• Lucas F. M. da Silva
• Raul D. S. G. Campilho
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

## Abstract

The analysis of adhesively bonded joints started in 1938 with the closed-form model of Volkersen. The equilibrium equation of a single lap joint led to a simple governing differential equation with a simple algebraic equation. However, if there is yielding of the adhesive and/or the adherends and substantial peeling is present, a more complex model is necessary. The more complete is an analysis, the more complicated it becomes and the more difficult it is to obtain a simple and effective solution. The finite element (FE) method, the boundary element (BE) method and the finite difference (FD) method are the three major numerical methods for solving differential equations in science and engineering. These methods have also been applied to adhesive joints, especially the FE method. This book deals with the most recent numerical modelling of adhesive joints. Advances in damage mechanics and extended finite element method are described in the context of the FE method with examples of application. The classical continuum mechanics and fracture mechanics approach are also introduced. The BE method and the FD method are also discussed with indication of the cases they are most adapted to. There is not at the moment a numerical technique that can solve any problem and the analyst needs to be aware of the limitations involved in each case.

## Keywords

Adhesive joints Finite element method Continuum mechanics Fracture mechanics Damage mechanics Extended finite element method Boundary element method Finite difference method

## References

1. Abaqus® Documentation (2009) Dassault Systèmes, Vélizy-VillacoublayGoogle Scholar
2. A. Abdul-Baqi, P.J.G. Schreurs, M.G.D. Geers, Fatigue damage modeling in solder interconnects using a cohesive zone approach. Int. J. Solids Struct. 42, 927–942 (2005)
4. R.D. Adams, R. Davies, Strength of lap shear joints, in The Mechanics of Adhesion, ed. by D.A. Dillard, A.V. Pocius (Elsevier, Amsterdam, 2002)Google Scholar
5. R.D. Adams, J.A. Harris, Strength prediction of bonded single lap joints by nonlinear finite element methods. Int J Adhes Adhes 4, 65–78 (1984)Google Scholar
7. R.D. Adams, V. Mallick, A method for the stress analysis of lap joints. J. Adhesion 38, 199–217 (1992)Google Scholar
8. R.D. Adams, N.A. Peppiatt, Effects of Poisson’s ratio strains in adherend on stresses of an idealized lap joint. J. Strain Anal. 8, 134–139 (1973)Google Scholar
9. R.D. Adams, N.A. Peppiatt, Stress analysis of adhesive-bonded lap joints. J. Strain Anal. 9, 185–196 (1974)Google Scholar
10. R.D. Adams, R.W. Atkins, J.A. Harris, A.J. Kinloch, Stress analysis and failure properties of carbon-fibre-reinforced-plastic/steel double-lap joints. J. Adhesion 20, 29–53 (1986)Google Scholar
11. R.D. Adams, J. Comyn, W.C. Wake, Structural Adhesive Joints in Engineering, 2nd edn. (Chapman & Hall, London, 1997)Google Scholar
12. G. Alfano, On the influence of the shape of the interface law on the application of cohesive-zone models. Compos. Sci. Technol. 66, 723–730 (2006)Google Scholar
13. G. Alfano, M.A. Crisfield, Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. Int. J. Numer. Methods Eng. 50, 1701–1736 (2001)
14. O. Allix, A. Corigliano, Modeling and simulation of crack propagation in mixed-modes interlaminar fracture specimens. Int. J. Fract. 77, 111–140 (1996)Google Scholar
15. T. Andersson, U. Stigh, The stress-elongation relation for an adhesive layer loaded in peel using equilibrium of energetic forces. Int. J. Solids Struct. 41, 413–434 (2004)Google Scholar
16. I. Ashcroft, Fatigue Load Conditions, in Handbook of Adhesion Technology, ed. by L.M.F. da Silva, A. Öchsner, R.D. Adams (Springer, Heidelberg, 2011)Google Scholar
17. ASTM D3433-99 Standard, Standard Test Method for Fracture Strength in Cleavage of Adhesives in Bonded Metal Joints. ASTM International, West Conshohocken (2005)Google Scholar
18. ASTM D4501-01, Standard Test Method for Shear Strength of Adhesive Bonds Between Rigid Substrates by The Block Shear Method (ASTM International, West Conshohocken, 2009)Google Scholar
19. M.G. Bader, I. Hamerton, J.N. Hay, M. Kemp, S. Winchester, Double cantilever beam of repaired carbon fibre composites. Compos Part A 31, 603–608 (2000)Google Scholar
20. M.D. Banea, L.F.M. da Silva, Adhesively bonded joints in composite materials: an overview. J. Mater. Design Appl. 223, 1–18 (2009)Google Scholar
21. M.D. Banea, L.F.M. da Silva, R.D.S.G. Campilho, Temperature dependence of the fracture toughness of adhesively bonded joints. J. Adhesion Sci. Technol. 24, 2011–2026 (2010)Google Scholar
22. M.D. Banea, L.F.M. da Silva, R.D.S.G. Campilho, Mode I fracture toughness of adhesively bonded joints as a function of temperature: experimental and numerical study. Int. J. Adhes. Adhes. 31, 273–279 (2011)Google Scholar
