= \tfrac{1}{2} \, \lambda \ln^2 J + \tfrac{1}{2} \, \mu \, [\b{F}:\b{F} - 3 - 2 \ln J] \\[8pt]
  &\text{Piola--Kirchhoff stress} && \b{P}(\b{F}) = \f{\p\Psi}{\p\b{F}} = \lambda \ln J \, \b{f}\trns + \mu \, [\b{F} - \b{f}\trns] \\[8pt]
  &\text{Piola stress tangent} && \v{A}(\b{F}) = \f{\p\b{P}}{\p\b{F}} = \lambda\, [\b{f}\trns \dyad \b{f}\trns + \ln J \, \v{D}] + \mu \, [\v{I} - \v{D}]
 \end{aligned}$$ $$\begin{aligned}
 \v{D} := \f{\p\b{f}\trns}{\p\b{F}} = - \b{f}\trns \, \u\dyad \, \b{f} \qquad , \qquad \v{I} := \f{\p\b{F}}{\p\b{F}} = \b{i} \, \o\dyad \, \b{I}\end{aligned}$$\
**Surface**: $$\begin{aligned}
 \h{J} := \h{\Det}{\h{\b{F}}} \quad \text{: Surface Jacobian determinant} && \h{\mu} \,, \h{\lambda} \quad  \text{: Surface Lam\'{e} constants} && \h{\gamma} \quad  \text{: Surface tension}  \end{aligned}$$ $$\begin{aligned}
  &\text{Surface Free energy} && \h{\Psi}(\h{\b{F}}) = \tfrac{1}{2} \, \h{\lambda} \ln^2 \h{J} + \tfrac{1}{2} \, \h{\mu} \, [\h{\b{F}}:\h{\b{F}} - 2 - 2 \ln \h{J}] + \h{\gamma} \, \h{J} \\[8pt]
  &\text{Surface Piola--Kirchhoff stress} && \h{\b{P}}(\h{\b{F}}) = \f{\p\h{\Psi}}{\p\h{\b{F}}} = \h{\lambda} \ln \h{J} \ \h{\b{f}}\trns + \h{\mu} \, [\h{\b{F}} - \h{\b{f}}\trns] + \h{\gamma}\, \h{J}\ \h{\b{f}}\trns \\[6pt]
  &\text{Surface Piola stress tangent} && \h{\v{A}}(\h{\b{F}}) = \f{\p\h{\b{P}}}{\p\h{\b{F}}} = \h{\lambda}\, [\h{\b{f}}\trns \dyad \h{\b{f}}\trns + \ln \h{J} \, \h{\v{D}}] + \h{\mu} \, [\h{\v{I}} - \h{\v{D}}] + \h{\gamma}\, \h{J}\ [ \h{\b{f}}\trns \dyad \h{\b{f}}\trns + \h{\v{D}}]
 \end{aligned}$$ $$\begin{aligned}
 \h{\v{D}} 