Nonlinear isotropic constitutive laws: choice of the three invariants, convex potentials and constitutive inequalities

The aim of this paper is a new formulation of nonlinear isotropic constitutive laws. Our main hypothesis claims that the eigenvalues of stress and strain tensors are classified in the same order (the eigenvector associated to the highest eigenvalue of the stress tensor is also associated to the highest eigenvalue of the strain tensor, etc.). Further, we assume the existence of a differentiable convex isotropic potential. By introducing three new invariants for each tensor (called X, Y, Z for the stress tensor S and x, y, z for the strain tensor E) a constitutive law is revealed to be a simple duality between the chosen invariants: (x, y, z) and (X, Y, Z) look like Cartesian coordinates of E and S. We look at several potentials chosen as polynomials of these invariants. Finally, first and third order isotropic elasticity laws are reviewed and convexity of the potentials is discussed.