The representation theorem for linear, isotropic tensor functions revisited
type: Item
instance of: scholarly article
title: The representation theorem for linear, isotropic tensor functions revisited
zbMATH Open document ID: 0965.74013
DOI: 10.1023/A:1007670704239
author: Yanyan Li
published in: Journal of Elasticity
publication date: 2001-08-05T00:00:00Z
review text: Let \(V\) be a three-dimensional vector space, and let Lin denote a nine-dimensional vector space of linear maps from \(V\) to \(V\). The author shows that if \({\mathbf C}:\text{Lin} \to\text{Lin}\) is linear, invariant (under Orth) and invertible, then there exist scalars \(\mu,\alpha\), and \(\lambda \) such that \({\mathbf C}[{\mathbf H}] =\mu({\mathbf H}+ {\mathbf H}^T)+ \alpha ({\mathbf H}-{\mathbf H}^T)+ \lambda(tr {\mathbf H}) \mathbf{1}\) for every \({\mathbf H}\in \text{Lin}\). The proof is based on the decomposition formula \(\text{Lin}= \text{Skw} \oplus\text{Sph}\oplus \text{Dev}\), and on a study of linear operators in the subspaces Skw, Sph, and Dev. Here, Skw, Sph, and Dev denote the space of skew, spherical and deviatoric tensors, respectively.
reviewed by: Józef Ignaczak
Mathematics Subject Classification ID: 74B05
Mathematics Subject Classification ID: 15A72
zbMATH DE Number: 1442291
zbMATH Keywords: representation theorem
zbMATH Keywords: linear isotropic tensor functions
zbMATH Keywords: skew tensors
zbMATH Keywords: spherical tensors
zbMATH Keywords: decomposition formula
zbMATH Keywords: linear operators
zbMATH Keywords: deviatoric tensors
MaRDI profile type: Publication
full work available at URL: a:1007670704239
OpenAlex ID: W66043116
description: scientific article; zbMATH DE number 1442291