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December  2017, 10(6): 1487-1517. doi: 10.3934/dcdss.2017077

Cohesive zone-type delamination in visco-elasticity

Marita Thomas 1,, and  Chiara Zanini 2,

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39,10117 Berlin, Germany
2. 

Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

* Corresponding author: Marita Thomas.

To the occasion of the 60th anniversary of Tomáš Roubíček

Received  September 2016 Revised  December 2016 Published  June 2017

Fund Project: This research has been carried out during several research stays of MT at Politecnico di Torino and of CZ at WIAS, Berlin. The hospitality of the two institutions is gratefully acknowledged. MT also acknowledges the partial financial support by GNAMPA 2014 and by the DFG Project “Finite element approximation of functions of bounded variation and application to models of damage, fracture, and plasticity” within the DFG Priority Programme SPP 1748 “Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis.” CZ acknowledges the partial financial support through the ERC Project No. 267802, “Analysis of Multiscale Systems Driven by Functionals”. CZ is also a member of the Progetto di Ricerca GNAMPA 2016 “Analisi di processi inelastici nella meccanica dei solidi e delle cellule: proprietà fini delle soluzioni” and of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)..

We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [32], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.

Due to the presence of multivalued and unbounded operators featuring non-penetration and the 'memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [41] and refined in [38].

Keywords: Cohesive zone delamination, weak formulation, rate-independent processes, semistable energetic solutions, non-smooth constraint, gradient systems, dynamics, irreversibility.
Mathematics Subject Classification: Primary: 35A15, 35Q74, 74H20, 74C10, 49J53, 49J45; Secondary: 74C05.
Citation: Marita Thomas, Chiara Zanini. Cohesive zone-type delamination in visco-elasticity. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1487-1517. doi: 10.3934/dcdss.2017077
References:
[1]

L. Adam, J. Outrata and T. Roubíček, Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains, A Journal of Mathematical Programming and Operations Research Latest Articles, (2015), 1-25.  doi: 10.1080/02331934.2015.1111364.

[2]

R. Alessi, J.-J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal., 214 (2014), 575-615.  doi: 10.1007/s00205-014-0763-8.

[3]

S. Almi, Energy release rate and quasistatic evolution via vanishing viscosity in a cohesive fracture model with an activation threshold, ESAIM: Control Optim. Calc. Var. Published online.

[4]

M. Artina, F. Cagnetti, M. Fornasier and F. Solombrino, Linearly constrained evolutions of critical points and an application to cohesive fractures, Math. Models Methods Appl. Sci., 27 (2017), 231-290.  doi: 10.1142/S0218202517500014.

[5]

H. Attouch, Variational Convergence for Functions and Operators Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[6]

G. Barenblatt, The mathematical theory of equilibrium of cracks in brittle fracture, Adv. Appl. Mech., 7 (1962), 55-129. 

[7]

E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064.  doi: 10.1002/mma.957.

[8]

E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731.  doi: 10.1088/0951-7715/22/11/007.

[9]

E. Bonetti, E. Rocca, R. Scala and G. Schimperna, On the strongly damped wave equation with constraint, WIAS-Preprint 2094.

[10]

G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems, J. Reine Angew. Math., 458 (1995), 1-18.  doi: 10.1515/crll.1995.458.1.

[11]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert North-Holland Publishing Co. , Amsterdam-London; American Elsevier Publishing Co. , Inc. , New York, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).

[12]

F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071.  doi: 10.1142/S0218202508002942.

[13]

F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measures approach, ESAIM Control Optim. Calc. Var., 17 (2011), 1-27.  doi: 10.1051/cocv/2009037.

[14]

V. Crismale, G. Lazzaroni and G. Orlando, Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue, SISSA-Preprint SISSA 40/2016/MATE 1508. 02965.

[15]

G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rat. Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.

[16]

G. Dal Maso, G. Orlando and R. Toader, Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation, Adv. Cal. Var., 10 (2017), 183-207.  doi: 10.1515/acv-2015-0036.

[17]

G. Dal Maso, G. Orlando and R. Toader, Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: The anticline case Calc. Var. Partial Differential Equations 55 (2016), 39pp. doi: 10.1007/s00526-016-0981-z.

[18]

G. Dal Maso and C. Zanini, Quasi-static crack growth for a cohesive zone model with prescribed crack path, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 253-279.  doi: 10.1017/S030821050500079X.

