From an adhesive to a brittle delamination model in thermo-visco-elasticity

Rossi, Riccarda; Thomas, Marita

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 1-59.

We address the analysis of a model for brittle delamination of two visco-elastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques.

Classification : 35K85, 47J20, 49J45, 49S05, 74F07, 74R10
Mots clés : Rate-independent evolution of adhesive contact, brittle delamination, Kelvin−Voigt viscoelasticity, nonlinear heat equation, Mosco-convergence, special functions of bounded variation, regularity of sets, lower density estimate

V. Acary and B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems. Springer (2008). | Zbl

G. Alberti, Variational models for phase transitions, an approach via $\Gamma$-convergence, in Calculus of variations and partial differential equations (Pisa 1996). Springer (2000) 95–114. | MR | Zbl

L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives. In vol. 108. Proc. of Amer. Math. Soc. (1990) 691–702. | MR | Zbl

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2005). | Zbl

H. Attouch, Variational convergence for functions and operators. Appl. Math. Ser. Pitman (Advanced Publishing Program), Boston, MA (1984). | MR | Zbl

J.F. Bell, Mechanics of Solids. Vol. 1: The Experimental Foundations of Solid Mechanics. Springer (1984). | MR

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149–169. | DOI | MR | Zbl

E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion. Math. Meth. Appl. Sci. 31 (2008) 1029–1064. | DOI | MR | Zbl

E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: Modelling and analysis. Nonlinearity 22 (2009) 2697–2731. | DOI | MR | Zbl

E. Bonetti, G. Schimperna and A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Differ. Equ. 218 (2005) 91–116. | DOI | MR | Zbl

B. Bourdin, G. Francfort and J.-J. Marigo, The variational approach to fracture. J. Elasticity 91 (2008) 5–148. | DOI | MR | Zbl

S. Campanato, Proprietà di hölderianità di alcune classi di funzioni. Annali della Scuola Normale Superiore di Pisa 17 (1963) 175–188. | Numdam | MR | Zbl

S. Campanato, Proprietà di una famiglia di spazi funzionali. Annali della Scuola Normale Superiore di Pisa 18 (1964) 137–160. | Numdam | MR | Zbl

V. Caselles, A. Chambolle and M. Novaga, Regularity for solutions of the total variation denoising problem. Revista Matemática Iberoamericana 27 (2011) 729–1098. | MR | Zbl

G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity. Arch. Rat. Mech. Anal. 176 (2005) 165–225. | DOI | MR | Zbl

G. Dolzmann, N. Hungerbühler and S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of $n$-Laplace type with measure valued right hand side. J. für die reine und angewandte Mathematik 520 (2000) 1–35. | DOI | MR | Zbl

L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press (1992). | MR | Zbl

H. Federer, Geometric Measure Theory. Springer (1969). | MR | Zbl

E. Feireisl, Dynamics of viscous compressible fluids, Vol. 26 of Oxford Lect. Ser. Math. Appl. Oxford University Press, Oxford (2004). | MR | Zbl

E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007) 461–490. | DOI | MR | Zbl

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009) 1345–1369. | DOI | MR | Zbl

E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals. Arch. Ration. Mech. Anal. 205 (2012) 651–672. | DOI | MR | Zbl

I. Fonseca and N. Fusco, Regularity results for anisotropic image segmentation models. Annali della Scuola Normale Superiore di Pisa 24 (1997) 463–499. | Numdam | MR | Zbl

G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies. J. reine angew. Math. 595 (2006) 55–91. | MR | Zbl

F. Freddi and M. Frémond, Damage in domains and interfaces: A coupled predictive theory. J. Mech. Mat. Struct. 1 (2006) 1205–1233. | DOI

M. Frémond, Contact with adhesion, in Topics in Nonsmooth Mechanics. Edited by J.J. Moreau, P.D. Panagiotopoulos and G. Strang. Birkhäuser (1988) 157–186. | MR | Zbl

M. Frémond, Non-Smooth Thermomechanics. Springer-Verlag, Berlin, Heidelberg (2002). | MR | Zbl

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33 (1996) 1083–1103. | DOI | MR | Zbl

Y.C. Fung, Foundations of solid mechanics. Prentice-Hall, Inc. (1965).

