# American Institute of Mathematical Sciences

April  2013, 6(2): 591-610. doi: 10.3934/dcdss.2013.6.591

## A quasistatic mixed-mode delamination model

 1 Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8 2 Group of Elasticity and Strength of Materials, Dept. of Continuum Mechanics, School of Engineering, University of Seville, Camino de los Descubrimientos s/n, ES-41092 Sevilla, Spain, Spain

Received  June 2011 Revised  November 2011 Published  November 2012

The quasistatic rate-independent evolution of delamination in the so-called mixed-mode, i.e. distinguishing opening (mode I) from shearing (mode II), devised in [45], is described in detail and rigorously analysed as far as existence of the so-called energetic solutions concerns. The model formulated at small strains uses a delamination parameter of Frémond's type combined with a concept of interface plasticity, and is associative in the sense that the dissipative force driving delamination has a potential which depends in a 1-homogeneous way only on rates of internal parameters. A sample numerical simulation documents that this model can really produce mode-mixity-sensitive delamination.
Citation: Tomáš Roubíček, V. Mantič, C. G. Panagiotopoulos. A quasistatic mixed-mode delamination model. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 591-610. doi: 10.3934/dcdss.2013.6.591
##### References:
 [1] E. C. Aifantis, On the microstructural origin of certain inelastic models,, ASME J. Eng. Mater. Technol., 106 (1984), 326. doi: 10.1115/1.3225725. Google Scholar [2] L. Banks-Sills and D. Ashkenazi, A note on fracture criteria for interface fracture,, Intl. J. Fracture, 103 (2000), 177. doi: 10.1023/A:1007612613338. Google Scholar [3] Z. Bažant and M. Jirásek, Nonlocal integral formulations of plasticity and damage: Survey of progress,, J. Engrg. Mech., 128 (2002), 1119. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119). Google Scholar [4] S. Bennati, M. Colleluori, D. Corigliano and P. Valvo, An enhanced beam-theory model of the asymmetric double cantilever beam (adcb) test for composite laminates,, Composites Science and Technology, 69 (2009), 1735. doi: 10.1016/j.compscitech.2009.01.019. Google Scholar [5] M. A. Biot, Thermoelasticity and irreversible thermodynamics,, J. Appl. Phys., 27 (1956), 240. doi: 10.1063/1.1722351. Google Scholar [6] M. A. Biot, "Mechanics of Incremental Deformations,", Wiley, (1965). doi: 10.1063/1.3047001. Google Scholar [7] E. Bonetti, G. Bonfanti and R. Rossi, Well-posedness and long-time behaviour for a model of contact with adhesion,, Indiana Univ. Math. J., 56 (2007), 2787. doi: 10.1512/iumj.2007.56.3079. Google Scholar [8] E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion,, Math. Meth. Appl. Sci., 31 (2008), 1029. doi: 10.1002/mma.957. Google Scholar [9] B. Bourdin, G. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture,, J. Mech. Phys. Solids, 48 (2000), 797. doi: 10.1016/S0022-5096(99)00028-9. Google Scholar [10] P. Cornetti and A. Carpinteri, Modelling the FRP-concrete delamination by means of an exponential softening law,, Engineering Structures, 33 (2011), 1988. doi: 10.1016/j.engstruct.2011.02.036. Google Scholar [11] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Rational Mech. Anal., 176 (2005), 165. doi: 10.1007/s00205-004-0351-4. Google Scholar [12] A. Evans, M. Rühle, B. Dalgleish and P. Charalambides, The fracture energy of bimaterial interfaces,, Metallurgical Transactions A, 21A (1990), 2419. doi: 10.1007/BF02646986. Google Scholar [13] M. Frémond, Dissipation dans l'adhrence des solides,, C. R. Acad. Sci., 300 (1985), 709. Google Scholar [14] M. Frémond, Contact with adhesion,, in, (1988). Google Scholar [15] M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002). Google Scholar [16] M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power,, Internat. J. Solids Structures, 33 (1996), 1083. doi: 10.1016/0020-7683(95)00074-7. Google Scholar [17] E. Fried and M. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales,, Arch. Rational Mech. Anal., 182 (2006), 513. doi: 10.1007/s00205-006-0015-7. Google Scholar [18] A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Royal Soc. London Ser. A. Math. Phys. Eng. Sci., 221 (1921), 163. doi: 10.1098/rsta.1921.0006. Google Scholar [19] W. Han and B. D. Reddy, "Plasticity (Mathematical Theory and Numerical Analysis),", Springer-Verlag, (1999). Google Scholar [20] J. W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials,, Advances in Applied Mechanics, 29 (1992), 63. doi: 10.1016/S0065-2156(08)70164-9. Google Scholar [21] M. Jirásek and J. Zeman, Localization study of non-local energetic damage model,, , (2008). Google Scholar [22] D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation,, Math. Models Meth. Appl. Sci. (M$^3$AS), 18 (2008), 1529. doi: 10.1142/S0218202508003121. Google Scholar [23] M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem,, Math. Mechanics Solids, 11 (2006), 423. Google Scholar [24] K. Liechti and Y. Chai, Asymmetric shielding in interfacial fracture under in-plane shear,, J. Appl. Mech., 59 (1992), 295. doi: 10.1115/1.2899520. Google Scholar [25] V. Mantič, Discussion on the reference length and mode mixity for a bimaterial interface,, J. Engr. Mater. Technology, 130 (2008), 045501. Google Scholar [26] A. Mielke, Evolution in rate-independent systems (Ch. 6),, in, (2005), 461. doi: 10.1016/S1874-5717(06)80009-5. Google Scholar [27] A. Mielke, Differential, energetic and metric formulations for rate-independent processes,, in, (2010), 87. doi: 10.1037/a0020489. Google Scholar [28] A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity,, Math. Models Meth. Appl. Sci., 16 (2006), 177. doi: 10.1142/S021820250600111X. Google Scholar [29] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity,, Math. Model. Numer. Anal. (M2AN), 43 (2009), 399. doi: 10.1051/m2an/2009009. Google Scholar [30] A. Mielke, T. Roubíček and M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains,, J. Elasticity, 109 (2012), 235. doi: 10.1007/s10659-012-9379-0. Google Scholar [31] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. Part. Diff. Eqns., 31 (2008), 387. doi: 10.1007/s00526-007-0119-4. Google Scholar [32] A. Mielke, T. Roubíček and J. Zeman, Complete damage in elastic and viscoelastic media and its energetics,, Computer Methods Appl. Mech. Engr., 199 (2009), 1242. doi: 10.1016/j.cma.2009.09.020. Google Scholar [33] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, in, (1999), 117. Google Scholar [34] A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151. Google Scholar [35] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Arch. Rational Mech. Anal., 162 (2002), 137. doi: 10.1007/s002050200194. Google Scholar [36] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion,, Math. Models Methods Appl. Sci., 18 (2008), 1895. doi: 10.1142/S0218202508003236. Google Scholar [37] C. Panagiotopoulos, "Open BEM Project,", 2010. Available from: , (). Google Scholar [38] C. Panagiotopoulos, V. Mantič and T. Roubíček, BEM solution of delamination problems using an interface damage and plasticity model,, Computational Mechanics, (). Google Scholar [39] F. París and J. Cañas, "Boundary Element Method,", Oxford University Press, (1997). Google Scholar [40] P. Podio-Guidugli and G. Vergara Caffarelli, Surface interaction potentials in elasticity,, Arch. Rat. Mech. Anal., 109 (1990), 343. doi: 10.1007/BF00380381. Google Scholar [41] N. Point, Unilateral contact with