23. G.I. Barenblatt, The formation of equilibrium cracks during brittle fracture. General ideas and hypothesis. Axisymmetrical cracks. J. Appl. Math. Mech. 23, 622–636 (1959)
24. G.I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962)
25. W.D. Bascom, R.L. Cottingham, Effect of temperature on the adhesive fracture behavior of an elastomer–epoxy resin. J. Adhesion 7, 333–346 (1976)Google Scholar
26. W.D. Bascom, R.L. Cottingham, R.L. Jones, P. Peyser, The fracture of epoxy- and elastomer-modified epoxy polymers in bulk and as adhesives. J. Appl. Polymer Sci. 19, 2545–2562 (1975)Google Scholar
27. A.J. Bell, A.J. Kinlock, The effect of the substrate material on the value of the adhesive fracture energy, G(c). J. Mater. Sci. Lett. 16, 150–1453 (1997)Google Scholar
28. T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing. Int. J. Fract. Mech. 45, 601–620 (1999)
29. M.L. Benzeggagh, M. Kenane, Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos. Sci. Technol. 56, 439–449 (1996)Google Scholar
30. B. Bhattacharya, B. Ellingwood, Continuum damage mechanics analysis of fatigue crack initiation. Int. J. Fatigue 20, 631–639 (1998)Google Scholar
31. A. Biel, Constitutive behaviour and fracture toughness of an adhesive layer. Licentiate of Engineering Dissertation, Chalmers University of Technology (2005)Google Scholar
32. A. Biel, U. Stigh, Effects of constitutive parameters on the accuracy of measured fracture energy using the DCB-specimen. Eng. Fract. Mech. 75, 2968–2983 (2008)Google Scholar
34. B.R.K. Blackman, H. Hadavinia, A.J. Kinloch, J.G. Williams, The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints. Int. J. Fract. 119, 25–46 (2003)Google Scholar
35. B.R.K. Blackman, A.J. Kinloch, M. Paraschi, The determination of the mode II adhesive fracture resistance, GIIC, of structural adhesive joints: an effective crack length approach. Eng. Fract. Mech. 72, 877–897 (2005)Google Scholar
36. D. Bushnell, B.O. Almroth, F. Brogan, Finite-difference energy method for nonlinear shell analysis. Comput. Struct. 1, 361–387 (1971)Google Scholar
37. R.D.S.G. Campilho, M.F.S.F. de Moura, J.J.M.S. Domingues, Modelling single and double-lap repairs on composite materials. Compos. Sci. Technol. 65, 1948–1958 (2005)Google Scholar
38. R.D.S.G. Campilho, M.F.S.F. de Moura, J.J.M.S. Domingues, Stress and failure analyses of scarf repaired CFRP laminates using a cohesive damage model. J. Adhes. Sci. Technol. 21, 855–970 (2007)Google Scholar
39. R.D.S.G. Campilho, M.F.S.F. de Moura, J.J.M.S. Domingues, Using a cohesive damage model to predict the tensile behaviour of CFRP single-strap repairs. Int. J. Solids Struct. 45, 1497–1512 (2008a)
40. R.D.S.G. Campilho, M.F.S.F. de Moura, J.J.M.S. Domingues, J.J.L. Morais, Computational modelling of the residual strength of repaired composite laminates using a cohesive damage model. J. Adhes. Sci. Technol. 22, 1565–1591 (2008b)Google Scholar
41. R.D.S.G. Campilho, M.F.S.F. de Moura, J.J.M.S. Domingues, Numerical prediction on the tensile residual strength of repaired CFRP under different geometric changes. Int. J. Adhes. Adhes. 29, 195–205 (2009a)Google Scholar
42. R.D.S.G. Campilho, M.F.S.F. de Moura, A.M.G. Pinto, J.J.L. Morais, J.J.M.S. Domingues, Modelling the tensile fracture behaviour of CFRP scarf repairs. Compos. Part B 40, 149–157 (2009b)Google Scholar
43. R.D.S.G. Campilho, M.F.S.F. de Moura, D.A. Ramantani, J.J.L. Morais, J.J.M.S. Domingues, Tensile behaviour of three-dimensional carbon-epoxy adhesively bonded single- and double-strap repairs. Int. J. Adhes. Adhes. 29, 678–686 (2009c)Google Scholar
44. R.D.S.G. Campilho, M.F.S.F. de Moura, D.A. Ramantani, J.J.L. Morais, A.M.J.P. Barreto, J.J.M.S. Domingues, Adhesively-bonded repair proposal for wood members damaged by horizontal shear using carbon-epoxy patches. J. Adhesion 86, 649–670 (2010)Google Scholar
45. R.D.S.G. Campilho, M.D. Banea, A.M.G. Pinto, L.F.M. da Silva, A.M.P. de Jesus, Strength prediction of single- and double-lap joints by standard and extended finite element modelling. Int. J. Adhes. Adhes. 31, 363–372 (2011a)Google Scholar
46. R.D.S.G. Campilho, A.M.G. Pinto, M.D. Banea, L.F.M. da Silva, Optimization study of hybrid spot welded-bonded single-lap joints. Int. J. Adhes. Adhes. (2011b) (accepted)Google Scholar
47. R.D.S.G. Campilho, M.D. Banea, F.J.P. Chaves, L.F.M. da Silva, Modelling of single-lap joints using cohesive zones models: effect of the cohesive parameters on the output of the simulations. J. Adhesion (2011c) (accepted)Google Scholar
48. R.D.S.G. Campilho, M.D. Banea, F.J.P. Chaves, L.F.M. da Silva, eXtended Finite Element Method for fracture characterization of adhesive joints in pure mode I. Comput. Mater. Sci. 50, 1543–1549 (2011d)Google Scholar
49. R.D.S.G. Campilho, Modelação da Execução de Reparações em Materiais Compósitos. M.Sc. Dissertation, Engineering Faculty of Porto University (2005)Google Scholar
50. R.D.S.G. Campilho, Repair of composite and wood structures. Ph.D. Dissertation, Engineering Faculty of Porto University (2009)Google Scholar
