Remarks on ellipticity for the generalized Blatz-Ko constitutive model for a compressible nonlinearly elastic solid

One of the most widely used constitutive models for compressible isotropic nonlinearly elastic solids is the generalized Blatz-Ko material for foam-rubber and its various specializations. For this model, a unified derivation of necessary and sufficient conditions for ellipticity of the governing three-dimensional displacement equations of equilibrium is provided. When the parameter f occurring in the generalized Blatz-Ko model is in the range 0 less than or equal to f < 1, it is shown that ellipticity is always lost at sufficiently large stretches, while for f = 1, the equilibrium equations are globally elliptic. The implications of these results for a variety of physical problems are discussed.