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

Cohesive zone-type delamination in visco-elasticity

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].

Citation: Marita Thomas, Chiara Zanini. Cohesive zone-type delamination in visco-elasticity. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1487-1517. doi: 10.3934/dcdss.2017077
References:
[1]

L. AdamJ. 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.  Google Scholar

[2]

R. AlessiJ.-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.  Google Scholar

[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. Google Scholar

[4]

M. ArtinaF. CagnettiM. 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.  Google Scholar

[5]

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

[6]

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

[7]

E. BonettiG. 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.  Google Scholar

[8]

E. BonettiG. 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.  Google Scholar

[9]

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

[10]

G. BouchittéA. 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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

G. Dal MasoG. 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.  Google Scholar

[16]

G. Dal MasoG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Di NezzaG. 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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

M. KružíkC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

A. MielkeT. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

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

[45]

T. RoubíčekC. 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.  Google Scholar

[46]

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

[47]

T. RoubíčekM. 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.  Google Scholar

[48]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

R. VodičkaV. 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.  Google Scholar

show all references

References:
[1]

L. AdamJ. 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.  Google Scholar

[2]

R. AlessiJ.-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.  Google Scholar

[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. Google Scholar

[4]

M. ArtinaF. CagnettiM. 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.  Google Scholar

[5]

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

[6]

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

[7]

E. BonettiG. 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.  Google Scholar

[8]

E. BonettiG. 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.  Google Scholar

[9]

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

[10]

G. BouchittéA. 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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

G. Dal MasoG. 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.  Google Scholar

[16]

G. Dal MasoG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Di NezzaG. 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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

M. KružíkC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

A. MielkeT. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

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

[45]

T. RoubíčekC. 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.  Google Scholar

[46]

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

[47]

T. RoubíčekM. 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.  Google Scholar

[48]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

R. VodičkaV. 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.  Google Scholar

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