\, , \quad  \h{g}_{\alpha\beta} = | [\h{\b{g}}_\alpha \times \h{\b{g}}_\beta]| = \sqrt{\, \h{g}}\; \h{e}_{\alpha\beta} \, , \quad  \h{g}\,{}^{\alpha\beta} = |[\h{\b{g}}{}^\alpha \times \h{\b{g}}{}^\beta]| = [\sqrt{\, \h{g}}]\inv\; \h{e}_{\alpha\beta}\\[18pt]
    \h{\b{u}} \times \h{\b{v}} = [ \h{\b{u}} \dyad \h{\b{v}}] : \h{\v{g}} \, , \, \h{\b{u}} \cdot \h{\b{v}} = [ \h{\b{u}} \dyad \h{\b{v}}] : \h{\b{i}} \, , \,
    \h{\b{i}} = {\delta}^\alpha_\beta \, \h{\b{g}}_\alpha\dyad\h{\b{g}}{}^\beta = \h{\b{g}}_\alpha\dyad\h{\b{g}}{}^\alpha = \h{\b{g}}_1\dyad\h{\b{g}}{}^1 + \h{\b{g}}_2\dyad\h{\b{g}}{}^2 = \b{i} - \b{n}\dyad\b{n} 
 \end{array}$$

Constitutive relations
----------------------

A hyperelastic, neo-Hookean material model is assumed for the constitutive response of the volume (see Summary \[constiutive\_relations\]). The free energy $\Psi = \Psi(\b{F})$ is parametrised by the Lamé moduli $\lambda$ and $\mu$. The constitutive relation on the surface is chosen to mimic that in the volume. In addition to the neo-Hookean type surface hyperelastic response, the surface free energy $\h{\Psi} = \h{\Psi}(\h{\b{F}})$ accounts for surface tension via the parameter $\h{\gamma}$. The contribution of surface tension renders the surface free energy non-zero in the material configuration. This has implications for the resulting numerical scheme. For a study of the admissible range for the surface material parameters, see @Javili2012c.

**Volume**: $$\begin{aligned}
 J := \Det{\b{F}} \quad \text{: Jacobian determinant} && \mu \,, \lambda \quad  \text{: Lam\'{e} constants} \end{aligned}$$ $$\begin{aligned}
  &\text{Free energy} && \Psi(\b{F}) 