhesive stress state are developed and extended to the case of reversing plastic deformation. Methods of predicting the stiffness, static strength, and rate of fatigue degradation of a bonded joint are presented, and subsequently applied in the prediction of the bond performance of a composite repair. In subsequent chapters, the results of this analysis will be used to predict the behavior of the patch adhesive system near the crack. 3.2 Classical Mechanics of Bonded Joints A bonded joint consists of two or more adherends, which are held together by layers of adhesive. For a single-sided reinforcement or repair patch, the equivalent bonded joint geometry used for analytical comparison is the single lap joint. Here, the adherends are the repair patch and the underlying cracked plate. The method of analysis for such a lap joint, first performed by Volkersen, [6] is well known. In addition, Hart-Smith [7, 8, 75] has provided solutions considering the elastic-plastic adhesive behavior for many joint configurations. Unfortunately, consideration of more complex adhesive behavior (e.g. power-law or visco-elastic) requires the application of a numerical solution. 28 The following figure shows a segment of a bonded joint and the free-body diagram for a differential element of a bonded joint. The local coordinate rj , shown on the figure, applies to an individual segment of the joint. F is the force intensity, or force/length acting on the joint. >T1 A T ^ reinforcement 7 ~7 -7—7 — T -l—l-l-Lt-l—l—L / F + A F -L-L-L-L.I-I—L plate X AT) U ^ ' • "'A F - l OT| orj F F + A F Figure 3-1: Lap Joint Free Body Diagram Adhesive shear strains may be approximated by ya = SI ta, where 8 is the relative displacement of the adherends. Considering equilibrium and compatibility, equation (3-1) may be derived, which characterizes the joint behavior. Note that X contains information regarding the stiffness and dimensions of the adherends and adhesive, and may be modified to account for joint geometry and the shear displacement of the adherends. d2ya drf - X2 r = 0, where A2 = 1 1 + \Ertr E t , p p/ (3-1) This solution is valid only for locally uniform adherends. Integration leads to the following general solutions for adhesive elastic and plastic response, respectively. rayn) = C, • sinh(2 • 77) + C 2 • cosh(2 • rj) Ta=ya- Ga T„ • X1 2-G„ n2+crTj+c2 (elastic) (plastic) (3-2) Here, C, and C 2are integration constants, and may be found by considering the boundary conditions of the adherends. Step changes in adherend dimensions or in the state of the adhesive require consideration of discrete segments of the joint. The full solution for a particular joint results in a system of equations solvable for the integration constants. For the single-lap joint, considering / to be long, and assuming elastic adhesive response, the boundary conditions » CO become F(oo) = 0 and F(0) = j ya(rj) Ga-dtj, and thus the adhesive shear strain is; 29 (3-3) At the onset of adhesive plasticity, a plastic zone of length lp will develop. Adhesive shear deformation is limited only by other sections of the joint. Here,ra = ry, F(lp) = Tylk, and Ya{lp) = Ty I Ga. It follows from equation 3-2 that; (3-4) 3.3 Fatigue and Fracture of Bonded Joints Using this analysis, one can now consider a lap joint under a cyclically applied loading alternating between Fx and F2 with|F,| > |F2|. When subjected to this fatigue loading, there are three possible scenarios for the response of the adhesive: (1) purely elastic adhesive response, (2) the peak load creates a monotonic plastic zone, (3) there is a reversed plastic zone. In the subsequent analysis of each scenario, note that the global coordinate x begins at the inner edge of the lap joint, and extends through all segments. —>X <— L "P — * reinforcement plate Figure 3-2: Lap Joint Showing Reversed Plastic Zone For scenario (1), a lap joint with purely elastic adhesive response, the strain may be found directly from equation 3-3. For scenario (2), a monotonic plastic zone of length / is formed by the peak load. As there is no reversed plasticity, the stress distribution will be the same as that for a lap joint under a monotonic load. Using equations (3-3) and (3-4) yields i - A - ( * - g h - f ( * - / , ) cosh^ -(x-lp^- sinh^A • (x - lp^j 0<x<L x>L (3-5) 30 where the length of the plastic zone is found through equilibrium to be 1 u Ty — F i > — X X (3-6) otherwise. The peak load results in plastic deformation and residual strains which must be considered when calculating the strain state under F2. Here, the application of force F2 causes only elastic strains, and thus the result follows through superposition to be: ra{x,F2) = ra{x,F]) + X-G„ • (cosh(A • x) - sinh(/l • x)) (3-7) For scenario (3), F2 causes reversed plasticity over the interval < x < lrp j and F{ has created a monotonic plastic zone corresponding to case (2). Considering the change in adhesive strain state moving from Fx to F2, the situation is analogous to a monotonically loaded joint with an equivalent adhesive yield stress of re =2 - T . The reversed plastic zone length and the change in adhesive shear strain in moving from F] to F2 may then be found, as follows; F2-Fx 2 T „ X otherwise 2 T „ X (3-8) Aya(x) = 2 T „ 2-r„ cosh(/l • (x - lrp)) - sinh(A • (x - lrp)} 0 < x < / rp X > /, rp (3-9) The dynamic component of the shear strain, Aya, may be superposed on the strain state at Fx such that ya{F2) = ya(Fi)- A/a. Here, /a(Fl) may be found using equations 3-5. Shear strains are directly related to the joint displacement, and hence the stiffness of a perfectly bonded joint. Of particular interest is the determination of yA(0). ra(o,Fi) = 2-G„ X^-X (3-10) otherwise 31 The alternating component of the shear strain is then found to be: Aya(0) 1 + 2-T y J F2-Fi -X F2-Fx > otherwise 2 T „ X (3-11) These shear strain results will be particularly important in subsequent sections, where the stiffness of the joint is critical to the calculation of the effect of crack-bridging. Analysis of a Single Lap Joint with a Disbond Also of interest is the material response and deflection of a lap joint or repair patch with a disbond of length b. Here, one must consider both the additional compliance of the disbonded portion of the reinforcement and the potential for dissipation of energy through the processes of irrecoverable work, i.e. crack formation and plastic damage. It should be noted that the force acting on the reinforcement is applied at the edge of the disbond. This corresponds to the means in which the load is transferred from the plate to the patch in the models developed in Chapter 4. < — —>x k — I reinforcement : -> F plate *—u Figure 3-3: Lap Joint With Disbond For this condition, the crack face and disbond-edge displacement are given by; S = ya(0)-ta+F-b-1 1 + • \Ertr EptpJ (3-12) ra(o)-ta+F- (3-13) The strain energy release rate, G„, may be found by considering an energy balance for the system. With an incremental increase in b, under a given load applied at the disbond edge, the energy available for irrecoverable work, i.e. the creation of new crack surfaces and plastic 32 damage, is given as follows. Here, W is the work done by the external load and U is the change in the stored elastic energy. (3-14) Noting the self-similarity of a very long lap joint with an incremental change in b, the energy terms may be found as follows; dW _ d i db ~ db 8 + -^—b V Ertr J Ertr dU _ d db db ( r-2 \ 2-Ej r J 2-Ertr (3-15) (3-16) The strain-energy release rate for monotonic loading follows by substitution; F2 G„ = 2-Ertr (3-17) The cyclic strain energy release rate, A G , may be found as follows, and is often correlated to disbonding growth rates in adhesive joints, although with mixed results [50]. Note that load reversals are accounted for by recovering the loading directions, which would otherwise be lost by blindly applying energy methods. FrF2 2-Ertr (3-18) Similarly, one may calculate the rate of energy consumed by plastic deformation. Again noting the self-similarity of a long lap joint, the result for monotonic loading is; dUP ~db 1 •' + 2 (3-19) Substituting the relationship for / from equation 3-6; dUP(Fx) _ db 0' r F ^ -1 X otherwise (3-20) 33 Considering load reversals, plastic work must be done to overcome both residual strains and develop reversed plastic strains. The result is dUP(F2) db 0 nF2 - F , | - ^ 2 2 • T„ j is - F,\> y- (3-21) otherwise Note that the rate of plastic work depends on the geometry of the lap joint. Strain energy release rates are typically 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 done is constant for an increment in crack growth [35]. The second condition holds for a lap joint experiencing gross plasticity, but due to the dependence on joint geometry, there should not be a discernible critical strain energy release rate as a material property for the adhesive. Indeed, Hart-Smith [75] has noted that joint strength under static loading depends mainly on the adhesive shear strain energy. In a similar manner, strain-energy release rates may not be useful for predicting disbonding of plastically deforming adhesives, as they do not characterize the cyclic plastic strains. Alternatively, the alternating plastic strains, or an approach based on the rate of plastic work with disbond growth may result in a considerable improvement. Either alternative method directly accounts for the effect of adhesive thickness. An 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. In the following chapters, the lap joint stiffness relationships illustrated above are used in conjunction with approximate analytical and boundary element methods for the analysis of patch mechanics. Despite criticisms raised above, strain-energy release rates will be applied to disbond growth rate predictions, corresponding to the approach most often employed in engineering practice. 3.4 Summary This section has presented the classical methods of analysis for an internally pressurized lap joint. It has been shown that these methods, based on the one-dimensional theory of joints, can 34 predict the stiffness, stress distribution, and shear strains in a lap joint. Several complex phenomena can be included, including: non-linear adhesive behavior, adherend shear deformation, and the thickness and stiffness parameters of the adhesive and adherends. These methods have been extended to the case of fatigue loading of an adhesive joint, where it has been shown that adhesive thickness and shear stress are the principal determinants of fatigue strength. In subsequent chapters, these techniques will be employed in the damage tolerance analysis of a composite patch and the effect of a repair on the stresses about a crack. The next chapter discusses fatigue and fracture analysis of a cracked plate, which is the other requirement for prediction of damage progression in a repair. 35 4. Crack Bridging Analysis 4.1 Overview To predict the damage progression in a repair, it is necessary to have a means of stress analysis of a crack under a repair, and to be able to find a parameter which may be compared to experiments to predict the rate of fatigue damage growth under the applied loading. For prediction of fatigue crack growth in the underlying plate, solutions for stress intensity will be found. In the previous chapter, the stiffness of a bonded joint was determined, based on the one dimensional theory of joints. To predict the rate of disbonding, the displacement of the cracked plate at the edge of the disbond will be found, allowing one to calculate the load carried by the springs bridging the crack. The strain energy release rate can then be found using the methods described in the previous chapter. 4.2 The Crack Bridging Problem - Interpolation Solution The crack-bridging problem appears often in engineering and materials science, and several analytical approaches exist. The reinforced crack opening displacement, spring stretch, and crack stress intensity are the parameters of interest. The governing integral equation is of the Fredholm type, with a displacement term in the kernel. For linear elastic springs bridging the crack faces, Rose has developed accurate approximations for the crack-bridging problem [64]. These have been extended to the case of elastic/perfectly-plastic springs [65] and disbonds of constant size [5, 48]. The solutions are based on an interpolation between a short-crack solution, where crack-bridging is negligible due to small crack opening displacements, and a long-crack limit case, where the crack opening is limited mainly by the springs bridging the crack. For the case of constant crack opening pressure and no disbonding, the interpolation method is as follows. First, the stress across the crack plane, cr0, which depends on the applied loading, bending, thermal residual stresses, and the dimensions and stiffness of the patch system, is determined. For short cracks, the crack opening displacement caused by this stress is small and 36 bridging has little effect. Ignoring load transfer about the crack, the upper bound stress intensity, and half-crack opening displacement are; K0 = (JQ-JK • a 2-Er (4-1) (4-2) For long cracks, bridging has an increased effect. The parameters approach a theoretical limit value when the crack length greatly exceeds the patch characteristic length, A . These limits are; KX = o-0V;r- A cr0 • n • A (4-3) (4-4) The patch characteristic length, A , may then be found using equation 4-5, below, which results from strain-energy release rate considerations as described by Rose [1, 41]. This characteristic length may be considered an effective crack length. Lap joint displacement relationships developed in Chapter 3 may be used to determine the patch characteristic length. This method allows consideration of elastic, plastic, and reversing plastic adhesive behavior and disbonding, for a repair configuration for which A « L ; A 7T-Gn (4-5) The effect of crack bridging on stress intensity and crack opening displacement can now be estimated by interpolation. \7t-a-K a + A 1 + \a-(\+v)-(l + K) n-K) (4-6) (4-7) With disbonds, it is not possible to find the adhesive shear strains directly from the crack-opening displacement. Instead, it is required to know the displacement u at the disbond location. 37 At a distance b from the crack plane, the displacements without considering bridging [53], and for an equivalent lap joint are given by; •(1 +V) 2-E, (K + \)-yJa2 +b2 +b- 2-b 4a2 +b2 • + \-K (4-8) « „ = f L ( * . A - 6 ) (4-9) Interpolating between these, the result for a reinforced crack is; Ja2+b2 1 + " 2-b2 +b-4ar+b~2(\-K) + (K + \)(a2 +b2) + * - A - b (4-10) This interpolation model combines bonded joint and fracture mechanics techniques in a direct manner, and provides accurate results for a patch without disbonds. Note that this method is typically applied only for the determination of the stress intensity, but, as will be shown in Section 4.4, it has been found equally useful for the analysis of crack face displacements and adhesive shear strains, with or without a disbond. The next section describes an alternative, and presumably more accurate method based on boundary element techniques. Subsequent to this, in Section 4.5, correction factors will be applied to correlate the interpolation model with boundary element results for a constant-size disbond. 4.3 The Crack Bridging Problem - Boundary Element Solution The interpolation model is a traditional model for composite repairs to cracked structures, often used to estimate the efficiency of the repair and for preliminary design of patches. Unfortunately, this model is not amenable to cracks under partially disbonded patches unless the disbond is assumed to extend across the entire length of the patch [5, 48], and cannot be used to accurately assess nonlinear adhesive response. A n arbitrary disbond shape or adhesive response requires a numerical solution. The following is a description of a boundary element crack-bridging model. A boundary element crack-bridging model is required to estimate the stress intensity, crack opening displacement, and adhesive shear strains for an arbitrarily shaped disbond. Numerical models based on boundary-element techniques have been applied to the crack repair problem 38 [42, 53, 55-62]. These models have been applied with some success, but cannot take advantage of the classical bonded joint analysis techniques outlined in Chapter 3. These classical techniques allow for a simple and direct inclusion of secondary effects such as adherend shear deformation and adhesive plasticity, and account for the complex distribution of stresses in the repair. The model presented here employs boundary-element methods, but uses the classical methods of analysis for bonded joints. Because the adhesive stress distribution does not have to be explicitly modeled in this boundary-element scheme, the model runs relatively quickly, allowing concurrent crack-growth predictions. Assessment of the crack bridging properties of a repair requires the solution of the boundary value problem of a cracked elastic region, where the boundary value is related to its displacement. The result is a Fredholm integral equation, in which the kernel includes a displacement term. This problem may be approached in several ways. For example, repeated substitution for the displacement may lead to a closed-form solution. Alternatively, numerical solutions can be found through discretization, resulting in a series of linear equations solvable for the displacements of the cracked plate. Considering u(z) as the displacement of the cracked plate with reinforcement, u0 (z) the displacement without reinforcement, and C(z,t) as the plate compliance, the problem may be expressed as follows. Here, C(z,t) is the displacement at zdue to a unit force applied at t -- the Green's function solution for a point load applied to a cracked plate [52-54]. A derivation of the appropriate Green's functions are included in Appendix C, and Figure 4-4 illustrates the loading configuration. (4-11) 39 $ u(z) Figure 4-1: Green's Function Point Load Configuration The patch stiffness is modeled by ks (t) and L is the contour corresponding to the edge of the disbond. Dividing L into intervals, the integral equation may be discretized and expressed in the following form; Rearranging and dropping the subscripts, the following system of linear equations results. The matrix C is full, and so the system is solvable by the Gauss-Siedell iteration technique. Essentially, the patch is divided into strips, each simulating a lap joint of width A . The following figure illustrates a patch divided into strip elements. It is apparent that each strip is essentially a bonded lap joint. This method of analysis is applicable in the case of a highly orthotropic reinforcement, where the stiffness of the patch transverse to the fibres may be considered low. This is not an onerous requirement because most unidirectional fibre-reinforced plastic composites have a longitudinal stiffness which is nearly an order of magnitude greater than the transverse stiffness. (4-12) (4-13) 40 Equation 4-14 relates the bonded joint characteristic length parameter, A , and the extent of disbonding, b, to the spring stiffness of a reinforcement strip element. A may be determined as shown in Chapter 3. F EP-t.-A K = ± = -L-f-r (4-14) o n-A-b Note that the Green's function describing the displacement will be unbounded at the location of the point force. Here, the model estimates the displacement by offsetting this point load by one-half of the element width. The resulting error decreases in proportion to A. No assumptions have been made about the disbond shape or nature of the loading. Again, A may be modified to account for patch configuration and shear deformation, adhesive plasticity, and disbonding. Nonlinear effects require repeated solution of the system of equations and updating of the spring stiffness'. The insertion displacements, {u}, allow calculation of adhesive shear strains and the forces closing the crack. Crack opening displacements and the stress intensity may then be found using the appropriate Green's functions. Here, one must again subtract the reduction caused by crack bridging from the nominal fracture mechanics solution. For example, the stress intensity may be found as follows; 41 n 1=1 (4-15) HereGA^ is the Green's function for stress intensity, derived in Appendix D. Fi is the crack-closing force imparted by the patch strip element. Appendix A presents a MathCad worksheet in which the model has been implemented using simplified Green's functions which cannot consider disbonding. Appendix B presents a MathCad worksheet which employs the full Green's function solution and allows consideration of disbonding. The F O R T R A N programs D A M A G E and C O M P A R E employ the full solution. 