zbMATH — the first resource for mathematics

On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. (English) Zbl 1078.74048
Summary: We consider a model describing the evolution of damage in viscoelastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinear parabolic variational inclusion, due to the presence of two maximal monotone graphs involving the phase parameter and its time derivative. Existence of a solution is proved in some subinterval of time in which the damage process is not complete. Uniqueness is established in the case when one of the two monotone graphs is assumed to be Lipschitz continuous.

74R20 Anelastic fracture and damage
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)
Full Text: DOI
[1] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. · Zbl 0561.49012
[2] Bonetti, E.; Frémond, M., Damage theory: microscopic effects of vanishing macroscopic motions, Comput. appl. math., 22, 3, 313-333, (2003) · Zbl 1213.74037
[3] Bonfanti, G.; Frémond, M.; Luterotti, F., Global solution to a nonlinear system for irreversible phase changes, Adv. math. sci. appl., 10, 1, 1-24, (2000) · Zbl 0956.35122
[4] Bonetti, E.; Schimperna, G., Local existence for frémond’s model of damage in elastic materials, Contin. mech. thermodyn., 16, 4, 319-335, (2004) · Zbl 1066.74048
[5] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50), North-Holland Publishing Co., Amsterdam, 1973. · Zbl 0252.47055
[6] Frémond, M., Non-smooth thermomechanics, (2002), Springer Berlin · Zbl 0990.80001
[7] Frémond, M.; Kuttler, K.L.; Shillor, M., Existence and uniqueness of solutions for a dynamic one-dimensional damage model, J. math. anal. appl., 229, 1, 271-294, (1999) · Zbl 0920.73328
[8] Frémond, M.; Kuttler, K.L.; Nedjar, B.; Shillor, M., One-dimensional models of damage, Adv. math. sci. appl., 8, 2, 541-570, (1998) · Zbl 0915.73041
[9] Frémond, M.; Nedjar, B., Damage, gradient of damage and principle of virtual power, Internat. J. solids struct., 33, 8, 1083-1103, (1996) · Zbl 0910.73051
[10] J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. I, Springer, New York, 1972 (translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181). · Zbl 0223.35039
[11] Nirenberg, L., On elliptic partial differential equations, Ann. scuola norm. sup. Pisa (3), 13, 115-162, (1959) · Zbl 0088.07601
[12] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. mat. pura appl. (4), 146, 65-96, (1987) · Zbl 0629.46031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.