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

Adnene Elyacoubi; Taieb Hadhri

- Volume: 39, Issue: 4, page 637-648
- ISSN: 0764-583X

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topElyacoubi, Adnene, and Hadhri, Taieb. "Characterization of the limit load in the case of an unbounded elastic convex." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.4 (2005): 637-648. <http://eudml.org/doc/245820>.

@article{Elyacoubi2005,

abstract = {In this work we consider a solid body $\Omega \subset \{\mathbb \{R\}\}^3$ 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 $\Omega $, denoted by K(x), is written in the form of $\mbox\{K\}^D (x) + \{\mathbb \{R\}\}\mbox\{I\}$, I is the identity of $\{\{\mathbb \{R\}\}^9\}_\{sym\}$, and the deviatoric component $\mbox\{K\}^D$ 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 $\bar\{\lambda \}$ 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.},

author = {Elyacoubi, Adnene, Hadhri, Taieb},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {elasticity; limit load},

language = {eng},

number = {4},

pages = {637-648},

publisher = {EDP-Sciences},

title = {Characterization of the limit load in the case of an unbounded elastic convex},

url = {http://eudml.org/doc/245820},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Elyacoubi, Adnene

AU - Hadhri, Taieb

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 4

SP - 637

EP - 648

AB - In this work we consider a solid body $\Omega \subset {\mathbb {R}}^3$ 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 $\Omega $, denoted by K(x), is written in the form of $\mbox{K}^D (x) + {\mathbb {R}}\mbox{I}$, I is the identity of ${{\mathbb {R}}^9}_{sym}$, and the deviatoric component $\mbox{K}^D$ 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 $\bar{\lambda }$ 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.

LA - eng

KW - elasticity; limit load

UR - http://eudml.org/doc/245820

ER -

## References

top- [1] R. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
- [2] H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983). Zbl0511.46001MR697382
- [3] P.G. Ciarlet, Lectures on the three-dimensional elasticity. Tata Institute of Fundamental Research, Bombay (1983). Zbl0542.73046MR730027
- [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. Zbl0831.73016
- [5] R. Temam, Mathematical Problems in Plasticity. Bordas, Paris (1985).

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