[19]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[20]

D. Dugdale, Yielding of steel sheets containing clits, J. Mech. Phys. Solids, 8 (1960), 100-104. 

[21]

M. Fr´emond, Contact with adhesion, in Topics in Nonsmooth Mechanics (eds. J. Moreau, P. Panagiotopoulos and G. Strang), Birkhäuser, 1988,157–186.

[22]

M. Frémond, Non-Smooth Thermomechanics Springer-Verlag Berlin Heidelberg, 2002.

[23]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems vol. 6 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam, 1979, Translated from the Russian by Karol Makowski.

[24]

A. Ioffe, On lower semicontinuity of integral functionals. I, SIAM J. Control Optimization, 15 (1977), 521-538.  doi: 10.1137/0315035.

[25]

M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447.  doi: 10.1177/1081286505046482.

[26]

R. Kregting, Cohesive Zone Models Towards a Robust Implementation of Irreversible Behavior Technical Report MT05. 11, TU Eindhoven, Materials Technology, 2005.

[27]

M. Kružík, C. Panagiotopoulos and T. Roubíček, Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment, Math. Mech. Solids, 20 (2015), 582-599.  doi: 10.1177/1081286513507942.

[28]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.

[29]

M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Functional Analysis, 33 (1979), 217-229.  doi: 10.1016/0022-1236(79)90113-7.

[30]

A. Mielke and T. Roubíček, Rate-independent Systems: Theory and Application vol. 193 of Applied Mathematical Sciences, Springer, 2015. doi: 10.1007/978-1-4939-2706-7.

[31]

A. Mielke, T. Roubíček and M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, J. Elasticity, 109 (2012), 235-273.  doi: 10.1007/s10659-012-9379-0.

[32]

M. Ortiz and A. Pandolfi, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, International Journal for Numerical Methods in Engineering, 44 (1999), 1267-1282. 

[33]

C. Panagiotopoulos, V. Mantič and T. Roubíček, Two adhesive-contact models for quasistatic mixed-mode delamination problems Mathematics and Computers in Simulation 2016. doi: 10.1016/j.matcom.2016.10.004.

[34]

K. Park and G. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces Applied Mechanics Reviews 64 (2013), 060802. doi: 10.1115/1.4023110.

[35]

J. Rice, Fracture, chapter Mathematical analysis in the mechanics of fracture, , (1968): 191-311. 

[36]

R. Rossi and T. Roubíček, Thermodynamics and analysis of rate-independent adhesive contact at small strains, Nonlinear Anal., 74 (2011), 3159-3190.  doi: 10.1016/j.na.2011.01.031.

[37]

R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound., 15 (2013), 1-37.  doi: 10.4171/IFB/293.

[38]

R. Rossi and M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, SIAM J. Math. Anal., 49 (2017), 1419-1494.  doi: 10.1137/15M1051567.

[39]

R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, ESAIM Control Optim. Calc. Var., 21 (2015), 1-59.  doi: 10.1051/cocv/2014015.

[40]

R. Rossi and M. Thomas, From adhesive to brittle delamination in visco-elastodynamics Math. Models Methods Appl. Sci. , 2017. doi: 10.1142/S0218202517500257.

[41]

T. Roubíček, Rate-independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862.  doi: 10.1002/mma.1069.

[42]

T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.  doi: 10.1137/080729992.

[43]

T. Roubíček, M. Kružík and J. Zeman, Delamination and adhesive contact models and their mathematical analysis and numerical treatment, in Mathematical methods and models in composites, vol. 5 of Comput. Exp. Methods Struct. , Imp. Coll. Press, London, 2014,349– 400. doi: 10.1142/9781848167858_0009.

[44]

T. Roubíček, V. Mantič and C. Panagiotopoulos, A quasistatic mixed-mode delamination model, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 591-610. 

[45]

T. Roubíček, C. Panagiotopoulos and V. Mantič, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, ZAMM Z. Angew. Math. Mech., 93 (2013), 823-840.  doi: 10.1002/zamm.201200239.

[46]

T. Roubíček, L. Scardia and C. Zanini, Quasistatic delamination problem, Continuum Mech. Thermodynam., 21 (2009), 223-235.  doi: 10.1007/s00161-009-0106-4.

[47]

T. Roubíček, M. Thomas and C. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Anal. Real World Appl., 22 (2015), 645-663.  doi: 10.1016/j.nonrwa.2014.09.011.

