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

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 de®nitions, a tensor operation algebra, di€erentiation 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 justi®es the use of absolute notation preferred in this work. As a most important application ®eld of fourth-order tensors, elastic moduli are formulated in a material and a spatial description and given for various hyperelastic material models. Ó 2000 Elsevier Science S.A. All rights reserved.