4.4 Comparison of Boundary Element and Interpolation Models In previous sections of this chapter, two separate models have been developed for predicting the critical parameters of a repair. The first model, based on interpolation between two limit cases, has the advantage of simplicity and speed of calculation. It is, however, limited in at least the following ways; it is only applicable where the stresses in the crack plane can be assumed to be constant, it assumes a constant-width disbond, and crack growth must be contained beneath the patch. Additionally, the interpolation model has not previously been verified for the case of a partially disbonded patch. The second model, a boundary element model based on Green's functions, while more complicated and computationally much slower than the interpolation model, does not have the same limitations. It can also be assumed to more accurately describe the stresses and displacements in a repair, as it has been derived from first principles without using mathematical conveniences such as an interpolation between limit states, i.e. it explicitly solves the stated problem (including assumptions), and can be considered exact when converged. In this section, the results of these two models are compared. They are first compared using the A M R L specimen geometry, with and without disbonding. Subsequently, the results are compared for a wide range of practical repair parameters, and correction factors are developed for the interpolation model which bring its results into agreement with the boundary element model. These correction factors combine the accuracy of the boundary element model with the simplicity of the interpolation model. 42 AMRL Specimen, Without Disbonding Figures 4-3 and 4-4 compare the stress intensity and crack face displacement results of the boundary element model to those predicted with the interpolation model. Note that the boundary element model diverges from the interpolation model results as the crack length increases. This is because a constant number of elements were used in each calculation. For a long crack, fewer elements exist in the critical region near the crack. Alternatively, the element length may be scaled to a characteristic length such as A , thereby avoiding such problems. H a l f C r a c k L e n g t h ( m m ) Numerical reinforced " " Interpolation reinforced — Long crack solution — Short crack solution Figure 4-3: Stress Intensity Factor Results Compared to Interpolation Model 43 0 I I 1 I I 1 0 20 40 60 80 100 Half Crack Length (mm) Numerical reinforced " " Interpolation reinforced — Long crack solution — Short crack solution Figure 4-4: Crack Face Displacement Results Compared with Interpolation Model Figure 4-5 shows the error between the boundary element and interpolation solutions. The error was calculated using the following formula; (4-16) 0 20 40 60 80 Half Crack Length (mm) Stress Intensity " " Displacement Figure 4-5: Percent Error Comparison 44 Note that for displacements, the error is small (<5%) for both the short and long crack limit. The error for the stress intensity factor increases for long cracks (>5%) because a constant number of elements was used regardless of crack length. For long cracks, too few elements exist near the crack tip to accurately estimate the stress intensity. AMRL Specimen, With Disbonding The disbond model was tested assuming an elliptical disbond shape, of the dimensions shown in the following figure. Material properties as well as plate and patch thicknesses match those of the A M R L specimen. Figure 4-6: Elliptic Disbond Shape Figure 4-7 shows stress intensity results for various disbond aspect ratios. This is compared to the results predicted by interpolation. Note that the boundary element model results are very similar to the interpolation results when the disbond is very small. Note also that in the presence of a disbond, the stress intensity can be well above the limiting upper bound for a patch without disbonds. 45 0 20 40 60 80 Half Crack Length (mm) ~B~ Numerical result, b/c = 0.5 Numerical result, b/c = 0.25 -°" Numerical result, b/c = 0.001 — Interpolation result, b/c = 0.5 — Interpolation result, b/c = 0.25 Interpolation result, b/c = 0.001 Interpolation, b/c = infinte Figure 4-7: Stress Intensity Factor Results with Disbonds Figure 4-8 shows the crack face displacement at the center of the crack for various disbond sizes. Figure 4-9 shows the displacement at the edge of the disbond, above the center of the crack. Note again that with disbonds the displacement greatly exceeds the interpolation prediction, but the error is not so large for the disbond-edge displacement. This displacement is a key parameter for determination of adhesive shear strains, and hence, the rate of disbonding. 46 Half Crack Length (mm) _ e ~ Numerical result, b/c = 0.5 Numerical result, b/c = 0.25 ~°" Numerical result, b/c = 0.001 — Interpolation result, b/c = 0.5 — Interpolation result, b/c = 0.25 • " Interpolation result, b/c = 0.001 Interpolation result, b/c = infinite Figure 4-8: Displacement Results with Disbonds 47 & 10 Half Crack Length (mm) Numerical result, b/c = 0.5 Numerical result, b/c = 0.25 Numerical result, b/c = 0.001 Interpolation result, b/c = 0.5 Interpolation result, b/c = 0.25 Interpolation result, b/c = 0.001 Figure 4-9: Disbond-Edge Displacement Figure 4-10 shows the shear stress distribution for an elliptical disbond with various amounts of disbonding, for a fixed crack length of 20 mm. Note that the shear stresses decrease with increasing disbond sizes. This is because the compliance of the reinforcement strips increases, resulting in lower crack bridging forces. Also, as expected, the adhesive shear stresses causing disbonding will decrease with disbond length. This means that near-crack disbond growth, while it does influence crack growth, is self-limiting and so is not likely to cause complete disbonding of the patch. However, this may not be true in the presence of significant thermal residual stresses. 48 10 15 20 Distance from Centre (mm) _ a _ Numerical result, b/c = 0.5 •®~ Numerical result, b/c = 0.25 Numerical result, b/c = 0.001 Figure 4-10: Adhesive Shear Stress Distribution with Various Amounts of Disbonding Figure 4-11 shows the adhesive shear stress distribution for a 160 mm long crack. Here, in the presence of a large disbond (— = 0.5), the location of the peak shear stress moves from the center c of the cracked plate towards the crack tip. This is because the disbond shape was assumed to be elliptical. In reality, the disbond will grow to a shape such that the shear stresses will be fairly even. 15 10 r~ 1 o—e—o—e—e—e—e—e—o o—e—e—a—&—e—e ©-o—©—©—©—©—©—©—©—©—©—©—©—©—e—e—o p.. - © — © . 40 Numerical result, b/c = 0.5 Numerical result, b/c = 0.25 Numerical result, b/c = 0.001 80 Distance from Centre (mm) 120 160 Figure 4-11: Adhesive Shear Stress Distribution for a Long Crack 49 Note that shear stresses decrease with increasing disbond sizes. This is because the compliance of the reinforcement strips increases, resulting in lower crack bridging forces. So, also as expected, the adhesive shear stresses causing disbonding will decrease with disbond length. This means that near-crack disbond growth, while it influences crack growth, is self-limiting and so is not likely to cause complete disbonding of the patch. This is not true in the presence of significant thermal residual stresses. Parametric Study The crack-bridging model above allows a parametric investigation to compare the boundary element model in Section 4.2 with the interpolation model in Section 4.3. Using the F O R T R A N program C O M P A R E , the investigation tested a practical range of the patch parameters a, b, and A . The parameters were tested in the range 1 / 25 < (b I a, A / a) < 1. The results apply for any length of crack. Combinations of parameters resulting in n- A < b + s were not included, as this would imply an infinite or negative patch stiffness, E I a was set at 1/25 and is included to further reduce the range of the analysis to remove extra cases for which the patch would be extremely stiff. The figures below compare the results for K, 5, and w in a state of plane stress. Stress Intensity Stress Intensity 50 1.45 e- 1.3 l.is h Half-COD Half-COD 1.45 lis r-_L 0.5 Lambda/a Disbond-Edge Displacement Disbond-Edge Displacement Figure 4-12: Comparison of Boundary Element and Interpolation Crack Bridging Models Comparing the two methods, the interpolation model leads to reasonable, although unconservative results in its present state, particularly for stress intensity and crack-opening displacement, Results accurate to within 30% can be expected over the range of parameters tested. The tend is for error to increase with b I a and Ala, i.e. with increasing disbond length and decreasing patch stiffness. Note that as b Ia -» 0 the error decreases to near zero, illustrating that the interpolation model and boundary element model provide very similar results for a repair without a disbond. The ratio between the boundary element and interpolation results can be seen to be highly dependent on b I a. For this reason, the author proposes a geometry correction factor, to be applied to the crack-bridging components of the stress intensity, crack-opening displacement, and disbond-edge displacement. In addition, there is a weaker dependence on A / a which must be accounted for to get accurate results. For a disbond of constant size, extending over the length of the crack, a non-linear least-squares fit using SYSTAT [77], a statistical analysis package, results in the following relationships; 51 Plane Strain: V f U \ 0 7 3 7 ( A N 0 5 3 4 -^ - = 1.048 + 0.123-1^] ± s , , s 0.993 / . > -0.048 — = 1.022+ 0.267-f-1 (4-17) . N 0.121 , . \ 0.230 U = 1 .265-0.292 ' M f A aJ \a Plane Stress: 0.844 / , \ 0.217 A . = 1 , 2 0 + 0.,49.W 0.965 / . x -0.310 A = 1.032 +0.305-f*V fA 8. \aJ \a (4-18) s , \ 0.425 , . % 0.520 ^ = 1.130-0.177-f*l M w„ Va/ l a With these correction factors, the difference between boundary element and interpolation model results is reduced. Table 4-1 shows average and mean absolute percent errors for each method. Interpolation Model .. with Correction Factors Plane Stress Plane Strain Plane Stress Plane Strain Stress Intensity maximum 13.9 13.8 4.6 7.4 mean 8.2 9.1 0.7 1.0 Half COD maximum 30.0 24.0 8.6 14.0 mean 16.9 13.1 1.1 1.4 Disbond-Edge maximum 10.2 10.7 13.0 13.5 Displacement mean 3.0 2.8 1.1 1.4 Table 4-1: Comparison of Interpolation and Boundary Element Models by Relative Error 52 The correction factors generally improve the accuracy of the interpolation model when compared to boundary element results. Caution should be used i f the correction factors are extended beyond the tested interval. For b I a < 1 / 25, the interpolation functions should be used without modification. These equations allow a simplified yet accurate damage tolerance assessment of a repair. Because these closed-form solutions are easy to program and fast to compute, they allow an engineer to perform crack- and disbond growth rate predictions on a simple spreadsheet, numerically optimize patch designs considering patch life, or incorporate statistical analysis into inspection planning. They are also useful as a check against finite-element calculations, and amenable to inclusion in design codes. 4.5 Summary These models are useful tools for damage tolerant design, disposition of inspection results, and research of bonded repairs. They can be extended to include non-linear adhesive response and thermal residual stresses. In addition, by implementing crack and disbond growth laws, a predictive model can be developed, as described in the last part of this section. By adding extra degrees of freedom and more springs, structures such as bonded or riveted stringers can be included. The front end of these models can be extended to include thermal residual stresses by changing the crack opening stress, cr0. This would only require the adoption of previously developed relationships. This model is simple enough that fatigue crack growth and disbond predictions can also be included, using the methods previously discussed. Note that such predictions will only be valid when the crack is within the "constant stress" region of the patch. These models are also applicable to problems such as cracking in fiber-reinforced metals, Dugdale strip-yield models for crack-tip plasticity, crack-closure models used for predictions of load-interaction in fatigue, adhesive infiltration into the crack [42], and non-linear bridging such as elastic-perfectly plastic, visco-elastic, and visco-plastic springs. For non-linear problems, one may apply load-stepping and updating of the spring stiffness or other techniques commonly used in finite element analysis. Methods have been presented which allow for an analysis of damage progression in a cracked plate, and considering the effects of crack bridging and an arbitrary loading, via Green's 53 functions. In the next chapter, these will be combined with the methods of analysis in Chapter 3, for bonded joints, to analyze the mechanics of a bonded repair and solve for the parameters relevant to prediction of fatigue damage. 54 5. Patch Fatigue Analysis 5.1 Overview In this chapter, the required elements for the fatigue analysis of a patch will be assembled. In previous chapters, methods have been presented which allow assessment of the state of a repair. These methods have the benefits of simplicity, and direct application of the theory of joints. The reinforcing effect of a repair occurs through load transfer from the weakened or cracked plate. Two separate load transfer mechanisms exist: load transfer around the edge of the patch, and load transfer around the edges of the crack. In Chapters 3 and 4, methods were established to analyze the load transfer near the crack and obtain the critical parameters determining patch degradation. In this chapter, basic techniques for obtaining the crack opening stress will be discussed, including the effects of bending and thermal residual stresses. The method of implementation of the crack bridging model will then be presented, including a description of the crack- and disbond- growth laws used to predict patch degradation. In addition, conventional fracture analysis techniques will be extended to the analysis of overload effects and the effect of the cure cycle on fatigue crack growth re-initiation following the application of a repair. 5.2 Crack-Opening Stress A repair consists of a boron/epoxy or graphite/epoxy composite patch, bonded to a weakened or cracked structure. The patch attracts load from the surrounding structure. The larger and stiffer the patch, the more load it attracts. Load transfer occurs through adhesive shear stresses near changes in compliance of the adherends. In a repair, load transfer predominantly occurs at the edges of the patch and around the crack, thus reducing stresses and restricting crack opening. These benefits may be partially offset by stresses arising from the mismatch in thermal expansion coefficients between the patch and cracked structure. 55 Load Transfer Regions Crack Figure 5-1: Patch Schematic Drawing Having established that load transfer to the plate occurs mostly in two regions, near the edges of the patch and near the crack, it is now necessary to estimate the stresses in the cracked plate under the patch. The state of the actual repair may then be determined by superposition of the solution for a doubler or reinforcement and the fracture mechanics solution considering the crack opening stresses. Figure 5-2 illustrates this process. Figure 5-2: Superposition and the Two-Step Method of Analysis. Only the upper half of the patch is shown. In this section, methods for determining the stresses in the plate under a reinforcement will be developed. The phenomena which must be considered are; attraction of load into the reinforcement from the surrounding structure, thermal residual stresses, and bending. The reinforcement attracts loads from the surrounding structure due to its high stiffness. Thermal residual stresses occur due to a mismatch in thermal expansion coefficients between the reinforcement and the plate, which, in the case of a boron/epoxy reinforcement, results in tensile 56 stresses in the plate. Bending occurs due to an offset in the neutral axis of the reinforced plate. These stresses are applied as crack-opening stresses when analyzing a repair. Attraction of Load The analytical approach developed by Rose [1, 41] allows one to calculate the stresses in the plate considering attraction of load. Here, an analogy is developed between a stiffened inclusion and a reinforcement, which allows the analytical solution for stresses in an elliptical inclusion to be applied to the case of a reinforcement. This analysis provides the strain in the reinforcement and plate, which are assumed to be equal in the center of the reinforced region. The stresses in the patch and plate may then be found from their respective elastic moduli. The effect of a reinforcement on the stresses in the underlying plate may be characterized by a stress reduction factor, Y0. Other methods which may be used to estimate this stress include a finite element analysis of the patch applied to an uncracked plate, or the classical bonded joint analysis of a doubler. For a patch of width W and height H , the result of Rose's analysis is that the stress in the plate under the patch, a0, under uni-axial tension may be found from equations 5-19 and 5-20. 1 + | { 1 + 2 ( 1 + S ) f - ^ ( 1 + 5 -^ 5 )} a0 = Y0ax = — ^ f - Lax (5-19) l + S where D = 3(1 + S)1 + 2(1 + + f + v,5) + l - v\S2 (5-20) The force intensity transmitted by the reinforcement is found to be F = Y0aJp(\ + S) (5-21) This analysis is sufficient to determine the stress state in a repair i f thermal residual stresses and bending are not issues. The next section will consider these effects. 57 Thermal Residual Stresses and Bending Thermal residual stresses occur due to a mismatch in thermal expansion coefficients between the reinforcement and the plate. During the cooling part of the cure cycle, the adhesive becomes solid at approximately its glass-transition temperature. During the remainder of the cooling cycle, the adhesive will resist a mismatch in thermal strains, resulting in residual stresses in the repair. In the case of a boron/epoxy reinforcement on an aluminum plate, the result is a tensile stress in the plate.. This stress may be characterized [48] by an effective stress free temperature, Tsf, which must be determined experimentally due to the non-linear response of the adhesive, particularly at temperatures near the glass-transition temperature. Rose [1] has developed simple analytical means to estimate the thermal residual stresses. With no restraint, the stress in the region of the crack can be estimated with the following equation [21]; Gr = E,T,[(l+v)a>-2a'] CT° ( l - v ) [ ( l + v) + 2/s] This solution is valid only for a patch which is restrained against bending, but not restrained against thermal expansion in the plane of the plate. Other solutions are available from Rose [1] which consider the effects of mechanical constraints against thermal expansion in the plane of the plate. As laboratory specimens are generally cured without restraint, these alternative solutions are not relevant to this thesis, but are readily available for analysis of in-service repairs. In a real patch repair, both the offset in neutral axis caused by the patch, and the thermal residual stresses result in bending. The bending stresses have been estimated by using simple beam bending theory [42] or a more complex analysis of rigidly bonded plates [1], where a properly-formed boundary value problem is posed and the resulting bending stresses are found. To the knowledge of the author, no analysis for bonded repairs has examined the combined effects of thermal residual stresses and bending. Where the patch is large compared to its load transfer length, i.e. [W, H\ » X, the behavior away from the edges of the patch may be characterized by considering the adhesive bond to be rigid. The patch may then be modeled using the bimetallic strip analysis developed by Timoshenko 58 [93,94]. This analysis is presented here using patch terminology, and extended to include an applied force and bending moment. Figure 5-3 shows a segment of a reinforced plate. F 4 M reinforcement plate > F M Interface or adhesive layer Figure 5-3: Coordinate system for Bimetallic Strip Analysis For an elongation of e0 and curvature K0 , the stresses in the plate and reinforcement are found given by equations 5-23 and 5-24. (5-23) °> = E P { £ o -K0-z-ap-AT) °V = Er{£o -*"o-z-a r-AT) (5-24) Considering equilibrium of both forces and moments, equations 5-25 and 5-26 must hold. F = J"°~'^z = j^r( £o ~Ko ~ arAT^dz + Ep(£o ~lco -ccpAT)dz (5-25) M = jz-a-dz = ^z• Er(eQ - K 0 -arAT)dz + z• Ep[s0 - K 0 -apAT^dz (5-26) After integration, defining Q = — , the result is; (1 + S)e0 + U\ - SQ)tPK0 = (ap + Sar)AT + - ^ -(1 - SQ)s0 +1 (l + SQ2 )tPKQ = (ap - SQar)AT -•J lit „ t 2M 2 P'P (5-27) (5-28) Under an external loading given by F, M, and AT, the state of the strip may now be fully determined in terms of £ 0and K 0 . There are two cases of interest here, first, unrestrained bending, and second, bending constrained to a specified curvature, K S P E C . The first case provides a lower limit on the stress state in the repair, and the second case allows consideration of an 59 arbitrary degree of constraint. With maximum constraint, K = 0 and an upper limit on the stress state of the repair may be found. For the first case, s0 and K0 may are given by equations 5-29 and 5-30. E„tr 2\ + SQ2E„ll v ; 4 l + SQ2 (ap+Sar)AT + -f--(\ + S) (5-29) (5-30) For a reinforced plate constrained to a curvature of K , the elongation and applied moment are given by equations 5-31 and 5-32. En = (ap+Sar)AT + ^- + U\-SQ)TPK spec \ + s Et M = - ^ [ K ~SQa r )AT-(\-SQ)e 0 (5-31) (5-32) These limit-state solutions are not particularly useful unless the curvature is specified. Additionally, geometric effects, for example when the curvature of the plate is restricted or enhanced by the applied force, are not included. To examine the effect of a finite patch and include geometric effects, the following geometry will be examined; 60 y=0 Y=H reinforcement Figure 5-4: A Bending Finite Reinforcement This structure may be considered in two parts, first, a strip of plate under a remote applied force with an initial curvature, and second, a finite patch under external loads, and with an initial curvature as found considering the infinite reinforced strip. For the first part of the analysis, it is required to know the moment distribution in the strip given an initial angle and a remote applied force, M(y,0,F). Equation 5-33 may be found by considering a moment balance. M(y -H) = Fw(y-H)+ M(H) (5-33) Considering the bending stiffness of the plate, the following differential equation results. M(H) .w(y - H) w(y -H) = dy2 V ' El. V ' EI (5-34) where — w(oo) = 0 and ^—^ W(H) = — — - are the boundary conditions. dy y } dy2 ^ } EI The solution is then given by equation 5-35, where %p = l S a characteristic length. w(y) = ^ % x p ( - Sp(y - H)) -1] + w(H) (5-35) 61 Differentiating, the moment distribution may be found. The moment at the edge of the reinforcement will be given by the following; M(H,9,F) = £pEIp0 (5-36) Now, for a finite strip of patch, with an initial deflection w0(y) as determined from unconstrained plate equations 5-29 and 5-30, geometric stiffening may be included by again considering a moment balance of the strip. Here, the moment applied at the base of the strip is M(0). M(y)=M(0)+F-[w(y)+w0(y)\ Again, considering the stiffness of the reinforced plate, and setting £ p with the boundary conditions w(o) = 0 and — w(6) = 0 . The solution is then given by equation 5-39 _F_ rp W-il"+^2M^)-1)-^ Considering a moment balance at y = 0 with w(//) known, M(0) = F • (w(H) + w0(H)) + e(H)ZpEIp From which M(0) may be found to be M(0) EI, rp cosh(^//) + ^ s i n h ( ^ / / ) -1 (5-37) (5-38) (5-39) (5-40) (5-41) The final curvature of the plate, considering a finite plate and geometric stiffening may now be found from equation 5-42. 