Analytical results on a model for damaging in domains and interfaces
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 955-974

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 detail the 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 the existence of global in time solutions (to a suitable variational formulation) of the related Cauchy problem by means of a Schauder fixed point argument, combined with 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 +∞.

DOI : https://doi.org/10.1051/cocv/2010033
Classification:  35K55,  74A15,  74M15
Keywords: damage, contact, adhesion, existence, asymptotic analysis
@article{COCV_2011__17_4_955_0,
     author = {Bonetti, Elena and Fr\'emond, Michel},
     title = {Analytical results on a model for damaging in domains and interfaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     pages = {955-974},
     doi = {10.1051/cocv/2010033},
     zbl = {1230.35034},
     mrnumber = {2859860},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_4_955_0}
}
Bonetti, Elena; Frémond, Michel. Analytical results on a model for damaging in domains and interfaces. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 955-974. doi : 10.1051/cocv/2010033. http://www.numdam.org/item/COCV_2011__17_4_955_0/

[1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden (1976). | MR 390843 | Zbl 0328.47035

[2] E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (2008) 1187-1208. | Numdam | MR 2466326 | Zbl 1152.35505

[3] E. Bonetti and M. Frémond, Collisions and fracture, a 1-D example: How to tear off a chandelier from the ceiling. J. Elast. 74 (2004) 47-66. | MR 2058195 | Zbl 1058.74071

[4] E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials. Contin. Mech. Thermodyn. 16 (2004) 319-335. | MR 2061321 | Zbl 1066.74048

[5] E. Bonetti, A. Segatti and G. Schimperna, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Diff. Equ. 218 (2005) 91-116. | MR 2174968 | Zbl 1078.74048

[6] E. Bonetti, G. Bonfanti and R. Rossi, Well-posedness and long-time behaviour for a model of contact with adhesion. Indiana Univ. Math. J. 56 (2007) 2787-2819. | MR 2375702 | Zbl 1145.35027

[7] E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion. Math. Meth. Appl. Sci. 31 (2008) 1029-1064. | MR 2419088 | Zbl 1145.35301

[8] E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: modelling and analysis. Nonlinearity 22 (2009) 2697-2731. | MR 2550692 | Zbl 1185.35122

[9] P. Colli, F. Luterotti, G. Schimperna and U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes. NoDEA Nonlinear Diff. Equ. Appl. 9 (2002) 255-276. | MR 1917373 | Zbl 1004.35061

[10] F. Freddi and M. Frémond, Damage in domains and interfaces: a coupled predictive theory. J. Mech. Mater. Struct. 7 (2006) 1205-1233.

[11] M. Frémond, Équilibre des structures qui adhèrent à leur support. C. R. Acad. Sci. Paris 295 (1982) 913-916. | MR 695554 | Zbl 0551.73096

[12] M. Frémond, Adhérence des solides. J. Méc. Théor. Appl. 6 (1987) 383-407. | Zbl 0645.73046

[13] M. Frémond, Non-smooth Thermomechanics. Springer-Verlag, Berlin (2002). | MR 1885252 | Zbl 0990.80001

[14] M. Frémond, Collisions. Edizioni del Dipartimento di Ingegneria Civile dell' Università di Roma Tor Vergata, Italy (2007).

[15] M. Frémond and N. Kenmochi, Damage problems for viscous locking materials. Adv. Math. Sci. Appl. 16 (2006) 697-716. | MR 2356296 | Zbl 1158.74310

[16] M. Frémond and B. Nedjar, Damage, gradient of damage and priciple of virtual power. Int. J. Solids Struct. 33 (1996) 1083-1103. | MR 1370124 | Zbl 0910.73051

[17] M. Frémond, K. Kuttler and M. Shillor, Existence and uniqueness of solutions for a dynamic one-dimensional damage model. J. Math. Anal. Appl. 229 (1999) 271-294. | MR 1664356 | Zbl 0920.73328

[18] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars, Paris (1969). | MR 259693 | Zbl 0189.40603

[19] J.J. Moreau, 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.

[20] N. Point, Unilateral contact with adherence. Math. Meth. Appl. Sci. 10 (1998) 367-381. | MR 958479 | Zbl 0656.73052

[21] J. Simon, Compact sets in the space Lp(0,T; B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR 916688 | Zbl 0629.46031