adherence,, Math. Methods Appl. Sci., 10 (1988), 367. doi: 10.1002/mma.1670100403. Google Scholar [42] N. Point and E. Sacco, A delamination model for laminated composites,, Math. Methods Appl. Sci., 33 (1996), 483. Google Scholar [43] N. Point and E. Sacco, Mathematical properties of a delamination model,, Math. Comput. Modelling, 28 (1998), 359. doi: 10.1016/S0895-7177(98)00127-7. Google Scholar [44] R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis,, Preprints: No.26/2011 at Univ. Brescia, (1110). Google Scholar [45] T. Roubíček, M. Kružík and J. Zeman, "Delamination and Adhesive Contact Models and Their Mathematical Analysis and Numerical Treatment,", in, (2013), 978. Google Scholar [46] T. Roubíček, L. Scardia and C. Zanini, Quasistatic delamination problem,, Cont. Mech. Thermodynam, 21 (2009), 223. Google Scholar [47] J. Simo and T. Hughes, "Computational Inelasticity,", Springer, (1998). Google Scholar [48] J. Swadener, K. Liechti and A. deLozanne, The intrinsic toughness and adhesion mechanism of a glass/epoxy interface,, J. Mech. Phys. Solids, 47 (1999), 223. doi: 10.1016/S0022-5096(98)00084-2. Google Scholar [49] L. Távara, V. Mantič, E. Graciani, J. Cañas and F. París, Analysis of a crack in a thin adhesive layer between orthotropic materials: An application to composite interlaminar fracture toughness test,, CMES - Computer Modeling in Engineering and Sciences, 58 (2010), 247. Google Scholar [50] L. Távara, V. Mantič, E. Graciani and F. París, BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model,, Engineering Analysis with Boundary Elements, 35 (2011), 207. Google Scholar [51] M. Thomas, "Rate-Independent Damage Processes in Nonlinearly Elastic Materials,", PhD thesis, (2010). Google Scholar [52] M. Thomas, Quasistatic damage evolution with spatial BV-regularization,, Disc. Cont. Dynam. Syst. - S, 6 (2013), 235. doi: 10.1097/FPC.0b013e32833d1011. Google Scholar [53] M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain - Existence and regularity results,, Z. angew. Math. Mech. (ZAMM), 90 (2010), 88. Google Scholar [54] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth,, Boll. Unione Matem. Ital., 2 (2009), 1. Google Scholar [55] R. Toupin, Elastic materials with couple stresses,, Arch. Rat. Mech. Anal., 11 (1962), 385. doi: 10.1007/BF00253945. Google Scholar [56] V. Tvergaard and J. Hutchinson, The influence of plasticity on mixed mode interface toughness,, J. Mech. Phys. Solids, 41 (1993), 1119. doi: 10.1016/0022-5096(93)90057-M. Google Scholar

show all references

##### References:
 [1] E. C. Aifantis, On the microstructural origin of certain inelastic models,, ASME J. Eng. Mater. Technol., 106 (1984), 326. doi: 10.1115/1.3225725. Google Scholar [2] L. Banks-Sills and D. Ashkenazi, A note on fracture criteria for interface fracture,, Intl. J. Fracture, 103 (2000), 177. doi: 10.1023/A:1007612613338. Google Scholar [3] Z. Bažant and M. Jirásek, Nonlocal integral formulations of plasticity and damage: Survey of progress,, J. Engrg. Mech., 128 (2002), 1119. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119). Google Scholar [4] S. Bennati, M. Colleluori, D. Corigliano and P. Valvo, An enhanced beam-theory model of the asymmetric double cantilever beam (adcb) test for composite laminates,, Composites Science and Technology, 69 (2009), 1735. doi: 10.1016/j.compscitech.2009.01.019. Google Scholar [5] M. A. Biot, Thermoelasticity and irreversible thermodynamics,, J. Appl. Phys., 27 (1956), 240. doi: 10.1063/1.1722351. Google Scholar [6] M. A. Biot, "Mechanics of Incremental Deformations,", Wiley, (1965). doi: 10.1063/1.3047001. Google Scholar [7] E. Bonetti, G. Bonfanti and R. Rossi, Well-posedness and long-time behaviour for a model of contact with adhesion,, Indiana Univ. Math. J., 56 (2007), 2787. doi: 10.1512/iumj.2007.56.3079. Google Scholar [8] E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion,, Math. Meth. Appl. Sci., 31 (2008), 1029. doi: 10.1002/mma.957. Google Scholar [9] B. Bourdin, G. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture,, J. Mech. Phys. Solids, 48 (2000), 797. doi: 10.1016/S0022-5096(99)00028-9. Google Scholar [10] P. Cornetti and A. Carpinteri, Modelling the FRP-concrete delamination by means of an exponential softening law,, Engineering Structures, 33 (2011), 1988. doi: 10.1016/j.engstruct.2011.02.036. Google Scholar [11] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Rational Mech. Anal., 176 (2005), 165. doi: 10.1007/s00205-004-0351-4. Google Scholar [12] A. Evans, M. Rühle, B. Dalgleish and P. Charalambides, The fracture energy of bimaterial interfaces,, Metallurgical Transactions A, 21A (1990), 2419. doi: 10.1007/BF02646986. Google Scholar [13] M. Frémond, Dissipation dans l'adhrence des solides,, C. R. Acad. Sci., 300 (1985), 709. Google Scholar [14] M. Frémond, Contact with adhesion,, in, (1988). Google Scholar [15] M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002). Google Scholar [16] M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power,, Internat. J. Solids Structures, 33 (1996), 1083. doi: 10.1016/0020-7683(95)00074-7. Google Scholar [17] E. Fried and M. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales,, Arch. Rational Mech. Anal., 182 (2006), 513. doi: 10.1007/s00205-006-0015-7. Google Scholar [18] A. Griffith, The phenomena of rupture and flow in solids,, Philos. Trans. Royal Soc. London Ser. A. Math. Phys. Eng. Sci., 221 (1921), 163. doi: 10.1098/rsta.1921.0006. Google Scholar [19] W. Han and B. D. Reddy, "Plasticity (Mathematical Theory and Numerical Analysis),", Springer-Verlag, (1999). Google Scholar [20] J. W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials,, Advances in Applied Mechanics, 29 (1992), 63. doi: 10.1016/S0065-2156(08)70164-9. Google Scholar [21] M. Jirásek and J. Zeman, Localization study of non-local energetic damage model,, , (2008). Google Scholar [22] D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation,, Math. Models Meth. Appl. Sci. (M$^3$AS), 18 (2008), 1529. doi: 10.1142/S0218202508003121. Google Scholar [23] M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem,, Math. Mechanics Solids, 11 (2006), 423. Google Scholar [24] K. Liechti and Y. Chai, Asymmetric shielding in interfacial fracture under in-plane shear,, J. Appl. Mech., 59 (1992), 295. doi: 10.1115/1.2899520. Google Scholar [25] V. Mantič, Discussion on the reference length and mode mixity for a bimaterial interface,, J. Engr. Mater. Technology, 130 (2008), 045501. Google Scholar [26] A. Mielke, Evolution in rate-independent systems (Ch. 6),, in, (2005), 461. doi: 10.1016/S1874-5717(06)80009-5. Google Scholar [27] A. Mielke, Differential, energetic and metric formulations for rate-independent processes,, in, (2010), 87. doi: 10.1037/a0020489. Google Scholar [28] A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity,, Math. Models Meth. Appl. Sci., 16 (2006), 177. doi: 10.1142/S021820250600111X. Google Scholar [29] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity,, Math. Model. Numer. Anal. (M2AN), 43 (2009), 399. doi: 10.1051/m2an/2009009. Google Scholar [30] A. Mielke, T. Roubíček and M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains,, J. Elasticity, 109 (2012), 235. doi: 10.1007/s10659-012-9379-0. Google Scholar [31] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. Part. Diff. Eqns., 31 (2008), 387. doi: 10.1007/s00526-007-0119-4. Google Scholar [32] A. Mielke, T. Roubíček and J. Zeman, Complete damage in elastic and viscoelastic media and its