62 L fmal cosh(^J/) +j^sinh(^tf) -1 (5-42) "7> This function implies that curvature will decrease under large tensile loads, increase under compressive loads, and will be equal to the unrestrained curvature, K0 , under zero load. The offset in neutral axis causes an additional moment which may be included during the analysis of the infinite reinforcement. 5.3 Fatigue Damage Progression Fatigue damage of materials is often characterized by a power-law fit relating applied loading to the rate of damage progression. In the case of cracked-body fatigue, crack growth is usually expressed in terms of the applied stress intensity at the root of the crack. In this section we apply this principle, and its extension to the prediction of disbonding of adhesive joints, such that the performance of a degrading composite repair may be predicted. Two crack bridging models have been developed in this thesis; a boundary element model and an interpolation model with geometry factors. Both have been incorporated into a predictive model for patch degradation in the F O R T R A N program D A M A G E , also written by the author. The boundary element model has the advantage of being able to analyze an arbitrary disbond shape with a finite patch, while the interpolation model has the advantage of computational speed and ease of use. Employing either of these models allows degradation of a patch to be predicted using conventional power-law relationships for crack and disbond growth rates. These growth rate calculations require the alternating stress intensity, AK,, and cyclic energy release rate, AG,, , as an input for the cracked plate and adhesive, respectively. The form of the growth-rate equations, and their coefficients are dependent on the materials used and the type of applied loading and environmental conditions. The power-law relationships can be modified to account for mean stress effects and threshold effects in the manner of Forman [36] and Priddle [37] to improve predictions for mean stress and threshold effects. Other predictive methods, such as crack-closure, are equally applicable. It has been noted earlier that AG,, may not be a suitable 63 parameter for predicting fatigue damage in the case of gross reversing plastic deformation of the adhesive. In this thesis, equations 5-1 and 5-2 were used to describe fatigue crack growth in 2024-T3 aluminum sheet [36], and disbonding of F M 73M adhesive [50], respectively. Mean stresses, due to a mismatch in thermal expansion coefficients between the plate and patch, are particularly important, and so the Forman method was employed. Note that the equation for the adhesive was arbitrarily chosen from the four different reported results. da dN 6.311-10" MP a • -Jm. mm I C AK MP a• 4m) V MP a • -Jm cycle aluminum (5-1) db_ dN = 4.3-10 -16 AG Jim2 4.3 mm cycle F M 73M adhesive (5-2) Patch degradation may then be predicted by repeatedly calculating an increment of crack growth and re-evaluating the crack-bridging parameters. Additionally, D A M A G E allows inclusion of the adherend shear lag effect, elastic-plastic adhesive behavior, patch bending, arbitrary load history, and Wheeler crack growth retardation. 5.4 Load Interaction and Overload Effects In the literature review, Section 2.3 covered methods of analysis of fatigue crack growth following an overload. The situation for a patched plate is unique from those covered in the literature. In this analysis, it is attempted to adapt the existing models to the prediction of fatigue crack re-initiation following application of a composite patch repair. The assumed process is; constant amplitude cyclic loading (Kmax<], c r m a x l , i?,) until the repair is applied, a cure cycle involving a constant temperature cure (T) for a specified length of time (tcure), followed by cyclic loading (K m a x 2 , cr, n a x 2 , R2). For constant amplitude cyclic loading, crack growth rates may be estimated using the Paris relation [33]. These crack growth rates are sensitive to the load ratio, R - rjrmin / cr m a x . This is 64 attributed to crack closure effects evident when the crack tip plastic zone and the plastic wake are compressed by the surrounding elastic material. The figure below shows these zones; Figure 5-5: Crack Tip Plastic Deformation Elber [40] popularized the use of crack closure models to estimate the effect of crack closure. Crack closure models are based on the assumption that a crack opening stress intensity, K , must be surpassed before a crack can grow. This leads to an effective stress intensity, Keff = Kmax - Kop. Once Keff is known, crack propagation may be calculated with the Paris equation using AK - Keff • (1 - R). Several empirical relationships exist to estimate Kop. Several have been reviewed [85], and compared to experimental results, and the following relations best correlate with the experiments. K 1 <? i K K ( Y * - ) * * = ( - ^ ) « = o <l-R)2+R (5-3) max max )K<O = )K=O ' — C"p~") ) max max .V Where Scp is the maximum applied compressive stress. 65 Assuming constant amplitude loading is applied up to the point of application of the repair, Kopl can be estimated from equations 5-3 using Kmax,, cr m a x , , and i?,. Kop{, corresponds to an external applied load large enough to both open the crack faces, and take up the compressive residual stresses at the tip of the crack. Typical analytical approaches to prediction of crack closure and its effect on fatigue crack growth are described in Section 2.3. Clayton [82] describes an advanced method based on superposition of Dugdale distributions of the crack-tip stress state for an elastic/perfectly-plastic material. This allows estimation of the crack tip residual stresses. Clayton's analysis considers the case of a single overload which causes enough plastic deformation to prevent the crack tip from closing. Here, crack face pressures can be neglected and the analysis is significantly simplified. This analysis may be extended to the case of a step change in applied load, with crack face pressures, by separating the contribution of crack face pressures from that caused by compressive residual stresses ahead of the crack tip. Kop=Kface+Kres (5-4) One can estimate the effect of subsequent crack growth on Kres by extending the crack length by Aa into a Dugdale crack tip plastic zone stress distribution. Expressing the results in the simplest form, without including thickness effects, constraint factors, or the plastic zone caused by loads applied after the change in load,Kres(Aa) is given by equation 5-4. This is the result presented by Clayton. ^max,i • V A a / r P 0 < Aa < 0.25 • rp ^max.l • (^Aa/rp -1), 0.25•r p<Aa<r p (5-5) rp <Aa So, for Aa = 0, the applied loading, 2 must surpass K , for crack extension to occur. Subsequent extension of the crack into the reversed plastic zone will result in a change in Kns, which is now a superposition of the crack opening stress intensities required to open the crack and to overcome the residual stresses. Cutting into the compressive residual stresses in this zone will cause an increase in Kop and a corresponding decrease in the crack growth rate. Once the crack has grown through the reversed plastic zone (Aa > 0.25 • rp), Kres will decrease, and the crack growth rate will increase until it reaches the steady-state rate under the loading in the patched condition. Crack growth into the plastic zone will also affect the stress intensity required 66 just to open the crack faces, Kface, which will also eventually reach the steady-state value for the patched condition. This will occur because the plastic wake will be of reduced height under the reduced loading. The literature does not contain any advice on how to estimate this decrease. Likely, it would be required to consider the plastic wake as springs of various lengths acting between the crack faces, and solve the crack-bridging problem using a boundary-element model similar to that described in Chapter 4. Here, it would be necessary to use a solution method which could solve the non-linear problem of crack face contact. Assuming that the stress intensity required to separate the crack faces remains at Kfacel while Aa < rp, one can find Kop, and hence the crack growth rate using equation 5-4. This simple model should be sufficient to estimate crack growth re-initiation neglecting the influence of the cure cycle. The effect of thickness, plastic constraint factors, and plastic zone caused by loads applied after the change in load can also be included in the model using the methods described by Clayton. However, the cure cycle has a demonstrated effect on fatigue crack re-initiation following patching [5]. The effect of the cure cycle is a reduction in residual stresses due to creep. The following is a typical creep law for a metal; ds -® ~£ = C-e -cr(t)" (5-6) To apply this law, it is required to know the stress state in the material during the cure cycle. Examining a Dugdale-type residual stress distribution, and considering the crack face pressures due to the constant amplitude crack-opening stress intensity we have the following; 67 Stress Peak Loading Sy <— Distance from Tip Crack Face Pressure -2Sy Reversed Loading Figure 5-6: Stresses Near the Crack Tip The stresses near the crack tip during the cure cycle are caused by compression of plastically-deformed material either in the plastic wake or in the crack tip plastic zone. The elastic material in the bulk of the component causes this compression. Unfortunately, current methods of analysis of the crack face pressure caused by residual stresses are difficult to apply. A new way to model the crack closing effect that is simpler than those provided in the literature is proposed here. When the loading goes below Kop, the crack has closed. Now, the crack tip is modeled as a center-cracked plate with a crack length equal to the reversed plastic zone. Applying a stress corresponding to Kface gives the crack face pressure directly, by using the Dugdale strip yield stress distribution. It also changes the stress distribution in the plastic zone. Now, examining stress-strain curves one can describe the situation for elements of material in several different locations. 68 reversed plastic z o n e remainder of plastic zone elastic regions Figure 5-7: Cyclic Stress/Strain Curves for Near the Crack Tip At the start of the cure cycle, material in the plastic wake and crack tip is loaded in compression by the surrounding material. This is a strain-controlled loading. It is the mismatch in size between the plastically deformed parts and the surrounding material that results in the compressive stresses. Here, creep will act in two ways; it will strain material in the plastic wake and plastic zone, making it smaller, and it will plastically deform highly stressed regions of the plate near the crack which have not yet been plastically strained. Both will act to relieve the mismatch in strains, lessen residual stresses, and reduce crack retardation. Modeling this situation is difficult. Simple models based on the Dugdale strip yield model do not provide the strain state near the crack, but instead examine stresses near the crack tip. Developing a simple model to look at creep effects will require the same sort of assumption. If it is assumed that the release of residual strains is related to the creep strain through Hooke's law, and that the initial stresses in the plastic zone are characterized by the flow stress, the following relationship is obtained; a{t) = cr0 -E-sc{t) (5-7) where <j{t) is a stress representative of residual stresses near the crack. Integrating with respect to time to get the total creep strain yields; 'cure Q a(t) = cr0-E- { C • e~J • a{t)" • dt (5-8) 0 69 By expressing equation 5-8 as a differential equation, the solution may be found by separation of variables. The result is equation 5-9; o-(f)= cr0 _" -(\-n)-ECe'r t l-n (5-9) As a first order approximation, noting generally that stress intensities are proportional to stress, we may apply cr(tcure) I a0directly as a factor to KresandKop,. This will result in a decrease in the retardation effect. A simpler implementation could apply this factor to the Wheeler retardation coefficient. Figure 5-6 shows the relaxation as a function of the cure cycle length. The result presented is not representative of any particular material. Note that the model results in an exponential decay, which is expected for an internally-driven creep process, where the creep relieves the stored energy that is driving the process. Cure Time (hours) Figure 5-8: Relaxation vs. Cure Time A proper model would need to include the effects of redistribution of the stresses. Existing creep models assume an outside load is driving the creep, and not the crack tip residual stresses. Since the creep law is of the same form as the Ramberg-Osgood power law one may use the Hutchinson-Rice-Rosengren [88, 89] crack-tip stress and strain field solutions developed using J-integral analysis. The model presented above will likely require a fit to experimentally determined parameters to give accurate results for a particular thickness, material, and loading range. 70 5.5 Summary With the addition of the crack bridging model and fatigue crack growth retardation models to those developed previously, analytic and simple numerical tools exist which may be applied at the critical locations in a composite patch repair, and can estimate the critical parameters that determine failure initiation times and rates of crack and disbond growth, such as stresses and stress intensity factors. Also, because of the large amount of data available in the literature for 2024-T3 aluminum, elaborate crack growth rate equations are available. However, determining which of these critical locations will initiate a failure, and when, is not possible without extensive material testing and analysis. Particularly, it is difficult to predict disbond initiation and growth rates due to the lack of an accepted model and material data for the behavior of the adhesive. A method has also been developed which allows consideration of the annealing effect of the cure cycle on re-initiation of crack growth. This model also requires experimental verification. In the next section, these models will be used to predict the behavior of a patch under fatigue loading. 71 6. Damage Progression: Comparison to Experiments 6.1 Overview In previous chapters, models have been developed which allow the prediction of damage progression in a repair. Here, the as-modeled results are compared to experiments, and the relative importance of adhesive plasticity, adhered shear-lag, bending, thermal residual stresses, and disbond size on the rate of crack growth are illustrated. The relevance of each of the above effects to estimation of patch life will be demonstrated, and the capabilities and shortcomings of the models will be highlighted. 6.2 AMRL Specimen A specimen geometry has been devised by the A M R L for testing of composite patches which minimizes bending. Here, two patched plates are bonded to a honeycomb core and tested simultaneously. This configuration represents the repair of panels reinforced against bending by underlying structure, or a double-sided repair. The patch is applied to edge-cracked plates of 3.25 mm thick 2024-T3 aluminum plate. The patch consists of seven layers of Textron 5521/4 tape bonded with 0.127 mm thick American Cyanamid FM-73M film adhesive. The cyclically applied loading was 138 MPa, with a 0.1 load ratio. The process-induced thermal residual stress in the plane of the crack [48] is approximately °'residual = 50 MPa . Baker [66] has reported results for patches with manufactured disbonds of varying size. The following figures compare the experimental crack growth rate results to those predicted using the modified interpolation model. The model predicted little disbond growth. 72 Interpolation Model Predictions, b=2 mm 300 C y c l e s ( t h o u s a n d s ) Interpolation model with correction factors .. with thermal residual stresses .. and shear lag effect .. and adhesive plasticity Experimental result (initially no disbond) Figure 6-1: Comparison of Interpolation Model and Experimental Results, With a Small Disbond Interpolation Model Predictions, b=20 mm 1 1 V/ 1 / / 1 o " /— / ^ • |0 1 1 1 1 1 -0 10 20 30 40 50 60 70 C y c l e s ( t h o u s a n d s ) Interpolation model with correction factors .. with thermal residual stresses .. and shear lag effect .. and adhesive plasticity Experimental result (initial 20 mm disbond) Figure 6-2: Comparison of Interpolation Model and Experimental Results, With a Large Disbond As shown in the figures above, the results for the patch with a 2 mm initial disbond compare very well to those predicted by the model. Also demonstrated is the importance of secondary effects; mean stress, adhesive plasticity, and shear deformation of the patch. Note that these effects are 73 oo© oo© not as important with a 20 mm disbond. Here, the compliance of the disbonded portion of the patch becomes the key parameter. The model results for a 20 mm disbond show a significant departure from what has been observed experimentally. One possible source of this error is that the model assumes a crack in an infinite plate. This assumption may no longer be supportable in the case of a large disbond in an edge-cracked plate. Here, the plate stiffness is reduced and the patch will carry more of the load, reducing the crack-growth rate. To test this explanation, the model has been applied to a different set of experiments performed on a center-cracked panel. 6.3 WPAFB Specimen Data is available for composite patch tests performed by Denney and Mall [69-71] at Wright-Patterson A F B (WPAFB). Here, patches were applied to thin centre-cracked panels with various sizes of full-width disbonds. A single thin patch is bonded to a 1 mm thick 2024-T3 aluminum sheet. The patch consists of three layers of Textron 5521/4 tape bonded with 0.127 mm thick AF-163-2 M film adhesive manufactured by 3M. The cyclically applied loading was 120 MPa with a stress ratio of 0.1. The process-induced thermal residual stress in the plane of the crack can be found to be approximately aresidual - 90 MPa. The following figure compares experimental crack growth rate results to those predicted using the modified interpolation model including adhesive plasticity, bending, and shear deformation of the patch. Again, the model predicted little disbond growth. 