Fourth-Order Tensors

  • Mikhail ItskovEmail author
Part of the Mathematical Engineering book series (MATHENGIN)


Fourth-order tensors play an important role in continuum mechanics where they appear as elasticity and compliance tensors. In this section we define fourth-order tensors and learn some basic operations with them. To this end, we consider a set \(\varvec{\mathcal {L}}\text {in}^n\) of all linear mappings of one second-order tensor into another one within \(\mathbf {L}\text {in}^n\). Such mappings are denoted by a colon as
$$ \mathbf {Y}=\varvec{\mathcal {A}} : \mathbf {X}, \quad \varvec{\mathcal {A}}\in \varvec{\mathcal {L}}\text {in}^n, \; \mathbf {Y}\in \mathbf {L}\text {in}^n, \; \forall \mathbf {X}\in \mathbf {L}\text {in}^n.$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Continuum MechanicsRWTH Aachen UniversityAachenGermany

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