energetics,, Computer Methods Appl. Mech. Engr., 199 (2009), 1242. doi: 10.1016/j.cma.2009.09.020. Google Scholar [33] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, in, (1999), 117. Google Scholar [34] A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151. Google Scholar [35] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Arch. Rational Mech. Anal., 162 (2002), 137. doi: 10.1007/s002050200194. Google Scholar [36] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion,, Math. Models Methods Appl. Sci., 18 (2008), 1895. doi: 10.1142/S0218202508003236. Google Scholar [37] C. Panagiotopoulos, "Open BEM Project,", 2010. Available from: , (). Google Scholar [38] C. Panagiotopoulos, V. Mantič and T. Roubíček, BEM solution of delamination problems using an interface damage and plasticity model,, Computational Mechanics, (). Google Scholar [39] F. París and J. Cañas, "Boundary Element Method,", Oxford University Press, (1997). Google Scholar [40] P. Podio-Guidugli and G. Vergara Caffarelli, Surface interaction potentials in elasticity,, Arch. Rat. Mech. Anal., 109 (1990), 343. doi: 10.1007/BF00380381. Google Scholar [41] N. Point, Unilateral contact with adherence,, Math. Methods Appl. Sci., 10 (1988), 367. doi: 10.1002/mma.1670100403. Google Scholar [42] N. Point and E. Sacco, A delamination model for laminated composites,, Math. Methods Appl. Sci., 33 (1996), 483. Google Scholar [43] N. Point and E. Sacco, Mathematical properties of a delamination model,, Math. Comput. Modelling, 28 (1998), 359. doi: 10.1016/S0895-7177(98)00127-7. Google Scholar [44] R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis,, Preprints: No.26/2011 at Univ. Brescia, (1110). Google Scholar [45] T. Roubíček, M. Kružík and J. Zeman, "Delamination and Adhesive Contact Models and Their Mathematical Analysis and Numerical Treatment,", in, (2013), 978. Google Scholar [46] T. Roubíček, L. Scardia and C. Zanini, Quasistatic delamination problem,, Cont. Mech. Thermodynam, 21 (2009), 223. Google Scholar [47] J. Simo and T. Hughes, "Computational Inelasticity,", Springer, (1998). Google Scholar [48] J. Swadener, K. Liechti and A. deLozanne, The intrinsic toughness and adhesion mechanism of a glass/epoxy interface,, J. Mech. Phys. Solids, 47 (1999), 223. doi: 10.1016/S0022-5096(98)00084-2. Google Scholar [49] L. Távara, V. Mantič, E. Graciani, J. Cañas and F. París, Analysis of a crack in a thin adhesive layer between orthotropic materials: An application to composite interlaminar fracture toughness test,, CMES - Computer Modeling in Engineering and Sciences, 58 (2010), 247. Google Scholar [50] L. Távara, V. Mantič, E. Graciani and F. París, BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model,, Engineering Analysis with Boundary Elements, 35 (2011), 207. Google Scholar [51] M. Thomas, "Rate-Independent Damage Processes in Nonlinearly Elastic Materials,", PhD thesis, (2010). Google Scholar [52] M. Thomas, Quasistatic damage evolution with spatial BV-regularization,, Disc. Cont. Dynam. Syst. - S, 6 (2013), 235. doi: 10.1097/FPC.0b013e32833d1011. Google Scholar [53] M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain - Existence and regularity results,, Z. angew. Math. Mech. (ZAMM), 90 (2010), 88. Google Scholar [54] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth,, Boll. Unione Matem. Ital., 2 (2009), 1. Google Scholar [55] R. Toupin, Elastic materials with couple stresses,, Arch. Rat. Mech. Anal., 11 (1962), 385. doi: 10.1007/BF00253945. Google Scholar [56] V. Tvergaard and J. Hutchinson, The influence of plasticity on mixed mode interface toughness,, J. Mech. Phys. Solids, 41 (1993), 1119. doi: 10.1016/0022-5096(93)90057-M. Google Scholar
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