$\partial \Omega = \partial \Omega_{\mathbf{u}} \cup \partial \Omega_{\sigma}$,
where
$\partial \Omega_{\mathbf{u}} \cap \partial \Omega_{\boldsymbol{\sigma}} = \emptyset$.
-The prescribed Cauchy traction, denoted $\mathbf{t}^\text{p}$, is applied to $ \partial \Omega_{\boldsymbol{\sigma}}$ while the motion is prescribed on the remaining portion of the boundary $\partial \Omega_{\mathbf{u}}$.
+The externally prescribed Cauchy traction, denoted $\mathbf{t}^\text{p}$, is applied to $ \partial \Omega_{\boldsymbol{\sigma}}$ while the motion (displacement) is prescribed on the remaining portion of the boundary $\partial \Omega_{\mathbf{u}}$.
The body force per unit current volume is denoted $\mathbf{b}^\text{p}$.
+(In these terms, the superscript $\text{p}$ is meant to suggest that a
+quantity is *prescribed*, not that it is related to the unknown pressure
+$\tilde p$.)
@f}
where
@f{align*}{
- D_{d \mathbf{u}} R( \mathbf{\Xi}; \delta \mathbf{\Xi})
+ D_{d \mathbf{u}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) \cdot d \mathbf{u}
&=
\int_{\Omega_0} \bigl[ \textrm{grad}\ \delta \mathbf{u} :
\textrm{grad}\ d \mathbf{u} [\boldsymbol{\tau}_{\textrm{iso}} + \boldsymbol{\tau}_{\textrm{vol}}]
\\
&\quad + \int_{\Omega_0} \delta \widetilde{p} J \mathbf{I} : \textrm{grad}\ d \mathbf{u} ~\textrm{d}V
\\
- D_{d \widetilde{p}} R( \mathbf{\Xi}; \delta \mathbf{\Xi})
+ D_{d \widetilde{p}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) d \widetilde{p}
&=
\int_{\Omega_0} \textrm{grad}\ \delta \mathbf{u} : J \mathbf{I} d \widetilde{p} ~\textrm{d}V
- \int_{\Omega_0} \delta \widetilde{J} d \widetilde{p} ~\textrm{d}V \, ,
\\
- D_{d \widetilde{J}} R( \mathbf{\Xi}; \delta \mathbf{\Xi})
+ D_{d \widetilde{J}} R( \mathbf{\Xi}; \delta \mathbf{\Xi}) d \widetilde{J}
&= -\int_{\Omega_0} \delta \widetilde{p} d \widetilde{J}~\textrm{d}V
+ \int_{\Omega_0} \delta \widetilde{J} \dfrac{\textrm{d}^2 \Psi_{\textrm{vol}}(\widetilde{J})}{\textrm{d} \widetilde{J}\textrm{d}\widetilde{J}} d \widetilde{J} ~\textrm{d}V \, .
@f}