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 [
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 [
Citation: |
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. | |
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. | |
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. | |
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. | |
H. Attouch, Variational Convergence for Functions and Operators Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. | |
G. Barenblatt , The mathematical theory of equilibrium of cracks in brittle fracture, Adv. Appl. Mech., 7 (1962) , 55-129. | |
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. | |
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. | |
E. Bonetti, E. Rocca, R. Scala and G. Schimperna, On the strongly damped wave equation with constraint, WIAS-Preprint 2094. | |
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. | |
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). | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
D. Dugdale , Yielding of steel sheets containing clits, J. Mech. Phys. Solids, 8 (1960) , 100-104. | |
M. Fr´emond, Contact with adhesion, in Topics in Nonsmooth Mechanics (eds. J. Moreau, P. Panagiotopoulos and G. Strang), Birkhäuser, 1988,157–186. | |
M. Frémond, Non-Smooth Thermomechanics Springer-Verlag Berlin Heidelberg, 2002. | |
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. | |
A. Ioffe , On lower semicontinuity of integral functionals. I, SIAM J. Control Optimization, 15 (1977) , 521-538. doi: 10.1137/0315035. | |
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. | |
R. Kregting, Cohesive Zone Models Towards a Robust Implementation of Irreversible Behavior Technical Report MT05. 11, TU Eindhoven, Materials Technology, 2005. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
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. | |
J. Rice, Fracture, chapter Mathematical analysis in the mechanics of fracture |
|
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. | |
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. | |
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. | |
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. | |
R. Rossi and M. Thomas, From adhesive to brittle delamination in visco-elastodynamics Math. Models Methods Appl. Sci. , 2017. doi: 10.1142/S0218202517500257. | |
T. Roubíček , Rate-independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009) , 825-862. doi: 10.1002/mma.1069. | |
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. | |
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. | |
T. Roubíček , V. Mantič and C. Panagiotopoulos , A quasistatic mixed-mode delamination model, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) , 591-610. | |
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. | |
T. Roubíček , L. Scardia and C. Zanini , Quasistatic delamination problem, Continuum Mech. Thermodynam., 21 (2009) , 223-235. doi: 10.1007/s00161-009-0106-4. | |
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. | |
R. Scala, A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint, WIAS-Preprint 2172. | |
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. | |
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. | |
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. | |
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. |
Typical cohesive laws. Solid lines: curves corresponding to maximal loading envelope, dashed lines: loading -unloading rules.