On the theory of fourth-order tensors and their applications in computational mechanics Original Research Article

Many problems concerned with the mathematical treatment of fourth-order tensors still remain open in the literature. In the present paper they will be considered in the framework of a complete theory involving a set of notations and definitions, a tensor operation algebra, differentiation rules, eigenvalue problems, applications of fourth-order tensors to isotropic tensor functions and some other relevant aspects. A tensor is understood to be an invariant quantity with respect to any co-ordinate system transformation, which justifies the use of absolute notation preferred in this work. As a most important application field of fourth-order tensors, elastic moduli are formulated in a material and a spatial description and given for various hyperelastic material models.