Certain constitutive inequalities introduced by Hill (1, 2) are considered for compressible isotropic elastic solids under finite strain, in particular in relation to Hadamard-Green materials. The inequalities are based on a class of conjugate stress and strain measures. It is shown that the inequality having the maximum degree of compatibility with these materials corresponds to logarithmic strain (so that the principal Kirchhoff stresses are jointly convex functions of logarithmic strain). A constitutive law which fits the observed behaviour of foam rubbers is shown to satisfy this inequality under restrictions confirmed by experiment. On the other hand, the Coleman-Noll inequality is violated.
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Ray W. Ogden (1970) studied this question.