Relation between decomposed stress measures

Question

In continuum mechanics a wide range of stress measures are used. The two which I am interested in are the Cauchy stress tensor and the second Piola-Kirchhoff stress tensor. These two stress measures can be related through:

$\mathbf{S} = \mathbf{F}^{-1} \cdot J\mathbf{\sigma}\cdot\mathbf{F}^{-T} $

Where $\mathbf{\sigma}$ is the Cauchy stress tensor, $\mathbf{F}$ is the deformation gradient, $J$ is its determinant and $\mathbf{S}$ is the second Piola-Kirchhoff stress tensor.

The Cauchy stress tensor can be decomposed into its isotropic/hydrostatic and deviatoric stress components such that:

$\mathbf{\sigma} = \mathbf{\sigma}' + \mathbf{\sigma}_{hyd}$

What I would like to confirm is that:

$\mathbf{S} = \mathbf{F}^{-1} \cdot J\mathbf{\sigma}\cdot\mathbf{F}^{-T} = \left(\mathbf{F}^{-1} \cdot J\mathbf{\sigma}'\cdot\mathbf{F}^{-T}\right)+ \left(\mathbf{F}^{-1} \cdot J\mathbf{\sigma}_{hyd}\cdot\mathbf{F}^{-T}\right) \neq \mathbf{S}' + \mathbf{S}_{hyd}$

I thought this inequality was true because, if calculated as shown above, the $\mathbf{S}_{hyd}$ term above will be a diagonal this but not a scalar matrix and therefore must contain some deviatoric components. Is this true?

Answer 1

It turns out that I was wrong. For a detailed reference see here (page 115, eq. 3.53). The equation above does give the values for the second Piola-Kirchhoff deviatoric and hydrostatic components. Though these do not have the same meanings that they do for the Cauchy stress tensor. What I mean by this is that the 2P-K deviatoric tensor is not necessarily traceless and they hydrostatic component is not necessarily a scalar matrix. This was the false assumption that I had originally made.