74 Interpolation Model Predictions 80 Cycles (thousands) 6 mm disbond, interpolation • " 3 mm disbond, interpolation — 1.5 mm disbond, interpolation i"*"t' 6 mm disbond, experimental 0 0 0 3 mm disbond, experimental 1.5 mm disbond, experimental Figure 6-3: Comparison of Interpolation Model and Experimental Results, WPAFB Specimen In the result presented in Figure 6-3, the significant under-estimate of fatigue life is not as evident as for the edge-cracked A M R L specimen. This suggests that the geometry of the cracked panel may indeed play a large role. This is particularly true with a large disbond, as the reduced patch stiffness and hence larger patch characteristic length, A , means that edge-effects will remain a factor for a larger period of crack growth. Fortunately, models based on a center-cracked sheet will tend to under-estimate how much the patch will restrict crack opening when applied to finite plate geometries. This is true because finite plates with cracks will be more compliant than an infinite center-cracked plate. So, conservative crack-growth results should be expected for patches with large disbonds, and patch life and inspection intervals may be safely planned. The models under-estimate crack growth rates for small disbonds. The source of this error is likely in the method used to predict disbonding. In the experiments, significant disbond growth was noted for patches with initially small disbonds, however, the damage tolerance model predicted very little disbond growth. As mentioned previously, the author is not convinced that the cyclic strain energy release rate is a suitable parameter for predicting disbonding. One would 75 expect the disbond growth rate to be largest for patches with a small disbond, as the load transferred to the patch is larger. One advantage of the full boundary element model over analytical means is the ability to model a finite width patch. The next figure compares such analyses for the three W P A F B specimens. This feature is particularly important for planning inspection intervals, because the crack tip will extend beyond the edge of the patch, where it can be more easily detected. Numerical Model Predictions 60 50 h 10 '•h o - - O _ •-'t- . O _ - ' O _ — — , . O . - O _ — — 20 40 60 80 100 Cycles (thousands) 120 140 160 6 mm disbond, numerical 3 mm disbond, numerical — 1.5 mm disbond, numerical 6 mm disbond, experimental 0 0 0 3 mm disbond, experimental 0 0 0 1.5 mm disbond, experimental Figure 6-4: Comparison of Boundary element Model and Experimental Results, WPAFB Specimen For crack growth under the patch, a < 25 mm, the results are similar to the interpolation model results. It should be noted again that in the experiments, significant disbond growth is apparent in ultrasonic C-scans. For example, the specimen with an initial 1.5 mm disbond appears to have a disbond larger than 3 mm at the end of the test. So, at this point it is difficult to say i f the predicted life for crack growth beyond the edges of the patch will be accurate, as the model does not yet accurately predict disbonding rates. The model also assumes that the stress in the crack plane is constant, when in fact it changes as the crack tip moves from under the patch to an unreinforced section of the structure. 76 In addition, using the Wheeler crack growth retardation model, cracking was found to be nearly completely retarded in the A M R L specimens, resulting in a many-fold increase in predicted patch life. In crack-growth experiments, significant retardation was observed, but not to such a large degree. 6.4 Summary In this chapter, the models developed within this thesis have been tested against experimental results from the literature. Limitations of the modeling techniques have been highlighted. It has been shown that prediction of the behavior of a disbonding composite patch is possible, but is currently limited principally by two issues. First, the inability to accurately model realistic patch configurations, including the edge-cracked plate, is a significant issue. Second, disbonding of adhesives is very difficult to predict, and it would appear that strain energy-release rate methods are not applicable in the case of significant adhesive plasticity. In the next chapter, the sources of any differences between as-modeled and experimental results will be discussed, and means of improving the models to overcome these difficulties will be formulated. 77 7. Discussion 7.1 Overview In this chapter, a discussion is presented on the trends observed in the modeling of patches, and on the comparison of these modeled results with experiments. Based on this, the limitations of the modeling techniques are highlighted, and possibility for improvement of predictive capability of the models is discussed. This is followed by a discussion on the barriers to certification of the composite patch technology, possible means by which these barriers may be ameliorated, the requirements for a damage tolerance analysis, and recommendations for future work on composite patches. 7.2 Results Several observations regarding trends in patch behavior and limitations of the models were noted in Chapter 6. These are listed and discussed below; • Elastic-plastic adhesive behavior and patch shear deformation are important with little or no disbonding, and less important for a greatly disbonded patch. Power-law disbond growth model based on strain-energy release rates appear to provide poor results. The large dependence on adhesive behavior for small disbonds means that accurate adhesive properties under different loading rates, load dwell, and temperature conditions will be required to predict crack and disbond initiation and growth rates under service conditions. The models in this thesis rely on the one-dimensional analysis of a bonded joint. Within this scheme, closed form solutions are possible for any piece-wise linear adhesive constitutive law. So, modeling of the adhesive can be improved using a multi-linear stress/strain curve. Temperature and strain rate effects could be included by modifying the properties upon which the stress/strain curve is based, i.e. the shear modulus and yield strength of the adhesive, i f necessary for the application. Criticism of the strain energy release rate method in this thesis and other papers suggests a need for a more rational approach to disbonding and failure analysis of bonded joints. This approach 78 might be based on the adhesive shear yield properties and joint shear energy/bond area, as suggested by Hart-Smith for the static strength of bonded joints. It should also be noted that there is no direct analogue between a patch and a lap joint for the determination of strain energy release rates, because an incremental change in the disbond length for a small section of a patch releases stored strain energy in the cracked plate and in all other parts of the patch. Models employed in this thesis and in the literature [21] have not considered this extra energy available for disbonding of a patch. • Edge effects and finite plate size are increasingly important considerations as disbonds get larger. This causes models to over-predict crack growth rate results for finite plates. Crack-bridging models based on analytical techniques for infinitely wide, flat plates cannot easily be made to model finite width or edge-cracked plates. While these models should over-predict crack growth, they will also under-predict the stresses causing disbonding of the patch. In addition, these models should tend to under-predict crack growth beyond the edges of the patch since a constant stress is typically applied in the plane of the crack. This crack-opening stress, cr0, is reduced from that in the surrounding plate due to the patch reinforcing effect. Beyond the edges of the patch, the crack opening stress should gradually increase until it is equal to the remotely applied stress. This effect could be included by going back to the original inclusion analogy and using the appropriate stress distribution, or by inputting the stress distribution from a finite element analysis of an equivalent doubler. • Thermal residual stresses are a dominant factor in crack growth. This analysis correctly predicts that thermal residual stresses are detrimental to patch performance, for both perfectly bonded and disbonded patches. This is a well-known phenomenon. It should be noted that during patch application in the field, the cracked plate is restrained against thermal expansion. Laboratory specimens are typically cured without restraint. As models and material properties used for analysis of patches are based on laboratory work, inclusion of the effects of mean stress on crack growth and disbonding is essential to get accurate predictions of patch behavior in the field. • The model predicts significant crack retardation effects that are not nearly as apparent in experiments. 79 The model predicts initial crack growth retardation of a degree which is not apparent in experiments. This disjoint between models and experimental results has previously been observed, and is accounted to the annealing effect of the cure cycle. Predictive models could be modified to include this annealing effect, possibly by correlating of the crack growth retardation coefficient to the temperature and time of the cure cycle. A theoretical derivation of such a model is presented in Chapter 5, but insufficient experimental data is available to calibrate it. 7.3 Experimental Validation Testing to validate the disbond models will involve the existing AMRL-type specimens. It is envisaged that while testing, potential drop probes will monitor crack length. The tests will be stopped at regular intervals to measure the accompanying disbond size. A thorough review of the open literature has not turned up any experiments tracking natural disbond initiation and growth for a composite patch repair using the A M R L specimen. Mall and Denney [52] have tracked natural disbond growth for a thick aluminum plate reinforced on only one side. Baker [5] has published results for natural disbonds following fatigue testing and destructive examination. Other experiments have monitored the growth of artificial disbonds [69-71]. It is essential to gather data which will validate the initiation and disbond models. Unfortunately there is no method for directly monitoring stress intensity or adhesive shear stresses. Validation is only possible by comparing crack and disbond growth rates, and by measuring the stresses in the patch near the crack. The best demonstration is to compare a patch with a large disbond to a patch with a much smaller one. Since the stress intensity and adhesive shear stresses will be proportional to growth rate, a ratio of growth rates between tests would illustrate the effect of disbonds. It should be possible to control disbonding of the patch. The method proposed here is to grow a crack in one plate at an elevated temperature, such that the adhesive modulus is reduced and hence the near-crack adhesive shear stresses are reduced. Evidence for such an effect is documented in [5]. Alternatively, sufficiently low loads can be applied so as to induce only minimal disbonding near the crack. It will be necessary to periodically change the load on the specimen with a small disbond to that used on the naturally disbonded patch, such that the crack-and disbond- growth rates could be compared directly. The crack growth rate at this load will be 80 measured, and then the patch will be returned to the nominal testing condition. This will require careful planning to ensure that retardation and load interaction effects do not alter the measured growth rates. Test Program The testing would be performed in two steps; Step one, static loading of an instrumented patch specimen to validate the stress and fracture mechanics models without disbonds. In addition, a clip gauge will be installed to measure the crack opening displacement. Step two, fatigue testing of two composite patch specimens within the limits of validity of the model. It is still required to decide on the magnitude of the cyclically applied loading for these specimens. Following testing, the naturally disbonded patch will be destructively examined. Further fatigue testing, destructive testing to determine failure loads, and fractographic examination of the specimen with a suppressed disbond should await the development of models that can predict these stages of patch degradation. This could be the basis for a subsequent research program - prediction of the remaining strength of the patches - and would go a long way toward a complete damage tolerance assessment of bonded patches. While non-destructive examination during testing will provide important information on crack and disbond initiation and growth, post-test fractographic examination will provide a concrete assessment of the failure mode. To this end, both specimens will be destructively examined. Of particular interest will be the mode by which disbonds initiate and propagate near the crack and at the edge of the patch. Observations in this vein will greatly influence the choice of models for these failure modes. Crack Opening Displacement The potential drop method will be used to monitor the crack length, and clip gauges will monitor the crack opening displacement. The MDT control software and potential drop monitoring system currently available for this research can be programmed to automatically take readings during fatigue testing. This will enable the accurate measurement of crack growth re-initiation 81 without continually stopping the test and taking readings with an eddy current probe. Figure 7-1 shows the potential drop probe configuration and crack opening displacement gauge. Figure 7-1: Clip Gauge and Potential Drop Probe Configuration Crack Length Johnson's equation [90] may be used to predict the change in voltage across a crack. For a simple edge-cracked panel, this equation takes the following form; The parameters in this equation are; y, the distance from the electrode to the crack center line; W, the width of the plate; a, the crack length; and , the initial crack length. Minimizing y will maximize the sensitivity. To do this, the electrodes will be applied very close to the crack opening. In previous work on A M R L specimen preparation [48], the size of the crack-tip plastic zone was estimated from the pre-cracking data. The final plastic zone size was 0.029 mm. Applying (7-10) 82 Johnson's equation, one can estimate the sensitivity required to detect re-initiation of crack growth. Setting y at 2 mm, W at 160 mm, and ao at 20 mm, the result is the following; V (ao+rp) = 1.00144 (7-11) So, detecting crack growth through the initial plastic zone will require instruments sensitive enough to detect a 0.14% change in the voltage signal. If this is not possible, re-initiation can be estimated by projecting crack growth curves back to the initial crack length. The following figure shows the voltage ratio vs. crack length predicted by Johnson's equation for crack growth under the patch. V 4 a mm Figure 7-2: Voltage Ratio Predicted by Johnson's Equation It is apparent that monitoring of crack growth following re-initiation will not require great probe sensitivity. Calibration of the potential drop system will be achieved by cracking a plate with the same geometry as the A M R L specimen face plates. Optical measurements will be taken at regular intervals to compare to the potential drop measurements and theoretical results. Calibration of the clip gauge is based on the standard procedure outlined in the MTS manuals. The potential drop specimen is made from 2024-T3 aluminum, i.e. from the same material supply as that used to manufacture the patch specimens. The dimensions are also the same. Fatigue pre-cracking will follow the same procedure as for the A M R L specimens, except that the initial notch will be sawn, i.e. not created by E D M . Two concerns are the effect of the honeycomb core, which may complete a circuit that bridges the crack, and contact between the machine and the specimen, which again may complete a circuit and interfere with the potential 83 drop. To investigate the effect of the honeycomb core, clamps and a piece of scrap aluminum will be pressed against the specimen near the crack. A digital voltmeter will also be used to estimate the resistance between the A M R L specimens. To investigate the problem with the end fixtures, the end conditions of the calibration specimen will be made in a manner similar to that of the patch specimens. Paper or some other material will insulate the specimen from the shims, as the adhesive will do in the patched specimens. Another concern is the effect of the surface treatment of the specimens. The composite patch specimens were grit blasted and primed. The outer layer of the clad 2024-T3 aluminum is a more pure aluminum than the alloyed inner material and may change the electrical properties. Also, the effect of the honeycomb core and the potential for a crack bridging circuit will be very dependent on the surface treatment. It is highly likely these effects are of only secondary importance. Disbonding It will be necessary to utilize a C-scan, or other non-destructive examination technique, to monitor disbonding of the patch. Acquisition of a suitable system is being investigated, however the development of a new facility may be necessary. Calibration will require scanning of test specimens with known disbonds. This will allow assessment of the accuracy of the system and the measurement technique. These may be available commercially, or may be developed at U B C . It will be necessary to do fulfill these requirements prior to specimen testing. Thermographic techniques [91, 92] are becoming more widely accepted for measurement of disbonding. The result of a thermographic investigation of a disbonding repair is similar to that of a C-scan i.e. a two-dimensional picture showing color contours which may be used to estimate the extent of disbonding. Thermography has the advantage of ease-of-use as there is no need for scanning equipment or removal of the specimen from the loading frame. The temperature profile may be detected using either a infrared camera or using temperature sensitive paints. Patch Stresses To validate patch structural mechanics models, it will be beneficial to monitor stresses in the patch in the area of disbonding using strain gauges. Here, it is desirable to obtain a stress distribution which would correlate with the crack-bridging stresses determined by models. Also 84 of interest w i l l be the non-linear response of the patch, which w i l l be indicative of the state of the adhesive. Lap Shear Specimens Testing of bonded joint coupons w i l l be required to develop or validate models for predicting the disbonding rate of the patch. Ideally, the specimen would have the same materials and through-thickness dimensions as the A M R L patch specimens, would be built with the same surface treatments and cure cycle, would be manufactured in such a way as to obtain similar thermal residual stresses, and would minimize bending effects. The specimens would need to be wide enough that edge effects would be negligible. Load hysteresis curves would be gathered such that the inelastic energy consumed in crack growth could be determined, allowing investigation of energy-based parameters for disbonding. 7.4 Certification It has been shown that current patch modeling techniques can be improved upon in several ways. It has also been demonstrated which experiments would be of use in the validation of patch structural mechanics models. However, only some of these are relevant to certification of a patch repair. Certification w i l l require an initial assessment of joint strength, followed by a full damage tolerance analysis to allow long-term use. The requirements of a full damage tolerance assessment include a proven predictive capability for patch degradation for a range of repair configurations and loading conditions, and the effect of the degradation on the strength of the repair. The state-of-the art, as presented in the open literature, does not yet allow a full damage tolerance assessment of a composite repair. Prediction of patch degradation is limited by several factors. Real repair configurations are difficult to model, particularly when considering disbonding of the patch or out-of-plane loading. Disbonding rates have proven difficult to predict. Environmental effects cannot yet be predicted to the point of examining the effects of the thermal and environmental cycling which a joint would experience in service. Regarding the joint strength, little has been done to address the strength of a degraded patch. Efforts should best be directed toward improving the range of patch 85 configurations analyzable, characterizing adhesive behavior, and predicting the strength of a degraded patch. A model capable of predicting the growth of edge cracks would be useful because of the large amount of information available for the A M R L test specimen, and because this is a practical application for a repair. In addition, a method of analysis of multiple cracks would be useful i f this technology is to replace riveted repairs which have left many holes for crack initiation, or be applied to structures with wide-spread fatigue damage. A method to input crack-plane stresses from a finite element model of a reinforcement without a crack would greatly extend the ability of Green's function-based damage-tolerance models to predict failure of complex patch configurations. Improvements in the characterization of adhesive behavior are needed to account for changes in material properties with temperature, environment, and loading rate. This will allow accurate disbonding predictions and residual strength assessment under field conditions. Characterizing the effects of the cure cycle on crack retardation is an interesting problem, and a useful one to analyze - as it will largely determine the actual life, and hence, feasibility, of a repair. But, it is not likely to ever be relied upon on for approval of a repair. It is envisaged that design codes will employ the extensions of the interpolation model developed in this thesis (equations 4-6, 4-7, 4-10). The correction factors developed in Section 4.4 should be considered by the code writers when developing design factors for applied loads or material properties, but are likely too confusing for direct inclusion in a design code. This will allow a straightforward and conservative assessment of the stress intensity and adhesive stresses for a disbonding repair. The crack-opening stress would be determined by the methods of analysis of bonded doublers. It is yet required to develop simple equations for A , the patch characteristic length, which allow consideration of more complex adhesive behavior than the linear-elastic case. A means of predicting the disbonding rate of a patch in a conservative manner is also required, and will be a difficult task given the observation made in Section 7-2 that the strain energy release rate depends on the strain energy of the entire repair and not just the equivalent lap joint. The next section describes how this process for fatigue damage progression can be extended into a full damage tolerance analysis of a repair. 