# zbMATH — the first resource for mathematics

Existence and uniqueness of solutions for a dynamic one-dimensional damage model. (English) Zbl 0920.73328
Summary: We consider a one-dimensional dynamic model that describes the evolution of damage caused by tension in a viscoelastic material. The process is modeled by a coupled set of two differential inclusions for the elastic displacement and damage fields. We establish the existence of local weak solutions. The existence result is derived from a priori estimates obtained for a sequence of regularized, truncated, and time-retarded approximations. We also establish the existence of unique weak solution in a simplified version of the model. $$\copyright$$ Academic Press.

##### MSC:
 74R99 Fracture and damage 74Hxx Dynamical problems in solid mechanics 35Q72 Other PDE from mechanics (MSC2000)
Full Text:
##### References:
 [1] Adams, R., Sobolev spaces, (1975), Academic Press · Zbl 0314.46030 [2] M. Frémond, K. L. Kuttler, B. Nedjar, M. Shillor, One dimensional models of damage, Adv. Math. Sci. Appl. · Zbl 0915.73041 [3] Frémond, M.; Nedjar, B., Endommagement et principe des puissances virtuelles, C. R. acad. sci. Paris ser. II, 317, 857-864, (1993) · Zbl 0780.73055 [4] Frémond, M.; Nedjar, B., Damage in concrete: the unilateral phenomenon, Nuclear engineering design, 156, 323-335, (1995) [5] Frémond, M.; Nedjar, B., Damage, gradient of damage, and principle of virtual power, Int. J. solids structures, 33, 1083-1103, (1996) · Zbl 0910.73051 [6] Kuttler, K.L., Time dependent implicit evolution equations, Nonlinear anal., 10, 447-463, (1986) · Zbl 0603.47038 [7] Kufner, A.; John, O.; Fucik, S., Function spaces, (1977), Noordhoff Leyden [8] Lions, J.L., Quelques methods de resolution des problemes aux limites non-lineaires, (1969), Dunod · Zbl 0189.40603 [9] Seidman, T.I., The transient semiconductor problem with generation terms, II, Nonlinear semigroups, partial differential equations and attractors, Springer lecture notes in math., 1394, (1989), p. 185-198
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.