A rigorous derivation of constitutive laws for nonlinear behavior of compressible and incompressible elasticity is presented. As a tensor-valued function with a single tensor variable, i.e. strain, stress is expressed as a tensor polynomial that contains high order elasticity tensors. Various symmetric features of these tensors are then introduced to simplify the expression. Finally, for isotropic material the constitutive laws and strain energy functions are obtained. The expressions satisfy the criterion known as representation theorem for tensor functions and are complete and irreducible.
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