51. T. Carlberger, U. Stigh, An explicit FE-model of impact fracture in an adhesive joint. Eng. Fract. Mech. 74, 2247–2262 (2007)Google Scholar
52. T. Carlberger, U. Stigh, Influence of layer thickness on cohesive properties of an epoxy-based adhesive—an experimental study. J. Adhesion 86, 814–833 (2010)Google Scholar
53. T. Carlberger, A. Biel, U. Stigh, Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive. Int. J. Fract. 155, 155–166 (2009)Google Scholar
54. M.N. Cavalli, M.D. Thouless, The effect of damage nucleation on the toughness of an adhesive joint. J. Adhesion 76, 75–92 (2001)Google Scholar
55. G. Cavallini, G. Davi, A. Milazzo, Boundary element modeling and analysis of adhesive bonded structural joints. Electron. J. Bound Elem. 4, 31–48 (2006)
56. H. Chai, Bond thickness effect in adhesive joints and its significance for mode I interlaminar fracture of composites. ASTM STP 893, 209–231 (1986a) Google Scholar
57. H. Chai, On the correlation between the mode I failure of adhesive joints and laminated composites. Eng. Fract. Mech. 24, 413–431 (1986b)Google Scholar
58. H. Chai, Shear fracture. Int. J. Fract. 37, 137–159 (1988)Google Scholar
59. H. Chai, Experimental evaluation of mixed-mode fracture in adhesive bonds. Experimental Mech. 32, 296–303 (1992)Google Scholar
60. N. Chandra, H. Li, C. Shet, H. Ghonem, Some issues in the application of cohesive zone models for metal–ceramic interfaces. Int. J. Solids Struct. 39, 2827–2855 (2002)
61. M. Charalambides, A.J. Kinloch, Y. Wang, J.G. Williams, On the analysis of mixed mode failure. Int. J. Fracture 54, 269–291 (1992)Google Scholar
62. J. Chen, Predicting progressive delamination of stiffened fibre-composite panel and by decohesion models. J. Thermopl. Compos. Mater. 15, 429–441 (2002)Google Scholar
63. Z. Chen, R.D. Adams, L.F.M. da Silva, The use of the J-integral to analyse adhesive bonds with and without a crack. Int. J. Adhes. Adhes. 31, 48–55 (2011a)Google Scholar
64. Z. Chen, R.D. Adams, L.F.M. da Silva, Prediction of crack initiation and propagation of adhesive lap joints using an energy failure criterion. Eng. Fract. Mech. 78, 990–1007 (2011b)Google Scholar
65. P.T. Cheuk, L. Tong, A.N. Rider, J. Wang, Analysis of energy release rate for fatigue cracked metal-to-metal double-lap shear joints. Int. J. Adhes. Adhes. 25, 181–191 (2005)Google Scholar
66. J.Y. Choi, H.J. Kim, J.K. Lim, Y.W. Mai, Numerical analysis of adhesive thickness effect on fracture toughness in adhesive-bonded joints. Key Eng. Mat. 270–273, 1200–1205 (2004)Google Scholar
68. J.D. Clarke, I.J. Mcgregor, Ultimate tensile stress over a zone: a new failure criterion for adhesive joints. J. Adhes. 42, 227–245 (1993)Google Scholar
69. L. Collatz, The Numerical Treatment of Differential Equations (Springer, Berlin, 1966)Google Scholar
70. R.D. Cook, Finite Element Modeling for Stress Analysis (Wiley, New York, 1995)
71. R. Courant, Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1–23 (1943)
72. J.H. Crews, K.N. Shivakumar, I.S. Raju, Factors influencing elastic stresses in double cantilever beam specimens. ASTM STP 981, 119–132 (1988)Google Scholar
73. A.D. Crocombe, Global yielding as a failure criteria for bonded joints. Int. J. Adhes. Adhes. 9, 145–153 (1989)Google Scholar
74. A.D. Crocombe, R.D. Adams, Influence of the spew fillet and other parameters on the stress distribution in the single lap joint. J. Adhes. 13, 141–155 (1981)Google Scholar
75. A.D. Crocombe, D.A. Bigwood, Development of a full elasto-plastic adhesive joint design. J. Strain Anal. Eng. Des. 27, 211–218 (1992)Google Scholar
76. A.D. Crocombe, Y.X. Hua, W.K. Loh, M.A. Wahab, I.A. Ashcroft, Predicting the residual strength for environmentally degraded adhesive lap joints. Int. J. Adhes. Adhes. 26, 325–336 (2006)Google Scholar
77. R.W. Clough, Second ASCE Conference on Electronic Computation, Pittsburgh, PA (1960)Google Scholar
78. W. Cui, M.R. Wisnom, A combined stress-based and fracture-mechanics-based model for predicting delamination in composites. Composites 24, 467–474 (1993)Google Scholar
79. L.F.M. da Silva, G.W. Critchlow, M.A.V. Figueiredo, Parametric study of adhesively bonded single lap joints by the Taguchi method. J. Adhes. Sci. Technol. 22(13), 1477–1494 (2008)Google Scholar
80. L.F.M. da Silva, R.J.C. Carbas, G.W. Critchlow, M.A.V. Figueiredo, K. Brown, Effect of material, geometry, surface treatment and environment on the shear strength of single lap joints. Int. J. Adhes. Adhes. 29, 621–632 (2009a)Google Scholar
81. L.F.M. da Silva, P.J.C. das Neves, R.D. Adams, J.K. Spelt, Analytical models of adhesively bonded joints—Part I: literature survey. Int. J. Adhes. Adhes. 29, 319–330 (2009b)Google Scholar
82. L.F.M. da Silva, P.J.C. das Neves, R.D. Adams, A. Wang, J.K. Spelt, Analytical models of adhesively bonded joints—part II: comparative study. Int. J. Adhes. Adhes. 29, 331–341 (2009c)Google Scholar
83. L.F.M. da Silva, R.F.T. Lima, R.M.S. Teixeira, Development of a software for the design of adhesive joints. J. Adhes. 85, 889–918 (2009d)Google Scholar
