Logarithmic strain, logarithmic spin and logarithmic rate
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 using the concept of observers and objectivity, the authors obtain a natural extension of Hill's work-conjugacy notion [\textit{R. W. Ogden}, Nonlinear elastic deformations (1984; Zbl 0541.73044)] for Lagrangian strain and stress measures to the Eulerian strain and stress measures and conclude that the Cauchy stress \(\sigma\) and logarithmic strain \(\ln V\) form a work-conjugate pair, i.e. \(\rho\dot w=Tr(\sigma D)=Tr(\sigma (\ln {\overset\circ V})^{\log})\), where \(\dot w\) is the stress power per unit mass density. \(3^\circ\). An explicit basis-free expression for \(\Omega^{\log}\) is derived in terms of \(D\), \(W= {1\over 2} (L-L^T)\), and the left Cauchy-Green tensor \(B=V^2\). \(4^\circ\). With respect to the rate-form constitutive models relating the same kind of objective corotational rates of Eulerian stress and strain measures, the authors show that the logarithmic rate is the only possible choice, and the strain measure must be the logarithmic strain \(\ln V\) if the stretching \(D\) is used as the measure of the rate of deformation. \(5^\circ\). To illustrate the obtained results, the authors show that all finite deformation responses of the zeroth-grade hypoelastic model based on the logarithmic rate are in agreement with those of a finite elastic model. In the reviewer's opinion, this is a nicely written and very important paper which contains new basic results on continuum mechanics.
scientific article; zbMATH DE number 1143322
scientific article