$\sth$ and analogously $\hsth$ denotes an arbitrary surface quantity. The surface quantity can be a vector, not necessarily tangent to the surface, or a tensor, not necessarily tangential or superficial to the surface. The (conventional) identity tensor in $\mathsf{E}^3$ is denoted as $\b{i}$ in the spatial configuration and $\b{I}$ in the material configuration. In what follows the identity tensors $\b{i}$ and $\b{I}$ are understood as the conventional identity tensors in $\mathsf{E}^3$, i.e. their matrix representation would be a $3 \times 3$ matrix with 1 in the main diagonal entries and 0 elsewhere. Although these identity tensors are invariant and $\b{i} = \b{I}$, we use different letters to indicate explicitly which configuration they belong to.

Summary of the problem of surface elasticity {#sec_summary}
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The objective of this section is to recall briefly the problem of surface elasticity at finite deformations. For additional details the reader is referred to @Gurtin1975 [@Javili-in-review; @Javili2013], among others.

Kinematics {#sec_kinematics}
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Consider a continuum body that takes the material configuration $\c{B}_0$ at time $t=t_0$ and is mapped via the (volume) non-linear deformation map $\b{\varphi}$ to the spatial configuration $\c{B}_t$ at time $t>0$, as shown in . Material points in the material and spatial configurations are 