Fourth‐order tensors – tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis

The present contribution provides a tensor formalism for fourth‐order tensors in the so‐called absolute notation and focusses in particular on the use of this notation in the process of tensor differentiation with respect to a second‐order tensor. Three tensor products, two new double contraction rules and a set of well‐defined notations are introduced which in combination with the tensor differentiation rules simplify analytical derivation procedures considerably and provide significant advantages for various tasks in continuum mechanics. The suitability of the proposed rules and definitions is demonstrated in a number of relevant problems of continuum mechanics such as linearization of the generalized midpoint‐rule and the exponential function. Special attention is given to the differentiation with respect to symmetric, skew‐symmetric and inverse second‐order tensors.

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