86 7.5 Damage Tolerance Assessment Method The models described above are the essential elements of a complete damage tolerance analysis of the composite patch. As the focus throughout has been to develop hand calculations and simple numerical models, this damage tolerance assessment will be easy to follow and easy to present. This makes it suitable for preliminary design and for implementation in codes or standards for composite patches. It also makes feasible the use of stochastic models for patch degradation - an analysis capability which is highly desirable when a structure sees random load patterns or includes materials with variable properties. The flow chart in Figure 7-3 shows the analysis process for a complete damage tolerance assessment procedure for a repair. Once degradation and final failure of the patch have been modeled, different inspection regimes may be tested to determine an appropriate method and interval. 87 Input Parameters Doubler Analysis Damage Initiation Damage Propagation Output Parameters Inclusion Analogy Crack Bridging Figure 7-3: Damage Tolerance Assessment Flow Chart Input Parameters Material properties, dimensions, and loading are input parameters. Also, an estimate of the accumulated damage to the aluminum at the edge of the patch is desirable. The model will update some of these parameters, crack length, for example, as it works through the analysis. Doubler Analysis Doubler analysis looks at an uncracked plate reinforced by a doubler. The result is an estimate of the shear and peel stresses at the edge of the patch, and the stresses at ply drop-offs. Once a disbond initiates, the doubler analysis will allow an estimate the fracture parameter governing damage propagation. Inclusion Analogy 88 The inclusion analogy will estimate the maximum stresses in the metal at the edges of the patch, and the reduced stress at the crack location. The reduced stress can also be estimated using a simple finite element analysis of an inclusion. This is an input to the crack bridging assessment. Crack Bridging Here, the program will calculate stress intensity, crack face displacement, and adhesive stress. Before near-crack disbonding initiates, the crack bridging analysis will use interpolation functions. After initiation, the model will either perform a full boundary element crack bridging analysis, or, i f feasible, use an interpolation function modified to account for disbonding of a specified shape. After a disbond initiates, the model will calculate an adhesive fracture parameter for use in disbond growth rate calculations. Damage Initiation For the first pass, the model will calculate the time to initiation for each failure mode. It will then flag the one that begins first as "initiated", and set the cycle count at the time to initiation for this mode. It will also calculate the damage accumulated for each of the other modes. In subsequent passes, it will increment the damage for each failure mode, and flag the mode as "initiated" when appropriate. Damage Propagation From here, the model will work with load blocks. Damage propagation will occur in all "initiated" modes. The initial crack length for disbonding will need to be set according to some arbitrary criteria. One possibility is to set it at the length of the plastically deformed region. Failure Criteria The model will check against specified failure criteria. Examples of failure criteria are; • complete disbonding of the patch • disbonding has compromised the elastic well, and complete disbonding is imminent • crack growth has exceeded a limit length 89 • joint strength is insufficient to resist a specified load • crack critical stress intensity exceeded at a specified load • damage has initiated in the metal outside of the patch Output Parameters The model will output the parameters that it has tracked through the modeling process. Results The result will be a model, based on relatively simple analytical methods, to predict the damage tolerance properties of a repair. It would be useful in design, inspection planning, and disposition of inspection results. Because of the emphasis on analytical techniques, this model should be amenable to the future addition of such effects such as temperature, bending, and hygrodynamic effects without the total model being cumbersome or unstable. For the same reason, the simplicity of the final resulting relations, this work is amenable to inclusion in future codes or standards for composite repairs. Limitations As stated earlier, a lack of fracture and fatigue properties for the adhesive system is a limitation on the model. Other limitations are: the assumption of a large, flat plate inherent in the Green's functions; temperature; load interaction, rate and dwell; and environmental effects. These are not included because they are outside of the scope of the current research program, however, upon completion of this program a good baseline will exist that can be extended. Additionally, little work has been done to date to assess the strength of a partially disbonded patch, which is required to assess when a peak load may fail the structure. 7.6 Summary Several shortcomings in current techniques for prediction of patch fatigue life have been identified. Also identified were means by which these shortcomings may be addressed - both 90 through improvement of the existing models, and by validation against experiments. The result, after completion of the experimental work, would be a validated model to predict; (a) stresses at critical locations, and the crack tip stress intensity, (b) re-initiation of crack growth following application of a composite repair, (c) crack growth within the "constant stress" region of the patch, and beyond, where the stress state will approach that of the nominal, unrepaired state of the structure, (d) the effect of near-crack disbond growth on patch efficiency. This will be useful in design and damage tolerance assessment of bonded composite repairs. Disbond data gathered periodically during the experiments will be valuable for the development of improved models and comparisons with different material systems and testing environments. The final expression of the models developed will be simple enough for inclusion in codes or standards for damage tolerance assessment of composite repairs, which it is hoped will aid in the certification of composite patch technology. The synthesis of available models will allow a full damage tolerance method in which stochastic analysis methods may be implemented. This would allow a full risk assessment of a repair. 91 8. Conclusions The objective of this thesis has been met: the structural mechanics of a composite patch have been modeled using Green's functions and classical methods for analysis of bonded joints, and a means of prediction of patch life in the presence of near-crack disbonding has been developed and compared to experiments. Additionally, this structural mechanics model has been used to develop and verify design equations suitable for rapid assessment of the degradation of a patch, which advances the University of British Columbia composite patch repair program toward its objective of airworthiness certification and widespread use of the technology. Several models for patch mechanics exist, either numerical or analytical, which can be used to predict the damage tolerance of the composite patch repair. Two models have been presented in this thesis, one a boundary element model based on Green's functions, and the other a closed-form, approximate solution based on interpolation techniques originally developed by the A M R L . In this thesis, these mechanical models have been combined with the analytical techniques necessary for prediction of patch life, and the influence of various physical phenomenon (adhesive plasticity, patch shear deformation, thermal residual stresses, and bending) have been investigated. The interpolation model has been verified against the boundary-element solution for the case of a partially disbonded patch, and has been extended to allow the prediction of adhesive shear stresses and rate of disbonding. The boundary-element model has been simplified by adapting classical bonded joint mechanics relationships. Both models may be used with good results within the limits of their validity. The models developed in this thesis allow prediction of damage progression in repairs while addressing the following factors, and deal with these within the limitations and criticisms described in this thesis. • Thermal residual stresses arising from patch curing. The specimen was assumed not to be restrained against thermal expansion in the plane of the patch. • Bending including both geometric stiffening and curvature resulting from thermal residual stresses. To the knowledge of the author, these two effects have not previously been combined in a patch analysis. 92 • Patch shear deformation, which has a significant impact on patch performance when using laminated composite patches with a small disbond. • Disbond shape and size, which both have a demonstrated effect on patch fatigue life. The interpolation and the boundary-element methods allow the analysis of a disbonding patch. • Elastic, elastic/perfectly plastic, and reversing elastic/perfectly plastic adhesive response and the effect of the adhesive response on the stiffness of the patch and its ability to restrain the opening of the crack. • Disbonding rate under constant amplitude loading applying strain energy release rate methods. • Crack growth rates with load interaction, threshold, and load ratio, using classical means for predicting fatigue crack growth. • Finite patch geometry, which allows the prediction of final failure after the crack has grown beyond the edge of the patch. By comparison with experimental results available in the literature, several limitations of these models have been identified. Ideally, a repair fatigue degradation model would be capable of addressing the following important factors: • Effect of fatigue crack growth retardation considering the effects of the cure cycle. Currently, no means exists to predict this effect. A method has been proposed in this thesis. • Accurate prediction of disbonding rate - effects of loading dwell time, interaction, and rate, which have been noted in experiments to have a significant effect. • Prediction of disbond mode - likely characterized by relative stiffness' of patch and adhesive. Experiments have observed different disbonding modes for different types of loading and different patch designs. • Extension to practical geometries - means of analysis are required which can account for curvature, multiple cracks, and panels of finite dimension. 93 • Effect of temperature and water ingress and on joint strength, compliance, and disbonding rate. The adhesive properties are sensitive to temperature and moisture content. These effects may be important determinants of patch life in service. • Mixed-mode loading and multi-directional patches. The models in this thesis and most models in the literature consider the case of a unidirectional patch under tension. A full damage tolerance assessment of a repair under service conditions will also require the prediction of the residual strength of the repaired structure and the possibility of crack arrest. To allow widespread use of the bonded composite repair technology, certification will require a means of determining the rate of fatigue damage progression in a repair, and a full damage tolerance assessment considering most of the effects above. In light of this requirement, the work in this thesis is an incremental but important step toward certification of the repair technology. The disparate elements required to analyze the fatigue failure of a patch have been refined and combined in one package. Improvements have been made to the estimation of thermal residual stresses in the presence of bending. The interpolation model has been extended and verified against numerical results. The limitations of these patch analysis techniques have been determined, and the need for experimental verification of patch stresses and of disbonding equations has been proven. As a result, a detailed experimental program and modeling improvements have been specified which will increase our predictive capability and move the technology significantly closer to certification. This work will be carried out in the next phase of the program, and is described in the next chapter. 94 9. Recommendations for Future Work The composite repair program at the University of British Columbia is directed toward addressing some of the shortcomings preventing certification. Primarily, the focus is on determination of the patch parameters critical to life assessment, developing predictive models, and validating them through experiments. This will ultimately be accomplished through the development of simple analytical tools and methods suitable for inclusion in codes and standards, which will hopefully bring the level of analysis for a conservative patch design down to a level comparable with other repair technologies, for example riveted repairs. To further the goals of this program, it is recommended that the analytical approaches illustrated in this thesis be validated through the testing of at least two AMRL-type specimens under constant amplitude loading. Currently, there is limited information on disbonding of patches and the stresses in the disbonded portion of the reinforcement for joints with unidirectional composite adherends. Experiments should be performed to verify the crack-bridging models by monitoring strain gauges at critical locations and systematically measuring disbond propagation. This should be done in addition to typical crack length and crack-opening displacement measurements. The analytical methods should also be extended to the cases of an edge crack and multiple cracks. Development and testing of a rational model for disbond growth, based on rate of plastic work, will also require the testing of lap-shear specimens with similar materials and dimensions to those in the A M R L specimens. In addition, comparisons should be made with published experimental results to hopefully show that the method is applicable to a wide range of material systems and joint configurations. Failing this, conventional approaches can be calibrated to the A M R L specimen results for the purpose of validating the damage tolerance model for this particular configuration. Based on these recommendations, currently planned work under this research program includes: critical evaluation of the damage tolerance methodology for key patch and material parameters, development of specific small-scale specimens to test these parameters under room temperature and moderate load rates; validation of the complete damage tolerance model through testing of the four remaining specimens; and work to follow up on this testing and improve the model. 95 There is scope for further work beyond the current research program. A full damage tolerance analysis of the patch would require the following; (a) disbond initiation properties of adhesives, (b) disbond rate properties of adhesives under diverse loading rates and environments, (c) load dwell, interaction, and rate effects on repair efficiency. A new specimen geometry, using a patched center crack would be more suitable for studies of near-crack disbonding. Because in a center-cracked configuration the adhesive shear stress varies across the crack in a well understood manner, a large amount of data on initiation and propagation of disbonds could be collected with relatively few specimens, addressing many of the above points. A further advantage of this test technique would be that the results would not be affected by the confounding load transfer that occurs near the free edges of the lap-joint specimens typically used in disbonding experiments. So, a test program involving development, standardization, and testing of this type of specimen is suggested as a possible future project. 96 10. Bibliography 1] Baker, A . A. , Jones, R. (editors), "Bonded Repair of Aircraft Structures", Martinus Nijhoff, Dordrecht, The Netherlands, 1988 2] Jones, R., Chiu, W. K. , Smith, R., "Airworthiness of Composite Repairs: Failure Mechanisms", Engineering Failure Analysis 2/2 (1995) 117-128 3] Jones, R., "Assessing and Maintaining Continued Airworthiness in the Presence of Wide Spread Fatigue Damage: An Australian Perspective", Engineering Fracture Mechanics 60 (1998)109-130 4] Jones, R., Chiu, W. K. , Paul, J., "Designing for Damage Tolerant Bonded Joints", Composite Structures 25 (1993) 201-207 5] Baker, A . A. , "Repair Efficiency in Fatigue-Cracked Aluminum Components Reinforced with Boron/Epoxy Patches", Fatigue and Fracture of Engineering Structures 16/7 (1993) 753-765 6] O. Volkersen, "Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschenquerschnitten", Luftfahrtforschung 15 (1938) 4-47 [7] Hart-Smith, L . J., "Adhesive-Bonded Scarf and Steeped-Lap Joints", NASA Technical Report CR-112237, 1973 [8] Hart-Smith, L . J., "Analysis and Design of Advanced Composite Bonded Joints", NASA Contractor Report CR-2218, 191A [9] Albat, A . M . and Romilly, D.P., "Stress Analysis of a Tapered Bonded Reinforcement", Unpublished as of September 1999. [10] Jones, R., Chiu, W.K, Tomas, J., and Trippit, J., "Lap Joint Theory Revisited", Polymer and Polymer Composites 3/1 (1995) 11-19 [11] Goland, M . , Reissner, E., Journal of Applied Mechanics 20 (1953) 355 [12] Wang, C. H. , Rose, L . R. F., "Determination of Triaxial Stresses in Bonded Joints", International Journal of Adhesion and Adhesives 17 (1997) 17-25 [13] Ignjativic, I., Chalkey, P., Wang, C , "The Yield Behavior of a Structural Adhesive Under Complex Loading", DSTO-TR-0728, Airframes and Engines Division, Aeronautical and Maritime Research Laboratory, Australia, 1998 97 [14] Russell, A . , "Fatigue Crack Growth in Adhesively Bonded Graphite/Epoxv Joints Under Shear Loading", Proceedings ASME Symposium on Advances in Adhesively Bonded Joints (1998) 87-92 [15] Minford, J. D., "Durability Evaluation of Adhesive Bonded Structures", Adhesive Bonding, Plenum Press, New York, U S A (1991) 239-290 [16] Andrews, E. FL, "Generalized Fracture Mechanics Approach to Adhesives", Adhesive Bonding, Plenum Press, New York, U S A (1991) 337-358 [17] Maugis, A . , "Adherence and Fracture Mechanics", Adhesive Bonding, Plenum Press, New York, U S A (1991) 303-336 [18] Sancaktar, S., "Fracture Aspects of Adhesive Joints, Material. Fatigue, Inter-phase, and Stress Concentration Considerations". Journal of Adhesion Science and Technology 9/2 (1995) 119-147 [19] McGrath, G. C , "The Fatigue Response of Adhesively Bonded Joints", Welding and Metal Fabrication 61/5 (1993) 233-238 [20] Clayton, J. Q., "Modeling Delay and Thickness Effects in Fatigue", Engineering Fracture Mechanics 32/2 (1989) 289-308 [21] Rose, L . R. F., "Basic Concepts in Assessing the Efficiency of Bonded Repairs", International Conference on Aircraft Damage Assessment and Repair, Melbourne, 26-28 (1991)202-208 [22] Shroff, S., Abdo, Z., Mohamed H. , Aglan, H. , "Fatigue Disbond Growth in Scrim Cloth Adhesively Bonded Joints", Proceedings of the 26tn International SAMPE Technical Conference, October 17-20 (1994) 245-257 [23] Aglan, Ft., Abdo, Z., "An Innovative Approach to Fatigue Disbond Propagation in Adhesive Joints", Journal of Adhesion Science and Technology 10/3 (1996) 183-198 [24] L. J. Hart-Smith, "Stress Analysis: A Continuum Mechanics Approach", Developments in Adhesives - 2, Applied Science Publishers (1981) 1-44 [25] Inglis, C. E., "Stresses in a Plate Due to the Presence of Cracks and Sharp Corners", Transactions of the Institution of Naval Architects 55 (1913) 219-230 [26] Muskhelishvilli, N . I., "Some Basic Problems in the Mathematical Theory of Elasticity", P. Noordhoff, 1953 [27] Westergaard, H . M . , "Bearing Pressures and Cracks". Journal of Applied Mechanics 6 (1939) 49-53 [28] Sneddon, I. N . , "The Stress Intensity Factor for a Griffith Crack in an Elastic Body in which Point Forces are Acting", International Journal of Fracture Mechanics 3 (1967) 317-331 98 29] Irwin, G. R., "Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate", Journal of Applied Mechanics 24 (1957) 361-364 30] Irwin, G. R., "Crack-Extension Force for a Part-Through Crack in a Plate", Journal of Applied Mechanics 29 (1962) 651-654 31] Irwin, G. R., "Fracture". Handbuch der Physik 6 (1958) 551-590 32] Griffith, A. A., "The Theory of Rupture". Proceedings of the First International Conference for Applied Mechanics (1925) 55-63 33] Paris, P., Erdogan, F., "A Critical Analysis of Crack Propagation Laws", Journal of Basic Engineering 85 (1963) 