Bulk Rheologies for Elasticity#
In this section we describe the mathematic formulations of the bulk rheologies for elasticity. The bulk rheologies include
Isotropic linear elasticity,
Isotropic linear (Maxwell) viscoelasticity,
Isotropic linear generalized Maxwell viscoelasticity, and
Isotropic power-law viscoelasticity.
For the viscoelastic rheologies, we assume the viscous deformation is incompressible; consequently, we separate the stress and strain fields into deviatoric and volumetric components.
Variable |
Symbol |
Definition |
|---|---|---|
Mean stress |
\(\mathit{P}\) |
\(\frac{\mathop{\mathrm{Tr}}(\boldsymbol{\sigma})}{3}\) |
Mean strain |
\(\mathit{\theta}\) |
\(\frac{\mathop{\mathrm{Tr}}(\boldsymbol{\epsilon})}{3}\) |
Deviatoric stress |
\(\boldsymbol{\sigma}^{\mathit{dev}}\) |
\(\boldsymbol{\sigma} - \mathit{P}\mathit{\boldsymbol{I}}\) |
Deviatoric strain |
\(\boldsymbol{\epsilon}^{\mathit{dev}}\) |
\(\boldsymbol{\epsilon} - \mathit{\theta}\mathit{\boldsymbol{I}}\) |
2nd deviatoric stress invariant |
\(\mathit{J}_{2}^{\prime}\) |
\(\frac{1}{2}\boldsymbol{\sigma}^{dev}:\boldsymbol{\sigma}^{dev}\) |
2nd deviatoric strain invariant |
\(\mathit{L}_{2}^{\prime}\) |
\(\frac{1}{2}\boldsymbol{\epsilon}^{dev}:\boldsymbol{\epsilon}^{dev}\) |