×

Conditions necessaires de faible fermeture et de 1-rang convexity en dimension 3. (Necessary conditions for weak closure and rank-one convexity in three-dimensional elasticity). (French) Zbl 0647.73017

In this interesting paper, motivated by minimization problems of nonlinear isotropic elasticity, authors study certain necessary conditions of sequentially weak closure for the domain where the candidate minimizers are sought; these conditions have the form of kinematical constraints on admissible stretches. Moreover, for rank-one convex stored energy functions of class \(C^ 2\), necessary conditions are found whose constitutive significance would perhaps deserve further investigation.
Reviewer: P.Podio-Guidugli

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74B20 Nonlinear elasticity
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aubert-Tahraoui.,Sur la faible fermeture de certains ensembles de contraintes en élasticité non linéaire plane. CRAS t. 290 série A 24 mars 1980, et à paraître dans Archive For Rational Mechanics. · Zbl 0434.35021
[2] Ball.,Existence theorems in non linear elastcity. Archive for Rational Mechanic 1976, Vol. 63.
[3] Tartar.,Compensated compactness and applications to partial differential equations. Non linear Analysis and Mechanics, Heriot Watt Symposium Vol. 4. · Zbl 0437.35004
[4] Morrey C. B.,Multiple integrals in the calculus of variations Springer Verlag. Quasi convexity and the lower semi continuity of multiple integrals. Pacific Journal of Maths2 (1952).
[5] Knowles & Sternberg.,On the failure of ellipticity of the equations of finite elastostatic plane strain. Archive For Rational Mechanics; 1976, vol. 63.
[6] Murat.; de Giorgi, E.; Magenes, E.; Mosco, U., Compacité par compensation 2, ≪Proc. Intern. Meeting on Recent Methods in Non-linear Analysis, Rome, 245-256 (1978), Bologna: Pitagora, Bologna · Zbl 0427.35008
[7] Aubert-Tahraoui.,Condition de Legendre-Hadamard en élasticité isotropique. 16° colloque d’Analyse Numérique (Guidel, 24-28 mai 1983).
[8] Aubert-Tahraoui.,Conditions nécessaires de faible fermeture et de 1-rang convexité en dimension 3. Prépublications mathématiques n^o 83 T 15—Université Paris Sud-Dép. Mathématiques. · Zbl 0647.73017
[9] Ball.,Differentiability properties of symmetric and isotropic functions Duke Mathematical Journal; 1984, vol 51, n^o 3. · Zbl 0566.73001
[10] Aubert.,Sur quelques théorèmes de caractérisation de la polyconvexité. (To appear)
[11] Tahraoui.,Existence theorems in the calculus of variations and application to non linear elasticity (To appear) · Zbl 0587.49008
[12] Ciarlet-Necas.,Unilateral problems in non linear three dimensional elasticity. Publication du laboratoire d’Analyse Numérique. Université Pierre et Marie Curie n^o 84003, 1984.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.