P. Gérard, Microlocal defect measures. Commun. Partial Differ. Equ. 16 (1991) 1761–1794. | DOI | MR | Zbl

A. Giacomini, Ambrosio−Tortorelli approximation of quasi-static evolution of brittle fracture. Calc. Var. Partial Differ. Equ. 22 (2005) 129–172. | DOI | MR | Zbl

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press (1983). | MR | Zbl

E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984). | MR | Zbl

J.A. Griepentrog, Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals. Math. Nachrichten 243 (2002) 19–42. | DOI | MR | Zbl

J.A. Griepentrog and L. Recke, Linear elliptic boundary value problems with non-smooth data: Normal solvability on Sobolev-Campanato spaces. Math. Nachrichten 225 (2001) 39–74. | DOI | MR | Zbl

B. Halphen and Q.S. Nguyen, Sur les matériaux standards généralisés. J. Mécanique 14 (1975) 39–63. | MR | Zbl

C.O. Horgan and J.K. Knowles, The effect of nonlinearity on a principle of Saint-Venant type. J. Elasticity 11 (1981) 271–291. | DOI | MR | Zbl

A.D. Ioffe and V.M. Tihomirov, Theory of extremal problems, vol. 6 of Stud. Math. Appl. Translated from the Russian by Karol Makowski. North-Holland Publishing Co., Amsterdam (1979). | MR | Zbl

D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18 (2008) 1529–1569. | DOI | MR | Zbl

J.K. Knowles, The finite anti-plane shear near the tip of a crack for a class of incompressible elastic solids. Int. J. Fracture 13 (1977) 611–639. | DOI | MR

M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423–447. | DOI | MR | Zbl

J.L. Lewis, Uniformly fat sets. Trans. Amer. Math. Soc. 308 (1988) 177–196. | DOI | MR | Zbl

F. Maggi, Sets of finite perimeter and geometric variational problems. Cambridge (2012). | MR | Zbl

M. Marcus and V.J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972) 294–320. | DOI | MR | Zbl

A. Mielke, Evolution in rate-independent systems, in vol. 2 of Handbook Differ. Equ. Evol. Equ., edited by C.M. Dafermos and E. Feireisl. Elsevier B.V., Amsterdam (2005) 461–559. | MR | Zbl

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16 (2006) 177–209. | MR | Zbl

A. Mielke and F. Theil, On rate-independent hysteresis models. Nonlin. Differ. Equ. Appl. 11 (2004) 151–189. | DOI | MR | Zbl

A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31 (2008) 387–416. | DOI | MR | Zbl

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 | MR | Zbl

L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123–142. | DOI | MR | Zbl

L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza. Boll. U. Mat. Ital. B 14 (1977) 285–299. | MR | Zbl

J. Naumann, On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids. Math. Methods Appl. Sci. 29 (2006) 1883–1906. | DOI | MR | Zbl

M. Negri and C. Ortner, Quasi-static crack propagation by Griffith’s criterion. Math. Models Methods Appl. Sci. 18 2008 1895–1925. | DOI | MR | Zbl

R. Rossi and T. Roubíček, Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlinear Anal. 74 (2011) 3159–3190. | DOI | MR | Zbl

R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis. Interfaces Free Bound. 14 (2013) 1–37. | DOI | MR | Zbl

T. Roubíček, Nonlinear Partial Differential Equations with Applications. Birkhäuser (2005). | MR | Zbl

T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains. SIAM J. Math. Anal. 42 (2010) 256–297. | DOI | MR | Zbl

T. Roubíček, L. Scardia and C. Zanini, Quasistatic delamination problem. Contin. Mech. Thermodyn. 21 (2009) 223–235. | DOI | MR | Zbl

J. Simon, Compact sets in the space $L^p(0,T;B)$. Ann. Mat. Pura Appl. (1987). | MR | Zbl

M. Thomas, Griffith formula for mode-III-interface-cracks in strainhardening compounds. Mech. Adv. Mater. Struct. 12 (2008) 428–437. | DOI

M. Thomas, Rate-independent damage processes in nonlinearly elastic materials. Ph.D. thesis, Humboldt-Universität zu Berlin (2010).

M. Thomas, Quasistatic damage evolution with spatial BV-regularization. Discrete Contin. Dyn. Syst. Ser. S 6 (2013) 235–255. | MR | Zbl

M. Thomas, Uniform Poincar*error*é−Sobolev and relative isoperimetric inequalities for classes of domains. Accepted for publication in Discrete Contin. Dyn. Syst. WIAS-Preprint 1797 (2013). | MR

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 | MR | Zbl

M. Thomas and R. Rossi, Rate-independent Systems with viscosity and inertia: existence and evolutionary T-convergence. In preparation (2014).