84. L.F.M. da Silva, A. Öchsner, R.D. Adams (eds.), Handbook of Adhesion Technology (Springer, Heidelberg, 2011)Google Scholar
85. L. Daudeville, P. Ladeveze, A damage mechanics tool for laminate delamination. Compos. Struct. 25, 547–555 (1993)Google Scholar
86. C. Daux, N. Moës, J. Dolbow, N. Sukumark, T. Belytschko, Arbitrary branched and intersecting cracks with the extended finite element method. Int. J. Numer. Meth. Eng. 48, 1741–1760 (2000)
87. M.F.S.F. de Moura, R.D.S.G. Campilho, J.P.M. Gonçalves, Crack equivalent concept applied to the fracture characterization of bonded joints under pure mode I loading. Compos. Sci. Technol. 68, 2224–2230 (2008)Google Scholar
88. M.F.S.F. de Moura, R.D.S.G. Campilho, J.P.M. Gonçalves, Pure mode II fracture characterization of composite bonded joints. Int. J. Solids Struct. 46, 1589–1595 (2009)Google Scholar
89. D.A. Dillard, H.K. Singh, D.J. Pohlit, Observations of decreased fracture toughness for mixed mode fracture testing of adhesively bonded joints. J. Adhes. Sci. Technol. 23, 515–1530 (2009)Google Scholar
90. J. Dolbow, N. Moës, T. Belytschko, An extended finite element method for modeling crack growth with frictional contact. Finite Elem. Anal. Des. 36, 235–260 (2000)
91. K. Duan, X.Z. Hu, F.H. Wittmann, Explanation of size effect in concrete fracture using non-uniform energy distribution. Mater. Struct. 35, 326–331 (2002)Google Scholar
92. K. Duan, X.Z. Hu, F.H. Wittmann, Boundary effect on concrete fracture and non-constant fracture energy distribution. Eng. Fract. Mech. 70, 2257–2268 (2003)Google Scholar
93. K. Duan, X. Hu, Y.W. Mai, Substrate constraint and adhesive thickness effects on fracture toughness of adhesive joints. J. Adhes. Sci. Technol. 18, 39–53 (2004)Google Scholar
94. P.A. DuBois, Crashworthiness Engineering Course Notes (Livermore Software Technology Corporation, Livermore, 2004)Google Scholar
95. F. Ducept, P. Davies, D. Gamby, Mixed mode failure criteria for a glass/epoxy composite and an adhesively bonded composite/composite joint. Int. J. Adhes. Adhes. 20, 233–244 (2000)Google Scholar
96. D.S. Dugdale, Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960)Google Scholar
97. T. Elguedj, A. Gravouil, A. Combescure, Appropriate extended functions for X-FEM simulation of plastic fracture mechanics. Comput. Methods Appl. Mech. Eng. 195, 501–515 (2006)
98. Engineering Sciences Data Unit, Inelastic shear stresses and strains in adhesive bonding lap joints loaded in tension or shear (Computer Program). Engineering Sciences Data Item Number 79016 (1979)Google Scholar
99. M. Fagerström, R. Larsson, Theory and numerics for finite deformation fracture modelling using strong discontinuities. Int. J. Numer. Methods Eng. 66, 911–948 (2006)
100. P. Feraren, H.M. Jensen, Cohesive zone modelling of interface fracture near flaws in adhesive joints. Eng. Fract. Mech. 71, 2125–2142 (2004)Google Scholar
102. G. Fernlund, M. Papini, D. McCammond, J.K. Spelt, Fracture load predictions for adhesive joints. Compos. Sci. Technol. 51, 587–600 (1994)Google Scholar
103. D.B. Flinn, C.S. Lo, W.F. Zok, A.G. Evans, Fracture resistance characteristics of a metal-toughened ceramic. J. Am. Ceram. Soc. 76, 369–375 (1993)Google Scholar
104. G.E. Forsythe, W.R. Wasow, Finite-Difference Methods for Partial Differential Equations (Wiley, New York, 1960)
105. D.M. Gleich, M.J.L. Van Tooren, A. Beukers, Analysis and evaluation of bondline thickness effects on failure load in adhesively bonded structures. J. Adhes. Sci. Technol. 15, 1091–1101 (2001)Google Scholar
106. M. Goland, E. Reissner, The stresses in cemented joints. J. Appl. Mech. 66, A17–A27 (1944)Google Scholar
107. J.P.M. Gonçalves, M.F.S.F. de Moura, P.T. de Castro, A.T. Marques, Interface element including point-to-surface constraints for three-dimensional problems with damage propagation. Eng. Comput. 17, 28–47 (2000)
108. J.P.M. Gonçalves, M.F.S.F. de Moura, P.M.S.T. de Castro, A three-dimensional finite element model for stress analysis of adhesive joints. Int. J. Adhes. Adhes. 22, 357–365 (2002)Google Scholar
109. L. Greenwood, The Strength of a Lap Joint, in Aspects of Adhesion-5, ed. by D. Alner (University of London Press, London, 1969)Google Scholar
110. H.L. Groth, Stress singularities and fracture at interface corners in bonded joints. Int. J. Adhes. Adhes. 8, 107–113 (1988)Google Scholar
111. A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth. Part I: yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 2–15 (1977)Google Scholar
112. L. Hamitouche, M. Tarfaoui, A. Vautrin, An interface debonding law subject to viscous regularization for avoiding instability: application to the delamination problems. Eng. Fract. Mech. 75, 3084–3100 (2008)Google Scholar
113. L.J. Hart-Smith, Adhesive-bonded single-lap joints. NASA Contract Report, NASA CR-112236 (1973)Google Scholar