528-534 34] Schijve, J., "Four Lectures on Fatigue Crack Growth". Engineering Fracture Mechanics 11/1 (1979) 167-221 35] Broek, D., "Elementary Engineering Fracture Mechanics", fourth ed., Kluwer Academic Publishers, Dotrecht, The Netherlands, 1986 36] Forman, R. G., Kearney, V. E., Engle, R. M., "Numerical Analysis of Crack Propagation in Cyclic-Loaded Structures", Journal of Basic Engineering 89 (1967) 459-464 '37] Priddle, E. K., "High Cycle Fatigue Crack Propagation Under Random and Constant Amplitude Loading", International Journal of Pressure Vessels and Piping 4 (1976) 89 [38] Wheeler, O. E., "Spectrum Loading and Crack Growth", Journal of Basic Engineering 94 (1972) 181-186 [39] Willemborg, J., Engle, R. M., Wood, H. A., "A Crack Growth Retardation Model Using an Effective Stress Concept". AFFDL-TM-71-l-FBR. 1971 40] Elber, W., "The Significance of Fatigue Crack Closure", Damage Tolerance in Aircraft Structures, ASTM STP 486 (1971) 230-242 [41] Rose, L. R. F., "An Application of the Inclusion Analogy for Bonded Reinforcements". International Journal of Solids and Structures 17 (1981) 827-838 [42] Ratwani, M. M., "Analysis of Cracked, Adhesively Bonded Laminated Structures". AIAA Journal 17/9 (1978) 988-994 [43] Naboulsi, S., Mall, S., "Modeling of Cracked Metallic Structure With Bonded Composite Patch Using Three Layer Technique". Composite Structures 35 (1996) 295-308 [44] Naboulsi, S., Mall, S., "Nonlinear Analysis of Bonded Composite Patch Repair of Cracked Aluminum Panels". Composite Structures 41 (1998) 303-313 99 [45] Park, J. H. , Ogiso, T., Atluri, S. N . , "Analysis of Cracks in Aging Aircraft Structures, With and Without Composite-Patch Repairs" Computational Mechanics 10 (1992) 169-201 [46] Lena, M . R., Klug, J. C , Sun, C. T., "Composite Patches as Reinforcements and Crack Arrestors in Aircraft Structures", Journal of Aircraft 35/2 (1998) 318-321 [47] Rybicki, E. F., Kanninen, E. F., " A Finite Element Calculation of Stress Intensity Factors by a Modified Crack-Closure Integral", Engineering Fracture Mechanics 9/9 (1977) 931-938 [48] Albat, A . M . , "Thermal Residual Stresses in Composite Repairs on Cracked Metal Structures", Ph. D. Thesis, University of British Columbia (1998) [49] Chandra, R., Guruprasad, K. , "Numerical Estimation of Stress Intensity Factors in Patched Cracked Plates", Engineering Fracture Mechanics 27/5 (1987) 559-569 [50] Lin, C , Lietchi, K. M . , "Similarity Concepts in the Fatigue Fracture of Adhesively Bonded Joints", Journal of Adhesion 21 (1987) 1-24 [51] Domingues, J., Portela, A. , de Castro, P, "Trefftz Boundary Element Methods Applied to Fracture Mechanics", Engineering Fracture Mechanics 64 (1999) 67-86 [52] Erdogan, F., "On the Stress Distribution in Plates with Collinear Cuts Under Arbitrary Loads", Proceedings of the 4tn U. S. National Congress on Applied Mechanics (1962) 547-553 [53] Erdogan, F., Arin, K. , " A Sandwich Plate with a Part-Through and Disbonding Crack", Engineering Fracture Mechanics 4 (1972) 449-458 [54] Roderick, G. L. , "Green's Functions for Stresses, Stress Intensity Factors, and Displacements in a Cracked, Infinite, Isotropic Sheet Under Symmetric Loads", Engineering Fracture Mechanics 13 (1980) 95-105 [55] Boyd, K. L. , et al., "Development of Structural Integrity Analysis Technologies for Aging Aircraft Structures: Bonded Composite Patch Repair and Weight Function Methods", Flight Dynamics Directorate, Wright Laboratory, Air Force Materiel Command, Wright-Patterson A F B , Ohio, WL-TR-97-3105, 1997 [56] Dowrick, G., Cartwright, D. J., and Rooke, D. P., "The Effects of Repair Patches on the Stress Distribution in a Cracked Sheet", Proceedings of the Second International Conference on Numerical Methods in Fracture Mechanics (1980) 763-775 [57] Young, A. , Cartwright, D. J., Rooke, D. P., "Influence of Tapering on the Stresses in Repair Patches", Proceedings of the Third International Conference on Numerical Methods in Fracture Mechanics (1984) 721-737 [58] Young, A. , Rooke, D. P., Cartwright, D. J., " A Numerical Study of Balanced Patch Repairs to Cracked Sheets", Aeronautical Journal (1989) 327-334 100 [59] Young, A. , Rooke, D. P., Cartwright, D. J., "The Boundary Element Method for Analyzing Repair Patches on Cracked Finite Sheets", Aeronautical Journal 92 (1988) 416-421 [60] Poole, P., Lock, D., Young, A . , "Composite Patch Repair of Thick Aluminum Alloy Sections", International Conference on Aircraft Damage Assessment and Repair (1991) 85-91 [61] Guo, Y . G., Wu, X . R., "Bridging Stress Distribution in Center-Cracked Fiber Reinforced Metal Laminates: Modeling and Experiment", Engineering Fracture Mechanics 63 (1999) 147-163 [62] Suzuki, T., Sakai, M . , " A Model for Crack-Face Bridging", International Journal of Fracture 65 (1994) 329-344 [63] Rose, L . R. F., " A Cracked Plate Repaired by Bonded Reinforcements", International Journal of Fracture 18/2 (1982) 135-144 [64] Rose, L . R. F., "Crack Reinforcement by Distributed Springs", Journal of the Mechanics and Physics of Solids 35/4 (1987) 383-405 [65] Cox, B. N . , Rose, L . R. F., " A Self-Consistent Approximation for Crack Bridging by Elastic/Perfectly Plastic Ligaments", Mechanics of Materials 22 (1996) 249-263 [66] Baker, A . , "Bonded Composite Repair of Metallic Aircraft Components - Overview of Australian Activities", A G A R D - 79 t h Meeting of the Structures and Materials Panel. Specialists Meeting: Composite Repairs of Military Aircraft Structures (1994) 1-14 [67] Baker, A. , "Fiber Composite Repair of Cracked Metallic Aircraft Components - Practical and Basic Aspects". Composites 18/4 (1987) 293-308 [68] Baker, A . , Roberts, J. D., Rose, L . R. F., "Experimental Study of Overlap Joint Parameters Relevant to the K Reduction Due to Crack Patching", 28tn National SAMPE Symposium (1983) 627-639 [69] Denney, J., Mall , S., "Fatigue Behaviour in Thick Aluminum Panels with a Composite Repair", Collection of Technical Papers, 39tn Annual AIAA-ASME-ASCE-AHS Structures, Structural Dynamics, and Materials Conference 3 (1998) 2434-2443 [70] Denney, J., Mall , S., "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 (1996) 14-21 [71] Denney, J., Mall , S., "Characterization of Disbond Effects on Fatigue Crack Growth Behavior in Aluminum Plate with Bonded Composite Patch", Engineering Fracture Mechanics 57 (1997) 507-525 101 [72] 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 20/4 (1997) 605-614 [73] Baker, A . , "Repair of Cracked or Defective Metallic Aircraft Components with Advanced Fibre Composites - an Overview of Australian Work", Composite Structures 2 (1984) 153-181 [74] Cain, K. , Tan, C , "Numerical Modeling of Fiber-Bridging in Cracked Fiber-Metal Laminates", Journal of Advanced Materials 28 (1996) 8-15 [75] Hart-Smith, "Stress Analysis: A Continuum Mechanics Approach", Developments in Adhesives 2 (\98\) 1-44 [76] Schultz, J., Nardin, M . , "Theories and Mechanisms of Adhesion", Handbook of Adhesive Technology, Marcel Dekker Inc, New York, New York, U S A (1994) 19-33 [77] "SYSTAT for Windows: Statistics, Version 5 Edition". Systat Inc, Evanston, Illinois (1992) [78] "Microsoft F O R T R A N Reference, Version 5.0, for MS OS/2 and MS-DOS Operating Systems", Microsoft Corporation (1989) [79] McCartney, L . N . , " A Theoretical Explanation of the Delaying Effects of Overloads on Fatigue Crack Propagation", International Journal of Fracture 14 (1978) 213-232 [80] Newman, J. C , "Prediction of Fatigue Crack Growth under Variable-Amplitude and Spectrum Loading Using a Closure Model", Design of Fatigue and Fracture Resistant Structures, A S T M STP 761 (1982) 255 [81] Dexter, R. J., Hudak, S. J., Davidson, D. L. , "Modeling and Measurement of Crack Closure and Crack Growth Following Overloads and Underloads", Engineering Fracture Mechanics 33/6 (1989) 855-870 [82] Clayton, J. Q., "Modeling Delay and Thickness Effects in Fatigue", Engineering Fracture Mechanics 32/2 (1989) 289-308 [83] Russell, S. G., " A New Model for Fatigue Crack Growth Retardation Following an Overload", Engineering Fracture Mechanics 33/6 (1989) 839-854 [84] Budiansky, B. , and Hutchinson, J. W., "Analysis of Closure in Fatigue Crack Growth", Journal of Applied Mechanics 45 (1978) 267-276 [85] Ibrahim, F. K. , "The Effects of Stress Ratio, Compressive Peak Stress and Maximum Stress Level on Fatigue Behaviour of 2024-T3 Aluminum Alloy", Fatigue and Fracture of Engineering Materials and Structures 12/1 (1989) 1-8 [86] Schivje, J., "Some Formulas for the Crack Opening Stress Level", Engineering Fracture Mechanics 14 (1981) 461-465 102 [87] Pantelakis, S. G., "Fatigue Crack Growth Retardation Assessment of 2024-T3 and 6061-T6 Aluminum Specimens", Theoretical and Applied Fracture Mechanics 22 (1995) 35-42 [88] Rice, J. R., Rosengren, G. F., "Plane Strain Deformation Near a Crack Tip in a Power-Law Hardening Material", Journal of the Mechanics and Physics of Solids 16 (1968) 1-12 [89] Hutchinson, J. W., "Singular Behavior at the End of a Tensile Crack In a Hardening Material", Journal of the Mechanics and Physics of Solids 16 (1968) 13-31 [90] Johnson, H . H , "Calibrating the Electric Potential Method for Studying Slow Crack Growth", Materials Research and Standards 5/9 (1965) 442-445 [91] White, W. F., Schueinberg, W. H. , Mullis, R. T., Armstrong, J. D., "Process for Testing Integrity of Bonds Between Epoxy Patches and Aircraft Structural Materials", United States Patent 5709469, January 20, 1998 [92] Schubbe, J. J., Mall , S., "Investigation of a Cracked Thick Aluminum Panel Repaired With a Bonded Composite Patch", Engineering Fracture Mechanics 63 (1999) 305-323 [93] Timoshenko, S. P., "Analysis of Bi-Metal Thermostats", Journal of the Optical Society of American (1925) 233-255 [94] Pionke, C. D., Wempner, G., "The Various Approximations of the Bimetallic Thermostatic Strip". Journal of Applied Mechanics 58 (1991) 1015-1020. [95] Hellan, K. , "Introduction to Fracture Mechanics", McGraw Hi l l , Inc., 1984 [96] Helitech Industries Pty Limited. "Bonded Composite Repair Technology for Metallic Aircraft Components", technical report, Aeronautical Research Laboratory, Defense Science & Technology Organization, Australia. [97] Textron Specialty Materials, "Boron Epoxy Prepreg Taper -5521", 1995 [98] U S A Department of Defense, "Military Standardization Handbook - Metallic Materials and Elements for Aerospace Vehicle Structures", 1987 [99] Cytec Engineering Materials Inc. " F M 73 Film Adhesive", 1996 [100] Raizenne, M . D., Berak, T. J., Heath, J. B. R., "Bonded Composite Repair of Thin Metallic Materials: Variable Load/Amplitude/Temperature Cycling Effects", LTR-ST-1979, National Research Council of Canada, 1994 [101] Gaudert, P. C , Raizenne, M . D., Bellinger, N . C , Heath, J. B. R . "Fractographic Investigation of Boron Epoxy 5521/4 Patches", LTR-ST-1973, National Research Council of Canada, 1994 103 Appendix A: Crack Bridging Model Without a Disbond Units MPa = 10 6 Pa Loading Remote Applied Stress Dimensions GPa := 10-Pa cK =40-MPa t p := 3.25-mm Thickness of Plate Thickness of Reinforcement t r =0.132-7-mm Material Properties Plate Elastic Modulus Patch Elastic Modulus E p :=72.4GPa E r =210GPa Adhesive Shear Modulus G a = 844 MPa N : = kg- m \im := 10 mm sec Stress Reduction Factor Y Q =0.6 Thickness of Adhesive Half-Crack Length t a : = 0.25-mm c = 20mm v =0.33 Poison's Ratio of Plate Shear Modulus of Plate Plane Stress (1) or Strain (0) Plane = 1 G p =27.2GPa Calculational Parameters # of Elements/Half Crack n = 100 Strip Element Counters i Strip Element Width Stress State K : = -Patch Characteristics Load Transfer Length P Characteristic Length i : = 0 . . ( n - 1) c A : = -n 3 - v Eliptical Coordinates Strip Element Counters j Strip Element Location l t v •Plane •+- (3 - 4-v) •( 1 - Plane) l a \ E p , , ; p "«4H E r-t r Stiffness Ratio Lap Shear Load Transfer Length 2 2 c - X a(x) j :=0..(n- 1) - 4 E r - t r E p t p K l s (1 + S) Crack Opening Displacement Calculation Modified Green's function GF(x,ij) Interaction matrix K l s ' A 2-7C 1 +-K • 1 + v -In o(x) + a (0) C I > J : = i f [ i - j , l + G F ( x i - ) - I > x j ) > G F ( x i , x j ; Unbridged stress and displacement O Q = Y Q ' 0 Bridged crack displacement 8 0 i : = 2 ^ p ' < X ( X i ) ' ( 1 ^ K ^ l 4 " V ^ 8 - C - ' - S Q 104 Calculate Stresses l P Patch Strip Stresses o r : = K | S - E p-S--— Crack Closing Stress a b : = K i S E p - 8 t r Calculate Stress Intensities f<jQ- ° b ; ) ' A / |c-t-x. | c - X Reinforced (numerical) K n r = ^ 1 A l ' C No Bridging (numerical) K n u =y^~-o n - A c + x . c - X ; c - Xj . c + Xj Reinforced (Interpolation) K r =o0- 71 ° ^  /\j c -i- A No Bridging K u : = a Q ' ^ C Upper Bound K f f l : = rjQ-\/ii-A Calculate Displacements, Adhesive Strains, Adhesive Shear Stresses a 0 a Q § - ( I + V ) - ( I + K ) Upper Bound Reinforced § M = Reinforced (Interp.) ,5 r . K 1 S ' E P P l - f K k 4 - ( l + v ) - ( l + K ) i s T 1 Peak Strain (Interp.) Tr =7~ Shear Strain (numerical) y; : = —' 8 r 8 i 8 r 8j Peak Shear Stress (Interp.) x r : = - — G a Shear Stress (numerical) tj : = — G a t 51 t o - a Results: Dimensions c=20'mm — = 5.61248 -mm A=3.95289'mm — = 12.41838 •mm P Kls Stress Intensities K u =6.01591 -MPa-Jin K n u = 5.85214'MPa- /^m K 1 0 = 2.67451 'MPa^/m K r = 2.44388 -MPa-^m K n r = 2.33704 -MPa-^m Displacements 8 r = 3.14133 ^m 80 =3.29683 •jim 5^=4.11659'^ Shear Strains y r = 0.01257 y 0 =0.01319 Shear Stresses t r = 10.60514 «MPa x0 = 11.1301-MPa 105 15 A 10 SOj Um Sr 5 um Crack Face Displacement 1 s -\ V 1 % 0 10 2 80 MPa 60 h MPa "0 MPa 40 20 l mm Contribution to SIF Reduction for a Reinforcement Strip Reinforcing Strip SIF Contribution K, Adhesive Shear Strain and Stress Near the Crack Adhesive Shear Strain o.oi 0.005 h 106 Appendix B: Crack Bridging Analysis with a Disbond Units MPa = 106-Pa Loading Remote Applied Stress Dimensions Thickness of Plate Thickness of Reinforcement GPa=10-Pa N = newton |im = 10 mm = 40-MPa tp := 3.25-mm t r = 0.132-7-mm Stress Reduction Factor Y Q =0.6 Thickness of Adhesive ta=0.25-mm Half-Crack Length c : = 40-mm Disbond Description Material Properties Plate Elastic Modulus Patch Elastic Modulus Adhesive Shear Modulus Calculational Parameters # of Strip Elements/Half Crack n := 100 Strip Element Counter Strip Element Width 500 n r disbond(x) : = OT-Vc E p =72.4-GPa E r =210GPa G a = 844 MPa Poison's Ratio of Plate Plate Shear Modulus Plane Stress (1) or Strain (0) v =0.33 G p =27.2GPa Plane := 1 Eliptical F'n I(z) = A / Z 2 - c2if(Re(z)>0,1, Strip Element Counter Strip Element Location x( =i-A 0 . . (n- 1) c Disbond length bj := disbondfxA Point Load Location j =0..(n- 1) A •b; Convenient Constants Stress State A Constant 3- v 1 + v -Plane -t- (3 - 4-v)-( 1 - Plane) 2-71- (1 -HK -t, Define Stress Intensity Green's Function 2i-C 0- 1-GKl(z 0 ) : = Kz 0 0 I z oj t c-(zo-zo I z. 0 • z 0 Z n - C GK(z 0 ) : = G K l ( z 0 ) + GK1 - z 0 - GK1 z 0 - G K l ( - z 0 Define Displacement Green's Function j3 rz,z o) =ln I(z 0)-I(z)-i-z 0-z-c 2 z-i-I(z) m J4fz,z 0) = Kz) \(z~i : 0 " z 0 107 J3fz,zi z 0 " \z o) + ! ( z ) - z (z- z 0 yi(z) J4fz,z 0) I z, :0 I(z) I z 0 -I(z) ; 0 " z 0 z - Z ( •t1 l ( z ' z 0) = l n Z Z ( $ i (z ,z 0 ) co J Z . Z Q ) :=ln , z - z o z o - z o fli(z-zo) : 0 ~ z 0 0 z - z (Z'Z0) : = I ' ( j 3 ( z ' z 0) - K'J3(z>zo) +J 4 ( Z - Z 0)) ^ 0( z ' z0) : = i ' ( J 3 ( z ' z 0) " K - J3 (z ,z 0 ) + J4(z,z 0^ <t>(z' o) : = i> l ( z ' z 0) <t> o( z ' z o) O(z,z 0) : = <D , (z ,z 0 ) + 4 5 0 ( z , z 0 ) co(z,z0) : = (B J ^ Z . Z Q J +(|)0^z,zo) C 0 n(z,z 0 ) : = Q i (z ,z 0 ) 4-Q50(z,z0) C P 1 ( z , z 0 ) =Im 2-G, i -K-(|)rz,zo) - i -co z , z 0 - (z- zj- i -®(Z,ZQ Cp(z ,z 0 ) - C P 1 ( z , z 0 ) + C P 1 z , - z 0 - C P 1 z , z 0 - C P 1 ( z , - z 0 Stiffness Calculations Load Transfer Length B a / 1 1 l a l^p'tp E r - t r Stiffness Ratio E r t r Ep'tp Characteristic Length A : = --(l H-4V-Disbond-Edge Displacement Calculation Interaction Matrix 1 B-S Elastic Lap Shear Load K i s : = n — Transfer Length + C u : = i f i=j,l-i-( ~ + p ^ - ) - A - C p f z ^ ^ z l f i + TX) •A-C P(z i ,z 1 J U l s-Ep-tp E r - t r j \' 2 J/Ui r -E„-t n E r - t r j V ' J ^K l s E p L p Unbridged stress and insertion displacement ° 0 : = Yo-°« 8 - 0 0 k-t- 1 •Im/'J7z!' • 2 ' b i - R t e 1 - K -b. Bridged Insertion Displacement 8 =C" -80 t, Patch Strip Stresses •5; 1 K l o - E n - t l s c p - l p r / 108 Bridged COD (x=0) ' b : = ^-( l h K)-I!CP( xo^ 0- 0 ( ) O I i m m ' Zj)- 0r j-A- tr P j Calculate Stress Intensities No Bridging No Patch K 0 : = om-<J*c Upper Bound, Reinforced K B :-c Q - V I - A Reinforced (Interpolation) K r -c Q- Tt-C-A c-i-A Reinforced, numerical K b : = g Q-^/TI-C - y^GK^z^-g r -A-t Calculate Displacements Upper Bound Reinforced 5 ls'^p COD, Unbridged 5 u n b ° 0 ' E p . ( l - v Patch Strip Stretch 5 i n . : = b i i r -1 j . ° 0 Reinforced br-^— (Interpolation) P r { l + V ) - ( l + K ) 1 - | - K ls '2 'V 1 +-v)-(l-I-K) Calculate Adhesive Shear Strains and Stresses Interpolation Result r t, V= » r G a Numerical Result 5 i " 5 rein: Results: Dimensions c=40'mm — = 5.61248 'mm A=3.95289,mm P Stress Intensities K u = 8.50778 -MPa-^m 1 = 12.41838-mm K ls =2.67451 -MPa-^/m K r = 2.55141 •MPa /^rn K b =4.51088 -MPa^m K 0 = 14.17963 •MPa-^ /m Displacements 8 r = 3.56344'nm Shear Strains y =0.01425 5 b = 13.43548 -\im y0 =0.0103 5 =4.11659'um Shear Stresses x r = 12.03016 -MPa T 0 =8.6968-MPa 109 Adhesive Shear Strain and Stress Near the Disbond Adhesive Shear Strain Adhesive Shear Stress 110 Appendix C: Green's Function - Point Load Applied to the Crack Face Following is the Green's function solution for the load configuration shown below. Four point loads act on the faces of a crack in an infinite plate. Two acts on the lower face, and two on the upper. The forces acting on a crack face act at an equal distance from the center of the crack. Figure C- 1: Loading Configuration Using the method of solution presented by Hellan [95], the y-direction displacement may be expressed with the following function, t is a temporary variable corresponding to x. K + l uy^ = ~A ^ R E L y 4 • n • G„ f 1 (acP(t)-I{t) J / ( x ) U x-t dt dx (C-l) Performing the first integration for a constant pressure applied over a width A , for A « c and centered at a distance h from the middle of the crack, we get; P(h) • A • (IT + 1) uyW = ~. ^ Rel y 4 • n -GP 1 I(x) x - h dx (C-2) Integrating, and removing spurious rigid body displacements 111 i»(J») • A • (* + 1) /(/*) • I(x) + x-h-c2 h - x (C-3) This is valid over the range -c < x < c. Adding an equal force applied at - h Uy(X,0) P(h) • A • ( K - + 1) 4•n• G, In ( / ( A ) • / ( x ) + x • h — c2) • (l(h) • I(x) - x h - c 2 ) (h - x) •(-/!- x) (C-4) Note that the argument of the logarithmic function has dimensions of length2. If we non-dimensional the coordinates, so that all lengths are divided by the crack length, and proceed through the calculations, we see that we should divide by the square of the crack length. The equation then reduces to />(/!)• A •(*:+ 1) «y(x,0) = -— In] 2-c2 -h2 - x2 - 2 • 1(h) • I(x) x2-h2 (C-5) Which further reduces to , v P(h) • A • (K + 1) ^ ( X ' ° ) = 4 - ^ . G , - l n l (l(x) + I(h)f h2 -x2 (C-6) To illustrate the use of this function, the Figure C-2 shows displacement results when we apply a constant crack opening pressure as point loads acting on 100 evenly space nodes. The contribution for a point load at its node is left out, as the equations predict an infinite displacement at this point. This is compared to the exact result given by u\x)= J.lm{l(x)) (C-7) 112 0.003 | 0.002 Q 0.001 h 20 40 60 Distance from Centre of Crack (mm) Greens function solution, 100 intervals " " Exact solution Figure C- 2: Displacement Result for Uniform Opening Pressure 80 Increasing the number of nodes improves the result. The stress intensity function is well known, and follows [35]; K = P(x) • A c + x jc — X + J \ \ c - x C + X J (C-8) Comparing the displacement results to finite element analysis, we get the following. 