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

Adnene Elyacoubi; Taieb Hadhri

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 39.4 (2010): 637-648. <http://eudml.org/doc/194280>.

@article{Elyacoubi2010,

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.
},

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

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Elasticity; limit load.; elasticity; limit load},

language = {eng},

month = {3},

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/194280},

volume = {39},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 4

SP - 637

EP - 648

AB -
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.

LA - eng

KW - Elasticity; limit load.; elasticity; limit load

UR - http://eudml.org/doc/194280

ER -

## References

top- R. Adams, Sobolev Spaces. Academic Press, New York (1975).
- H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983).
- P.G. Ciarlet, Lectures on the three-dimensional elasticity. Tata Institute of Fundamental Research, Bombay (1983). Zbl0542.73046
- 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
- R. Temam, Mathematical Problems in Plasticity. Bordas, Paris (1985). Zbl0457.73017

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