114. S. Hashemi, A.J. Kinloch, J.G. Williams, Corrections needed in double cantilever beam tests for assessing the interlaminar failure of fibre composites. J. Mater. Sci. Lett. 8, 125–129 (1989)Google Scholar
115. J.L. Högberg, U. Stigh, Specimen proposals for mixed mode testing of adhesive layer. Eng. Fract. Mech. 73, 2541–2556 (2006)Google Scholar
116. J.L. Högberg, B.F. Sørensen, U. Stigh, Constitutive behaviour of mixed mode loaded adhesive layer. Int. J. Solids Struct. 44, 8335–8354 (2007)
117. M. Hojo, T. Ando, M. Tanaka, T. Adachi, S. Ochiai, Y. Endo, Modes I and II interlaminar fracture toughness and fatigue delamination of CF/epoxy laminates with self-same epoxy interleaf. Int. J. Fatigue 28, 1154–1165 (2006)
118. Y. Hua, A.D. Crocombe, M.A. Wahab, I.A. Ashcroft, Continuum damage modelling of environmental degradation in joints bonded with EA9321 epoxy adhesive. Int. J. Adhes. Adhes. 28, 302–313 (2008)Google Scholar
119. D.L. Hunston, A.J. Kinloch, S.J. Shaw, S.S. Wang, Characterization of fracture behavior of adhesive joints, in Characteristics Formation Testing, Adhesive Joints, ed. by K. Mittal (Plenum Press, New York, 1984)Google Scholar
120. D.L. Hunston, A.J. Kinloch, S.S. Wang, Micromechanics of fracture in structural adhesive bonds. J. Adhes. 28, 103–114 (1989)Google Scholar
121. J.W. Hutchinson, Singular behavior at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids 16, 13–31 (1968)
122. T. Ikeda, A. Yamashita, N. Miyazaki, Elastic-plastic analysis of crack in adhesive joint by combination of boundary element and finite element methods. Comput. Mech. 21, 533–539 (1998)
123. T. Ikeda, A. Yamashita, D. Lee, N. Miyazaki, Failure of a ductile adhesive layer constrained by hard adherends. J. Eng. Mater. Technol. 122, 80–85 (2000)Google Scholar
124. K. Ikegami, T. Takeshita, K. Matsuo, T. Sugibayashi, Strength of adhesively bonded scarf joints between glass fibre-reinforced plastics and metal. Int. J. Adhes. Adhes. 10, 199–206 (1990)Google Scholar
125. M. Imanaka, T. Hamano, A. Morimoto, R. Ashino, M. Kimoto, Fatigue damage evaluation of adhesively bonded butt joints with a rubber-modified epoxy adhesive. J. Adhes. Sci. Technol. 17, 981–994 (2003)Google Scholar
126. G. Ji, Z. Ouyang, G. Li, S. Ibekwe, S.S. Pang, Effects of adhesive thickness on global and local Mode-I interfacial fracture of bonded joints. Int. J. Solids Struct. 47, 2445–2458 (2010)
127. J. Jing, F. Gao, J. Johnson, F.Z. Liang, R.L. Williams, J. Qu, Simulation of dynamic fracture along solder-pad interfaces using a cohesive zone model. Eng. Failure Anal. 16, 1579–1586 (2009)Google Scholar
128. S.J. John, A.J. Kinloch, F.L. Matthews, Measuring and predicting the durability of bonded fibre/epoxy composite joints. Composites 22, 121–127 (1991)Google Scholar
129. A. Karac, B.R.K. Blackman, V. Cooper, A.J. Kinloch, S.R. Sanchez, W.S. Teo, A. Ivankovic, Modelling the fracture behavior of adhesively-bonded joints as a function of test rate. Eng. Fract. Mech. 78, 973–989 (2011)Google Scholar
130. M.S. Kafkalidis, M.D. Thouless, The effects of geometry and material properties on the fracture of single lap-shear joints. Int. J. Solids Struct. 39, 4367–4383 (2002)
131. P.I. Kattan, G.Z. Voyiadjis, Damage Mechanics with Finite Elements (Springer, Heidelberg, 2005)Google Scholar
132. A.R. Khoei, M. Nikbakht, Contact friction modeling with the extended finite element method (X-FEM). J. Mater. Process. Technol. 177, 58–62 (2006)Google Scholar
133. H. Khoramishad, A.D. Crocombe, K.B. Katnam, I.A. Ashcroft, Predicting fatigue damage in adhesively bonded joints using a cohesive zone model. Int. J. Fatigue 32, 1146–1158 (2010)Google Scholar
136. A.J. Kinloch, S.J. Shaw, The fracture resistance of a toughened epoxy adhesive. J. Adhesion 12, 59–77 (1981)Google Scholar
137. A.J. Kinloch, J.G. Williams, Crack blunting mechanisms in polymers. J. Mater. Sci. 15, 987–996 (1980)Google Scholar
138. A.J. Kinloch, R.J. Young, Fracture Behaviour of Polymers (Applied Science Publishers, London, 1983)Google Scholar
139. L. Lammerant, I. Verpoest, Modelling of the interaction between matrix cracks and delaminations during impact of composite plates. Compos. Sci. Technol. 56, 1171–1178 (1996)Google Scholar
140. S.S. Lee, Boundary element analysis of the stress singularity at the interface corner of viscoelastic adhesive layers. Int. J. Solids Struct. 35, 1385–1394 (1998)
141. S.J. Lee, D.G. Lee, Development of a failure model for the adhesively bonded tubular single lap joint. J. Adhes. 40, 1–14 (1992)Google Scholar
142. D.B. Lee, T. Ikeda, N. Miyazaki, N.S. Choi, Fracture behavior around a crack tip in rubber-modified epoxy adhesive joint with various bond thicknesses. J. Mater. Sci. Lett. 22, 229–233 (2003)Google Scholar
143. D.B. Lee, T. Ikeda, N. Miyazaki, N.S. Choi, Effect of bond thickness on the fracture toughness of adhesive joints. J. Eng. Mater. Technol. 126, 14–18 (2004)Google Scholar
144. M.J. Lee, T.M. Cho, W.S. Kim, B.C. Lee, J.J. Lee, Determination of cohesive parameters for a mixed-mode cohesive zone model. Int. J. Adhes. Adhes. 30, 322–328 (2010)Google Scholar
