# Simulation of Three Constitutive Behaviors Based on Nonlinear Ultrasound
## 2 Classical Nonlinear Elasticity
### Nonlinear Wave Equation  
This subsection will begin with [PERSON]'s law and simply study the source of classical acoustic nonlinearity through theoretical analysis.  
This is the general expression for the free energy of a deformed isotropic body [27]:  
\[F=F_{0}+\frac{1}{2}\lambda\varepsilon_{ii}^{2}+\mu\varepsilon_{ik}^{2} \tag{1}\]  
where, \(\varepsilon_{ik}\) is the strain tensor whose linear expression is [27]:  
\[\varepsilon_{ik}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{k}}+\frac{ \partial u_{k}}{\partial x_{i}}) \tag{2}\]  
When nonlinearity is considered, the strain tensor should be expanded [27]:  
\[\varepsilon_{ik}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{k}}+\frac{ \partial u_{k}}{\partial x_{i}}+\frac{\partial u_{l}}{\partial x_{k}}\frac{ \partial u_{l}}{\partial x_{i}}) \tag{3}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline Free energy & \(F\) \\ Lamé coefficients & \(\lambda\,,\,\mu\) \\ Strain tensor & \(\varepsilon_{ik}\) \\ Displacement component & \(u_{i}\) \\ Cartesian coordinate & \(x_{i}\) \\ Quadratic nonlinear perturbation coefficient & \(\beta\) \\ Cubic nonlinear perturbation coefficient & \(\epsilon\) \\ Hysteresis & \(\alpha\) \\ Simplified first perturbation coefficient & \(C_{1}(=-\frac{1}{2}E_{0}\beta)\) \\ Simplified second perturbation coefficient & \(C_{2}(=-\frac{1}{2}E_{0}e)\) \\ Simplified nonlinear hysteretic coefficient & \(C_{3}(=\frac{1}{2}E_{0}\alpha)\) \\ Young’s modulus & \(\dot{E}\) \\ Kronecker symbol & \(\delta_{ij}\) \\ Strain rate & \(\dot{\epsilon}=d\epsilon/dt\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main symbols used in the equation.  
If only a one-dimensional situation is considered, the linear wave equation:  
\[\frac{\partial^{2}u}{\partial t^{2}}-c_{0}\frac{\partial^{2}u}{\partial x^{2}}=0 \tag{4}\]  
should be amended to:  
\[\frac{\partial^{2}u}{\partial t^{2}}-c_{0}(1-\beta\frac{\partial u}{\partial x} +\epsilon(\frac{\partial u}{\partial x})^{2})\frac{\partial^{2}u}{\partial x^ {2}}=0 \tag{5}\]  
There is another simpler theoretical explanation that when the material is undamaged, the relationship between stress and strain is linear:  
\[\sigma=E\epsilon \tag{6}\]  
When the material is damaged, however, the relationship between stress and strain is no longer linear:  
\[\sigma=E\epsilon(1+\beta\epsilon+\cdots) \tag{7}\]  
Substituting this relation into the one-directional wave equation:  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=\frac{\partial\sigma}{\partial x} \tag{8}\]  
Then the nonlinear wave equation is obtained:  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=A\frac{\partial^{2}u}{\partial x^{2 }}+B\frac{\partial u}{\partial x}\frac{\partial^{2}u}{\partial x^{2}}+\cdots \tag{9}\]  
[PERSON] [7; 29] et al. also established a similar second nonlinear ultrasonic wave equation:  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=K_{2}\frac{\partial^{2}u}{\partial x ^{2}}+(3K_{2}+K_{3})\frac{\partial u}{\partial x}\frac{\partial^{2}u}{\partial x ^{2}} \tag{10}\]  
Therefore, it is found that when the material constitutive relation changes, the wave equation is no longer linear. Based on this theory, when the material is damaged, the input of the system is monochromatic, and the output must contain harmonics.  
The derivation of wave equation above does not consider the nonlinear hysteresis for simplicity.