# Nondestructive Surface Crack Detection of Laser-Repaired Components by Laser Scanning Thermography
## 2 Detection Principle  
In order to perform theoretical analyses of laser spot thermography, a semi-infinite model is used with heat excitation on the surface by a circular Gaussian heat source, and the resultant heat conduction follows [PERSON]'s law. Ignoring the effects of heat convection and external radiation, [PERSON]'s law of heat conduction can be expressed as [26]:  
\[\frac{\partial T(x,y,z,t)}{\partial t}=q+k\bigg{(}\frac{\partial^{2}T(x,y,z,t)} {\partial x^{2}}+\frac{\partial^{2}T(x,y,z,t)}{\partial y^{2}}+\frac{\partial ^{2}T(x,y,z,t)}{\partial z^{2}}\bigg{)} \tag{1}\]  
where \(k\) is the thermal diffusivity, \(k=\lambda/\rho c\), is the material density, and \(c\) is the specific heat capacity. \(T(x,\ y,\ z,\ t)\) is the temperature distribution of the model under thermal excitation, the temperature change rate \(\partial T/\partial t\) of each point after being heated is proportional to \(\
abla^{2}T\), \(q\) is the heat input per unit time, and if the effect of thermal excitation is removed, then \(q=0\). For the semi-infinite model without crack, the surface temperature distribution along the \(y\)-axis direction under the excitation of gaussian pulse heat source is [22]:  
\[T_{0}(0,y,0,t)=\frac{2Q_{0}\eta}{\varepsilon\sqrt{\pi^{3}t}}\frac{\exp\bigg{(} -\frac{2(y-d)^{2}}{\alpha^{2}+2\mu^{2}}\bigg{)}}{a^{2}+2\mu^{2}} \tag{2}\]  
Consider the semi-infinite model, which contains infinite surface vertical cracks. On the model surface, \(y\) axis is perpendicular to the crack direction and \(x\) axis is parallel to the crack direction, and circular Gaussian pulse heat source excitation is applied at a distance of d from the crack, as shown in Figure 2. The total energy obtained by the model is \(Q_{0}\), and based on the heat conduction Equation (2) and the boundary conditions, the temperature distribution along \(y\)-axis can be obtained [22]:  
\[T(0,y,0,t)=T_{0}+T_{I}=\frac{2Q_{0}\eta}{\varepsilon\sqrt{\pi^{3}t}}\frac{ \exp\bigg{(}-\frac{2(y-d)^{2}}{a^{2}+2\mu^{2}}\bigg{)}}{a^{2}+2\mu^{2}}+sign(y) \frac{Q_{0}}{\varepsilon\pi^{2}a\sqrt{t}\mu\sqrt{a^{2}+2\mu^{2}}}\int_{- \infty}^{\infty}\varphi_{1}dy_{0} \tag{3}\]  
Figure 1: The research framework and technical route. \[\varphi_{1}(0,y,0,t)=sign(y_{0})\exp\left(-\frac{2(y_{0}-d)^{2}}{a^{2}}-\frac{u^{2} }{\mu^{2}}\right)\left[1-\frac{\sqrt{\pi}\mu}{\lambda R_{th}}\exp\left(\frac{\mu }{\lambda R_{th}}+\frac{u}{\mu}\right)^{2}\text{erfc}\left(\frac{\mu}{\lambda R _{th}}+\frac{u}{\mu}\right)\right] \tag{4}\]  
where \(T_{I}\) is the interference term of the crack on the temperature distribution, \(a\) is the radius of the laser spot (at \(1/e^{2}\) of the intensity), \(\mu=\sqrt{4k}t\) is the thermal diffusion length, and \(\varepsilon=\lambda/\sqrt{k}\) is the thermal effusivity. Air is the main filling medium inside the crack; its thermal conductivity is \(\lambda_{air}\), and the width of the crack contained in the model is \(L\), so the thermal resistance coefficient of the crack is expressed as \(R_{th}=L/\lambda_{air}\). \(R_{th}\) reflects the resisting effect of the crack, and with the increase in \(R_{th}\), the blocking effect on heat conduction becomes more significant. \(\eta\) is the heat absorption energy fraction of the model, which denotes the energy absorption efficiency. Then, \(u=|y|+|y_{0}|\), where \(y_{0}\) is the \(y\)-axis coordinate of the laser spot center [22]. Based on Equations (3) and (4), we perform the calculations by MATLAB, and the thermal response of the cracked model under pulse excitation is obtained, as shown in Figure 3. Thus, the influence of material properties and laser power on the distribution of surface temperature and temperature gradient can be analyzed, which provides reference for the simulation and experiment.