## 2 On the Steigmann-Ogden Surface Elasticity  
Let us briefly introduce the Steigmann-Ogden surface elasticity in the case of infinitesimal deformations. We introduce the displacement vector as a differentiable vector function of the position vector \(\mathbf{x}\)  
\[\mathbf{u}=\mathbf{u}(\mathbf{x}).\]  
The constitutive relations of a hyperelastic solid can be expressed through the bulk \(\mathcal{W}\) and surface \(\mathcal{U}\) strain energies as follows  
\[\mathcal{W}=\mathcal{W}(\mathbf{e}),\quad\mathcal{U}=\mathcal{U}(\mathbf{\varepsilon },\mathbf{\kappa}). \tag{2}\]  
Here, the strain measures in the bulk are given by  
\[\mathbf{e}=\mathbf{e}(\mathbf{u})= \tfrac{1}{2}(\
abla\mathbf{u}+\
abla\mathbf{u}^{T}),\]  
whereas their surface counterparts take the form  
\[\mathbf{\varepsilon}=\mathbf{\varepsilon}(\mathbf{u})= \tfrac{1}{2}(\
abla_{s}\mathbf{u}\cdot\mathbf{P}+\mathbf{P}\cdot \
abla_{s}\mathbf{u}^{T}),\quad\mathbf{\kappa}=\mathbf{\kappa}(\mathbf{u})=\tfrac{1}{ 2}(\
abla_{s}\mathbf{\vartheta}\cdot\mathbf{P}+\mathbf{P}\cdot\
abla_{s}\mathbf{ \vartheta}^{T}),\quad\mathbf{\vartheta}=\mathbf{\vartheta}(\mathbf{u})=\
abla_{s}w+ \mathbf{B}\cdot\mathbf{u}.\]  
Here, we introduce the spatial \(\
abla\) and surface \(\
abla_{s}\) nabla operators, \(\
abla_{s}=\mathbf{P}\cdot\
abla\), \(\mathbf{P}=\mathbf{I}-\mathbf{n}\otimes\mathbf{n}\), where \(\mathbf{I}\) is the 3D unit tensor, \(``(...)^{T}\"\) denotes the transpose of a second-order tensor \((...)\), and \(``\cdot\"\) and \(``\otimes\"\) stand for the scalar and dyadic products, respectively. In addition, \(w=\mathbf{u}\cdot\mathbf{n}\), \(\mathbf{B}=-\
abla_{s}\mathbf{n}\) is the tensor of curvature and \(\mathbf{n}\) is the normal outward unit to the body boundary \(S\equiv\partial V\). Hereinafter, we use the direct (coordinate-free) tensor calculus as in [23; 24]. In what follows we consider isotropic solids, so \(\mathcal{W}\) and the stress tensor \(\sigma\) are defined by  
\[\mathcal{W}=\tfrac{1}{2}\mathbf{e}:\mathbf{C}:\mathbf{e},\quad\mathbf{\sigma}= \frac{\partial\mathcal{W}}{\partial\mathbf{e}}=\mathbf{C}:\mathbf{e},\]where \(\mathbf{C}\) is a forth-order elastic moduli tensor given by  
\[\mathbf{C}=C_{ijkl}\mathbf{i}_{i}\otimes\mathbf{i}_{j}\otimes\mathbf{i}_{k} \otimes\mathbf{i}_{l},\quad C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu\left( \delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right), \tag{3}\]  
where \(\delta_{ij}\) is the Kronecker symbol, \(\mathbf{i}_{k}\) are unit Cartesian base vectors, \(\lambda\) and \(\mu\) are the Lame elastic moduli, and \"\(\cdot\)\" stands for the inner product in the space of second-order tensors. The surface strain energy can be written in the form [9; 10; 25]  
\[\mathcal{U}=\mu^{S}\epsilon:\epsilon+\tfrac{1}{2}\lambda^{S}\text{tr}\,^{2} \epsilon+\eta\kappa:\kappa+\tfrac{1}{2}\lambda\text{tr}\,^{2}\kappa, \tag{4}\]  
where \(\lambda^{S}\) and \(\mu^{S}\) are the surface Lame moduli, \(\chi\) and \(\eta\) are additional stiffness moduli related to the bending stiffness of the material surface with surface stresses, and \(\text{tr}\) is the trace operator. By considering the determination of the surface elastic moduli in (4), we can use such estimations as  
\[\mu^{S}\approx\bar{\mu}h_{S},\quad\eta\approx\bar{\mu}h_{S}^{3},\quad\text{etc.},\]  
where \(\bar{\mu}\) is a shear modulus of a material in the vicinity of a free interface or an interface and \(h_{S}\) is the thickness of the surface/interfacial layer. In other words, we have here the so-called static characteristic length scale parameter, defined as \(\mu^{S}/\mu\). For more detail, we refer readers to [17; 19], where recent discussions on the evaluation of surface elastic moduli are provided and relative references are presented. In particular, in [17; 19] it was mentioned that the linear Gurtin-Murdoch model corresponds to so-called stiff interfaces discussed earlier in [26; 27] or plates with rigid skins [28].