Eulerian conjugate stress and strain

TL;DR

This paper introduces new results for Eulerian conjugate stress and strain measures, highlighting the dependence on arbitrary quantities and establishing a unique spin that approximates the Cauchy stress, with implications for continuum mechanics.

Contribution

It provides a unified formulation for Eulerian conjugate stresses related to arbitrary strain measures and defines a unique f-spin that optimally approximates the Cauchy stress.

Findings

Existence of a unique f-spin for each strain measure

The f-spin reduces to the logarithmic spin for Hencky strain

Formulation emphasizes similarities between Eulerian and Lagrangian conjugate stresses

Abstract

New results are presented for the stress conjugate to arbitrary Eulerian strain measures. The conjugate stress depends on two arbitrary quantities: the strain measure f(V) and the corotational rate defined by the spin \Omega. It is shown that for every choice of f there is a unique spin, called the f-spin, which makes the conjugate stress as close as possible to the Cauchy stress. The f-spin reduces to the logarithmic spin when the strain measure is the Hencky strain log(V). The formulation and the results emphasize the similarities in form of the Eulerian and Lagrangian stresses conjugate to the strains f(V) and f(U), respectively. Many of the results involve the solution to the equation AX-XA=Y, which is presented in a succinct format.