## 2 Materials and Methods
\right| \tag{2}\]  
From the Onsager principle of irreversible process thermodynamics, the thermal conductivity component \(\mathrm{k_{i,j}}\) is subject to reciprocity:  
\[\mathrm{k_{ij}}=\mathrm{k_{ji}} \tag{3}\]  
In addition, according to irreversible thermodynamics, the coefficients \(\mathrm{k_{11}},\ \mathrm{k_{22}},\ \mathrm{k_{33}}\) are all positive:  
\[\mathrm{k_{ij}}(\mathrm{i=j})>0 \tag{4}\]  
And the size of the coefficient \(\mathrm{k_{ij}}\) (i \(\
eq\) j) is limited by the following requirements:  
\[\mathrm{k_{ij}}\mathrm{k_{ji}}-\mathrm{k_{ij}^{2}}>0\qquad\mathrm{i\
eq j} \tag{5}\]  
\(\mathrm{Q_{L}}\) is the heat source term of long-pulsed laser, whose expression can be written as:  
\[\mathrm{Q_{L}}(\mathrm{x,y,z,T,t})=\frac{\left[\mathrm{I-R(T)}\right]\times \mathrm{\alpha(T)}\times\mathrm{E}\mathrm{\alpha(t)}}{\mathrm{\pi r_{\mathrm{ los}}^{2}\mathrm{\alpha}}\tau}\times 2\exp\left(\frac{-2\sqrt{x^{2}+y^{2}}}{\mathrm{r_{ \mathrm{los}}^{2}}}\right)\exp[-\mathrm{\alpha(T)}z] \tag{6}\]  
Figure 1: Schematic diagram of experimental set-up. where, E is the single pulse energy of the laser, r\({}_{\text{las}}\) is the laser spot radius located on the material surface, \(\tau\) is the pulse width of the laser R(T), \(\alpha\)(T) are reflection coefficient and absorption coefficient of material, g(t) is time distribution of laser beam, respectively. They are defined as:  
\[\text{g(t)}=\left\{ \begin{array}{ll}1\,&0<t\leq\tau\\ 0\,&t>\tau\end{array} \right. \tag{7}\]  
The initial condition is the temperature field distribution when the laser does not act on the material  
\[\text{T}|_{t-0}=\text{T}_{0}(\text{x},\text{y},\text{z}) \tag{8}\]  
where T is the initial condition of room temperature, T\({}_{0}=293\)K. In the process of laser cutting CFRP, there are thermal convection and thermal radiation at the same time, which are the third type of boundary conditions. The expression of the boundary conditions is  
\[-\lambda\frac{\partial\text{T}}{\partial\text{m}}\Big{|}_{\text{r}_{1}}=\text {h}(\text{T}-\text{T}_{0}) \tag{9}\]  
where h is the heat transfer coefficient. Carbon fiber (CF) is a new type of fiber material with high strength and high modulus containing more than 95% carbon [21], and the elastic modulus and tensile strength of the carbon fiber reinforced composite are greater than 210 GPa and 3.5 GPa, respectively. It is made of flake graphite microcrystalline and other organic fibers stacked along the axial direction of the fibers. The microcrystalline graphite material is obtained by carbonization and graphitization. Carbon fiber reinforced polymers (CFRP) in this reias an extremely strong and light material which consist of T700 carbon fiber and MTM28 resin. The prepreg cloth resembles a sandwich structure up to several millimeters thick. The Prepreg sheet would stick together after the solidification, but the resin is a liquid state during the layup process. Table 1 shows the properties of laser and CFRP material using for both simulation and experiment. It is a 1064 nm wavelength laser with 1 ms pulse length. WE chose both single and multiple modes to ablate the composite material. For the CFRP material, the braided structure contains both the carbon fiber and epoxy.