NECESSARY AND SUFFICIENT CRITERIA FOR ELLIPTICITY OF THE EQUILIBRIUM EQUATIONS OF A NONLINEARLY ELASTIC MEDIUM

Necessary and sufficient conditions are found for ellipticity of the e quilibrium equations of a homogeneous isotropic compressible elastic m aterial. These conditions comprise a finite system of elementary inequ alities imposing explicit constraints on the strain energy density of the material and the principal relative elongations, as well as a seri es of relationships for domains in which certain polynomials of one re al variable, whose coefficients are determined by the energy density f unction and the principal elongations, remain constant in sign. The de grees of the polynomials are one, two and six, respectively. An effect ive sufficient condition is formulated that guarantees ellipticity of the equilibrium equations and does not contain any auxiliary parameter s. It is shown that if the material satisfies certain physically plaus ible and not overly restrictive conditions, the ellipticity criterion admits of a simpler formulation, obviating the need to investigate pol ynomials.