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

Abstract

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In this work we consider a solid body Ω 3 constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces λ f and a density of forces λ g acting on the boundary where the real λ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by λ ¯ 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 K D ( x ) + I , I is the identity of 9 s y m , and the deviatoric component K D is bounded regardless of x Ω , we show under the condition “Rot f 0 or g is not colinear to the normal on a part of the boundary of Ω", that the Limit Load λ ¯ 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 λ = 1 ; moreover we show that this infimum is reached in a suitable function space.

How to cite

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Elyacoubi, 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 -

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