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A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints



We consider a viscoelastic body occupying a smooth bounded domain $\Omega\subset \mathbb{R}^3$ under the effect of a volumic traction force g . The macroscopic displacement vector from the equilibrium configuration is denoted by u. Inertial effects are considered; hence, the equation for u contains the second-order term u tt . On a part ΓD of the boundary of Ω, the body is fixed and no displacement may occur; on a second part Γ N ⊂ ∂Ω, the body can move freely; on a third portion Γ C ⊂ ∂Ω, the body is in adhesive contact with a solid support. The boundary forces acting on ΓC due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a non-linear ordinary differential equation settled on ΓC and describing the evolution of the delamination order parameter z. Following the lines of a new approach outlined in Bonetti et al. (2015, arXiv:1503.01911) and based on duality methods in Sobolev–Bochner spaces, we define a suitable concept of weak solution to the resulting system of partial differential equations. Correspondingly, we prove an existence result on finite-time intervals of arbitrary length.



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[1] Attouch, H. (1984) Variational Convergence for Functions and Operators, Pitman, London.
[2] Barboteu, M., Bartosz, K. & Kalita, P. (2015) A dynamic viscoelastic contact problem with normal compliance, finite penetration and nonmonotone slip rate dependent friction. Nonlinear Anal. Real World Appl. 22, 452472.
[3] Barbu, V., Colli, P., Gilardi, G. & Grasselli, M. (2000) Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation. Differ. Integral Equ. 13, 12331262.
[4] Barbu, V. (1976) Nonlinear Semigroups and Differential Equations in Banach Spaces, Noord-hoff, Leyden.
[5] Blanchard, D., Damlamian, A. & Ghidouche, H. (1989) A nonlinear system for phase change with dissipation. Differ. Integral Equ. 2, 344362.
[6] Bonetti, E., Bonfanti, G. & Rossi, R. (2008) Global existence for a contact problem with adhesion. Math. Methods Appl. Sci. 31, 10291064.
[7] Bonetti, E., BonfantiG. & Rossi, R. (2012) Analysis of a unilateral contact problem taking into account adhesion and friction. J. Differ. Equ. 235, 438462.
[8] Bonetti, E., Rocca, E., Scala, R. & Schimperna, G. (2015) On the strongly damped wave equation with constraint, arXiv:1503.01911, submitted.
[9] Brézis, H. (1973) Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Studies, Vol. 5, North-Holland, Amsterdam.
[10] Brézis, H. (1972) Intégrales convexes dans les espaces de Sobolev. Israel J. Math. 13, 923.
[11] Cocou, M., Schryve, M. & Raous, M. (2010) A dynamic unilateral contact problem with adhesion and friction in viscoelasticity. Z. Angew. Math. Phys. 61, 721743.
[12] Cocou, M. (2015) A class of dynamic contact problems with Coulomb friction in viscoelasticity. Nonlinear Anal. Real World Appl. 22, 508519.
[13] Colli, P., Luterotti, F., Schimperna, G. & Stefanelli, U. (2002) Global existence for a class of generalized systems for irreversible phase changes. NoDEA Nonlinear Differ. Equ. Appl. 9, 255276.
[14] Frémond, M. (2012) Phase Change in Mechanics, Springer-Verlag, Berlin, Heidelberg.
[15] Grun-Rehomme, M. (1977) Caractérisation du sous-différentiel d'intégrandes convexes dans les espaces de Sobolev (French). J. Math. Pures Appl. (9), 56, 149156.
[16] Ioffe, A. D. (1977) On lower semicontinuity of integral functionals. I. SIAM J. Control Optim. 15, 521538.
[17] Kuttler, K. L., Menike, R. S. R. & Shillor, M. (2009) Existence results for dynamic adhesive contact of a rod. J. Math. Anal. Appl. 351, 781791.
[18] Raous, M., Cangémi, L. & Cocu, M. (1999) A consistent model coupling adhesion, friction, and unilateral contact. Comput. Methods Appl. Mech. Eng. 177, 383399.
[19] Rossi, R. & Roubíček, T. (2011) Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlinear Anal. 74, 31593190.
[20] Rossi, R. & Roubíček, T. (2013) Adhesive contact delaminating at mixed mode, its thermodynamics and analysis. Interfaces Free Bound. 15, 137.
[21] Rossi, R. & Thomas, M. (2015) From an adhesive to a brittle delamination model in thermo-visco-elasticity. ESAIM Control Optim. Calc. Var. 21, 159.
[22] Roubíček, T. (2005) Nonlinear Partial Differential Equations with Applications, Birkhäuser, Springer, Basel.
[23] Roubíček, T. (2013) Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J. Math. Anal. 45, 101126.
[24] Scala, R. (2014) Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish, to appear on ESAIM:COCV.
[25] Schimperna, G. & Pawłow, I. (2013) On a class of Cahn-Hilliard models with nonlinear diffusion. SIAM J. Math. Anal. 45, 3163.
[26] Simon, J. (1987) Compact sets in the space Lp (0, T;B). Ann. Mat. Pura Appl. 146, 6596.


A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints



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