Fourth-order tensor calculus operations and application to continuum mechanics

Abstract

This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.