# A Point Crack Source Location Method without Velocity Information in Anisotropic Plates
## 3 Experiments
### Numerical Experiment Based on a Physical Model  
Numerical AE data were used to test the location estimation method. In order to create numerical AE data, the following elastic equation had to be solved:  
\[\rho\partial_{tt}u_{i}=\partial_{j}(c_{ijkl}\partial_{l}u_{k})+f_{i}\qquad \text{in }\Omega\times[0,T] \tag{8}\]  
where \(t\) is the time variable, \(i,j,k,l\in\{1,2,3\}\), \(\rho\) is the gravitational density, \(\partial_{i}\) is the partial derivative on \(i\), \(f_{i}\) is the wave source caused by cracking, and \(c_{ijkl}\) is the stiffness tensor of the object. The crack becomes a source of vibration \(f_{i}\) in the elastic equation, and the measured AE signal can be obtained from the solution \(\mathbf{u}=(u_{1},u_{2},u_{3})\) of the elastic equation at the measurement location. Note that the stiffness tensor \(C=c_{ijkl}\) can be obtained from the elastic properties of [PERSON]'s modulus \(E\), the Lame parameter \(\lambda\), the shear modulus \((G,\mu)\), and the Poisson ratio \(\
u\)[31] for a CFRP plate which has the orthotropic linear elasticity property:  
\[C= \begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{22}&C_{23}&0&0&0\\ C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix} .\]  
where  
\[ \begin{array}{rcl}\text{C}_{11}&=&-\frac{E_{1}^{2}E_{3}(-E_{3}v_{23}^{2}+E_{2})}{ E_{1}^{2}E_{2}v_{31}^{2}+2E_{1}E_{2}E_{3}v_{12}v_{23}v_{31}-E_{1}E_{2}E_{3}+E_{1}E_{3}^{2} v_{23}^{2}+E_{2}^{2}E_{3}v_{12}^{2}}\\ \text{C}_{12}&=&-\frac{E_{1}E_{2}E_{3}(E_{2}v_{12}+E_{1}v_{23}v_{31})}{E_{1}^{2} E_{2}v_{31}^{2}+2E_{1}E_{2}E_{3}v_{12}v_{23}v_{31}-E_{1}E_{2}E_{3}+E_{1}E_{3}^{2} v_{23}^{2}+E_{2}^{2}E_{3}v_{12}^{2}}\\ \text{C}_{13}&=&-\frac{E_{1}E_{2}E_{3}(E_{1}v_{31}+E_{3}v_{12}v_{23})}{(E_{1}^{2 }E_{2}v_{31}^{2}+2E_{1}E_{2}E_{3}v_{12}v_{23}v_{31}-E_{1}E_{2}E_{3}+E_{1}E_{3}^{ 2}v_{23}^{2}+E_{2}^{2}E_{3}v_{12}^{2}}\\ \text{C}_{22}&=&-\frac{E_{1}E_{2}^{2}(-E_{1}v_{31}^{2}+E_{3})}{E_{1}^{2}E_{2}v_ {31}^{2}+2E_{1}E_{2}E_{3}v_{12}v_{23}v_{31}-E_{1}E_{2}E_{3}+E_{1}E_{3}^{2}v_{23} ^{2}+E_{2}^{2}E_{3}v_{12}^{2}}\\ \text{C}_{23}&=&-\frac{E_{1}E_{2}E_{3}(E_{3}v_{23}+E_{2}v_{12}v_{31})}{E_{1}^{2 }E_{2}v_{31}^{2}+2E_{1}E_{2}E_{3}v_{12}v_{23}v_{31}-E_{1}E_{2}E_{3}+E_{1}E_{3}^ {2}v_{23}^{2}+E_{2}^{2}E_{3}v_{12}^{2}}\\ \text{C}_{33}&=&-\frac{E_{2}E_{3}^{2}(-E_{2}v_{12}^{2}+E_{1})}{E_{1}^{2}E_{2} v_{31}^{2}+2E_{1}E_{2}E_{3}v_{12}v_{23}v_{31}-E_{1}E_{2}E_{3}+E_{1}E_{3}^{2}v_{23} ^{2}+E_{2}^{2}E_{3}v_{12}^{2}})}\end{array} \]  
when we follow [PERSON] notation [32]. The experimental domain is a rectangular CFRP plate \(9\times 9\) cm\({}^{2}\) in size. The elastic properties of the CFRP were obtained from [33], which are  
\[E= \begin{bmatrix}E_{1}\\ E_{2}\\ E_{3}\end{bmatrix} = \begin{bmatrix}192.44\\ 23.39\\ \end{bmatrix} ,\quad G= \begin{bmatrix}G_{12}\\ G_{23}\\ G_{31}\end{bmatrix} = \begin{bmatrix}9.08\\ 10.75\\ 9.08\end{bmatrix} ,\quad v= \begin{bmatrix}v_{12}\\ v_{23}\\ v_{31}\end{bmatrix} = \begin{bmatrix}0.23\\ 0.37\\ 0.23\end{bmatrix} \]  
when units of the \(E\) and \(G\) are GPa. The AE source, to create the AE data, is defined as below at a single point, and we had four AE source candidate positions (red dots in Figure 4):  
\[f(t)= \begin{cases}\frac{1}{2}(1-\cos(2\pi\cdot pt/5))\sin(2\pi\cdot pt),&0<t< 5/p\\ 0,&\text{otherwise}\end{cases} \]  
where \(p=140\times 10^{3}\). Note that the excitation signal \(f(t)\) is a Hanning-windowed, 5-count, 1400-kHz tone burst signal, which was shown to be suitable for many industrial applications of guided waves [34, 35]. This source function gives the measured signal a similar shape to the actual AE signal. We used Comsol Multiphysics (Comsol, Inc., Burlington, MA, USA) to solve the elastic equation (Equation (8)). By solving Equation (8), the propagated wave was measured at eight positioned sensors (blue dots in Figure 4), and we obtained eight arrival times at each position accordingly.