Some stability conditions for a compressible elastic material

Abstract

Two sets of restrictions on the strain-energy function for a compressible isotropic elastic material are obtained which are necessary conditions for stability of the material. These arise from the following considerations. (i) A rectangular block is subjected to a finite pure homogeneous deformation, and an infinitesimal pure homogeneous deformation with arbitrary principal directions is superposed. The dimensions in two of these principal directions are held constant. Then the incremental modulus associated with the third principal direction must be positive for stability to obtain. (ii) In the initial pure homogeneous deformation one pair of faces of the block is force-free. The superposed inifinitesimal pure homogeneous deformation has one of its principal directions normal to these faces, which remain force-free, and the principal extension ratio corresponding to another is unity. The incremental modulus corresponding to the third principal direction must be positive to obtain stability.