## 3 Laminate Plate  
In what follows, using relations (6) and (7), we consider the equilibrium conditions of an \(N\)-layered plate; see Figure 1. For the plate, we introduce \(2N+1\) constitutive relations for \(N\) layers, for two faces and for \(N-1\) interfaces  
\[\mathcal{W}=\mathcal{W}_{i}(\mathbf{e}(\mathbf{u}_{i})),\quad\mathcal{U}= \mathcal{U}_{0}(\varepsilon(\mathbf{u}_{0}),\kappa(\mathbf{u}_{0})),\quad \mathcal{U}=\mathcal{U}_{i}(\varepsilon(\mathbf{u}_{i}),\kappa(\mathbf{u}_{i })),\qquad i=1,\ldots,N,\]  
where \(\mathbf{u}_{k}\) and \(k=0,\ldots,N\), are corresponding vectors of displacements. Let the \(i\)-th layer be of thickness \(h_{i}\), so the total thickness \(h\) of the plate is given by  
\[h=\sum_{i=1}^{N}h_{i}.\]  
In addition to the Cartesian coordinates \(x\), \(y\), \((x,y)\in\Omega\), \(z\in[0,h]\), with the unit base vectors \(\mathbf{i}_{1}\), \(\mathbf{i}_{2}\), \(\mathbf{i}_{3}\), for the \(i\)-th layer, we introduce a local coordinate \(\zeta_{i}\) such that  
\[\zeta_{i}=z-z_{i},\]  
where \(z_{i}\) is the coordinate of the middle plane of the \(i\)-th layer,  
\[z_{1}=\tfrac{1}{2}h_{1},\quad z_{j}=\tfrac{1}{2}h_{j}+\sum_{k=1}^{j-1}h_{k}, \quad j>1.\]  
Thus, we get \(\zeta_{i}\in[-h_{i}/2,h_{i}/2]\).  
As a result, functionals \(\mathbb{B}\) and \(\mathbb{S}\) take the form  
\[\mathbb{B}[\mathbf{u}]= \int_{V}\mathcal{W}(\mathbf{e}(\mathbf{u}))\,\,\mathrm{d}V=\int_ {0}^{h}\int_{\Omega}\mathcal{W}\,\mathrm{d}S\,\mathrm{d}z=\sum_{i=1}^{N}\int \limits_{2i-h_{i}/2}^{z_{i}+h_{i}/2}\biggl{(}\int_{\Omega}\mathcal{W}_{i}\, \mathrm{d}S\biggr{)}\,\,\mathrm{d}z\] \[= \sum_{i=1}^{N}\int\limits_{-h_{i}/2}^{h_{i}/2}\biggl{(}\int_{ \Omega}\mathcal{W}_{i}\,\mathrm{d}S\biggr{)}\,\,\mathrm{d}\zeta_{i}, \tag{8}\] \[\mathbb{S}[\mathbf{u}]= \int_{A}\mathcal{U}(\varepsilon(\mathbf{u}),\kappa(\mathbf{u}))\, \,\mathrm{d}S=\sum_{i=0}^{N}\int_{\Omega}\mathcal{U}_{i}\,\mathrm{d}S. \tag{9}\]  
Figure 1: \(N\)th layered plate of thickness \(h\). Here \(N=4\), and the \(i\)th layer has a thickness of \(h_{i}\), \(i=1,\ldots,N\).  
Let us note that, for a plate geometry, we make significant simplifications to the formulae. In particular, here we get \(H=0,\;\mathrm{d}S=\;\mathrm{d}x\,\mathrm{d}y\) and  
\[\
abla=\mathbf{i}_{1}\frac{\partial}{\partial x}+\mathbf{i}_{2}\frac{\partial}{ \partial y}+\mathbf{i}_{3}\frac{\partial}{\partial z},\quad\
abla_{s}=\mathbf{ i}_{1}\frac{\partial}{\partial x}+\mathbf{i}_{2}\frac{\partial}{\partial y}.\]  
In other words, for each surface with surface stresses, we can apply the same 2D nabla-operator. As a result, for a laminate plate, we get the equilibrium equations  
\[\
abla\cdot\mathbf{\sigma}_{i}+\rho_{i}\mathbf{f}=\mathbf{0}, \tag{10}\]  
boundary conditions on faces \(z=0\) and \(z=h\) in the simplified form of (7)  
\[\mathbf{n}\cdot\mathbf{\sigma}_{0}= \
abla_{s}\cdot[\mathbf{T}_{0}-(\
abla_{s}\cdot\mathbf{M}_{0}) \mathbf{n}]+\mathbf{t}_{0},\quad\mathbf{n}\cdot\mathbf{\sigma}_{N}=\
abla_{s} \cdot[\mathbf{T}_{N}-(\
abla_{s}\cdot\mathbf{M}_{N})\mathbf{n}]+\mathbf{t}_{N},\]  
and the compatibility conditions on the interfaces \(z=h_{i}\), \(i=1,\ldots N-1\)  
\[\mathbf{n}\cdot\llbracket\mathbf{\sigma}\rrbracket= \
abla_{s}\cdot[\mathbf{T}_{i}-(\
abla_{s}\cdot\mathbf{M}_{i}) \mathbf{n}]\,.\]  
Here, \(\llbracket(\ldots)\rrbracket\) means the discontinuity jump across an interface.  
These equations constitute an exact 3D boundary value problem for the laminate plate. For 3D-to-2D reduction, we can use various techniques resulting in a system of 2D governing equations; see, e.g., the recent review [31]. On the other hand, let us recall that the main aim of this paper is to determine the effective total bending stiffness of a laminate plate while considering surface elasticity. To this end, it is enough to consider a particular class of deformations related to pure bending. By considering \(\mathbb{L}=\mathbb{B}+\mathbb{S}\) on these deformations, we can estimate the bending stiffness as a stiffness coefficient related to changes in the mean curvature of the plate.