[48]

R. Scala, A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint, WIAS-Preprint 2172.

[49]

R. Scala and G. Schimperna, A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints, European J. Appl. Math., 28 (2017), 91-122.  doi: 10.1017/S0956792516000097.

[50]

M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 235-255.  doi: 10.3934/dcdss.2013.6.235.

[51]

M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain: Existence and regularity results, Zeit. angew. Math. Mech., 90 (2010), 88-112.  doi: 10.1002/zamm.200900243.

[52]

R. Vodička, V. Mantič and T. Roubíček, Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model, Meccanica, 49 (2014), 2933-2963.  doi: 10.1007/s11012-014-0045-4.

show all references

To the occasion of the 60th anniversary of Tomáš Roubíček

References:
[1]

L. Adam, J. Outrata and T. Roubíček, Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains, A Journal of Mathematical Programming and Operations Research Latest Articles, (2015), 1-25.  doi: 10.1080/02331934.2015.1111364.

[2]

R. Alessi, J.-J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal., 214 (2014), 575-615.  doi: 10.1007/s00205-014-0763-8.

[3]

S. Almi, Energy release rate and quasistatic evolution via vanishing viscosity in a cohesive fracture model with an activation threshold, ESAIM: Control Optim. Calc. Var. Published online.

[4]

M. Artina, F. Cagnetti, M. Fornasier and F. Solombrino, Linearly constrained evolutions of critical points and an application to cohesive fractures, Math. Models Methods Appl. Sci., 27 (2017), 231-290.  doi: 10.1142/S0218202517500014.

[5]

H. Attouch, Variational Convergence for Functions and Operators Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[6]

G. Barenblatt, The mathematical theory of equilibrium of cracks in brittle fracture, Adv. Appl. Mech., 7 (1962), 55-129. 

[7]

E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064.  doi: 10.1002/mma.957.

[8]

E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731.  doi: 10.1088/0951-7715/22/11/007.

[9]

E. Bonetti, E. Rocca, R. Scala and G. Schimperna, On the strongly damped wave equation with constraint, WIAS-Preprint 2094.

[10]

G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems, J. Reine Angew. Math., 458 (1995), 1-18.  doi: 10.1515/crll.1995.458.1.

[11]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert North-Holland Publishing Co. , Amsterdam-London; American Elsevier Publishing Co. , Inc. , New York, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).

[12]

F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071.  doi: 10.1142/S0218202508002942.

[13]

F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measures approach, ESAIM Control Optim. Calc. Var., 17 (2011), 1-27.  doi: 10.1051/cocv/2009037.

[14]

V. Crismale, G. Lazzaroni and G. Orlando, Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue, SISSA-Preprint SISSA 40/2016/MATE 1508. 02965.

[15]

G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rat. Mech. Anal., 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.

[16]

G. Dal Maso, G. Orlando and R. Toader, Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation, Adv. Cal. Var., 10 (2017), 183-207.  doi: 10.1515/acv-2015-0036.

[17]

G. Dal Maso, G. Orlando and R. Toader, Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: The anticline case Calc. Var. Partial Differential Equations 55 (2016), 39pp. doi: 10.1007/s00526-016-0981-z.

[18]

G. Dal Maso and C. Zanini, Quasi-static crack growth for a cohesive zone model with prescribed crack path, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 253-279.  doi: 10.1017/S030821050500079X.

[19]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[20]

D. Dugdale, Yielding of steel sheets containing clits, J. Mech. Phys. Solids, 8 (1960), 100-104. 

[21]

M. Fr´emond, Contact with adhesion, in Topics in Nonsmooth Mechanics (eds. J. Moreau, P. Panagiotopoulos and G. Strang), Birkhäuser, 1988,157–186.

[22]

M. Frémond, Non-Smooth Thermomechanics Springer-Verlag Berlin Heidelberg, 2002.

[23]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems vol. 6 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam, 1979, Translated from the Russian by Karol Makowski.

[24]

A. Ioffe, On lower semicontinuity of integral functionals. I, SIAM J. Control Optimization, 15 (1977), 521-538.  doi: 10.1137/0315035.

[25]

M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem, Math. Mech. Solids, 11 (2006), 423-447.  doi: 10.1177/1081286505046482.

