Tensorial inversion of fourth-order material tensor: orthotropy and transverse isotropy

In the theory of anisotropic linear elasticity, there are different approaches to define the elasticity in a coordinate-dependent relation by matrices—commonly, the Voigt-notation—or tensorial expressions using fourth-order tensors. In view of numerical treatment, for example, the finite element method, the stress state is defined by the strain state via the fourth-order elasticity tensor, \(\textbf{T} = {\pmb {\mathcal {{C}}}} \textbf{E}\). In view of analytical considerations required, for instance, for parameter identification purposes, the inverse relation \(\textbf{E} = {\pmb {\mathcal {{C}}}}^{-1} \textbf{T}\) is necessary. In this paper, the inversion of the fourth-order representation is developed in a coordinate-free representation. using the concept of invariant theory, which is based on the principal invariants and the so-called mixed invariants of the strain tensor/stress tensor. The mixed invariants are defined in terms of the structural tensors, which represent the preferred directions of the material under consideration. The advantage here is that the constitutive equations for the anisotropic material (which are invariant under the elements of the material symmetry group) can be represented as isotopic tensor functions. Thus, the compliance tensor \({\pmb {\mathcal {{C}}}}^{-1}\) can be obtained for any orientation of the anisotropy axes. We limit ourselves here to the case of transverse isotropy and orthotropy in a coordinate invariant representation.