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Comptes Rendus Mathématique
Volume 351, n° 5-6
pages 241-245 (mars 2013)
Doi : 10.1016/j.crma.2013.03.005
Received : 9 November 2011 ;  accepted : 11 Mars 2013
Problèmes capacitaires en viscoplasticité avec effets de torsion
Capacitary problems in viscoplasticity with torsion effects
 

Michel Bellieud
 LMGC, UMR–CNRS 5508, université Montpellier-2, case courrier 048, place Eugène-Bataillon, 34095 Montpellier cedex 5, France 

Résumé

Nous étudions lʼhomogénéisation de problèmes du type :
(1){minuε∈Wb1,p(Ω;R3)Fε(uε)−∫Ωf.uεdx,f∈Lp′(Ω;R3),(1p+1p′=1),Wb1,p(Ω;R3)={ψ∈W1,p(Ω;R3),ψ=0 sur Ωˆ×{0}},Fε(uε):=∫Ω∖Trεf(e(uε))dx+kε∫Trεg(e(uε))dx,e(uε)=12(∇uε+∇Tuε), lorsque f , g sont des fonctions strictement convexes satisfaisant des conditions de croissance dʼordre  , g est positivement homogène de degré p ,   tend vers +∞, et   est une distribution ε -périodique de fibres parallèles de section de taille  . Le problème (1) correspond à un modèle simplifié dʼélasticité non linéaire en petites déformations, décrivant par exemple les petites déformations dʼun matériau de Ogden (1972) [[8]]. Lorsque  , il est aussi employé pour décrire le fluage à haute température dʼun composite métallique dans le modèle de Norton–Hoff (Friaâ, 1979 [[7]]). Dans ce cas,   représente le champ des vitesses. Nous prouvons que, lorsque  , une concentration dʼénergie élastique apparaît dans une petite zone entourant les fibres. Elle sʼexprime en fonction de la densité locale des sections des fibres relativement à une capacité spécifique tenant compte, si  , des angles de rotation des fibres par rapport à leur axe principal. Ce comportement rotatoire génère, en parallèle, une concentration dʼénergie de torsion à lʼintérieur des fibres.

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Abstract

We study the homogenization of elasticity problems like (1) when f , g are strictly convex functions satisfying a growth condition of order  , g is positively homogeneous of degree p ,  , and   consists of an ε -periodic distribution of parallel fibers of cross sections of size  . The problem (1) corresponds to a simplified model of small deformation nonlinear elasticity describing, for instance, the small deformations of an Ogdenʼs material (Ogden, 1972 [[8]]). When  , it may also characterize the viscoplastic creep experienced, at high temperatures, by a metallic composite governed by the Norton–Hoff model (Friaâ, 1979 [[7]]). In this case,   represents the velocity vector field. We show that if  , a concentration of strain energy appears in a small region of space surrounding the fibers. This extra contribution is characterized by a local density of the sections of the fibers with respect to some appropriate capacity depending, if  , on the angles of rotation of the fibers with respect to their principal axis. This rotating behavior generates, in parallel, the emergence of torsional strain energy within the fibers.

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