# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 2 Theoretical Basis  
### Nonlinear Ultrasonic Guided Waves  
When the elastic solid deforms, the strain energy will be produced in the solid. The relationship between the strain energy, \(W\), and the second Piola-Kirchhoff stress tensor, \(T_{ij}\), is given by the following:  
\[T_{ij}=\frac{\partial W}{\partial E_{ij}} \tag{1}\]  
where \(E_{ij}=\frac{1}{2}\Big{(}u_{i,j}+u_{j,i}+u_{k,i}u_{k,j}\Big{)}\), \(u_{i,j}=\frac{\partial u_{i}}{\partial x_{j}}\), \(u\) is the particle displacement vector, \(x\) is the material spatial coordinates, and \(i,j,k=x,y,z\). Substitute the widely used constitutive model \(W\) that was proposed by [PERSON] and [PERSON] [33; 34] into Equation (1), and the expression of \(T_{ij}\) about \(E_{ij}\) is as follows:  
\[T_{ij}=\lambda E_{ik}\delta_{ij}+2\mu E_{ij}+\delta_{ij}(CE_{ik}E_{ll}+BE_{il}E _{lk})+2 BE_{ik}E_{ij}+AE_{jk}E_{ki} \tag{2}\]  
where \(\lambda,\mu\) are [PERSON]'s constants, \(A,B,C\) are the third-order elastic constants, \(\delta_{ij}\) is the Kronecker delta, and \(T_{ij}\) can be divided into two parts, \(T_{ij}{}^{L}\) and \(T_{ij}{}^{NL}\):  
\[T_{ij}{}^{L}=\lambda u_{k,k}\delta_{ij}+\mu(u_{i,j}+u_{j,i}) \tag{3}\]  
\[T_{ij}{}^{NL}=T_{ij}(E_{ij})-T_{ij}{}^{L} \tag{4}\]  
where \(T_{ij}{}^{L}\) and \(T_{ij}{}^{NL}\) represent the stress tensor due to the linear and nonlinear parts of the strain tensor.  
The nonlinear wave equation of ultrasonic waves propagates in homogeneous isotropic hyper-elastic solid [35]:  
\[(\lambda+\mu)u_{j,i}+\mu u_{i,jj}+f_{i}=\rho\tilde{u_{i}} \tag{5}\]  
where  
\[\begin{array}{c}f_{i}=\Big{(}\mu+\frac{\lambda}{4}\Big{)}\Big{(}u_{l,jk}u_{ l,i}+u_{l,jk}u_{i,l}+2u_{i,jk}u_{l,k}\Big{)}+\Big{(}\lambda+\mu+\frac{\lambda}{4}+B \Big{)}\Big{(}u_{l,jk}u_{l,k}+u_{k,l}u_{i,l}\Big{)}\\ \qquad+(\lambda+B)\Big{(}u_{i,jk}u_{l,l}\Big{)}+\Big{(}\frac{\lambda}{4}+B \Big{)}\Big{(}u_{k,l}u_{l,i}+u_{l,jk}u_{k,l}\Big{)}+\big{(}B+2C\Big{)}\Big{(}u _{k,ik}u_{l,l}\Big{)}\end{array} \tag{6}\]  
where \(f_{i}\) represents the body force, which includes all nonlinear terms. The stress-free boundary condition is as follows:  
\[S_{ij}{}^{L}n_{j}=-S_{ij}{}^{NL}n_{j}\text{ on }S \tag{7}\]  
where \(n_{j}\) is the unit vector normal to the surface of the waveguide, and \(S\) is the surface of the waveguide. \(S_{ij}{}^{L}\) and \(S_{ij}{}^{NL}\) represent the linear and nonlinear parts of the surface force \(S_{ij}\) on the material, and the expressions are as follows:  
\[S_{ij}{}^{L}=T_{ij}{}^{L} \tag{8}\]  
\[S_{ij}{}^{NL}=\Big{(}\frac{\lambda}{2}u_{k,l}u_{k,l}+Cu_{k,k}u_{l}\Big{)}\delta _{ij}+Bu_{k,k}u_{j,i}+\frac{\lambda}{4}u_{j,k}u_{k,i}+\frac{\beta}{2}\Big{(}u_ {k,l}u_{k,l}+u_{k,l}u_{l,k}\Big{)}\delta_{ij} \tag{9}\]  
The perturbation method is used to get the particle displacement. Assume the material particle displacement is as follows:  
\[u_{i}=u_{i}{}^{(1)}+u_{i}{}^{(2)} \tag{10}\]  
where \(u_{i}{}^{(1)}\) and \(u_{i}{}^{(2)}\) are the displacements caused by material linearity and nonlinearity, respectively; and the values of \(u_{i}{}^{(1)}\) and \(u_{i}{}^{(2)}\) are the linear sum of the amplitudes of the correlated guided waves modes, \(u_{i}{}^{(1)}>>u_{i}{}^{(2)}\). Substitute Equation (10) into Equations (5) and (7), and the equation of motion and boundary condition of nonlinear displacement components are obtained [36] as follows:  
\[(\lambda+\mu)u_{j,i}{}^{(2)}+\mu u_{i,jj}{}^{(2)}+f_{i}{}^{(1)}=\rho\tilde{u_{ i}}{}^{(2)} \tag{11}\]\[{S_{ij}}^{L(2)}n_{j}=-{S_{ij}}^{NL(1)}n_{j}\text{ on }S \tag{12}\]  
where \({f_{i}}^{(1)}=f_{i}\!\!\left({u_{i}}^{(1)}\right)\), \({S_{ij}}^{L(2)}={S_{ij}}^{L}\!\!\left({u_{i}}^{(2)}\right)\), and \({S_{ij}}^{NL(1)}={S_{ij}}^{NL}\!\!\left({u_{i}}^{(1)}\right)\). If we solve Equations (11) and (12), \({u_{i}}^{(2)}\) can be obtained, which represents the particle displacement caused by second harmonic modes.