145. K. Leffler, K.S. Alfredsson, U. Stigh, Shear behaviour of adhesive layers. Int. J. Solids Struct. 44, 530–545 (2007)
146. J. Lemaitre, Local approach of fracture. Eng. Fract. Mech. 25, 523–537 (1986)Google Scholar
147. J. Lemaitre, J.M. Chaboche, Mechanics of Solid Materials (Cambridge University Press, Cambridge, 1985)Google Scholar
148. J. Lemaitre, R. Desmorat, Engineering Damage Mechanics (Springer, Heidelberg, 2005)Google Scholar
149. S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, P.D. Zavattieri, Use of Mode-I cohesive-zone models to describe the fracture of an adhesively-bonded polymer–matrix composite. Compos. Sci. Technol. 65, 281–293 (2005a)Google Scholar
150. S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, P.D. Zavattieri, Use of a cohesive-zone model to analyze the fracture of a fiber reinforced polymer–matrix composite. Compos. Sci. Technol. 65, 537–549 (2005b)Google Scholar
151. C.D.M. Liljedahl, A.D. Crocombe, M.A. Wahab, I.A. Ashcroft, Damage modelling of adhesively bonded joints. Int. J. Fract. 141, 147–161 (2006)Google Scholar
152. X. Liu, G. Wang, Progressive failure analysis of bonded composite repairs. Compos. Struct. 81, 331–340 (2007)Google Scholar
153. H. Liu, X.L. Zhao, R. Al-Mahaidi, Boundary element analysis of CFRP reinforced steel plates. Compos. Struct. 91, 74–83 (2009)Google Scholar
154. A.G. Magalhães, M.F.S.F. de Moura, J.P.M. Gonçalves, Evaluation of stress concentration effects in single-lap bonded joints of laminate composite materials. Int. J. Adhes. Adhes. 25, 313–319 (2005)Google Scholar
155. S. Maiti, P.H. Geubelle, A cohesive model for fatigue failure of polymers. Eng. Fract. Mech. 72, 691–708 (2005)Google Scholar
156. A.V. Mello, K.M. Liechti, The effect of self-assembled monolayers on interfacial fracture. J. Appl. Mech. 73, 860–870 (2006)
157. Y. Mi, M.A. Crisfield, G.A.O. Davies, H.B. Hellweg, Progressive delamination using interface elements. J. Compos. Mater. 32, 1246–1272 (1998)Google Scholar
158. A.R. Mitchell, D.F. Griffiths, The Finite Difference Method in Partial Differential Equations (Wiley, New York, 1980)
159. N. Moës, T. Belytschko, Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 69, 813–833 (2002)Google Scholar
160. N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 46, 131–150 (1999)
161. S. Mohammadi, Extended Finite Element Method for Fracture Analysis of Structures (Blackwell Publishing, New Jersey, 2008)
162. F. Moroni, A. Pirondi, A procedure for the simulation of fatigue crack growth in adhesively bonded joints based on the cohesive zone model and different mixed-mode propagation criteria. Eng. Fract. Mech. 78, 1808–1816 (2011)Google Scholar
163. F. Mortensen, O.T. Thomsen, Analysis of adhesive bonded joints: a unified approach. Compos. Sci. Technol. 62, 1011–1031 (2002)Google Scholar
164. J.J. Munoz, U. Galvanetto, P. Robinson, On the numerical simulation of fatigue driven delamination with interface elements. Int. J. Fatigue 28, 1136–1146 (2006)
165. J.A. Nairn, Energy release rate analysis for adhesive and laminate double cantilever beam specimens emphasizing the effect of residual stresses. Int. J. Adhes. Adhes. 20, 59–70 (2000)Google Scholar
166. E.P. O’Brien, T.C. Ward, S. Guo, D.A. Dillard, Strain energy release rates of a pressure sensitive adhesive measured by the shaft-loaded blister test. J. Adhes. 79, 69–97 (2003)Google Scholar
167. A. Öchsner, Special Numerical Techniques, in Handbook of Adhesion Technology, ed. by L.M.F. da Silva, A. Öchsner, R.D. Adams (Springer, Heidelberg, 2011)Google Scholar
168. K.C. Pandya, J.G. Williams, Measurement of cohesive zone parameters in tough polyethylene. Polymer Eng. Sci. 40, 1765–1776 (2000)Google Scholar
169. S.K. Panigrahi, B. Pradhan, Three dimensional failure analysis and damage propagation behavior of adhesively bonded single lap joints in laminated FRP composites. J. Reinf. Plast. Compos. 26, 183–201 (2007)Google Scholar
170. T. Pardoen, T. Ferracin, C.M. Landis, F. Delannay, Constraint effects in adhesive joint fracture. J. Mech. Phys. Solids 53, 1951–1983 (2005)
172. Z. Petrossian, M.R. Wisnom, Prediction of delamination initiation and growth from discontinuous plies using interface elements. Compos. Part A 29, 503–515 (1998)Google Scholar
173. A.M.G. Pinto, A.G. Magalhães, R.D.S.G. Campilho, M.F.S.F. de Moura, A.P.M. Baptista, Single-lap joints of similar and dissimilar adherends bonded with an acrylic adhesive. J. Adhes. 85, 351–376 (2009)Google Scholar
174. V.P. Premchand, K.S. Sajikumar, Fracture analysis in adhesive bonded joints with centre crack. NCTT09 10th national conference on technological trends, Trivandrum, India (2009)Google Scholar
175. M. Quaresimin, M. Ricotta, Life prediction of bonded joints in composite materials. Int. J. Fatigue 28, 1166–1176 (2006)
176. R.S. Raghava, R. Cadell, G.S.Y. Yeh, The macroscopic yield behavior of polymers. J. Mater. Sci. 8, 225–232 (1973)Google Scholar
