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

Elyacoubi, Adnene; Hadhri, Taieb

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 637-648.

In this work we consider a solid body $\Omega \subset {ℝ}^3$ 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 $\overline{\lambda }$ 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 $\Omega$, denoted by K(x), is written in the form of ${\text{K}}^D\left(x\right)+ℝ\text{I}$, I is the identity of ${{ℝ}^9}_{sym}$, and the deviatoric component ${\text{K}}^D$ is bounded regardless of x $\in \Omega$, we show under the condition “Rot f $\ne 0$ or g is not colinear to the normal on a part of the boundary of $\Omega$”, that the Limit Load $\overline{\lambda }$ 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.

[1] R. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983). | MR | Zbl

[3] P.G. Ciarlet, Lectures on the three-dimensional elasticity. Tata Institute of Fundamental Research, Bombay (1983). | MR | Zbl

[4] H. El-Fekih and T. Hadhri, Calcul des charges limites d'une structure élastoplastique en contraintes planes. RAIRO: Modél. Math. Anal. Numér. 29 (1995) 391-419. | EuDML | Numdam | Zbl

[5] R. Temam, Mathematical Problems in Plasticity. Bordas, Paris (1985).