Use of some theorems related with the tensor equation AX + XA = H for some classes of implicit constitutive relations

The implicit function theorem is used to obtain some necessary and sufficient conditions, for which an implicit constitutive relation can be solved in terms of the stresses and strains, in the sense of expressing, for example, the strains in terms of the stresses or vice versa. For an isotropic body some exact solutions for the strains in term of the stresses are presented, using the theory of tensor equations of the form AX + XA = H. For the exact solutions considered, conditions for the existence and the uniqueness are given.