Objective corotational rates and unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures

Abstract

By virtue of objective corotational rates and related corotating frames, a unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures is established, which is a natural extension of Hill's work-conjugacy relation between Lagrangean strain and stress measures. It turns out that the latter is the particular case of the former one when a corotating frame with the well-known spin Ω^R = RR^T is concerned, where R is the rotation tensor defined by the polar decomposition of the deformation gradient. The work-conjugate stress measure of an arbitrary Hill's strain measure (either Eulerian or Lagrangean) with regard to any kind of objective corotational rate is determined in the sense of the introduced unified work-conjugacy relation. The result is presented both in the principal component form and explicit basis-free form valid for all cases of the principal stretches. In particular, the intrinsic, unique relationship between Hencky's logarithmic strain measures ã‘V and ã‘U and the fundamental mechanical quantities, i.e. the Eulerian and Lagrangean stretching tensors D and R^TDR and Eulerian and Lagrangean Kirchhoff stress measures σ and R^TσR, are disclosed. It is shown that there are objective corotational rates of ã‘V and ã‘U that are identical with the Eulerian and Lagrangean stretching tensors D and R^TDR. respectively, and further that only ã‘V and ã‘U enjoy the just-stated favourable properties. As a result, the two pairs of strain and stress measures, (ã‘V, σ) and (ã‘U, R^TσR), form a work-conjugate Eulerian strain-stress pair and a work-conjugate Lagrangean strain-stress pair, respectively, in the sense of the introduced work-conjugacy relation. Finally, application of the unified work-conjugacy notion in formulating the rate - type constitutive relations is indicated.