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.
Keywords: 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
@article{COCV_2015__21_1_1_0, author = {Rossi, Riccarda and Thomas, Marita}, title = {From an adhesive to a brittle delamination model in thermo-visco-elasticity}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1--59}, publisher = {EDP-Sciences}, volume = {21}, number = {1}, year = {2015}, doi = {10.1051/cocv/2014015}, zbl = {1323.35101}, mrnumber = {3348414}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014015/} }
TY - JOUR AU - Rossi, Riccarda AU - Thomas, Marita TI - From an adhesive to a brittle delamination model in thermo-visco-elasticity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 1 EP - 59 VL - 21 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014015/ DO - 10.1051/cocv/2014015 LA - en ID - COCV_2015__21_1_1_0 ER -
%0 Journal Article %A Rossi, Riccarda %A Thomas, Marita %T From an adhesive to a brittle delamination model in thermo-visco-elasticity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 1-59 %V 21 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014015/ %R 10.1051/cocv/2014015 %G en %F COCV_2015__21_1_1_0
Rossi, Riccarda; Thomas, Marita. From an adhesive to a brittle delamination model in thermo-visco-elasticity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 1, pp. 1-59. doi : 10.1051/cocv/2014015. http://archive.numdam.org/articles/10.1051/cocv/2014015/
V. Acary and B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems. Springer (2008). | Zbl
G. Alberti, Variational models for phase transitions, an approach via -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
Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149–169. | DOI | MR | Zbl
and ,Global existence for a contact problem with adhesion. Math. Meth. Appl. Sci. 31 (2008) 1029–1064. | DOI | MR | Zbl
, and ,Thermal effects in adhesive contact: Modelling and analysis. Nonlinearity 22 (2009) 2697–2731. | DOI | MR | Zbl
, and ,On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Differ. Equ. 218 (2005) 91–116. | DOI | MR | Zbl
, and ,The variational approach to fracture. J. Elasticity 91 (2008) 5–148. | DOI | MR | Zbl
, and ,Proprietà di hölderianità di alcune classi di funzioni. Annali della Scuola Normale Superiore di Pisa 17 (1963) 175–188. | Numdam | MR | Zbl
,Proprietà di una famiglia di spazi funzionali. Annali della Scuola Normale Superiore di Pisa 18 (1964) 137–160. | Numdam | MR | Zbl
,Regularity for solutions of the total variation denoising problem. Revista Matemática Iberoamericana 27 (2011) 729–1098. | MR | Zbl
, and ,Quasistatic crack growth in nonlinear elasticity. Arch. Rat. Mech. Anal. 176 (2005) 165–225. | DOI | MR | Zbl
, and ,Uniqueness and maximal regularity for nonlinear elliptic systems of -Laplace type with measure valued right hand side. J. für die reine und angewandte Mathematik 520 (2000) 1–35. | DOI | MR | Zbl
, and ,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
Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007) 461–490. | DOI | MR | Zbl
,Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009) 1345–1369. | DOI | MR | Zbl
, and ,A new approach to non-isothermal models for nematic liquid crystals. Arch. Ration. Mech. Anal. 205 (2012) 651–672. | DOI | MR | Zbl
, , and ,Regularity results for anisotropic image segmentation models. Annali della Scuola Normale Superiore di Pisa 24 (1997) 463–499. | Numdam | MR | Zbl
and ,Existence results for a class of rate-independent material models with nonconvex elastic energies. J. reine angew. Math. 595 (2006) 55–91. | MR | Zbl
and ,Damage in domains and interfaces: A coupled predictive theory. J. Mech. Mat. Struct. 1 (2006) 1205–1233. | DOI
and ,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
Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33 (1996) 1083–1103. | DOI | MR | Zbl
and ,Y.C. Fung, Foundations of solid mechanics. Prentice-Hall, Inc. (1965).
Microlocal defect measures. Commun. Partial Differ. Equ. 16 (1991) 1761–1794. | DOI | MR | Zbl
,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
Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals. Math. Nachrichten 243 (2002) 19–42. | DOI | MR | Zbl
,Linear elliptic boundary value problems with non-smooth data: Normal solvability on Sobolev-Campanato spaces. Math. Nachrichten 225 (2001) 39–74. | DOI | MR | Zbl
and ,Sur les matériaux standards généralisés. J. Mécanique 14 (1975) 39–63. | MR | Zbl
and ,The effect of nonlinearity on a principle of Saint-Venant type. J. Elasticity 11 (1981) 271–291. | DOI | MR | Zbl
and ,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
On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18 (2008) 1529–1569. | DOI | MR | Zbl
, and ,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
,A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423–447. | DOI | MR | Zbl
,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
Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972) 294–320. | DOI | MR | Zbl
and ,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
On rate-independent hysteresis models. Nonlin. Differ. Equ. Appl. 11 (2004) 151–189. | DOI | MR | Zbl
and ,T. Roubíček and U. Stefanelli, -limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31 (2008) 387–416. | DOI | MR | Zbl
,From damage to delamination in nonlinearly elastic materials at small strains. J. Elasticity 109 (2012) 235–273. | DOI | MR | Zbl
, and ,The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123–142. | DOI | MR | Zbl
,Un esempio di -convergenza. Boll. U. Mat. Ital. B 14 (1977) 285–299. | MR | Zbl
and ,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
,Quasi-static crack propagation by Griffith’s criterion. Math. Models Methods Appl. Sci. 18 2008 1895–1925. | DOI | MR | Zbl
and ,Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlinear Anal. 74 (2011) 3159–3190. | DOI | MR | Zbl
and ,Adhesive contact delaminating at mixed mode, its thermodynamics and analysis. Interfaces Free Bound. 14 (2013) 1–37. | DOI | MR | Zbl
and ,T. Roubíček, Nonlinear Partial Differential Equations with Applications. Birkhäuser (2005). | MR | Zbl
Thermodynamics of rate-independent processes in viscous solids at small strains. SIAM J. Math. Anal. 42 (2010) 256–297. | DOI | MR | Zbl
,Quasistatic delamination problem. Contin. Mech. Thermodyn. 21 (2009) 223–235. | DOI | MR | Zbl
, and ,J. Simon, Compact sets in the space . Ann. Mat. Pura Appl. (1987). | MR | Zbl
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).
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
Damage of nonlinearly elastic materials at small strain: existence and regularity results. Zeit. angew. Math. Mech. 90 (2010) 88–112. | DOI | MR | Zbl
and ,M. Thomas and R. Rossi, Rate-independent Systems with viscosity and inertia: existence and evolutionary T-convergence. In preparation (2014).
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