[26]

R. Kregting, Cohesive Zone Models Towards a Robust Implementation of Irreversible Behavior Technical Report MT05. 11, TU Eindhoven, Materials Technology, 2005.

[27]

M. Kružík, C. Panagiotopoulos and T. Roubíček, Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment, Math. Mech. Solids, 20 (2015), 582-599.  doi: 10.1177/1081286513507942.

[28]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.

[29]

M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Functional Analysis, 33 (1979), 217-229.  doi: 10.1016/0022-1236(79)90113-7.

[30]

A. Mielke and T. Roubíček, Rate-independent Systems: Theory and Application vol. 193 of Applied Mathematical Sciences, Springer, 2015. doi: 10.1007/978-1-4939-2706-7.

[31]

A. Mielke, T. Roubíček and M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, J. Elasticity, 109 (2012), 235-273.  doi: 10.1007/s10659-012-9379-0.

[32]

M. Ortiz and A. Pandolfi, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, International Journal for Numerical Methods in Engineering, 44 (1999), 1267-1282. 

[33]

C. Panagiotopoulos, V. Mantič and T. Roubíček, Two adhesive-contact models for quasistatic mixed-mode delamination problems Mathematics and Computers in Simulation 2016. doi: 10.1016/j.matcom.2016.10.004.

[34]

K. Park and G. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces Applied Mechanics Reviews 64 (2013), 060802. doi: 10.1115/1.4023110.

[35]

J. Rice, Fracture, chapter Mathematical analysis in the mechanics of fracture, , (1968): 191-311. 

[36]

R. Rossi and T. Roubíček, Thermodynamics and analysis of rate-independent adhesive contact at small strains, Nonlinear Anal., 74 (2011), 3159-3190.  doi: 10.1016/j.na.2011.01.031.

[37]

R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound., 15 (2013), 1-37.  doi: 10.4171/IFB/293.

[38]

R. Rossi and M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, SIAM J. Math. Anal., 49 (2017), 1419-1494.  doi: 10.1137/15M1051567.

[39]

R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, ESAIM Control Optim. Calc. Var., 21 (2015), 1-59.  doi: 10.1051/cocv/2014015.

[40]

R. Rossi and M. Thomas, From adhesive to brittle delamination in visco-elastodynamics Math. Models Methods Appl. Sci. , 2017. doi: 10.1142/S0218202517500257.

[41]

T. Roubíček, Rate-independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862.  doi: 10.1002/mma.1069.

[42]

T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.  doi: 10.1137/080729992.

[43]

T. Roubíček, M. Kružík and J. Zeman, Delamination and adhesive contact models and their mathematical analysis and numerical treatment, in Mathematical methods and models in composites, vol. 5 of Comput. Exp. Methods Struct. , Imp. Coll. Press, London, 2014,349– 400. doi: 10.1142/9781848167858_0009.

[44]

T. Roubíček, V. Mantič and C. Panagiotopoulos, A quasistatic mixed-mode delamination model, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 591-610. 

[45]

T. Roubíček, C. Panagiotopoulos and V. Mantič, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, ZAMM Z. Angew. Math. Mech., 93 (2013), 823-840.  doi: 10.1002/zamm.201200239.

[46]

T. Roubíček, L. Scardia and C. Zanini, Quasistatic delamination problem, Continuum Mech. Thermodynam., 21 (2009), 223-235.  doi: 10.1007/s00161-009-0106-4.

[47]

T. Roubíček, M. Thomas and C. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Anal. Real World Appl., 22 (2015), 645-663.  doi: 10.1016/j.nonrwa.2014.09.011.

[48]

R. Scala, A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint, WIAS-Preprint 2172.

[49]

R. Scala and G. Schimperna, A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints, European J. Appl. Math., 28 (2017), 91-122.  doi: 10.1017/S0956792516000097.

[50]

M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 235-255.  doi: 10.3934/dcdss.2013.6.235.

[51]

M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain: Existence and regularity results, Zeit. angew. Math. Mech., 90 (2010), 88-112.  doi: 10.1002/zamm.200900243.

[52]

R. Vodička, V. Mantič and T. Roubíček, Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model, Meccanica, 49 (2014), 2933-2963.  doi: 10.1007/s11012-014-0045-4.

Figure 1.  Typical cohesive laws. Solid lines: curves corresponding to maximal loading envelope, dashed lines: loading -unloading rules.
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