An adhesive unilateral contact between visco-elastic bodies at small strains and in a Kelvin–Voigt rheology is scrutinized, neglecting inertia. The flow-rule for debonding the adhesive is considered rate independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely Mode I (opening) needs (= dissipates) less energy than Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An efficient semi-implicit-in-time FEM discretization leading to recursive quadratic mathematical programs is devised. Its convergence and thus the existence of weak solutions is proved. Computational experiments implemented by BEM illustrate the modeling aspects and the numerical efficiency of the discretization.
CamanhoPPDávilaCGDe MouraMF. Numerical simulation of mixed-mode progressive delamination in composite materials. J Composite Mater2003; 37: 1415–1438.
10.
HutchinsonJWWuTY. Mixed mode cracking in layered materials. In HutchinsonJWWuTY (eds) Advances in Applied Mechanics. New York: Academic Press, 1992, pp.63–191.
11.
TávaraLMantičVGracianiECanasJParísF. Analysis of a crack in a thin adhesive layer between orthotropic materials: An application to composite interlaminar fracture toughness test. CMES2010; 58: 247–270.
12.
TávaraLMantičVGracianiEParísF. BEM analysis of crack onset and propagation along fiber-matrix interface under transverse tension using a linear elastic-brittle interface model. Eng Anal Bound Elem2011; 35: 207–222.
13.
FrémondM. Equilibre des structures qui adhèrent à leur support. Comptes Rendus Acad Sci Paris1982; 295: 913–916.
14.
FrémondM. Adhèrence des solides. J Méch Théorique Appliq1987; 6: 383–407.
15.
KočvaraMMielkeARoubíčekT. A rate-independent approach to the delamination problem. Math Mech Solids2006; 11: 423–447.
16.
RoubíčekTKružíkMZemanJ. Delamination and adhesive contact models and their mathematical analysis and numerical treatment. In MantičV (ed) Mathematical Methods and Models in Composites. London: Imperial College Press, 2013, pp.349–400.
RoubíčekTPanagiotopoulosCGMantičV. Local-solution approach to quasistatic rate-independent mixed-mode delamination. Math Mod Meth Appl Sci, submitted.
19.
RossiRRoubíčekT. Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlin Anal2011; 74: 3159–3190.
20.
MielkeA. Differential, energetic and metric formulations for rate-independent processes. In AmbrosioLSavaréG (eds) Nonlinear PDE’s and Applications. Heidelberg, New York: Springer, 2011, pp. 87–170.
21.
ToaderRZaniniC. An artificial viscosity approach to quasistatic crack growth. Boll Unione Matem Ital2009; 2: 1–36.
22.
RoubíčekTSoučekOVodičkaR. A model of rupturing lithospheric faults with re-occurring earthquakes. SIAM J Appl Math2013; 73: 1460–1488.
23.
BonettiEBonfantiGRossiR. Global existence for a contact problem with adhesion. Math Methods Appl Sci2008; 31: 1029–1064.
24.
ThomasM. Rate-independent damage processes in nonlinearly elastic materials. PhD Thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, 2010.
25.
ThomasMMielkeA. Damage of nonlinearly elastic materials at small strain—existence and regularity results. ZAMM-Z Angew Math Mech2010; 90: 88–112.
26.
MielkeA. Evolution in rate-independent systems. In DafermosCMFeireislE (eds) Handbook of Differential Equations, Evolutionary Equations. Amsterdam: Elsevier, 2005, Vol. 2, pp.461–559.
RoubíčekT. Maximally-dissipative local solutions to rate-independent systems and application to damage and delamination problems. Nonlin Anal, Th Meth Appl, submitted.
29.
RoubíčekT. Rate independent processes in viscous solids at small strains. Math Methods Appl Sci2009; 32: 825–862.
CornettiPCarpinteriA. Modelling the FRP-concrete delamination by means of an exponential softening law. Engineering Structures2011; 33: 1988–2001.
32.
MukherjeeS. On boundary integral equations for cracked and for thin bodies. Math Mech Solids2001; 6: 47–64.
33.
ParísFCanasJ. Boundary Element Method, Fundamentals and Applications. Oxford: Oxford University Press, 1997.
34.
RoubíčekTPanagiotopoulosCGMantičV. Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity. ZAMM-Z Angew Math Mech2013; 93: 823–840.
35.
RoubíčekT. Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J Math Anal2013; 45: 101–126.
36.
MielkeARoubíčekT. Numerical approaches to rate-independent processes and applications in inelasticity. Math Model Numer Anal2009; 43: 399–428.
37.
Dal MasoGFrancfortGToaderR. Quasistatic crack growth in nonlinear elasticity. Arch Rational Mech Anal2005; 176: 165–225.
38.
VisintinA. Models of Phase Transitions. Boston, MA: Birkhäuser, 1996.
39.
Dal MasoGi DeSimoneAMoraMGMoriniM. A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch Rational Mech Anal2008; 189: 469–544.
40.
KneesDMielkeAZaniniC. On the inviscid limit of a model for crack propagation. Math Models Meth Appl Sci2008; 18: 1529–1569.
