Logarithmic strain, logarithmic spin and logarithmic rate
type: Item
instance of: scholarly article
title: Logarithmic strain, logarithmic spin and logarithmic rate
zbMATH Open document ID: 0909.73006
DOI: 10.1007/BF01213020
published in: Acta Mechanica
publication date: 1999-04-12T00:00:00Z
review text: To point out the results obtained by the authors in this paper, we need to mention the following rudiments of continuum mechanics. If \(F\) is the deformation gradient of a solid deformable body, \(F=VR\) its left polar decomposition, \(\{\lambda_1,\lambda_2,\lambda_3\}\) the eigenvalues of \(V\), and \(\{n_1,n_2,n_3\}\) the corresponding subordinate orthonormal eigenvectors, then the general class of Eulerian strain measures has the form \(e=f(V)= \sum_{i=1}^3 f(\lambda_i) n_i\otimes n_i\), where the scale function \(f(\lambda)\) is a smooth monotone function satisfying the conditions \(f(1)=0\), \(f'(1)=1\). In particular, if \(f(\lambda)= \ln\lambda\), we obtain the Eulerian logarithmic strain \(\ln V=\sum^3_{i=1} (\ln\lambda_i) n_i\otimes n_i\). For a spin tensor \(\Omega^* =\dot Q^{^*T} Q^*\) defined in a rotating frame relative to a fixed background frame, and for an objective second order tensor \(G\), the tensor \({\overset \circ G}^*= \dot G+G \Omega^* -\Omega^*G\), where \(\dot G\) is the material time derivative of \(G\), is called the corotational rate of \(G\). Now we can formulate the main results of the paper: \(1^\circ\).For arbitrary choice of \(e\) and \(\Omega^*\), the equation (1) \(\dot e+e \Omega^* -\Omega^*e =D\Leftrightarrow (1)'\) \({\overset\circ e}^* =D\), where \(D={1\over 2} (L+L^T)\) is the stretching and \(L=\dot FF^{-1}\) is the velocity gradient, is consistent iff \(f(\lambda) =\ln \lambda\), i.e. iff the strain measure \(e\) is Eulerian logarithmic strain \(\ln V\). The solution for the spin of (1) with \(e=\ln V\) is referred to as the logarithmic spin, is noted by \(\Omega^{\log}_0\), and its defining equation is \((\ln^\bullet V)+(\ln V)\Omega^{\log} -\Omega^{\log} \ln V=D\Leftrightarrow (\ln^\circ V)^{\log} =D\). The corotational rate of \(G\) defined by \({\overset \circ G}^{\log}: =\dot G+G \Omega^{\log} -\Omega^{\log}G\) is referred to as the logarithmic rate of \(G\), and it is shown to be objective rate. \(2^\circ\). By usi…
Mathematics Subject Classification ID: 74A99
Mathematics Subject Classification ID: 74A20
zbMATH DE Number: 1143322
zbMATH Keywords: deformation gradient
zbMATH Keywords: left polar decomposition
zbMATH Keywords: Eulerian strain and stress measures
zbMATH Keywords: scale function
zbMATH Keywords: spin tensor
zbMATH Keywords: corotational rate
zbMATH Keywords: logarithmic spin
zbMATH Keywords: concept of observers and objectivity
zbMATH Keywords: Hill's work-conjugacy concept
zbMATH Keywords: Cauchy stress
zbMATH Keywords: left Cauchy-Green tensor
zbMATH Keywords: rate-form constitutive models
zbMATH Keywords: zeroth-grade hypoelastic model
author: Heng Xiao
author: Otto T. Bruhns
author: Albert Meyers
reviewed by: Gheorghe Gr. Ciobanu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01213020
OpenAlex ID: W1988852711
description: scientific article; zbMATH DE number 1143322