0.003 Distance from Centre of Crack (mm) Greens function solution Finite element solution Figure C- 3: Comparison to Finite Element Analysis 113 Appendix D: Green's Function - Point Load Applied at an Arbitrary Location Solutions for an arbitrary point loading in a cracked plate are available in the literature. A l l such analyses are based on either complex stress functions, using the approach developed in [26], or on integral transforms [28]. Integral transforms are a less intuitive technique and appear not to have been widely adopted. In [52, 53], the complex stress function approach is examined for point loading in the first quadrant. Results are left in terms of the complex stress functions and not reduced to Green's functions. In addition, there is a minor error in determination of stress intensity. In [54], for a point-symmetric loading scheme, a symbolic manipulation program has been used to express Green's Functions in a simplified form, however, finite element verification performed during work during the course of the work for this thesis did not agree with the results. Figure D-1: Loading Configuration In this Appendix, we derive Green's Functions required for the disbonding model (K^ and u (z,z0)). The results have been validated by comparisons to finite element analysis and to the Green's functions for point loads applied to the crack face. The derivation closely follows the development in [52,53], which itself follows closely from Muskhellishvilli's solution [28] to the general problem of a plate with collinear straight cuts. A full, validated, error-free, and clear 114 solution to the problem is not available in the literature, and is valuable in analysis of crack-patching technology. The focus here is to show enough of the development that any transcription errors may be found. In addition to the in-plane point load solutions, anti-plane solutions would also be valuable to investigate the effect of crack bridging adhesive shear stresses which are offset from the neutral axis of the patched plate. Stress Functions The problem of a cracked plate under arbitrary point loading is the superposition of the solutions for two distinct problems. The first case is the problem involving a point load in an uncracked sheet. Variables related to the problem will be denoted by the subscript 1. The second is the problem of a crack opening pressure, equal to the stresses in the crack plane from case 1, which is applied to the crack faces. This problem will be denoted by the subscript 0. Case 1 The complex stress functions for a point load in an infinite plane are -1 <D,(z) = F z - z n (D-l) Q,(z) = F z - z n + F-( z - z 0 ) 2 (D-2) Case 2 From [47], the Koslov-Muskhellishvilli stress function, O 0 ( z ) , for the load configuration shown above is a>0(z) = Q 0 (z) = 2-7T- I(Z) J\(z)-Jl(zQ) K-[j\(z)- J1(Z 0)] z - z n z - z n F z0-z0 \ J\(z) - J\(z0) - I -= J2(z 0 ) | 2 • n • I(z) z - z0 z-zn (D-3) where 115 r\c — t Jl(z) = dt = n • Hz) - n-z J t - z J2(z) = J \c -1 i it - zy n • z dt - —— - 7T I(z) Introducing some convenient functions, J3(z,z 0 ) J4(z,z 0 ) Z~ Z n 1 (l I(z0)] 1 V z - z 0 I(z)J I(z) z-zn The stress functions become O 0 (z,z 0) = Q 0 (z,z 0) = 1 [F • (J3(z,z0) - * • J3(z,z0 )) - F • J4(z,z 0 ) (D-4) (D-5) (D-6) (D-7) (D-8) Stresses and Displacements The following are relationships for the stresses and displacements [49]; u + i-v = T - ^ - Z J • [K • <t>{z) - tu{z) -(z-z)- O(z)] CT* - * • °~,y = 2 • + ~ + (z-z)-0' (z) <jy-i- crxy = <D(z) + Q(z) + (z - z) • O'(z) (D-9) Where ®(z) = O 0 (z) + <D,(z)and Q(z) = Q 0 (z) + Q,(z) . The solution is now in a useful form for use in programs or a symbolic processor. To get the point-symmetric solution required for the disbonding model, we superpose the solutions for a single point load applied in each quadrant. The following figure displays the y-direction displacement field under point-symmetric loading applied at z0 = (10 + 10 • /) • mm for an A M R L faceplate. 116 Y-Displacement (mm) 0.001—, 5-10 Y-Force(lOON) 40-" X (mm) Figure D- 2: Y-Displacement Under Point-Symmetric Loading Stress Intensity Factor The stress intensity factor for a plane problem [52] may be expressed as the following lim + i • K2 = 2-s/2 • *Jz-a • (<jv O) - i • cr, v (x)) x -> al V y y 1. (D-10) Substitution from above gives the following, which is a minor correction over the solutions [52-55]. lim K.+i-K2=2J2- Ux-a -(2-OJx)) Kl+i.K2 =-j= rI(zQ) 7(z 0) N K • = - 1 Vz0-c za-c J + F ( D - l l ) (D-12) 117 Derivatives and Integrals of Stress Functions In addition to the functions above, the following functions are required to evaluate the stresses and strains; (f>x (z) = J^j (z) • dz = F • In \ z - z j (D-13) <D!'(z) = F (D-14) ox (z) — F • K • ln(z - z 0) + F • z - z n (D-15) = F - = + F z - z n ( z " z o ) 2 (D-16) ( I(z0) • I(z) + z0 • z - c2^ j3(z) = ln| 2- 0 0 V z + /(z) (D-17) J3'(z) = /(z) *o -/ (z 0 ) + / ( z ) - z ( z 0 - z ) - / ( z ) 2 + (z0 - z) 2 • / (z) 2 (D-18) 74(z) = / ( f ) 1 U ( z 0 ) , z - z n (D-19) J4'(z) = — 2 ~ 2 — / — \ z 0-/(z) + Z - / ( Z Q ) +Z0-Z-\Z-Z0 (z-z 0 ) - / (z 0 ) . / (z) 3 ( Z _ Z ( ^ ^ Q ) /(z) -1 z0 ~z0 Z—Zr (D-20) 118 Appendix E: The FORTRAN Program COMPARE The F O R T R A N program C O M P A R E determines the stress intensity, crack opening displacement and disbond-edge displacement for a wide variety of patch parameters. The parameters are expressed as nondimensionalized ratios of characteristic lengths. This results in a significant savings in computational time and leads to results which are more directly applicable in design equations. Source Code ************************ "LISTING OF COMPARE.FOR $FREEFORM $STORAGE:2 $NODEBUG «*************** PROGRAM COMPARE "*************** INCLUDE "COMMON2.FOR" COMPLEX Z(NLIMIT), CALK REAL D(NLIMIT), CM(NLIMIT,NLIMIT+1), C(NLIMIT,NLIMIT+1) REAL COD, SPRING CHARACTER*6 PLANE CHARACTER*10 FILENAME N = 200 VP = 0.33 WRITE (*,*) 'Plane Stress or Plane S t r a i n ? ' READ (*,'(A)') PLANE . IF (PLANE.EQ.'STRAIN') THEN KAP = 3.0-4.0*VP ELSE IF (PLANE.EQ.'STRESS') THEN KAP = (3.0-VP)/(1.0+VP) END IF FILENAME=PLANE//'.PRN' CALL OPENFILE( FILENAME , 1) WRITE (1,*) ' A, B, L, Kn/S, Kt/S, CODn*E/S, CODt*E/S, D*E/S' WRITE (1,*) 'PLANE ', PLANE DO A=50,50 DO B=2,50,2 DO 1=1,N Z(I)=CMPLX((REAL(I)-0.5)*(A+B)/N,B) 119 END DO CALL STIFF(CM, Z) CALL LOADING(CM,Z) DO L=2,50,2 SPRING=(A+B)/(N*(PI*L-B)) IF (PI*L).GE.(B+2.0)) THEN DO 1=1,N DO J=1,N+1 IF (I.EQ.J) THEN C(I , J)=CM(I,J)*SPRING+1.0 ELSE IF (J.EQ.(N+l)) THEN C ( I , J) =CM ( I , J) ELSE C(I,J)=CM(I,J)*SPRING END IF END DO END DO CALL SOLVE (C,D) WRITE ( 1 , ' C ' ' ' , 3(F5.1,IX) ,5(F8.4,IX) ,F8.4) ' ) -A, B, L, REAL(CALK(D,Z,SPRING)), COD(D,Z,SPRING), D(l) WRITE (*,'('' ' ' , 3(F5.1,IX),F8.4) ' )-A, B, L, D(l) ELSE WRITE (*,*) 'SKIP' END IF END DO END DO END DO CLOSE(1) WRITE(*,*) 'DONE' END " * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE OPENFILE(NAME,NUM) "***************************** CHARACTER*10 NAME INTEGER NUM LOGICAL CHECK INQUIRE (FILE=NAME, EXIST=CHECK) IF (CHECK) THEN OPEN( NUM, FILE=NAME, STATUS='OLD') CLOSE (NUM, STATUS='DELETE') END IF OPEN (NUM, FILE=NAME, STATUS='NEW') END " * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE STIFF(CM,Z) "************************ INCLUDE 'COMMON2.FOR' REAL CM(NLIMIT,NLIMIT+1) COMPLEX Z(NLIMIT) COMPLEX GD DO 1=1,N DO J=1,N IF (I.EQ.J) THEN 120 CM(I,J)= AIMAG(GD(Z(I) - ( A + B ) / ( 2 . 0 * N ) , Z ( J ) ) ) ELSE CM(I,J)= AIMAG(GD(Z(I),Z(J))) END IF END DO END DO END SUBROUTINE LOADING(CM,Z) ************************* " THIS SUBROUTINE INSERTS THE UNBRIDGED DISPLACEMENTS IN THE LAST " COLUMN OF C. THIS ESSENTIALLY PROVIDES THE LOADING. HERE, THE RESULT HAS BEEN MULTIPLIED BY (EP/STRESS). INCLUDE 'COMMON2.FOR' REAL CM(NLIMIT,NLIMIT+1) COMPLEX Z(NLIMIT), Z l , EZI, AZI, AEZI COMPLEX E DO 1 = 1 , N ZI = Z (I) AZI=AIMAG(Zl) EZI=E(ZI) AEZI=AIMAG(EZI) C M ( I , N+ 1 ) = ( 1 + V P ) * ( ( K A P+ 1 ) * A E Z I + A Z I * ( ( 1 - K A P ) - 2 * R E A L ( Z l / E Z I ) ) ) / 2 END DO END SUBROUTINE SOLVE(CM,D) " THIS SUBROUNTINE SOLVES A LINEAR SET OF EQUATIONS. " CM(N,N+1) IS THE COEFFICIENT MATRIX " D(N) IS THE OUTPUT VECTOR " CMX=0, BECOMES IX=D INCLUDE 'C0MM0N2.FOR' REAL CM(NLIMIT,NLIMIT+1), D(NLIMIT), TEM • INTEGER I,K,J " TRIANGULARIZE THE MATRIX DO K=1,N DO I=K,N TEM=CM(I,K) DO J=K,N+1 CM(I,J)=CM(I,J)/TEM END DO END DO DO I=K+1,N DO J=K,N+1 CM(I,J)=CM(I,J)-CM(K,J) END DO END DO END DO " BACK SUBSTITUTION D(N)=CM(N,N+1) DO K=N- 1 , 1 , - 1 121 D(K)=CM(K/N+1) DO I=N,K+1,-1 D(K)=D(K)-CM(K,I)*D(I) END DO END DO END * * * * * * * * * * * * * * * * * * * * * * * * * * COMPLEX FUNCTION GD(Z,Z0) >' ************************* " THIS FUNCTION COMPUTES THE DISPLACEMENT AT Z DUE TO A POINT " SYMMETRIC FORCE F AT ZO. IT USES SUPERPOSITION OF THE SOLUTION " FOR INDIVIDUAL POINT FORCES IN EACH QUADRANT. INCLUDE 'COMMON2.FOR' COMPLEX F, Z, ZO, CF, CZO COMPLEX ED F=CMPLX(0.0,1.0) CF=CONJG(F) CZ0=CONJG(ZO) GD=ED(F, Z, ZO)+ED(CF,Z,CZO)+ED(-F,Z,-ZO)+ED(-CF,Z,-CZO) END COMPLEX FUNCTION E(Z) "********************* " THIS FUNCTION CONVERTS A COMPLEX POSITION Z INTO AN ELIPTICAL " COORDINATE E(Z). INCLUDE 'COMMON2.FOR' INTRINSIC REAL COMPLEX Z E=CSQRT(Z**2-A**2)*SIGN(1.0,REAL(Z)) END "*************************** COMPLEX FUNCTION ED(F,Z,Z0) » *************************** " THIS FUNCTION COMPUTES THE DISPLACEMENT AT Z DUE TO A " FORCE F AT ZO. " NOTE THAT YOU NEED TO SUPERPOSE FORCES TO ACHIEVE EQUILIBRIUM. INCLUDE 'COMMON2.FOR1 COMPLEX S, CS, F, Z, ZO COMPLEX E ! LOADING AND COORDINATES ! EXTERNAL FUNCTIONS COMPLEX CZ, CZO COMPLEX EZ, EZO, ECZ, ECZO COMPLEX J3,J4 COMPLEX SJ3, SJ4 COMPLEX CSJ3,CSJ4 COMPLEX CCSJ3 COMPLEX C0J3, COSJ3 COMPLEX CSPHIO,SPHIO,PHIO INTERMEDIATE VARIABLES TO SPEED UP THE CALCULATION CONVENIENT FUNCTIONS INTEGRAL OF J3, J4 Z=CZ DENOTED BY C Z=CZ, Z0=CZO DENOTED BY CC Z0=CZO DENOTED BY CO STRESS F'N FOR INT. PRES. 122 COMPLEX SPHI1,PHIl,CSOMEGAl COMPLEX SPHI,CSOMEGA, PHI ! STRESS F'N FOR POINT FORCE ! TOTAL STRESS F'N S=F/(2*PI*(1+KAP)) CS=CONJG(S) CZ=CONJG(Z) CZ0=CONJG(ZO) EZ=E(Z) ' . . EZ0=E(Z0) ECZ=E(CZ) ECZ0=E(CZO) SJ3=CLOG((EZ0*EZ+Z0*Z-A**2)/(500*(Z+EZ))) SJ4=(EZ/ECZO-1.0)*(CZO-ZO)/(Z-CZO) C0SJ3=CLOG((ECZ0*EZ+CZ0*Z-A**2)/(500*(Z+EZ))) SPHI0=0.5*(S*(SJ3-KAP*C0SJ3)-CS*SJ4) CSJ3=CLOG ( (EZ0*ECZ+Z0*CZ-A**2) / (500* (CZ+ECZ).) ) CSJ4=(ECZ/ECZO-1.0)*(CZO-ZO)/(CZ-CZO) CCSJ3=CLOG((ECZ0*ECZ+CZ0*CZ-A**2)/(500*(CZ+ECZ))) CSPHI0=0.5*(S*(CSJ3-KAP*CCSJ3)-CS*CSJ4) J3=(ZO-EZO+EZ-Z)/((Z-ZO)*EZ) J4=((1-ECZO/EZ)/(Z-CZO)-CZO/(ECZ0*EZ))*((CZO-ZO)/(Z-CZO)) C0J3=(CZO-ECZO+EZ-Z)/((Z-CZO)*EZ) PHI0=0.5*(S*(J3-KAP*C0J3)-CS*J4) SPHIl=S*CLOG(1000/(Z-ZO)) PHI1=-S/(Z-ZO) CS0MEGA1=S*KAP*CL0G((CZ-CZO)/1000)+CS*(ZO-CZO)/(CZ-CZO) SPHI=SPHI0+SPHI1 CSOMEGA=CSPHI0+CSOMEGAl PHI=PHI0+PHI1 ED=(KAP*SPHI-CSOMEGA-(Z-CZ)*CONJG(PHI))*(1.O+VP) END INCLUDE 'C0MM0N2.FOR' COMPLEX GD REAL D(NLIMIT) COMPLEX Z(NLIMIT) REAL SPRING COD =0.0 DO 1=1,N COD=COD+D(I)*SPRING*AIMAG(GD((0.001,0.001), Z(I) ) ) END DO COD=(1.0+VP)*A*(1.0+KAP)/2.0 - COD END FUNCTION COD(D,Z,SPRING) 123 COMPLEX FUNCTION CALK(D,Z,SPRING) << ********************************* INCLUDE 'COMMON2.FOR' COMPLEX F, GK REAL D(NLIMIT) COMPLEX Z(NLIMIT) REAL SPRING CALK =0.0 DO 1=1,N F=CMPLX(0.0,D(I)*SPRING) CALK= CALK-+GK(F,Z (I) ) -+GK(-F,-Z(I))-+GK(-CONJG(F),-CONJG(Z(I)))-+GK(CONJG(F),CONJG(Z(I))) END DO CALK=CMPLX(SQRT(PI*A),0.0)-CALK END " * * * * * * * * * * * * * * * * * * * * * * * * COMPLEX FUNCTION GK(F,Z) "************************ " THIS FUNCTION COMPUTES THE STRESS INTENSITY DUE TO A Y " FORCE F AT Z. " NOTE THAT YOU NEED TO SUPERPOSE FORCES TO ACHIEVE EQUILIBRIUM. INCLUDE 'COMMON2.FOR COMPLEX S, F l , F2, F3, Z, CZ, F, E S=(F*(1.0-VP**2.0))/(2.0*PI*(1.0+KAP)) CZ=CONJG(Z) F1=(E(Z)/(Z-A))-1.0 F2=(E(CZ)/(CZ-A))-1.0 F3=(A*(CZ-Z) )/(E(CZ)*(CZ-A) ) GK=2*(S*(F1-KAP*F2)+CONJG(S)*F3)/SQRT(A) END ^********************** "LISTING OF COMMON2.FOR PARAMETER(NLIMIT=200) PARAMETER(PI=3.1415927464) IMPLICIT REAL (A-H, O-Z) IMPLICIT INTEGER (I-N) REAL KAP, L COMMON /PRPS [NEAR]/A, B, L, VP COMMON /PRMS [NEAR]/ KAP COMMON /CONT [NEAR]/ N, INCOUT, TOL 124 Appendix F: The FORTRAN Program DAMAGE The F O R T R A N program D A M A G E estimates the fatigue life of a repair patch. Input files control the patch properties, load levels, and format of the program output. Source Code "LISTING FOR DAMAGE.FOR "********************** $FREEFORM $DEBUG "************* PROGRAM DAMAGE "************* " PROGRAM STRUCTURE " THIS PROGRAM CONSISTS OF 5 FILES; II " DAMAGE.FOR - THIS FILE " COMMON3.FOR - COMMON VARIABLE DECLARATIONS " PATCH.FOR - CRACK BRIDGING ANALYSIS " CRACK.FOR - CRACK GROWTH PREDICTION MODULE " DISBOND.FOR - DISBOND GROWTH PREDICTION " COMFUN.FOR - CONTAINS SEVERAL USER-DEFINED FUNCTIONS PROGRAM DAMAGE INITIALIZES FILES AND VARIABLES, AND CONTROLS LOAD STEPPING. INCLUDE 'COMMON3.FOR' COMPLEX Z(NLIMIT) REAL DMAX(NLIMIT), DELD(NLIMIT) INTEGER INC INTEGER COUNT INTEGER CYREM REAL CM(NLIMIT,NLIMIT) INSERTION LOCATIONS INSERTION DISPLACEMENTS INCREMENT FOR COUNTING CYCLES COUNTER FOR CM(I,J) UPDATE CYCLES REMAINING IN LOAD BLOCK CRACKED PLATE COMPLIANCE MATRIX CALL OPENFILES CALL INPUT CALL INITIALIZE(Z, DMAX, DELD) CALL PLATE(CM,Z) DO WHILE (FLAG) DO WHILE (FLAG.AND.(.NOT.EOF(NHIS))) READ (NHIS,*) SMAX, SMIN, TAPP, CYREM DO WHILE ((CYREM.GT.0).AND.FLAG) IF (CYREM.GT.MAXINC) THEN INC = MAXINC ELSE INC = CYREM END I F CYREM = CYREM - INC CYCLE = CYCLE + INC LOOP TIL FAIL LOOP T I L FAIL OR END OF HISTORY READ A LOAD BLOCK LOOP TIL FAIL OR END OF BLOCK SET INC ! INCREMENT THE # OF CYCLES 1 2 5 IF ((COUNT+INC).GT.UPDATE) THEN ! UPDATE CM(I,J) CALL PLATE(CM,Z) WRITE (*,*) 'UPDATING PLATE COMPLIANCE MATRIX' COUNT = 0 ELSE COUNT = COUNT+INC END IF SMAX=OPENSTRESS(SMAX*Y0*TP*(1+SR),TAPP,-TP/2) ! FIND CRACK OPENING STRESS SMIN=OPENSTRESS(SMIN*Y0*TP*(1+SR),TAPP,-TP/2) SALT=SMAX-SMIN CALL PATCH(Z,DMAX,SMAX,CM,TYA) CALL PATCH(Z,DELD,SALT,CM,2*TYA) CALL CRACK(Z,DMAX,DELD,INC) CALL DISBOND(Z,DMAX,DELD,INC) ! CRACK BRIDGING SUBROUTINE ! CALCULATE CRACK GROWTH ! CALCULATE DISBOND GROWTH IF (.NOT.INTER) THEN ! REMESH THE CRACK CALL REMESH(Z) END IF DO 1=1,N ! FIND THE NEW VALUE FOR B B=MAX(B,AIMAG(Z(I))) END DO CALL CHECKFAIL ! CHECK AGAINST FAILURE CRITERIA END DO DO I=l,NOUT CALL OUTPUT(LDH(I),I+NOFF,.FALSE.,Z,DMAX,DELD) END DO END DO DO I=l,NOUT CALL OUTPUT(BLK(I),I+NOFF,.FALSE.,Z,DMAX,DELD) END DO CALL OUTPUT('SCREEN ',6,.FALSE.,Z,DMAX,DELD) BLOCK = BLOCK + 1 ! INCREMENT BLOCK # REWIND (NHIS) ! REWIND LOAD HISTORY FILE END DO DO I=l,NOUT CALL OUTPUT(ENR(I),I+NOFF,.FALSE.,Z,DMAX,DELD) CLOSE (I+NOFF) END DO CALL OUTPUT('ENDOFRUN',6,.FALSE.,Z,DMAX,DELD) ! DUMP SUMMARY TO SCREEN END INCLUDE 'COMFUN.FOR' INCLUDE 'PATCH.FOR' INCLUDE 'CRACK.FOR' INCLUDE 'DISBOND.FOR' " * * * * * * * * * * * * * * * * * * * SUBROUTINE CHECKFAIL INCLUDE 'COMMON3.FOR' IF (B.GT.DISLIM) THEN ! CHECK DISBONDING MODE='DISBONDING' FLAG=.FALSE. 126 ELSE IF (A.GT.CRALIM) THEN MODE='CRACKING' FLAG=.FALSE. ELSE IF (KMAX-. GT.K1C) THEN MODE='FRACTURE' FLAG=.FALSE. ELSE IF (SMAX.GT.SY) THEN MODE='YIELD' FLAG=.FALSE. END IF ! CHECK CRACKING ! CHECK FRACTURE ! CHECK YIELDING• END SUBROUTINE OPENFILES * * * * * * * * * * * * * * * * * * * * THIS SUBROUTINE ASKS FOR THE INPUT FILE NAME AND SETS UP THE INPUT AND OUTPUT FILES. INCLUDE 'COMMON3.FOR' CHARACTER*1 QUERY, HEADER CHARACTER*8 TITLE CHARACTER*4 EXT WRITE (*,*) 'NAME OF INPUT FILE? (?.INP)' READ (*,'(A)') TITLE INQUIRE(FILE=TITLE//'.INP', EXIST=CHECK) DO WHILE (CHECK.EQV..FALSE.) WRITE (*,*) TITLE//'.INP DOES NOT EXIST. TRY AGAIN (Y/N) READ (*,'(A)') QUERY IF ((QUERY.EQ.'Y').OR.(QUERY.EQ.'y')) THEN WRITE (*,*) 'NAME OF INPUT FILE? (?.INP)' READ (*, ' (A) ') TITLE INQUIRE(FILE=TITLE//'.INP', EXIST=CHECK) ELSE CHECK=.TRUE. END IF OPEN (NINP, FILE = TITLE//'.INP', STATUS = 'OLD') INQUIRE(FILE=TITLE//'.HIS', EXIST=CHECK) DO WHILE (CHECK.EQV..FALSE.) WRITE (*,*) TITLE//'.HIS DOES NOT EXIST. TRY AGAIN (Y/N)' READ (*,'(A)') QUERY IF ((QUERY.EQ.'Y').OR.(QUERY.EQ.'y')) THEN WRITE (*,*) 'NAME OF HISTORY FILE? (?.HIS)' READ (*,'(A)') TITLE INQUIRE(FILE=TITLE//'.HIS', EXIST=CHECK) ELSE CHECK=.TRUE. END IF OPEN (NHIS, FILE = TITLE//'.HIS', STATUS = 'OLD') READ(NINP,*) HEADER READ(NINP,*) NOUT IF (NOUT.GT.MOUT) THEN NOUT=MOUT END IF DO 1=1,NOUT LOGICAL CHECK END DO END DO 127 EXT = '.OU'//CHAR(4 8+1) INQUIRE(FILE=TITLE//EXT, EXIST=CHECK) DO WHILE (CHECK) WRITE (*,*) TITLE//EXT//' EXISTS. OVERWRITE? (Y/N)' READ (*, ' (A) ') QUERY IF ((QUERY.EQ.'Y').OR.(QUERY.EQ.'y')) THEN OPEN (I+NOFF, FILE=TITLE//EXT, STATUS = 'OLD') CLOSE (I+NOFF, STATUS='DELETE') ELSE WRITE (*,*) 'PLEASE GIVE A NEW OUTPUT FILE NAME. (*'//EXT//')' READ (*,'(A)') TITLE END IF INQUIRE(FILE=TITLE//EXT, EXIST=CHECK) END DO OPEN (I+NOFF, FILE = TITLE//EXT, STATUS = 'NEW') END DO END SUBROUTINE INPUT "*************** " THIS SUBROUTINE READS THE INPUT FILE. INCLUDE 'COMMON3.FOR' CHARACTER*1 HEADER READ READ READ READ READ READ READ READ READ READ READ READ READ READ READ READ • READ READ READ READ READ READ READ READ READ READ READ READ DO 1 = 1 READ READ END DO END NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* NINP,* , NOUT (NINP, (NINP, HEADER EP, ER, GR, GA, VP, NONLIN, LAG HEADER TYA, SY, K1C, KTH, DISLIM, CRALIM, FINITE HEADER AP, AR, TSF HEADER TP, TR, TA, PLANE, W, H HEADER AO, RPO, BO, TYPE, CONSOPT HEADER N, INCOUT, MAXINC, UPDATE, TOL, INTER, MOD, BEND HEADER CLAW, RLAW, DLAW HEADER CI, C2, C3, C4 HEADER CL1, CL2 HEADER R l , R2, R3, R4 HEADER RL1, RL2 HEADER Dl, D2, D3, D4, NEWDISLEN HEADER DL1, DL2 HEADER Y l *) HEADER *) LDH(I), BLK(I), ENR(I) 128 SUBROUTINE I N I T I A L I Z E ( Z , DMAX, DELD) " * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * " THIS SUBROUTINE INI T I A L I Z E S SEVERAL COMMON VARIABLES. INCLUDE 'COMMON3.FOR' COMPLEX Z(NLIMIT) REAL DMAX(NLIMIT) REAL DISLEN REAL SIG REAL D REAL SR DELD(NLIMIT) ! DISBOND SHAPE ! DISPLACEMENTS ! TEMPORARY VARIABLES ! LOAD RATIO ! INCL ANALOGY VARIABLE ! PATCH STIFFNESS RATIO SIG 0.0 ! AXIAL TENSION FLAG=.TRUE. CYCLES=0 COUNT=0 BLOCK=l A=A0 RP=RP0 I F (RLAW.EQ.'WHEELER') THEN R2=A0 R3=RP0 END I F B=B0 SR=ER*TR/(EP*TP) D=3*((1+SR)**2)+ 2* (1+SR)*(H/W+W/H+VP*SR)+1-(VP*SR)**2 Y0=(1+(SR/D)*(1+2*(1+SR)*(H/W)*(1-VP*SIG)+(1+SR-VP*SR)*(SIG-VP)))/(1+SR) I F (INTER) THEN N = 1 TYPE = "CONS" END I F ! OVERRIDE USER INPUT I F USING ! INTERPOLATION METHOD. I F (((.NOT.INTER).AND.(TYPE.EQ.'CONS')).AND.(CONSOPT)) THEN DEL=1.5*A/N ELSE DEL=A/N END I F DO 1=1,N X=DEL*(REAL(I)-0.5) I F (TYPE.EQ.'SPEC) THEN READ (1,*) DISLEN Z(I)=CMPLX(X,DISLEN) ELSE I F (TYPE.EQ.'NONE') THEN Z(I)=CMPLX(X,0.001) ELSE I F (TYPE.EQ.'TRIA') THEN Z(I)=CMPLX(X,B*((A-X)/A)) ELSE I F (TYPE.EQ.'CONS') THEN Z(I)=CMPLX(X,B) ELSE I F (TYPE.EQ.'ELIP') THEN Z(I)=CMPLX(X,(B/A)*SQRT(A**2-X**2) END I F END DO DISBOND SHAPE GP=EP/(2*(1+VP)) I F (PLANE.EQ.'STRESS') THEN ! ACCOUNT FOR STRESS STATE 129 K A P = ( 3 . 0 - V P ) / ( 1 . 0 + V P ) E L S E I F (PLANE.EQ.'STRAIN') THEN : KAP=3.0-4.0*VP E L S E MODE='ERROR4' FLAG=.FALSE. END I F I F (.NOT.LAG) THEN ! FIND LOAD TRANSFER LENGTH S L A M = S Q R T ( ( G A / T A ) * ( 1 / ( E P * T P ) + 1 / ( E R * T R ) ) ) E L S E S L A M = S Q R T ( ( 1 / ( E P * T P ) + 1 / ( E R * T R ) ) / ( T A / G A + 0 . 3 7 5 * ( T P / G P + T R / G R ) ) ) END I F DO I=l,NOUT ! WRITE OUTPUT F I L E HEADERS I F (LDH(I).EQ.'NONENONE') THEN CA L L OUTPUT(BLK(I),I+NOFF,.TRUE.,Z,DMAX,DELD) E L S E C A L L OUTPUT(LDH(I),I+NOFF,.TRUE.,Z,DMAX,DELD) END I F , END DO END SUBROUTINE REMESH(Z0) i'******************** " THIS SUBROUTINE CHANGES THE MESHING SCHEME BETWEEN " INCREMENTS OF CRACK GROWTH. INCLUDE 'COMMON3.FOR' COMPLEX Z ( N L I M I T ) ! TEMPORARY V A R I A B L E FOR DISBOND SHAPE COMPLEX Z 0 ( N L I M I T ) ! DISBOND SHAPE REAL DELOLD ! OR I G I N A L DEL DELOLD=DEL I F ( ( ( . N O T . I N T E R ) . A N D . ( T Y P E . E Q . ' C O N S ' ) ) . A N D . ( C O N S O P T ) ) THEN DEL=1.5*A/N I F ( ( F I N I T E ) . A N D . ( ( N * D E L ) . G T . W ) ) THEN DEL=W/N END I F E L S E DEL=A/N END I F I F ( (.NOT.FINITE) .OR. ( F I N I T E . A N D . (A.LT.W) ) ) THEN DO 1=1,N X = D E L * ( 1 - 0 . 5 ) ! FIND THE NEW X-COORDINATE J = I N T ( X / D E L O L D - 0 . 5 ) + 1 ! FIND I N WHICH OLD I N T E R V A L T H I S L I E S I F ( J . L E . N ) THEN ! FIND THE INTERPOLATION I N T E R V A L Y L E F T = A I M A G ( Z 0 ( J ) ) X L E F T = R E A L ( Z 0 ( J ) ) I F ( J . L T . N ) THEN Y R I G H T = A I M A G ( Z 0 ( J + l ) ) X R I G H T = R E A L ( Z 0 ( J + l ) ) ELS E YRIGHT=NEWDISLEN X R I G H T = R E A L ( Z 0 ( J + l ) ) + D E L END I F Y = Y L E F T + ( X - X L E F T ) * ( Y R I G H T - Y L E F T ) / ( X R I G H T - X L E F T ) ! INTERPOLATE E L S E Y = NEWDISLEN 130 END I F Z (I) = CMPLX (X, Y) END DO DO 1=1,N Z0(I)=Z(I) END DO END IF END SUBROUTINE OUTPUT(OPT,FN,HEAD,Z,DMAX,DELD) "***************************************** " THIS SUBROUTINE WRITES TO SCREEN AND OUTPUT FILES. INCLUDE 'COMMON3.FOR' INTEGER FN ! FILE NUMBER COMPLEX Z(NLIMIT) REAL DMAX(NLIMIT), DELD(NLIMIT) CHARACTER*8 OPT ! TYPE OF OUTPUT LOGICAL HEAD ! PRINT HEADER IF (OPT.EQ.'SCREEN THEN WRITE(FN, WRITE(FN, WRITE(FN, WRITE(FN, WRITE(FN, WRITE(FN,*) ***BLOCK A/B= KMAX/KMIN= DMAX/DMIN= SMAX/SMIN= 13) BLOCK . 2(F7.2)! . 2(IX,F7, . 2 (IX, F7 . . 2 ( l x , F 7 . ' ) 4) ) 4) ) 3 ) ) ' ) A, B KMAX,KMIN DMAX(1) ,DMAX(1) • SMAX,SMIN •DELD(1) ELSE IF (OPT.EQ.'ENDOFRUN') THEN WRITE(FN,*) ' ' WRITE (FN,*) '•* WRITE(FN,•('' WRITE(FN,'('' WRITE(FN,'('' WRITE(FN,'('' WRITE(FN,'('' WRITE(FN,'('' WRITE(FN,'('' WRITE(FN,*) WRITE(FN,'( END OF RUN' INITIAL CRACK LENGTH = FINAL CRACK LENGTH = INITIAL DISBOND SIZE = FINAL DISBOND SIZE = CYCLES TO FAILURE = LOAD BLOCKS TO FAILURE = ' FAILURE MODE = ' STRESS REDUCTION FACTOR = ',F7.3)') ',F7.3) ') ',F7.3) ') \F7.3) ') ,18) ' ) ,18) ') ,A) ' ) •,F7.3) ') AO A BO B CYCLE (BLOCK-1) MODE YO IF (MODE.EQ.'ERROR1') THEN WRITE(FN,*) 'ERROR1: SPECIFIED CRACK GROWTH LAW DOES NOT EXIST' ELSE I F (MODE.EQ.'ERROR2') THEN WRITE(FN,*) 'ERROR2: FORMAN RELATION USED WITH NEGATIVE R-RATIO' ELSE IF (MODE.EQ.'ERROR3') THEN WRITE(FN,*) 'ERROR3: SPECIFIED DISBOND GROWTH LAW DOES NOT EXIST' ELSE IF (MODE.EQ.'ERROR4') THEN WRITE(FN,*) 'ERROR4: PLANE STATE MUST BE STRESS OR STRAIN' ELSE IF (MODE.EQ.'ERROR5') THEN WRITE(FN,*) 'ERROR5: REMESHING ALGORITHM ERROR' END IF ELSE IF (OPT.EQ.'BASICPAR') THEN IF (HEAD) THEN WRITE(FN,*) 'CYCLE, CRACK, DISBOND, KMAX, KMIN, DELD' ELSE WRITE (FN, ' ( ' ' ' ' , 17, 2 (IX, F7 . 3) , 3 (IX, F8 . 4) ) ' ) -CYCLE, A, B, KMAX, KMIN, DELD(1) END IF 131 ELSE IF (OPT.EQ.'DISBONDS') THEN IF (HEAD) THEN WRITE(FN,*) 'X, B, DMAX, LAMBDAD' ELSE DO 1=1,N,INCOUT WRITE (FN, ' ( ' ' ' ' , 2 (IX, F6. 3) , IX, F8 . 1, IX, F8 . 4 ) ' ) -REAL(Z(I) ), IMAG(Z(I)), DMAX(I)* 1000, -EP*TP/(PI/COMPLIANCE(AIMAG(Z(I)),DMAX(I),TYA)) + B / P I END DO END IF ELSE IF (OPT.EQ.'CRACKING') THEN IF (HEAD) THEN WRITE (FN,*) 'CYCLE, RP, R3, A, KMAX, KMIN' ELSE WRITE (FN, ' ( ' ' ' ' , 17, IX, F7 . 5, 4 (IX, F7 . 3) ) ' ) CYCLE, RP, R3, A, KMAX, KMIN END IF ELSE IF (OPT.EQ.'ENDBLOCK') THEN WRITE(FN,'('' ***BLOCK '',13,'', CRACK '',F5.2,'', DISBOND ' ' , F 5 . 