Characterization of the limit load in the case of an unbounded elastic convex

Abstract

In this work we consider a solid body $\Omega\subset{\Bbb R}^3$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f $ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda $ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by $\bar{\lambda}$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri, RAIRO: Modél. Math. Anal. Numér. 29 (1995) 391–419]. Then assuming that the convex of elasticity at the point x of Ω, denoted by K(x), is written in the form of $\mbox{K}^D (x) + {\Bbb R}\mbox{I}$, I is the identity of ${{\Bbb R}^9}_{sym}$, and the deviatoric component $\mbox{K}^D$ is bounded regardless of x $\in\Omega$, we show under the condition “Rot f $\not= 0$ or g is not colinear to the normal on a part of the boundary of Ω", that the Limit Load $\bar{\lambda}$ searched is equal to the inverse of the infimum of the gauge of the Elastic convex translated by stress field equilibrating the unitary load corresponding to $\lambda =1$; moreover we show that this infimum is reached in a suitable function space.