## 2 Methods  
### Viscoelastic Model for Acoustic Wave Propagation  
#### 2.1.1 Viscoelastic Model Construction of Fluororubber  
The mechanical properties of rubber are commonly described using phenomenological approaches based on continuum mechanics, treating rubber as an isotropic hyperelastic material. Existing rubber constitutive models, such as Mooney-Rivlin, neo-Hookean, Yeoh, Ogden, and Arruda-Boyce, describe the stress-strain relationship of rubber under large strains (\(\varepsilon>30\%\)). However, the elastic response of rubber under small strains (\(\varepsilon<5\%\)) is essentially Hookean [6; 20]. Therefore, hyperelastic constitutive models are not suitable for harmonic vibrations at ultrasonic frequencies. The direction of the long molecular chains of fluororubber is randomly distributed, so it is also regarded as an isotropic material. Its stress-strain relationship is described according to the generalized Hooke's law as:  
\[ \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{23}\\ \sigma_{31}\end{bmatrix} = \begin{bmatrix}K+\frac{4}{3}G&K-\frac{2}{3}G&K-\frac {2}{3}G&0&0&0\\ K-\frac{2}{3}G&K+\frac{4}{3}G&K-\frac{2}{3}G&0&0&0\\ K-\frac{2}{3}G&K-\frac{2}{3}G&K+\frac{4}{3}G&0&0&0\\ 0&0&0&G&0\\ 0&0&0&0&G&0\\ 0&0&0&0&0&G\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{12}\\ \gamma_{23}\\ \gamma_{31}\end{bmatrix}  \tag{1}\]  
where \(K\) represents the material bulk modulus and \(G\) represents the material shear modulus. Ultrasonic waves in fluororubber experience rapid attenuation, dispersion, and backscatter phenomena. Therefore, the constitutive model needs to account for the viscosity coefficient and the relaxation processes of the internal microstructure. Considering all factors, as shown in the Figure 1a, this paper treated it as a viscoelastic medium that is uniform in the depth direction and follows the generalized Maxwell model. The choice of a modified Maxwell model consisting of a spring, dashpot, and a parallel spring-dashpot combination was made to better capture the viscoelastic behavior of fluororubber under small-strain conditions. This model effectively accounts for both the elastic and viscous components of the material's response, which is essential to accurately model ultrasonic wave propagation and attenuation. The parallel spring-dashpot configuration is more suitable than the standard series arrangement as it allows for a detailed representation of the frequency-dependent damping and elasticity observed in fluororubber during ultrasonic testing. This modification ensures that the model can better describe the material's behavior in high-frequency, small-strain scenarios typically encountered in ultrasonic nondestructive evaluation. The governing equation for the stress-strain response is then given by:  
\[\eta_{0}\eta_{1}\frac{d^{2}\varepsilon}{dt^{2}}+(k_{1}\eta_{1}+k_{2}\eta_{1}+ k_{1}\eta_{0})\frac{d\varepsilon}{dt}+k_{1}k_{2}\varepsilon=\eta_{1}\frac{d \sigma}{dt}+k_{1}\sigma \tag{2}\]  
where \(k_{1}\) is the stiffness coefficient of spring 1 in Figure 1a, \(k_{2}\) is the stiffness coefficient of spring 2, and \(\eta_{0}\) and \(\eta_{1}\) correspond to the viscosity coefficients of viscous element 0 and viscous element 1, respectively. The equation describes a viscoelastic system involving a fluororubber mechanical structure that is subject to both elastic and viscous forces. The first derivative of strain \(\frac{dr}{dt}\) indicates the rate of deformation, while the second derivative \(\frac{d^{2}\varepsilon}{dt^{2}}\) shows how the material's deformation is accelerating or decelerating. The term involving stress \(\frac{d\sigma}{dt}\) reflects how changes in stress impact the system over time. The model includes both the shear viscosity coefficient \(\eta^{\prime}\) and volumetric viscosity coefficient \(\eta^{\prime\prime}\), and these are unified and represented as \(\eta_{1}\). According to Equation (1), this can be expressed as \(\eta_{1}=\frac{4}{3}\eta^{\prime}+\eta^{\prime\prime}\).