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Sur la faible fermeture de certains ensembles de contraintes en élasticité non-linéaire plane. (Weak closure of certain sets of constraints in plane nonlinear elasticity). (French) Zbl 0619.73014
The paper is concerned with plane-strain problems of elastostatics for hyperelastic materials and the attention is focused on such questions as the sequentially weak semicontinuity from below of the energy functional in the appropriate Sobolev space \((W^{1,p}(\Omega))^ 2\) and the properties of weak closedness of subsets of admissible stresses. While the first question was answered by J. M. Ball [ibid. 63, 337-403 (1977; Zbl 0368.73040)] and others [L. Tartar, unpublished], the second question is addressed in the paper, at least in a necessary form. Let K be the two-dimensional set of \(R^ 2\) spanned by the eigenvalues \((\lambda_ 1,\lambda_ 2)\) of the deformation gradient tensor. If the set E(K) of looked for stresses is weakly closed in \((W^{1,p}(\Omega))^ 2\), then any increasing part (resp. decreasing part) of the boundary of K by the application \((\lambda_ 1,\lambda_ 2)\to (\lambda_ 1.\lambda_ 2,\lambda_ 1+\lambda_ 2)\)- resp. \((\lambda_ 1.\lambda_ 2,\lambda_ 1-\lambda_ 2)\)- has for image the graph of a concave function. The sufficient part of the condition is left open but it might be conjectured that the given condition is also a sufficient one for ellipticity in the nonregular case. This is suggested by the fact that when the boundary of K is given by equipotentials of \(C^ 2\)-regular functions, then the result yields the necessary conditions for ellipticity due to J. K. Knowles and E. Sternberg [ibid. 63, 321-336 (1977; Zbl 0351.73061)] and the latter are known to be also sufficient. The proof uses previous results by Ball (already cited) and the notion of rank-one convexity.
Reviewer: G.A.Maugin

MSC:
74B20 Nonlinear elasticity
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
76N20 Boundary-layer theory for compressible fluids and gas dynamics
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[1] J. M. Ball, Existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis, 63, 337–403, 1976. · Zbl 0368.73040 · doi:10.1007/BF00279992
[2] L. Tartar, Cours Peccot Collège de France 1977, et cours 3ème cycle Fac. Sciences, Orsay, 1977–78.
[3] J. K. Knowles & E. Sternberg, On the failure of ellipticity of the equation of the finite elastostatic plane strain. Archive for Rational Mechanics and Analysis, 63, 321–336, 1976. · Zbl 0351.73061 · doi:10.1007/BF00279991
[4] G. Aubert & R. Tahraoui, Conditions nécessaires de faible fermeture et de 1-rang convexité en dimension 3. (to appear in Annali della Scuola Normale di Pisa). · Zbl 0647.73017
[5] L. Tartar, Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics (Proceedings of Heriott-Watt Symposium, IV, 136–212, R. J. Knops Editor, 1979.) · Zbl 0437.35004
[6] G. Aubert & R. Tahraoui, Sur la faible fermeture de certains ensembles de contrainte en élasticité non linéaire plane. C.R. Acad. Sci, Paris, 290, 537–540, 1980. · Zbl 0434.35021
[7] G. Aubert & R. Tahraoui, Conditions nécessaires de faible fermeture et de 1-rang convexité en dimension 3. Prépublications mathématiques n83T15, Université Paris Sud, Départment de Mathématiques, France.
[8] G. Aubert & R. Tahraoui, Condition de Legendre – Hadamard en élasticité isotropique. 16ème colloque d’Analyse Numérique (Guidel, Mai 1983) France.
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