## 2 Fundamental Theory
### A Generalized Fractional-Order Model for Viscoelastic ABH Composite Beams  
The structure consists of equal-thickness viscoelastic and inhomogeneous layers, as shown in Figure 2. \(L\) is length of the ABH part, \(L_{0}\) is the length of the uniform part, \(x_{0}\) represents the uncoated length on the ABH part, \(h\) denotes the terminal thickness of the ABH part, \(h_{\text{f}}\) is the thickness of an initial section of the ABH part, and \(h_{0}(x)\) is the coating thickness. Since fractional-order constitutive models can more accurately capture the properties of viscoelastic materials, a generalized fractional dynamic model for viscoelastic ABH composite beams is proposed. Figure 1: The ABH structure. Figure 2: Viscoelastic ABH beam with fixed-thickness film coverage. The deformation form of the two layers can be approximately considered to be consistent because the upper membrane surface is soft relative to the base. In the modeling process, the whole structure is divided into a series of micro-segments, each of which is a uniform structure. It is assumed that the approximate deformation form of a micro-fracture of the structure is shown in Figure 3.  
According to the classical theory, the displacement field of the composite beam under small deformation is as follows:  
\[ \begin{array}{ll}u_{1}(x,z,t)=-z\frac{\partial w}{\partial x},&\frac{h_{2}} {2}-D\leq z\leq\ \frac{h_{2}}{2}-D+h_{1}\\ u_{2}(x,z,t)=-z\frac{\partial w}{\partial x},&-\frac{h_{2}}{2}-D\leq z\leq\ \frac{h_{2}}{2}-D \end{array} \, \tag{3}\]  
where \(u_{1}(x,z,t)\) is the displacement of the coating layer, \(u_{2}(x,z,t)\) represents the thickness of the base layer, \(w\) is the lateral displacement of the midplane of the beam, \(z\) denotes the distance from the plane to the neutral plane, and \(D\) is the distance from the neutral plane to the midplane of the base. The corresponding strain relationship can be obtained from the displacement of each layer:  
\[\epsilon_{i}(x,z,t)=\frac{\partial u_{i}(x,z,t)}{\partial x},\ \ i=1,2, \tag{4}\]  
Since the material of the upper surface is viscoelastic material, fractional calculus is very suitable for describing the properties of it due to its genetic memory characteristic. In addition, many studies [27; 28] show that fractional calculus theory is very suitable to describe the properties of viscoelastic materials. It is assumed that the stress-strain relationship of viscoelastic material satisfies the fractional Kelvin-Voigt constitutive model, and the stress-strain relationship of the base material satisfies [PERSON]'s law; the equations of the strain-stress relationship are as follows:  
\[ \begin{array}{ll}\sigma_{1}(x,z,t)=E_{1}\epsilon_{1}(x,z,t)+\mathbf{\eta}_{1} \frac{\alpha^{\prime}\epsilon_{2}(x,z,t)}{dt^{\alpha}},&\frac{h_{2}}{2}-D\leq z \leq\ \frac{h_{2}}{2}-D+h_{1}\\ \sigma_{2}(x,z,t)=E_{2}\epsilon_{2}(x,z,t),&-\frac{h_{2}}{2}-D\leq z\leq\ \frac{h_{2}}{2}-D\end{array} \, \tag{5}\]  
where \(E_{1}\) and \(E_{2}\) are the elastic modulus of the viscoelastic material and the base material, respectively; \(\eta_{1}\) is the comprehensive damping coefficient and \(\alpha\) is the fractional order. The \(\alpha\)-order (0 < \(\alpha\) < 1) fractional derivative satisfied the _Riemann-Liouville_ definition. The entire membrane-covered beam has no axial external force, and the following formula is obtained:  
\[\Sigma F=\int_{-h_{2}/2-D}^{h_{2}/2-D}\sigma_{1}dA_{1}+\int_{h_{2}/2-D}^{h_{2} /2-D+h_{1}}\sigma_{2}dA_{2}=0, \tag{6}\]  
Figure 3: Deformation form of a micro-fracture of the film-covered beams.