Invariant Tensor-to-Matrix Mappings for Evaluation of Tensorial Expressions

A class of mappings is presented, parameterized by a variable η, that operates on tensorial expressions to yield equivalent matrical expressions which are then easily evaluated, either numerically or symbolically, using standard matrix operations. The tensorial expressions considered involve scalar, second- and fourth-order tensors, double contractions, inversion and transposition. Also addressed is coordinate transformation and eigenanalysis of fourth-order tensors. The class of mappings considered is invariant, meaning that for a given η the corresponding mapping depends only on the order of the tensor upon which it acts and not, for example, on its physical interpretation (e.g., stress vs. strain, or stiffness vs. compliance). As a result the proposed mappings avoid ad hoc definitions like that of engineering shear strain (i.e., γ_ij := 2ε_ij for i ≠ j) which is inconsistent with an invariant mapping. Two convenient choices for the parameter η are presented. Appendix B presents a convenient summary for instructional purposes.