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Stress Measures: Miehe (2002) Finite strain plasticity paper

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Hi all,

Im currently reading a very interesting paper by Miehe [I]et al.[/I] [URL= ](that can be found here)[/URL] on anisotropic finite plasticity modelling. In the paper, finite deformation meaures are transformed to logarithmic strains. The constitutive reponse is obtained using the log strain. Then the resulting stresses and strains are transformed back to the finite strain space. Or at least thats what I gather.

One problem that Im having interpreting the paper is that the definition its stress measures are different to those I am used to. For example in the paper, the nominal stress is defined as [B]P[/B] and the 2nd Piola-Kirchhoff stress is defined as [B]S=F[SUP]-1[/SUP] P[/B], these are defined in Fig. 2 of the paper. In any text I have seen (example Belytschko) 2PK is defined as [B]S= PF[SUP]-T[/SUP][/B]. The definition of the Eulerian Kirchhoff stress (\tau) is also different and Im not sure where the disconnect is.

The one hint that I have is that there is mention of [I]tangent [/I]and [I]co-tangent[/I] spaces where the measures are calculated in both Eulerian and Lagrangian space (Fig 2). Im not sure if this also has something to do with covariance and contra-variance? Those are concepts which I currently dont have a firm grasp of yet.

Any help or ideas on where to look to elucidate this issue are greatly appreciated.