177. P.G.S. Raghavan, A continuum damage mechanics model for unidirectional composites undergoing interfacial de-bonding. Mech. Mater. 37, 955–977 (2005)Google Scholar
178. J.R. Rice, G.F. Rosengren, Plane strain deformation near a crack tip in a powerlaw hardening material. J. Mech. Phys. Solids 16, 1–12 (1968)
179. M. Ridha, V.B.C. Tan, T.E. Tay, Traction-separation laws for progressive failure of a bonded scarf repair of composite panel. Compos. Struct. 93, 1239–1245 (2010)Google Scholar
180. P. Robinson, U. Galvanetto, D. Tumino, G. Bellucci, D. Violeau, Numerical simulation of fatigue-driven delamination using interface elements. Int. J. Numer. Methods Eng. 63, 1824–1848 (2005)
181. K.L. Roe, T. Siegmund, An irreversible cohesive zone model for interface fatigue crack growth simulation. Eng. Fract. Mech. 70, 209–232 (2003)Google Scholar
182. M. Sabsabi, E. Giner, F.J. Fuenmayor, Experimental fatigue testing of a fretting complete contact and numerical life correlation using X-FEM. Int. J. Fatigue 33, 811–822 (2011)Google Scholar
183. N.K. Salgado, M.H. Aliabad, The boundary element analysis of cracked stiffened sheets, renforced by adhesively bonded patches. Int. J. Numer. Meth. Eng. 42, 195–217 (1998)
184. E.M. Sampaio, F.L. Bastian, H.S.C. Mattos, A simple continuum damage model for adhesively bonded butt joints. Mech. Res. Commun. 31, 443–449 (2004)
185. C. Schuecker, B.D. Davidson, Effect of friction on the perceived mode II delamination toughness from three and four point bend end notched flexure tests. ASTM STP 1383, 334–344 (2000)Google Scholar
186. K. Shahin, F. Taheri, The strain energy release rates in adhesively bonded balanced and unbalanced specimens and lap joints. Int. J. Solids Struct. 45, 6284–6300 (2008)
187. V. Shenoy, I.A. Ashcroft, G.W. Critchlow, A.D. Crocombe, M.M. Abdel Wahab, An investigation into crack initiation and propagation behaviour of bonded single-lap joints using backface strain. Int. J. Adhes. Adhes. 29, 361–371 (2009)Google Scholar
188. V. Shenoy, I.A. Ashcroft, G.W. Critchlow, A.D. Crocombe, Unified methodology for the prediction of the fatigue behavior of adhesively bonded joints. Int. J. Fatigue 32, 1278–1288 (2010a)Google Scholar
189. V. Shenoy, I.A. Ashcroft, G.W. Critchlow, A.D. Crocombe, Fracture mechanics and damage mechanics based fatigue lifetime prediction of adhesively bonded joints subjected to variable amplitude fatigue. Eng. Fract. Mech. 77, 1073–1090 (2010b)Google Scholar
190. A.G. Solana, A.D. Crocombe, I.A. Ashcroft, Fatigue life and backface strain predictions in adhesively bonded joints. Int. J. Adhes. Adhes. 30, 36–42 (2010)Google Scholar
191. S.H. Song, G.H. Paulino, W.G. Buttlar, A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material. Eng. Fract. Mech. 73, 2829–2848 (2006)Google Scholar
192. B.F. Sørensen, Cohesive law and notch sensitivity of adhesive joints. Acta Mater. 50, 1053–1061 (2002)Google Scholar
193. B.F. Sørensen, T.K. Jacobsen, Determination of cohesive laws by the J integral approach. Eng. Fract. Mech. 70, 1841–1858 (2003)Google Scholar
194. S. Srinivas (1975) Analysis of bonded joints. NASA Technical Note, NASA TN D-7855Google Scholar
195. N. Sukumar, J.H. Prevost, Modeling quasi-static crack growth with the extended finite element method part i: computer implementation. Int. J. Solids Struct. 40, 7513–7537 (2003)
196. N. Sukumar, N. Moës, B. Moran, T. Belytschko, Extended finite element method for three-dimensional crack modeling. Int. J. Numer. Meth. Eng. 48, 1549–1570 (2000)
197. V. Tamuzs, S. Tarasovs, U. Vilks, Delamination properties of translaminar-reinforced composites. Compos. Sci. Technol. 63, 1423–1431 (2003)Google Scholar
198. D. Tumino, F. Cappello, Simulation of fatigue delamination growth in composites with different mode mixtures. J. Compos. Mater. 41, 2415–2441 (2007)Google Scholar
199. M.J. Turner, R.W. Clough, H.C. Martin, J.L. Topp, Stiffness and deflection analysis of complex structures. J. Aerosp. Sci. 23, 805–823 (1956)
200. A. Turon, J. Costa, P.P. Camanho, C.G. Dàvila, Simulation of delamination in composites under high-cycle fatigue. Compos. Part A 38, 2270–2282 (2007a)Google Scholar
201. A. Turon, C.G. Dávila, P.P. Camanho, J. Costa, An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Eng. Fract. Mech. 74, 1665–1682 (2007b)Google Scholar
202. V. Tvergaard, J.W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J. Mech. Phys. Solids 40, 1377–1397 (1992)
203. V. Tvergaard, J.W. Hutchinson, The influence of plasticity on the mixed-mode interface toughness. J. Mech. Phys. Solids 41, 1119–1135 (1993)
204. V.I. Tvergaard, A. Needleman, Analysis of cup one fracture in round tensile bar. Acta Metall. 32, 157–169 (1984)Google Scholar
205. M. Vable, Stress analysis of bonded joints by boundary element method, in Modeling of Adhesively Bonded Joints, ed. by L.F.M. da Silva, A. Öchsner (Springer, Heidelberg, 2008)Google Scholar
