Direct relationship between the Lagrangian logarithmic strain, the Lagrangian stretching and the Lagrangian Kirchhoff stress (Q1269821)

The Lagrangian stretching tensor \(R^TDR\) cannot be represented as the material time rate of a strain measure, and the Lagrangian Kirchhoff (L-K) stress measure \((\det F)\cdot R^T\sigma R\) is not conjugate to any strain measure. The stretching tensor is directly related to the Hencky logarithmic strain measure only in some special cases, and generally \(R^TDR\) can be approximated only by the time rate of a strain measure. In a constitutive description of elastoplastic responses one has to use the logarithmic strain rate and its conjugate stress as the approximations of the Lagrangian stretching tensor and the L-K stress measure. The paper shows that a corotational rate of the Lagrangian strain measure \(\ln U\) can be identical with the stretching tensor \(R^TDR\), and that the L-K stress measure \((\det F)\cdot R^T\sigma R\) is conjugate to the logarithmic strain measure \(\ln U\) in the sense of an extended work-conjugacy notion, and that in all possible strain tensor measures, only the logarithmic strain measure has the stated favourable feature. The paper gives also a new formulation of rate-type constitutive models in terms of Lagrangian description.