# Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties
## 2 The Model and Method
### Uncertain Density Modeled as an Interval Field  
In the deterministic metamaterial beam structure, the basic beam (aluminum beam) is considered to be homogeneous. Obviously, the basic beam may have spatially varying uncertainties under the influence of many uncertain factors. In order to study the effect of spatial uncertainties on the band gap, an interval field model based on KL expansion is used to deal with the spatially varying uncertainties. In the interval field model, the correlation between physical parameters at different positions is defined by the spatial dependency function. Then, the spatial dependency function is expanded into the form of the superposition of base functions by KL decomposition, which guarantees the spatial continuity of the physical parameters and reduces the number of interval parameters. It is assumed that the density of the basic beam has spatially varying uncertainties, and its form is:  
\[\rho^{I}(x)=[\underline{\rho}(x),\overline{\rho}(x)],x\in[0,L_{\text{beam}}] \tag{6}\]  
Here, the superscript \(I\) represents the interval quantities; \(\underline{\rho}(x)\) and \(\overline{\rho}(x)\) denote the lower and upper bounds of the interval field, respectively; and \(L_{\text{beam}}=N_{u}a\) represents the length of the basic beam. The midpoint \(\rho_{0}\) and deviation amplitude \(\Delta\rho(x)\) of the interval field can be expressed by Equation (7).  
\[\rho_{0}=\frac{\underline{\rho}(x)+\overline{\rho}(x)}{2};\Delta\rho(x)=\frac{ \overline{\rho}(x)-\underline{\rho}(x)}{2} \tag{7}\]  
The interval field can also be written in dimensionless form:  
\[\rho^{I}(x)=\rho_{0}\Big{[}1+B^{I}(x)\Big{]},x\in[0,L_{\text{beam}}] \tag{8}\]  
Here, \(B^{I}(x)\) represents a dimensionless interval function whose midpoint is zero. The spatial dependency function \(\Gamma_{B}\big{(}x_{i},x_{j}\big{)}\) of \(B^{I}(x)\) at different positions \(x_{i}\) and \(x_{j}\) can be defined as:  
\[\Gamma_{B}\big{(}x_{i},x_{j}\big{)}=\text{mid}\Big{\{}B^{I}(x_{i})B^{I}\big{(} x_{j}\big{)}\Big{\}}=\frac{\text{mid}\big{\{}\rho^{I}(x_{i})\rho^{I}\big{(}x_{j} \big{)}\big{\}}}{\rho_{0}^{2}}-1 \tag{9}\]  
Here, \(\text{mid}\{\bullet\}\) indicates the midpoint symbol. \(B^{I}(x)\) can be expanded using KL expansion technology:  
\[\Gamma_{B}\big{(}x_{i},x_{j}\big{)}=\sum_{n=1}^{+\infty}\lambda_{n}\psi_{n}(x _{i})\psi_{n}\big{(}x_{j}\big{)}\Rightarrow\Gamma_{B}(x_{i},x_{i})=\text{mid }\bigg{\{}\Big{(}B^{I}(x_{i})\Big{)}^{2}\bigg{\}}=\sum_{n=1}^{+\infty}\lambda_ {n}\psi_{n}(x_{i})^{2} \tag{10}\]  
where \(\lambda_{n}\) represents the \(n\)th eigenvalue of \(\Gamma_{B}\big{(}x_{i},x_{j}\big{)}\) and \(\psi_{n}(x)\) is the corresponding eigenfunction. According to Equations (9) and (10), and retaining the first \(N\) terms in KL expansion, we can obtain:  
\[B^{I}(x)=\sum_{i=1}^{N}\sqrt{\lambda_{i}}\psi_{i}(x)[\varepsilon_{i}] \tag{11}\]where \([e_{i}]\) is an interval parameter \([-1,1]\), whose deviation amplitude \(\Delta\varepsilon_{i}\) is equal to 1. Substituting Equation (11) into Equation (8) gives  
\[\rho^{I}(x)=\rho_{0}(1+\sum_{i=1}^{N}\sqrt{\lambda_{i}}\psi_{i}(x)[e_{i}]) \tag{12}\]  
In this paper, we assume that \(\Gamma_{B}\big{(}x_{i},x_{j}\big{)}\) has the following exponential form:  
\[\Gamma_{B}\big{(}x_{i},x_{j}\big{)}=C_{B}^{2}\exp\left(-\frac{\mid x_{i}-x_{j} \mid}{l_{B}}\right) \tag{13}\]  
Here, \(C_{B}\) and \(l_{B}\) are parameters affecting the deviation amplitude and the spatial dependency of the interval field, respectively.