## 4 Effective Bending Stiffness
### One-Layered Plate with Surface Stresses  
First, we recall the linear theory of plate bending without surface stresses [8]. The bending deformation of an homogeneous plate (\(N=1\)) is given by  
\[\mathbf{u}(x,y,z)=v_{1}(x,y,z)\mathbf{i}_{1}+v_{2}(x,y,z)\mathbf{i}_{2}+w(x,y )\mathbf{i}_{3}, \tag{11}\]  
where  
\[v_{1}=-z\frac{\partial w}{\partial x},\quad v_{2}=-z\frac{\partial w}{ \partial y}\]  
are in-plane displacements and \(w\) is a deflection. Under plane stress conditions, the 2D strain energy density takes the form  
\[\mathbb{E}_{B}\equiv\int\limits_{-h/2}^{h/2}\mathcal{W}\,\mathrm{d}\zeta= \frac{1}{2}D\left[\left(\frac{\partial^{2}w}{\partial x^{2}}\right)^{2}+\left( \frac{\partial^{2}w}{\partial y^{2}}\right)^{2}+2w\frac{\partial^{2}w}{ \partial x^{2}}\frac{\partial^{2}w}{\partial y^{2}}\right], \tag{12}\]  
where \(D\) is defined in (1).  
By assuming cylindrical bending with \(w=w(x)=\frac{1}{2}\kappa x^{2}\), we have the formulae  
\[\mathbf{u}= -z\frac{\partial w}{\partial x}\mathbf{i}_{1}+w\mathbf{i}_{3}=-z xx\mathbf{i}_{1}+\frac{1}{2}\kappa x^{2}\mathbf{i}_{3}, \tag{13}\] \[\
abla\mathbf{u}= -\frac{\partial w}{\partial x}\mathbf{i}_{3}\otimes\mathbf{i}_{1 }-z\frac{\partial^{2}w}{\partial x^{2}}\mathbf{i}_{1}\otimes\mathbf{i}_{1}+ \frac{\partial w}{\partial x}\mathbf{i}_{1}\otimes\mathbf{i}_{3}=-\kappa x \mathbf{i}_{3}\otimes\mathbf{i}_{1}-z\kappa\mathbf{i}_{1}\otimes\mathbf{i}_{1 }+\kappa x\mathbf{i}_{1}\otimes\mathbf{i}_{3},\] (14) \[\mathbf{e}= \varepsilon=-z\frac{\partial^{2}w}{\partial x^{2}}\mathbf{i}_{1} \otimes\mathbf{i}_{1}=-\kappa z\mathbf{i}_{1}\otimes\mathbf{i}_{1},\] (15) \[\boldsymbol{\vartheta}= \frac{\partial w}{\partial x}\mathbf{i}_{1}=\kappa x\mathbf{i}_ {1},\quad\boldsymbol{\kappa}=\frac{\partial^{2}w}{\partial x^{2}}\mathbf{i}_{ 1}\otimes\mathbf{i}_{1}=\kappa\mathbf{i}_{1}\otimes\mathbf{i}_{1}. \tag{16}\]  
With these formulae, we get the simplified formula for \(\mathbb{E}_{B}\)  
\[\mathbb{E}_{B}=\frac{1}{2}D\kappa^{2}.\]  
Thus, \(D\) can be calculated as follows  
\[D=\frac{\partial^{2}\mathbb{E}_{B}}{\partial\kappa^{2}}. \tag{17}\]  
Now let us consider surface energy functional \(\mathbb{S}\) given by (9) for \(N=1\). We get  
\[\mathbb{E}_{S}= \mathcal{U}_{0}\bigg{|}_{z=-h/2}+\mathcal{U}_{1}\bigg{|}_{z=h/2}\] \[= \frac{1}{2}\left[(2\mu_{0}^{S}+\lambda_{0}^{S})\frac{h^{2}}{4}+2 \eta_{0}+\chi_{0}\right]\kappa^{2}+\frac{1}{2}\left[(2\mu_{1}^{S}+\lambda_{1} ^{S})\frac{h^{2}}{4}+2\eta_{1}+\chi_{1}\right]\kappa^{2}. \tag{18}\]  
As a result, the total 2D surface strain density takes the following form  
\[\mathbb{E}\equiv\mathbb{E}_{B}+\mathbb{E}_{S}=\frac{1}{2}D_{\text{eff}}\kappa ^{2}, \tag{19}\]  
where the effective bending stiffness is given by  
\[D_{\text{eff}}=D+(2\mu_{0}^{S}+\lambda_{0}^{S})\frac{h^{2}}{4}+2\eta_{0}+\chi _{0}+(2\mu_{1}^{S}+\lambda_{1}^{S})\frac{h^{2}}{4}+2\eta_{1}+\chi_{1}. \tag{20}\]  
Obviously, the surface bending stiffness parameters \(\eta_{i}\) and \(\chi_{i}\) make a contribution to the effective bending stiffness, as it should be. Equation (20) contains also the expression of the effective bending stiffness obtained within the Gurtin-Murdoch surface elasticity [32]. Dimensionless bending stiffness as function of the plate thickness \(h\) is given in Figure 2. Here, \(\overline{D}=D_{\text{eff}}/D\) and we define the characteristic length \(l\) as in the case of the Gurtin-Murdoch model \(l=(\lambda^{S}+2\mu)h^{2}/D\). Obviously, the surface bending stiffness, taken into account within the Steigmann-Ogden model, leads to an increase in the effective bending stiffness in comparison with the Gurtin-Murdoch model or the classic Kirchhoff bending stiffness. This influence is essential when \(h\lesssim 5l\).