Linear Isotropic Relations in Finite Hyperelasticity: Some General Results

The form of the classical stress–strain relations of linear elasticity are considered here within the context of nonlinear elasticity. For both Cauchy and symmetric Piola-Kirchhoff stresses, conditions are obtained for the associated strain fields so that they are independent of the material constants and compatible with existence of a strain–energy function. These conditions can be integrated in both cases to obtain the most general strain field that satisfies these conditions and the corresponding strain–energy function is obtained. In both cases, a natural choice of form of solution is suggested by the special case of the compatibility conditions being satisfied identically. It will be shown that some strain–energy functions previously introduced in the literature are special cases of the results obtained here. Some recent linear stress–strain relations, proposed in the context of Cauchy elasticity, are examined to see if they are compatible with hyperelasticity.