2 ) ' ) -BLOCK, A, B END IF END ' " LISTING FOR COMMON3.FOR - COMMON VARIABLE DECLARATIONS PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER PARAMETER (PI=3.1415927464; (NLIMIT=200) (MOUT=9) (NOFF=4 0) (NINP=1) (NHIS=2) MAXIIMUN # OF ELEMENTS MAXIMUM NUMBER OF OUTPUT FILES OUTPUT FILE SPECIFIER OFFSET INPUT FILE SPECIFIER HISTORY FILE SPECIFIER IMPLICIT REAL (A-H, O-Z) IMPLICIT INTEGER (I-N) REAL KAP, NEWDISLEN, K1C, KTH, KMAX, KMIN, H INTEGER CYCLE, BLOCK, UPDATE CHARACTER*8 CLAW, RLAW, DLAW CHARACTER*8 MODE, TYPE, PLANE CHARACTER*8 LDH, BLK, ENR LOGICAL CL1, CL2, RL1, RL2, DL1, DL2, FINITE LOGICAL NONLIN, FLAG, INTER, LAG, MOD, CONSOPT, BEND *** ELASTIC PROPERTIES /PRPS/ EP PLATE ELASTIC MODULUS (MPa) ER REINFORCEMENT ELASTIC MODULUS (MPa) GR REINFORCEMENT SHEAR MODULUS (MPa) GP PLATE SHEAR MODULUS (MPa) GA ADHESIVE SHEAR MODULUS (MPa) VP PLATE POISSONS RATIO KAP PLANE STRESS/STRAIN PARAMETER PLANE TEXT VARIABLE 'STRESS' OR 'STRAIN' SR PATCH STIFFNESS RATIO *** THERMAL PROPERTIES /THER/ AR REINFORCEMENT THERMAL EXPANSION COEFFICIENT (1/DEGREES C) AP PLATE THERMAL EXPANSION COEFFICIENT (1/DEGREES C) TSF STRESS FREE TEMPERATURE (DEGREES C) 132 *** STRENGTH PROPERTIES AND FAILURE MODE /PRPS/ TYA ADHESIVE SHEAR STRENGTH (MPa) SY PLATE YIELD STRENGTH (MPa) K1C • PLATE FRACTURE TOUGHNESS (MPa*m/s0.5) KTH PLATE FATIGUE THRESHOLD (MPa*mA0.5) MODE FAILURE MODE DISLIM PERMISSIBLE DISBOND SIZE (nun) CRALIM PERMISSIBLE CRACK LENGTH (mm) *** DIMENSIONS /DIMS/ TP PLATE THICKNESS (mm) TR REINFORCEMENT THICKNESS (mm) TA ADHESIVE THICKNESS (mm) A CRACK LENGTH (mm) AO INITIAL CRACK LENGTH (mm) A l CRACK LENGTH FROM PREVIOUS STEP (mm) RP PLASTIC ZONE SIZE (mm) RPO INITIAL PLASTIC ZONE SIZE (mm) RP1 PLASTIC ZONE SIZE FROM PREVIOUS STEP (mm) B DISBOND SIZE (mm) BO INITIAL DISBOND (mm) TYPE INITIAL DISBOND SIZE (mm) W PATCH HALF-LENGTH PARALLEL TO FIBERS (mm) H PATCH HALF-LENGTH NORMAL TO FIBERS (mm) *** LAP JOINT PARAMETERS /PRMS/ SLAM 1/LOAD TRANSFER LENGTH (1/mm) DEL WIDTH OF A STRIP (mm) LAG INCLUDE SHEAR LAG CORRECTION *** PROGRAM CONTROL PARAMETERS /CONT/ N NUMBER OF STRIP ELEMENTS INCOUT INCREMENT BETWEEN OUTPUT FOR STRIP ELEMENTS MAXINC MAX # OF LOAD CYCLES BETWEEN CRACK BRIDGING ANALYSES UPDATE MAX # OF CYCLES BETWEEN COMPLIANCE MATRIX UPDATES NONLIN USE NONLINEAR CRACK BRIDGING MODEL (TRUE/FALSE) TOL TOLERANCE FOR NONLINEAR CRACK BRIDGING SOLUTIONS CLAW STRING SPECIFYING THE LINEAR CRACK GROWTH LAW RLAW STRING SPECIFYING THE LOAD INTERACTION LAW DLAW STRING SPECIFYING THE DISBOND GROWTH LAW FLAG INDICATES A FAILURE OR AN ERROR INTER USE THE INTERPOLATION MODEL MOD USE THE MODIFICATION FACTORS CONSOPT FOR TYPE='CONS', EXTEND PATCH PAST CRACK TIP BY A/2. BEND LET THE PLATE BEND FINITE INDICATES A FINITE WIDTH PATCH *** CRACK GROWTH LAW PARAMETERS /CGPR/ CI PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER C2 PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER C3 PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER C4 PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER CL1 LOGICAL PARAMETER SPECIFED BY GROWTH LAW PROGRAMMER CL2 LOGICAL PARAMETER SPECIFED BY GROWTH LAW PROGRAMMER *** CRACK GROWTH INTERACTION PARAMETERS /RGPR/ R l PARAMETER SPECIFIED BY INTERACTION LAW PROGRAMMER R2 PARAMETER SPECIFIED BY INTERACTION LAW PROGRAMMER R3 PARAMETER SPECIFIED BY INTERACTION LAW PROGRAMMER R4 PARAMETER SPECIFIED BY INTERACTION LAW PROGRAMMER RL1 LOGICAL PARAMETER SPECIFED BY INTERACTION LAW PROGRAMMER RL2 LOGICAL PARAMETER SPECIFED BY INTERACTION LAW PROGRAMMER 133 *** DISBOND GROWTH PARAMETERS /DGPR/ Dl PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER D2 PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER D3 PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER D4 PARAMETER SPECIFIED BY GROWTH LAW PROGRAMMER DL1 LOGICAL PARAMETER SPECIFED BY GROWTH LAW PROGRAMMER DL2 LOGICAL PARAMETER SPECIFED BY GROWTH LAW PROGRAMMER NEWDISLEN LENGTH OF NEWLY CREATED DISBOND *** LOAD PARAMETERS /LOAD/ SMAX MAXIMUM APPLIED STRESS (MPa) SMIN MINIMUM APPLIED STRESS (MPa) SALT ALTERNATING APPLIED STRESS (MPa) YO PATCH STRESS REDUCTION FACTOR YI CRACKED PLATE GEOMETRY FACTOR TAPP APPLIED TEMPERATURE (DEGREES C) CYCLE CURRENT LOAD CYCLE NUMBER BLOCK CURRENT LOAD BLOCK NUMBER KMAX PEAK STRESS INTENSITY KMIN MINIMUM STRESS INTENSITY *** OUTPUT FILE PARAMETERS /OUTP/ LDH(I) OPTION FOR OUTPUT FILE I, END OF LOAD HISTORY BLK(I) OPTION FOR OUTPUT FILE I, END OF LOAD BLOCK ENR(I) OPTION FOR OUTPUT FILE I, END OF RUN NOUT NUMBER OF OUTPUT FILES COMMON /PRPS [NEAR]/ EP, ER, GR, GP, GA, VP, KAP, PLANE, SR COMMON /PRPS [NEAR]/ TYA, SY, K1C, KTH, MODE, DISLIM, CRALIM COMMON /DIMS [NEAR]/ TP, TR, TA, A, AO, A l , RP, RPO, RP1, B, BO, TYPE, W, H COMMON /THER [NEAR]/ AP, AR, TSF COMMON /PRMS [NEAR]/ SLAM, DEL, LAG, OFFSET COMMON /CONT [NEAR]/ N, INCOUT, MAXINC, TOL, CLAW, RLAW, DLAW, FLAG,-NONLIN, UPDATE, INTER, MOD, CONSOPT, BEND, FINITE COMMON /CGPR [NEAR]/ CI, C2, C3, C4, CL1, CL2 COMMON /RGPR [NEAR]/ R l , R2, R3, R4, RL1, RL2 COMMON /DGPR [NEAR]/ D l , D2, D3, D4, DL1, DL2, NEWDISLEN COMMON /LOAD [NEAR]/ SMAX, SMIN, SALT, YO, YI, TAPP, CYCLE, BLOCK, KMAX, KMIN COMMON /OUTP [NEAR]/ LDH(MOUT), BLK(MOUT), ENR(MOUT), NOUT ************************************************ " LISTING FOR PATCH.FOR - CRACK BRIDGING ANALYSIS " ********************************** SUBROUTINE PATCH(Z,D,STRESS,CM,TYL) "********************************** INCLUDE 'COMMON3.FOR' COMPLEX Z(NLIMIT) REAL CM(NLIMIT,NLIMIT), C(NLIMIT,NLIMIT+1) REAL DO(NLIMIT), D(NLIMIT) REAL STRESS, TYL LOGICAL CONVERGED DO 1=1,N D0(I)=10E-6 END DO CONVERGED = .FALSE. 134 DO WHILE (INTER.AND.(.NOT.CONVERGED)) D( l ) = PATCHD(DO(1),STRESS,TYL) IF (D(l).LT.DO(1)) THEN D(l) = (D0(1)+D(1))/2 END IF IF (ABS((D(1)-DO(1))/DO(1)).LT.TOL) THEN CONVERGED=.TRUE. ELSE D0(1) = D( l ) . END IF END DO DO WHILE ((.NOT.INTER).AND.(.NOT.CONVERGED)) CALL STIFF(CM,C,Z,DO,TYL) CALL LOADING(C,Z,STRESS) CALL SOLVE(C,D) IF (NONLIN) THEN CALL CONVRG(D,DO,CONVERGED) ELSE CONVERGED= .TRUE. END IF DO 1=1,N D0(I)=D(I) END DO END DO END ^*********************** SUBROUTINE PLATE(CM,ZIN) ************************ INCLUDE 'COMMON3.FOR' REAL CM(NLIMIT,NLIMIT+1) COMPLEX Z, ZO, ZIN(NLIMIT) COMPLEX GD DO 1=1,N DO J=1,N Z0=ZIN(J) IF (I.EQ.J) THEN Z=ZIN(I)+DEL/(2.0) ELSE Z=ZIN(I) END IF CM(I,J)=AIMAG(GD(Z, ZO) ) END DO END DO END ********************************* SUBROUTINE STIFF(CM,C,ZIN,DO,TYL) INCLUDE 'COMMON3.FOR' REAL CM(NLIMIT,NLIMIT+1), C(NLIMIT,NLIMIT+1) REAL SPR(NLIMIT), DO(NLIMIT) REAL TYL, COMPLIANCE COMPLEX ZIN(NLIMIT) DO 1=1,N SPR(I)=DEL/COMPLIANCE(AIMAG(ZIN(I)),DO(I),TYL) 135 END DO DO 1=1,N DO J=1,N IF (I.EQ.J) THEN C ( I , J ) = 1.0+SPR(J)*CM(I,J) ELSE C ( I , J ) = SPR(J)*CM(I,J) END IF END DO END DO END SUBROUTINE LOADING(C,Z,STRESS) " THIS SUBROUTINE INSERTS THE UNBRIDGED DISPLACEMENTS IN THE LAST " COLUMN OF C. THIS ESSENTIALLY PROVIDES THE LOADING. INCLUDE 'COMMON3.FOR' REAL C(NLIMIT,NLIMIT+1) COMPLEX Z(NLIMIT), ZI, EZI, AZI, AEZI COMPLEX E DO 1=1,N ZI=Z(I) AZI=AIMAG(ZI) EZI=E(ZI) AEZI=AIMAG(EZI) C(I,N+1)=STRESS*((KAP+1)*AEZI+AZI*((1-KAP)-2*REAL(ZI/EZI)))/(4*GP) END DO END » ********************* SUBROUTINE SOLVE(CM,D) " THIS SUBROUNTINE SOLVES A LINEAR SET OF EQUATIONS. " CM(N,N+1) IS THE COEFFICIENT MATRIX " D(N) IS THE OUTPUT VECTOR " CMX=0, BECOMES IX=D INCLUDE 'COMMON3.FOR' REAL CM(NLIMIT,NLIMIT+1), D(NLIMIT), TEM TRIANGULARIZE THE MATRIX DO K=1,N DO I=K,N TEM=CM(I,K) DO J=K,N+1 CM(I,J)=CM(I,J)/TEM END DO END DO DO I=K+1,N DO J=K,N+1 CM(I,J)=CM(I,J)-CM(K,J) END DO END DO END DO BACK SUBSTITUTION D(N)=CM(N,N+1) DO K=N-1,1,-1 D(K)=CM(K,N+l) 136 DO I=N,K+1,-1 D(K)=D(K)-CM(K,I)*D(I) END DO END DO END COMPLEX FUNCTION GD(Z,Z0) " THIS FUNCTION COMPUTES THE DISPLACEMENT AT Z DUE TO A POINT " SYMMETRIC FORCE F AT ZO. IT USES SUPERPOSITION OF THE SOLUTION " FOR INDIVIDUAL POINT FORCES IN EACH QUADRANT. INCLUDE 'COMMON3.FOR' COMPLEX F, Z, ZO, CF, CZO COMPLEX ED F=CMPLX(0.0,1.0) CF=CONJG(F) CZ0=CONJG(ZO) GD=ED(F, Z,ZO)+ED(CF,Z,CZO)+ED(-F,Z,-ZO)+ED(-CF,Z,-CZO) END COMPLEX FUNCTION E(Z) " THIS FUNCTION CONVERTS A COMPLEX POSITION Z INTO AN ELIPTICAL " COORDINATE E ( Z ) . INCLUDE 'COMMON3.FOR' COMPLEX Z REAL X X=REAL(Z) E=CSQRT(Z* *2-A* *2)*SIGN(1.0,X) END COMPLEX FUNCTION ED(F,Z,Z0) » ************************** " THIS FUNCTION COMPUTES THE DISPLACEMENT AT Z DUE TO A " FORCE F AT ZO. " NOTE THAT YOU NEED TO SUPERPOSE FORCES TO ACHIEVE EQUILIBRIUM. INCLUDE 'COMMON3.FOR' COMPLEX S, CS, F, Z, ZO COMPLEX E COMPLEX CZ, CZO COMPLEX EZ, EZO, ECZ, ECZO COMPLEX J3,J4 COMPLEX SJ3, SJ4 COMPLEX CSJ3,CSJ4 COMPLEX CCSJ3 COMPLEX C0J3, COSJ3 COMPLEX CSPHIO,SPHIO,PHIO COMPLEX SPHI1,PHI1,CSOMEGA1 COMPLEX SPHI,CSOMEGA, PHI S=F/(2*PI*TP*(1+KAP)) CS=CONJG(S) LOADING AND COORDINATES EXTERNAL FUNCTIONS INTERMEDIATE VARIABLES TO SPEED UP THE CALCULATION CONVENIENT FUNCTIONS INTEGRAL OF J3, J4 Z=CZ DENOTED BY C Z=CZ, Z0=CZ0 DENOTED BY CC Z0=CZ0 DENOTED BY CO STRESS F'N FOR INT. PRES. CRACK STRESS F'N FOR POINT FORCE TOTAL STRESS F'N 137 CZ=CONJG(Z) CZ0=CONJG(ZO) EZ=E(Z) EZO=E(ZO) ECZ=E(CZ) ECZO=E(CZO) SJ3=CLOG((EZ0*EZ+Z0*Z-A**2)/(500*(Z+EZ))) SJ4=(EZ/ECZO-l.0)*(CZO-ZO)/(Z-CZO) C0SJ3=CLOG((ECZ0*EZ+CZ0*Z-A**2)/(500*(Z+EZ))) SPHIO=0.5*(S*(SJ3-KAP*C0SJ3)-CS*SJ4 ) CSJ3=CL0G((EZ0*ECZ+Z0*CZ-A**2)/(500*(CZ+ECZ))) CSJ4=(ECZ/ECZO-1.0)*(CZO-ZO)/(CZ-CZO) CCSJ3=CL0G((ECZ0*ECZ+CZ0*CZ-A**2)/(500*(CZ+ECZ))) CSPHI0=0.5*(S*(CSJ3-KAP*CCSJ3)-CS*CSJ4) J3=(ZO-EZO+EZ-Z)/((Z-ZO)*EZ) J4=((1-ECZO/EZ)/(Z-CZO)-CZO/(ECZO*EZ))*((CZO-ZO)/(Z-CZO)) C0J3=(CZO-ECZO+EZ-Z)/((Z-CZO)*EZ) PHI0=0.5*(S*(J3-KAP*C0J3)-CS*J4) SPHI1=S*CL0G(1000/(Z-ZO) ) PHI1=-S/(Z-ZO) CS0MEGA1=S*KAP*CL0G((CZ-CZO)/1000)+CS*(ZO-CZO)/(CZ-CZO) SPHI=SPHI0+SPHI1 CSOMEGA=CSPHIO+CSOMEGA1 PHI=PHI0+PHI1 ED=(KAP*SPHI-CSOMEGA-(Z-CZ)*CONJG(PHI))/(2*GP) END SUBROUTINE CONVRG(D,DO,CONVERGED) " * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * INCLUDE 'COMMON3.FOR' REAL D(NLIMIT), DO(NLIMIT), NSUB, CHECK, CHECKO LOGICAL CONVERGED CHECKO = CHECK CHECK = ABS(NSUB(D)/NSUB(DO)-1.0) IF(CHECK.LE.TOL) THEN CONVERGED=.TRUE. ELSE IF (CHECK.GT.CHECKO) THEN DO 1=1,N D(I)=(D(I)+D0(I))/2.0 END DO CONVERGED=.FALSE. ELSE CONVERGED=.FALSE. END IF END 138 » * * * * * * * * * * * * * * * * * * * * * * REAL FUNCTION NSUB(VEC) "********************** INCLUDE 'COMMON3.FOR' REAL VEC(NLIMIT) NSUB = 0 .0 DO 1=1,N NSUB = NSUB + VEC(I) ** 2.0 END DO NSUB = SQRT(NSUB) END II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * " LISTING FOR CRACK.FOR - CRACK GROWTH PREDICTION MODULE ^******************************************************* II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE CRACK(Z, DMAX, DELD, INC) II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * INCLUDE 'COMMON3.FOR' COMPLEX Z(NLIMIT), STRINT REAL DMAX(NLIMIT), DELD(NLIMIT) IF (INTER) THEN KMAX = PATCHK(DMAX,(SMAX*(YO+BC(SMAX))+SRES),TYA) KMIN = KMAX - PATCHK(DELD,(YO+BC(SMAX-SMIN))*(SMAX-SMIN),2*TYA) ELSE KMAX = REAL(STRINT(DMAX,Z,(SRES+(YO+BC(SMAX))*SMAX),TYA)) KMIN = KMAX - REAL(STRINT(DELD,Z,(YO+BC(SMAX-SMIN))*(SMAX-SMIN),2*TYA)) END IF RATE = CRACKRATE(KMIN,KMAX) IF (RLAW.EQ.'WHEELER') THEN ! USE WHEELER RETARDATION RP=PLASTIC(KMAX,2*SY,TP) ! BASED ON RPZ. IF ((R2+R3).LT.(A+RP)) THEN R2=A R3=RP END IF IF ((A+RP).LT.(R2+R3)) THEN RATE=RATE*((RP/(R2+R3-A))**R1) END IF END IF A=A+ RAT E *INC END "************************************** COMPLEX FUNCTION STRINT(D,Z,STRESS,TYL) INCLUDE 'COMMON3.FOR' COMPLEX F, GK REAL D(NLIMIT) COMPLEX Z(NLIMIT) REAL COMPLIANCE, STRESS, TYL, SPR 139 STRINT=CMPLX(Y1*STRESS*SQRT(PI*A),0.0) DO 1=1,N SPR=DEL/COMPLIANCE(AIMAG(Z(I)),D(I),TYL) F=D(I)*CMPLX(0.0,SPR) STRINT=STRINT-GK(F,Z(I))-- G K ( - F , - Z ( I ) ) --GK(-CONJG(F),-CONJG(Z(I)))--GK(CONJG(F),CONJG(Z(I))) END DO STRINT=STRINT/SQRT(1000.0) END "*********************** COMPLEX FUNCTION GK(F,Z) " THIS FUNCTION COMPUTES THE STRESS INTENSITY DUE TO A Y " FORCE F AT Z. " NOTE THAT YOU NEED TO SUPERPOSE FORCES TO ACHIEVE EQUILIBRIUM. INCLUDE 'COMMON3.FOR' COMPLEX S, F l , F2, F3, Z, CZ COMPLEX F COMPLEX E S= (F*(1-VP**2))/(2*PI*TP*(1+KAP)) CZ=CONJG(Z) F 1 = ( E ( Z ) / ( Z - A ) ) - l F2=(E(CZ)/(CZ-A))-1 F3=(A*(CZ-Z))/(E(CZ)*(CZ-A)) GK=2*(S*(F1-KAP*F2)+CONJG(S)*F3)/SQRT(A) END * * * * * * * * * * * * * * * * * * * * * * * * * * * FUNCTION CRACKRATE(KLO,KHI) "************************** " THIS SUBROUTINE CALCULATE THE STEADY STATE CRACK GROWTH RATE GIVEN " LOADING KLO, KHI. INCLUDE 'COMMON3.FOR' REAL KHI, KLO, DK, R R=KLO/KHI DK=KHI-KLO IF (CLAW.EQ.'PARIS') THEN CRACKRATE = C1*(DK**C2) ELSE IF (CLAW.EQ.'FORMAN') THEN CRACKRATE = CI*(DK**C2)/((1.0-R)*K1C-DK) IF (R.LT.(0.0)) THEN FLAG=.FALSE. MODE='ERROR2' END IF ELSE IF (CLAW.EQ.'PRIDDLE') THEN CRACKRATE=C1*((DK-KTH)**C2)/((1.0-R)*K1C-DK) IF (R.LT.(0.0)) THEN FLAG=.FALSE. MODE='ERROR2 ' END IF ELSE MODE='ERROR1' 140 FLAG=.FALSE. END IF END REAL FUNCTION PLASTIC(K,SY,TP) REAL K, SY, TP PLASTIC = (1/3.1415)*( (K/SY)**2) IF (PLASTIC.LT.(2.5*TP/1000)) THEN PLASTIC = PLASTIC/3.0 END IF END " ************************************************* " LISTING FOR DISBOND.FOR - DISBOND GROWTH PREDICTION " *************************************************** * *< *********************************** SUBROUTINE DISBOND (Z,DMAX,DELD,INC) INCLUDE 1 COMMON3.FOR1 REAL DMAX(NLIMIT), DELD(NLIMIT) COMPLEX Z(NLIMIT) INTEGER INC REAL COMPLIANCE, FMAX, FMIN DO 1=1,N FMAX=DMAX(I)/COMPLIANCE(AIMAG(Z(I)),DMAX(I),TYA) FMIN=FMAX-DELD(I)/COMPLIANCE(AIMAG(Z(I)),DELD(I),2*TYA) IF (DLAW.EQ.'STRAINEN 1) THEN ! PICK GROWTH LAW CALL STRAINEN (Z(I),FMAX,FMIN,INC) ELSE IF (DLAW.EQ.'STRAINLFE') THEN CALL ALTPLAS (FMAX,FMIN,INC) ELSE MODE='ERROR3' FLAG=.FALSE. END IF END DO END "************************************ SUBROUTINE STRAINEN(Z,FMAX,FMIN,INC) " THIS SUBROUTINE CALCULATES AN INTERVAL OF DISBOND GROWTH GIVEN " LAP JOINT DISPLACEMENTS DMAX, DMIN. " THE GROWTH RELATION IS OF THE FOLLOWING FORM; dB/dN=Dl(GMAX-GMIN)**D2 INCLUDE 'COMMON3.FOR' REAL FMAX, FMIN, RATE, GMAX, GMIN COMPLEX Z INTEGER INC GMAX=(FMAX**2)/(2*ER*TR) GMIN=(FMIN**2)/(2*ER*TR) RATE=D1*((GMAX-GMIN)*1O0O.O)**D2 Z = Z + CMPLX(0.0,RATE*INC) END 141 SUBROUTINE ALTPLAS(DMAX,DMIN,INC) "********************************** " THIS SUBROUTINE CALCULATES AN INTERVAL OF DISBOND GROWTH GIVEN " LAP JOINT DISPLACEMENTS DMAX, DMIN. " THE GROWTH RELATION IS OF THE FOLLOWING FORM; dB/dN= F(DG) INCLUDE 'COMMON3.FOR' REAL DMAX, DMIN, RATE INTEGER INC RATE=D1*((DMAX-DMIN)**D2) A=A+RATE*INC END " LISTING FOR COMFUN.FOR - CONTAINS SEVERAL USER-DEFINED FUNCTIONS ii****************************************************************** " ****************************** REAL FUNCTION OPENSTRESS(F,T,Z) 'i ****************************** " THIS FUNCTION CALCULATES THE STRESS IN AN INFINITE LAMINATED BEAM " GIVEN AN APPLIED FORCE INTENSITY, TEMPERATURE, AND OFFSET FROM THE " INTERFACE. THIS IS BASED ON THE TIMOSHENKO ANALYSIS OF A BIMETALLIC " STRIP, AND IS SUBJECT TO THE LIMITATIONS IMPLIED BY A BERNOULLI BEAM. INCLUDE 'COMMON3.FOR' REAL ELO, CUR REAL F,M REAL T, DT REAL Z REAL Q REAL NA REAL EIRP, EIP ELONGATION AND CURVATURE OF THE INTERFACE APPLIED FORCE AND MOMENT/LENGTH CURRENT TEMPERATURE DISTANCE FROM INTERFACE (REINFORCEMENT IS +) ! PATCH NEUTRAL AXIS ! PATCH, PLATE BENDING STIFFNESS' Q = TR/TP DT = T•- TSF ! TEMPERATURE CHANGE FROM STRESS FREE TEMP IF (BEND) THEN NA = (SR*(TP+TR/2)+(TP/2))/(SR+1) EIRP = ER*((TR**3)/12+TR*((TR/2+TP-NA)**2)) EIRP = EIRP + EP*((TP**3)/12+TP*((TP/2-NA)**2)) EIP = EP*(TP**3)/12 M = (NA - TP/2) * F ELO = ((AP+SR*AR)-0.75*(AP-SR*Q*AR)*(1-SR*Q)/(1+SR*Q*Q))*DT ELO = ELO + F/(EP*TP) ELO = ELO + 1.5*M*(1-SR*Q)/((1+SR*Q*Q)*(EP*(TP**2))) ELO = ELO / (l+SR-0.75*((1-SR*Q)**2)/(1+SR*Q*Q)) CUR = 2.0*((1+SR*AR/AP)*AP*DT+F/(EP*TP)-(1+SR)*ELO)/(TP-SR*TR) " INCLUDE GEOMETRIC STIFFENING (FINITE PATCH) CUR=CUR/(COSH(SQRT(F/EIRP)*H)+SQRT(EIRP/EIP)*SINH(SQRT(F/EIRP)*H)) ELO=((AP-SR*Q*AR)*DT+F/(EP*TP)-0.5*(1-SR*Q)*TP*CUR)/(1+SR) ELSE ELO=((AP+SR*AR)*DT+F/(EP*TP))/(1+SR) CUR=0.0 END IF IF (Z.GT.(0.0)) THEN OPENSTRESS=ER*(ELO-CUR*Z-AR*DT) 142 ELSE OPENSTRESS=EP*(ELO-CUR*Z-AP*DT) END IF END " * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * REAL FUNCTION COD(D,Z,STRESS,TYL) "******************************** " THIS FUNCTION CALCULATES THE CRACK OPENING DISPLACEMENT " BASED IN THE INSERTION DISPLACEMENTS. INCLUDE 'COMMON3.FOR1 COMPLEX GD, Z(NLIMIT) REAL D(NLIMIT), F, COMPLIANCE, STRESS, TYL COD=(STRESS*A*(1.0+KAP))/(4*GP) DO 1=1,N F=D(I)*DEL/COMPLIANCE(AIMAG(Z(I)),D(I),TYL) COD=COD-F*AIMAG(GD((0.001,0.001),Z(I))) END DO COD=COD*1000 END » * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * REAL FUNCTION COMPLIANCE(BB,ULAP,TYL) ************************************* " THIS FUNCTION CALCULATES THE COMPLIANCE/LENGTH GIVEN " THE DISPLACEMENT OF A DISBONDED LAP JOINT. INCLUDE 'COMMON3.FOR' REAL ULAP,TYL,BB,UPLA,Tl,T2 COMPLIANCE = SLAM*TA/GA + BB/(ER*TR) IF (LAG) COMPLIANCE = COMPLIANCE + SLAM*0.375*(TP/GP+TR/GR) UPLA = TYL*COMPLIANCE/SLAM IF (NONLIN.AND.(ULAP.GT.UPLA)) THEN T l = 1/(1/(ER*TR)+1/(EP*TP)) T2 = TYL*BB/(ER*TR*T1) COMPLIANCE = ULAP/(SQRT(T2**2+2*(ULAP-UPLA)*TYL/T1)-T2+TYL/SLAM) END IF END "*********************************** REAL FUNCTION PATCHK(U, STRESS, TYL) "*********************************** " THIS FUNCTION CALCULATES INTERPOLATION MODEL STRESS INTENSITY FACTORS. INCLUDE 1 COMMON3.FOR' REAL STRESS, U, COMPLIANCE, LD LD = (EP*TP*COMPLIANCE(B,U,TYL) + B)/PI PATCHK=STRESS*Y1*SQRT(PI*A*LD/(A+LD))/SQRT(1000) IF (MOD) THEN IF ((PLANE.EQ.'STRESS').AND.((B/A).GT.(1.0/25.0))) THEN PATCHK=PATCHK*(1.020+0.14 9*((B/A)**0.844)*(LD/A)**0.217) ELSE IF ((PLANE.EQ.'STRAIN').AND.((B/A).GT.(1.0/25.0))) THEN PATCHK=PATCHK*(1.04 8+0.123*((B/A)**0.737)*((LD/A)**0.534)) END IF END IF END 143 " * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * REAL FUNCTION PATCHD(U,STRESS,TYL) " THIS FUNCTION CALCULATES INTERPOLATION MODEL INSERTION DISPLACEMENTS INCLUDE 'COMMON3.FOR' REAL STRESS, U, COMPLIANCE, LD, TEMP LD = (EP*TP*COMPLIANCE(B,U,TYL)+B)/PI TEMP=SQRT(A**2+B**2) TEMP=(1+VP)*((KAP+1)*TEMP+B*(2*B/TEMP+1-KAP))12 PATCHD=(STRESS/EP)*(1/(1/TEMP+l/(PI*LD-B))) END Sample Input File "************************************************* " LISTING OF 3FULL.INP - WPAFB SPECIMEN INPUT FILE " ************************************************* '* number o f output f i l e s ' 3 '* s t i f f n e s s p r o p e r t i e s : EP, ER, GR, GA, VP, NONLIN, LAG' 71020.0, 208000.0, 7200.0, 965.0, 0.33, .TRUE., .TRUE. '* f a i l u r e p r o p e r t i e s : TYA, SY, K1C, KTH, DISLIM, CRALIM, FINITE' 17.0, 331.0, 74.721, 2.0, 40.0, 60.0, .TRUE. '* t h e r m a l p r o p e r t i e s : AP, AR, TSF' 22.7E-6, 4.5E-6, 101.2 '* d i m e n s i o n s : TP, TR, TA, PLANE, W, H' 1.0, 0.381, 0.127, 'STRESS', 25.0, 35.0 '* i n i t i a l c r a c k , d i s b o n d : AO, RP0, B0, TYPE, CONSOPT' 12.7, 0.34, 3.0, 'CONS', .TRUE. '* c a l c u l a t i o n a l parameters: N,INCOUT,MAXINC,UPDATE,TOL,INTER,MOD,BEND 100, 5, 1000, 3000, 0.001, .FALSE., .TRUE., .TRUE. '* growth law s e l e c t i o n : CLAW, RLAW, DLAW' 'FORMAN','NONE','STRAINEN' '* c r a c k growth parameters: CI, C2, C3, C4' 6.311E-6, 3.0, 0.0, 0.0 '* CL1, CL2' .FALSE.,.FALSE. '* r e t a r d a t i o n parameters: R l , R2, R3, R4' 0.569, 0.0, 0.0, 0.0 • * RL1, RL2' .FALSE.,.FALSE. '* d i s b o n d growth parameters: D l , D2, D3, D4, NEW DISBOND SIZE' 1.3E-18, 4.86, 0.0, 0.0, 3.0 '* DL1, DL2' .FALSE., .FALSE. '* CRACK geometry f a c t o r : Y0' 1.00 '* f i r s t o u tput f i l e : change i n l o a d h i s t o r y , end o f b l o c k , end o f run 'BASICPAR', 'NONENONE', 'ENDOFRUN' '* second output f i l e ' 'NONENONE', 'DISBONDS', 'NONENONE' '* t h i r d output f i l e ' 'CRACKING', 'NONENONE', 'NONENONE' 144 Appendix G: AMRL Specimen Dimensions The following figure [48] shows the A M R L specimen dimensions. \ / 4(H* mm 2Ui mm r-t-J-M-'-f + BctKjf (X>mb Cure EDM Nflicb I'liu (Vara* A'twht'K coetm*! ? Layer • 1 l.a>>M'l;W.\ I J.ajwr r>l?« (12.70 HMD OMH AlAUojr2lM4.TJ Figure E-1: A M R L Specimen Dimensions 145 Appendix H: Material Properties Boron/Epoxy 5521/4 Laminate The following are the material properties for Textron Specialty Materials Boron 5521/4. Note material properties in the fiber direction are for dry environmental conditions [96, 97]. Thickness Resin content (wt.) Cure cycle Cure pressure Test temperature Longitudinal Tensile Modulus Transverse Tensile Modulus Longitudinal Compressive Modulus Longitudinal Poisson's Ratio Transverse Poisson's Ratio Ultimate Tensile Stress Ultimate Compressive Stress Ultimate Interlaminar Shear Stress Coef. Of Thermal Expansion Density 0.132 mm (0.0052 in) 33% 2.2-3.3°C (4-6°F) per minute to 120°C (250°F) 60 minutes at 120°C (250°F) 345-586 kPa (50-85 psi) Room temperature 210GPa.(30msi) 25 GPa 210 GPa (30msi) 0.21 0.019 1520 MPa (220 msi) 2930 MPa (425 msi) 97 MPa (14.1 msi) 4.5 PPM/°C (2.5 PPM/°F) 2 g/cm3 (0.072 lbm/in3) 146 Aluminum 2024-T3 (A-Basis) Mechanical properties for 2024-T3 aluminum in the rolling direction [98] are; Thickness Test temperature Tensile Modulus Compressive Modulus Shear Modulus Poisson's Ratio Tensile Yield Stress Compressive Yield Stress Shear Yield Stress Ultimate Tensile Stress 3.175 mm (0.125 in) Room temperature 72.4 GPa (10.5 msi) 73.8 GPa (10.7 msi) 27.6 GPa 0.33 331 MPa (48 ksi) 276 MPa (40 ksi) 276 MPa (40 ksi) 448 MPa (65 ksi) 147 Adhesive FM 73M The properties for Cytec's F M 73M follow. Note the properties were measured without pre- or post-bond environmental exposure [99]. Thickness Nominal weight Primer Cure cycle Cure pressure Test temperature Shear Modulus Elastic Shear Stress Elastic Shear Strain Ultimate Shear Stress Ultimate Shear Strain 0.25 mm (0.010 in) 300 g/m2 (0.06 psf) B R 127 30 minutes to 120°C 60 minutes at 120°C 280 MPa (40 psi) 24°C 842 MPa (122 ksi) 17.3 MPa (2510 psi) 0.021 40.9 MPa (5.93 ksi) 0.873 148 


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