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A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain. (English) Zbl 0759.73013
This article gives a concise and simple proof of the necessary and sufficient conditions of J. K. Knowles and Eli Sternberg [Arch. Ration. Mech. Anal. 63, 321-336 (1977; Zbl 0351.73061)], see also G. Aubert and R. Tahraoui [Arch. Ration. Mech. Anal. 97, 33- 58 (1987; Zbl 0619.73014)], for strong ellipticity (or Legendre-Hadamard condition) of isotropic hyperelastic materials in two dimensions (i.e., under plane strain). The proof consists in using the isotropy assumption to reduce the case of a general gradient matrix to that of the diagonal matrix of its singular values. In this case, the second derivatives of the stored energy function can be explicitly computed in terms of the derivatives of the symmetric function of the singular values that represents it due to isotropy. The strong ellipticity of the material is then shown to be equivalent to the positive definiteness of a certain $$2\times 2$$ symmetric matrix, i.e., to the positivity of its principal invariants, which imply the aforementioned necessary and sufficient conditions after a few simple algebraic manipulations.
Reviewer: H.Le Dret (Paris)

##### MSC:
 74B20 Nonlinear elasticity 35Q72 Other PDE from mechanics (MSC2000)
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##### References:
 [1] G. Aubert and R. Tahraoui, Sur la faible fermeture de certains ensembles de contraintes en elasticite non-lineaire plane, Arch. Rat. Mech. Anal. 97 (1987) 33-58. · Zbl 0619.73014 · doi:10.1007/BF00279845 [2] J.M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51 (1984) 699-728. · Zbl 1077.74507 · doi:10.1215/S0012-7094-84-05134-2 [3] J.K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rat. Mech. Anal. 63 (1977) 321-336. · Zbl 0351.73061 · doi:10.1007/BF00279991 [4] J.K. Knowles and E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics, J. Elast. 8 (1978) 329-379. · Zbl 0422.73038 · doi:10.1007/BF00049187 [5] H.C. Simpson and S.J. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Rat. Mech. Anal. 84 (1983) 55-68. · Zbl 0526.73026 · doi:10.1007/BF00251549
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