Necessary and sufficient criteria for ellipticity of the equilibrium equations of a nonlinearly elastic medium
The necessary and sufficient conditions for ellipticity of the equilibrium equations of a nonlinear elastostatics are obtained in terms of constraints on the strain-energy function and on the principal elongations. In particular, to get an elliptic elasticity theory, the strain energy and elongations are to comply with a number of algebraic inequalities subject to constant-sign polynomial conditions. The results cover the case of an arbitrary anisotropic elastic body, and for particular material symmetries reduce to those discussed before by: (i) \textit{L. Zee} and \textit{E. Sternberg} [Arch. Ration. Mech. Anal. 83, 53-90 (1983; Zbl 0553.73011); (ii) \textit{P. K. Blatz} and \textit{W. L. Ko} [Trans. Soc. Rheol. 6, 223-251 (1962)]; and (iii) \textit{M. Baker} and \textit{J. L. Ericksen} [J. Washington Acad. Sci. 44, No. 2, 33-35 (1954)].
scientific article; zbMATH DE number 1077477