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Analytical results on a model for damagingin domains and interfaces*

Published online by Cambridge University Press:  18 August 2010

Elena Bonetti  and
Michel Frémond
Elena Bonetti
Affiliation:
Dipartimento di Matematica – Laboratoire Lagrange, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy. elena.bonetti@unipv.it
Michel Frémond
Affiliation:
Centre de Mathématiques et de leurs Applications – Laboratoire Lagrange, École Normale Supérieure de Cachan, France. fremond@cmla.ens-cachan.fr
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Abstract

This paper deals with a model describing damage processes in a (nonlinear) elastic body which is in contact with adhesion with a rigid support. On the basis of phase transitions theory, we detailthe derivation of the model written in terms of a PDE system, combined with suitable initial and boundary conditions. Some internal constraints on the variables are introduced in the equations and on the boundary, to get physical consistency. We prove theexistence of global in time solutions (to a suitable variational formulation) of therelated Cauchy problem by means of a Schauder fixed point argument, combinedwith monotonicity and compactness tools. We also perform an asymptotic analysis of the solutions as the interfacial damage energy (between the body and the contact surface) goes to +∞.

Keywords

Damagecontactadhesionexistenceasymptotic analysis
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 955 - 974
Copyright
© EDP Sciences, SMAI, 2010

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References

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden (1976).
Bonetti, E. and Bonfanti, G., Well-posedness results for a model of damage in thermoviscoelastic materials. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (2008) 1187–1208. CrossRef
Bonetti, E. and Frémond, M., Collisions and fracture, a 1-D example: How to tear off a chandelier from the ceiling. J. Elast. 74 (2004) 47–66. CrossRef
Bonetti, E. and Schimperna, G., Local existence for Frémond's model of damage in elastic materials. Contin. Mech. Thermodyn. 16 (2004) 319–335. CrossRef
Bonetti, E., Segatti, A. and Schimperna, G., On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Diff. Equ. 218 (2005) 91–116. CrossRef
Bonetti, E., Bonfanti, G. and Rossi, R., Well-posedness and long-time behaviour for a model of contact with adhesion. Indiana Univ. Math. J. 56 (2007) 2787–2819.
Bonetti, E., Bonfanti, G. and Rossi, R., Global existence for a contact problem with adhesion. Math. Meth. Appl. Sci. 31 (2008) 1029–1064. CrossRef
Bonetti, E., Bonfanti, G. and Rossi, R., Thermal effects in adhesive contact: modelling and analysis. Nonlinearity 22 (2009) 2697–2731. CrossRef
Colli, P., Luterotti, F., Schimperna, G. and Stefanelli, U., Global existence for a class of generalized systems for irreversible phase changes. NoDEA Nonlinear Diff. Equ. Appl. 9 (2002) 255–276. CrossRef
Freddi, F. and Frémond, M., Damage in domains and interfaces: a coupled predictive theory. J. Mech. Mater. Struct. 7 (2006) 1205–1233. CrossRef
Frémond, M., Équilibre des structures qui adhèrent à leur support. C. R. Acad. Sci. Paris 295 (1982) 913–916.
Frémond, M., Adhérence des solides. J. Méc. Théor. Appl. 6 (1987) 383–407.
M. Frémond, Non-smooth Thermomechanics. Springer-Verlag, Berlin (2002).
M. Frémond, Collisions. Edizioni del Dipartimento di Ingegneria Civile dell' Università di Roma Tor Vergata, Italy (2007).
Frémond, M. and Kenmochi, N., Damage problems for viscous locking materials. Adv. Math. Sci. Appl. 16 (2006) 697–716.
Frémond, M. and Nedjar, B., Damage, gradient of damage and priciple of virtual power. Int. J. Solids Struct. 33 (1996) 1083–1103. CrossRef
Frémond, M., Kuttler, K. and Shillor, M., Existence and uniqueness of solutions for a dynamic one-dimensional damage model. J. Math. Anal. Appl. 229 (1999) 271–294. CrossRef
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars, Paris (1969).
Moreau, J.J., Sur les lois de frottement, de viscosité et plasticité. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 271 (1970) 608–611.
Point, N., Unilateral contact with adherence. Math. Meth. Appl. Sci. 10 (1998) 367–381. CrossRef
J. Simon, Compact sets in the space Lp(0,T; B). Ann. Mat. Pura Appl. 146 (1987) 65–96.