1
$\begingroup$

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?

$\endgroup$

1 Answer 1

0
$\begingroup$

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 This was the false assumption that I had originally made.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.