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Archive of Applied Mechanics

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21.09.2021 | Original

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

verfasst von: David C. Kellermann, Mario M. Attard, Daniel J. O’Shea

Erschienen in: Archive of Applied Mechanics | Ausgabe 12/2021

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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.

Vorheriger Artikel On the appearance of horseshoe chaos in a nonlinear hysteretic systems with negative stiffness

Nächster Artikel An integral shear and normal deformation theory for bending analysis of functionally graded sandwich curved beams

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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.

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

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

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Titel: Fourth-order tensor algebraic operations and matrix representation in continuum mechanics
verfasst von: David C. Kellermann
Mario M. Attard
Daniel J. O’Shea
Publikationsdatum: 21.09.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Archive of Applied Mechanics / Ausgabe 12/2021
Print ISSN: 0939-1533
Elektronische ISSN: 1432-0681
DOI: https://doi.org/10.1007/s00419-021-01926-0

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