:= \f{\p\h{\b{f}}\trns}{\p\h{\b{F}}} = - \h{\b{f}}\trns \, \u\dyad \, \h{\b{f}} + [\b{i}-\h{\b{i}}]\, \o\dyad \, [\h{\b{f}}\cdot\h{\b{f}}\trns] \qquad , \qquad \h{\v{I}} := \f{\p\h{\b{F}}}{\p\h{\b{F}}} = \b{i} \, \o\dyad \ \h{\b{I}}\end{aligned}$$

Governing equations {#sec_governing_equations}
-------------------

The strong form of the equations governing the response of the volume and the surface (i.e. the quasi-static balances of linear momentum) are given by $$\begin{aligned}
\Div \b{P} + \b{b}_0^\text{p} = \b{0} && \text{in } \c{B}_0 \, , \label{strong_bulk} \\
\h{\Div} \h{\b{P}} + \h{\b{b}}_0^\text{p} - \b{P}\cdot\b{N} = \b{0} && \text{on } \c{S}_0 \, , \label{strong_surface}\end{aligned}$$ where $\b{b}_0^\text{p}$ is the (prescribed) body force per unit reference volume in the material configuration, and $\h{\b{b}}_0^\text{p}$ is the (prescribed) force per unit reference area of the material configuration. Note that in the absence of a surface with an independent free energy (i.e. an *energetic surface*), (\[strong\_surface\]) defines the standard Neumann boundary condition on the Piola traction, i.e. $\h{\b{b}}_0^\text{p} \equiv \b{P}\cdot\b{N}$.

The weak form is obtained by respectively testing (\[strong\_bulk\]) and (\[strong\_surface\]) with arbitrary test functions $\delta \b{\varphi} \in H^1_0(\c{B}_0)^3$ and $\delta \h{\b{\varphi}} \in H^1_0(\c{S}_0)^3$, applying the divergence theorem in its extended form [see e.g. @Javili2013], exploiting the orthogonality properties of the surface Piola–Kirchhoff stress measure (i.e. $\h{\b{P}}\cdot \b{N} = \b{0}$), and the kinematic constraint 