207. O. Volkersen, Die nietkraftoerteilung in zubeanspruchten nietverbindungen mit konstanten loschonquerschnitten. Luftfahrtforschung 15, 41–47 (1938)Google Scholar
208. G.Z. Voyiadjis, P.I. Kattan, Damage Mechanics (Marcell Dekker, New York, 2005)
209. M.M.A. Wahab, I.A. Ashcroft, A.D. Crocombe, S.J. Shaw, Prediction of fatigue thresholds in adhesively bonded joints using damage mechanics and fracture mechanics. J. Adhes. Sci. Technol. 15, 763–781 (2001)Google Scholar
210. M.M.A. Wahab, I.A. Ashcroft, A.D. Crocombe, P.A. Smith, Numerical prediction of fatigue crack propagation lifetime in adhesively bonded structures. Int. J. Fatigue 24, 705–709 (2002)
211. R.X. Wang, J. Cui, A.N. Sinclair, J.K. Spelt, Strength of adhesive joints with adherend yielding: I. Analytical model. J. Adhes. 79, 23–48 (2003)Google Scholar
212. Y. Wei, J.W. Hutchinson, Nonlinear delamination mechanics for thin films. J. Mech. Phys. Solids 45, 1137–1159 (1997)
213. Y. Wei, J.W. Hutchinson, Interface strength, work of adhesion and plasticity in the peel test. Int. J. Fract. 93, 315–333 (1999)Google Scholar
214. P.H. Wen, M.H. Aliabadi, A. Young, Boundary element analysis of flat cracked panels with adhesively bonded patches. Eng. Fract. Mech. 69, 2129–2146 (2002)Google Scholar
215. P.H. Wen, M.H. Aliabadi, A. Young, Boundary element analysis of curved cracked panels with adhesively bonded patches. Int. J. Numer. Methods Eng. 58, 43–61 (2003)
216. M.L. Williams, The stresses around a fault or crack in dissimilar media. Bull. Seism. Soc. Am. 49, 199–204 (1959)Google Scholar
217. D. Xie, A.M. Waas, Discrete cohesive zone model for mixed-mode fracture using finite element analysis. Eng. Fract. Mech. 73, 1783–1796 (2006)Google Scholar
218. D. Xie, A.G. Salvi, C. Sun, A.M. Waas, A. Caliskan, Discrete cohesive zone model to simulate static fracture in 2D triaxially braided carbon fiber composites. J. Compos. Mater. 40, 2025–2046 (2006)Google Scholar
219. W. Xu, G. Li, Finite difference three-dimensional solution of stresses in adhesively bonded composite tubular joint subjected to torsion. Int. J. Adhes. Adhes. 30, 191–199 (2010)Google Scholar
220. Y. Xu, H. Yuan, Computational analysis of mixed-mode fatigue crack growth in quasi-brittle materials using extended finite element methods. Eng. Fract. Mech. 76, 165–181 (2009)Google Scholar
221. J.Q. Xu, Y.H. Liu, X.G. Wang, Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients. Eng. Fract. Mech. 63, 775–790 (1999)Google Scholar
222. C. Yan, Y.W. Mai, L. Ye, Effect of bond thickness on fracture behaviour in adhesive joints. J. Adhes. 75, 27–44 (2001a)Google Scholar
223. C. Yan, Y.W. Mai, Q. Yuan, L. Ye, J. Sun, Effects of substrate materials on fracture toughness measurement in adhesive joints. Int. J. Mech. Sci. 43, 2091–2102 (2001b)
224. Q.D. Yang, M.D. Thouless, Mixed-mode fracture analyses of plastically deforming adhesive joints. Int. J. Fract. 110, 175–187 (2001)Google Scholar
225. Q.D. Yang, M.D. Thouless, S.M. Ward, Numerical simulations of adhesively-bonded beams failing with extensive plastic deformation. J. Mech. Phys. Solids 47, 1337–1353 (1999)
226. Q.D. Yang, M.D. Thouless, S.M. Ward, Analysis of the symmetrical 90º peel test with extensive plastic deformation. J. Adhes. 72, 115–132 (2000)Google Scholar
227. Q.D. Yang, M.D. Thouless, S.M. Ward, Elastic-plastic mode-II fracture of adhesive joints. Int. J. Solids Struct. 38, 3251–3262 (2001)
228. H. Yoshihara, Simple estimation of critical stress intensity factors of wood by tests with double cantilever beam and three-point end-notched flexure. Holzforschung 61, 182–189 (2007)
229. A. Young, D.J. Cartwright, D.P. Rooke, Boundary element method for analysing repair patches on cracked finite sheets. Aeronaut. J. 92, 416–421 (1988)Google Scholar
230. A. Young, D.P. Rooke, D.J. Cartwright, Analysis of patched and stiffened cracked panels using the boundary element method. Int. J. Solids Struct. 29, 2201–2216 (1992)
231. X. Zhao, R.D. Adams, L.F.M. da Silva, Single lap joints with rounded adherend corners: Stress and strain analysis. J. Adhes. Sci. Technol. 25, 819–836 (2011a)Google Scholar
232. X. Zhao, R.D. Adams, L.F.M. da Silva, Single lap joints with rounded adherend corners: experimental results and strength prediction. J. Adhes. Sci. Technol. 25, 837–856 (2011b)Google Scholar
233. Y. Zhu, K.M. Liechti, K. Ravi-Chandar, Direct extraction of rate-dependent traction-separation laws for polyurea/steel interfaces. Int. J. Solids Struct. 46, 31–51 (2009)Google Scholar
234. O.C. Zienkiewicz, Y.K. Cheung, The Finite Element Method in Structural and Continuum Mechanics (McGraw-Hill, London, 1967)
235. O.C. Zienkiewicz, R. Taylor, The Finite Element Method: Vol 2, Solid Mechanics, 5th edn. (Butterworth-Heinemann, Oxford, 2001)Google Scholar
236. G.P. Zou, K. Shalin, F. Taheri, An analytical solution for the analysis of symmetric composite adhesively bonded joints. Compos. Struct. 65, 499–510 (2004)Google Scholar

## Authors and Affiliations

• Lucas F. M. da Silva
• 1
Email author
• Raul D. S. G. Campilho
• 2