Necessary and sufficient criteria for ellipticity of the equilibrium equations of a non-linearly elastic medium

• Published in: Journal of applied mathematics and mechanics 1995, Vol.59 (2), p.197-208
• Main Authors: Zubov, L.M., Rudev, A.N.
• Format: Article
• Language: English
• ISSN: 0021-8928; 0021-8928
• DOI: 10.1016/0021-8928(95)00023-I

Necessary and sufficient conditions are found for ellipticity of the equilibrium equations of a homogeneous isotropic compressible elastic material. These conditions comprise a finite system of elementary inequalities imposing explicit constraints on the strain energy density of the material and the principal relative elongations, as well as a series of relationships for domains in which certain polynomials of one real variable, whose coefficients are determined by the energy density function and the principal elongations, remain constant in sign. The degrees of the polynomials are one, two and six, respectively. An effective sufficient condition is formulated that guarantees ellipticity of the equilibrium equations and does not contain any auxiliary parameters. It is shown that if the material satisfies certain physically plausible and not overly restrictive conditions, the ellipticity criterion admits of a simpler formulation, obviating the need to investigate polynomials.