#### Vol. 3, No. 2, 2008

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Eulerian conjugate stress and strain

### Andrew N. Norris

Vol. 3 (2008), No. 2, 243–260
##### 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 $f\left(V\right)$ 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 $lnV$. The formulation and the results emphasize the similarities in form of the Eulerian and Lagrangian stresses conjugate to the strains $f\left(V\right)$ and $f\left(U\right)$, respectively. Many of the results involve the solution to the equation $AX-XA=Y$, which is presented in a succinct format.

##### Keywords
conjugate, Eulerian, stress, logarithmic strain rate, Hencky, corotational