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Fourth-order tensor algebraic operations and matrix representation in continuum mechanics

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Abstract

This paper presents a system of cyclic tensor algebra for operations involving fourth-order tensors. The advantages are that the system is objectively and consistently defined in three ways that each fall into one of three universal classes. Operators within a given class are called conjugate operators such that many familiar and fundamental identities of scalars and second-order tensors are maintained in fourth order; this provides greater insight along with anthropological and pedagogical advantages over current systems, while also revealing new identities and solutions. The relationship between operators of a different class is such that a property of cyclic symmetry arises whereby mixed-class product operators can be cycled around without invalidating an equation. In defining this system, we have considered the following: preservation from identities in zeroth- (scalar) and second-order tensor identities to fourth-order tensor identities; possible permutations of definitions and subsequent logical restrictions; the visual notational consistency throughout the system; and maintenance to legacy definitions and operator symbols. Additionally, we present many new and useful algebraic identities and provide a comparison to some selected contemporary systems used in the literature. We also provide, to complete at least a basic exposition of our proposed system, a set of identities for matrix-equivalent operations, which facilitate programming for numerical computing. This article is designed to be used as a reference work for anyone choosing to adopt this system of tensor operations in continuum mechanics theory involving fourth-order tensors.

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Notes

  1. Author’s note: we put exhaustive effort into maintaining this consistency, and it required the re-definition of the circledot operator for the tensor product.

  2. Note that Itskov’s circledot is our circlestar for previously detailed reasons.

  3. Having chosen the right-handed rather than the left-handed definitions (not to be confused with left and right contractions).

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  1. Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, 2052, Australia

    Mario M. Attard & Daniel J. O’Shea

  2. School of Mechanical and Manufacturing Engineering, UNSW Sydney, Sydney, Australia

    David C. Kellermann

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Kellermann, D.C., Attard, M.M. & O’Shea, D.J. Fourth-order tensor algebraic operations and matrix representation in continuum mechanics. Arch Appl Mech 91, 4631–4668 (2021). https://doi.org/10.1007/s00419-021-01926-0

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