The following table lists conversion rules for the three stress measures
  • first Piola-Kirchhoff stress $\tnsr P$,
  • second Piola-Kirchhoff stress $\tnsr S$, and
  • Cauchy stress $\tnsr \sigma$,
based on the multiplicative decomposition of the deformation gradient $\tnsr F$ into elastic $\tnsr F_\text e$ and plastic part $\tnsr F_\text p$.

  $\tnsr P$ $\tnsr S$ $\tnsr\sigma$
$\tnsr P=$ $\tnsr P$ $\operatorname{det}(\tnsr F_\text p) \tnsr F_\text e \tnsr\sigma {\tnsr F_\text p}^{-\text T}$ $\operatorname{det}(\tnsr F) \tnsr\sigma {\tnsr F}^{-\text T}$
$\tnsr S=$ $\frac{1}{\operatorname{det}(\tnsr F_\text p)} {\tnsr F_\text e}^{-1} \tnsr P {\tnsr F_\text p}^{\text T}$ $\tnsr S$ $\operatorname{det}(\tnsr F_\text e) {\tnsr F_\text e}^{-1} \tnsr\sigma {\tnsr F_\text e}^{-\text T}$
$\tnsr\sigma=$ $\frac{1}{\operatorname{det}(\tnsr F)} \tnsr P {\tnsr F}^{\text T}$ $\frac{1}{\operatorname{det}(\tnsr F_\text e)} \tnsr F_\text e \tnsr S {\tnsr F_\text e}^{\text T}$ $\tnsr\sigma$
Topic revision: r1 - 12 Dec 2013, ChristophKords

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