Tensorial Character of Stress or Strain and the Experimental Constitutive Equation

The well known classical approach of tensorial constitutive laws, in the isotropic case, shows a quadratic constitutive law as a possible relation between stress and strain tensors [1]. This law is obtained as a direct consequence of the Hamilton-Cayley theorem for 3 × 3 matrices and the hypothesis of existence of an energy function [2]. Now, from the experimental point of view, the most reliable facts are, undoubtedly, those related to uniaxial loading tests. These experiments only indicate a certain functionality between stress and strain, having usually a representation by integral functions. Such a representation is quadratic only in special cases, and these cases are extremely useful in discerning some special materials.