# Thermoelastic Investigation of Carbon-Fiber-Reinforced Composites Using a Drop-Weight Impact Test
## 3 Finite Difference Method (MATLAB(r))  
The FDM is a numerical method for solving differential equations such as the two-dimensional wave [34; 35] and heat equation [36; 37; 38], as given in Equations (1) and (4). This method approximates the differentials by discretizing the dependent variables (strain and temperature) in the independent variable domains (space and time) [39; 40; 41]. Each discretized value of the dependent variable is referred to as a nodal value. In this case, 2D wave and heat equations in the cylindrical coordinate system are discretized using FTCS FDM. The discretized equations are given in Equations (3) and (5).  
The two-dimensional wave equation in the cylindrical coordinate system is shown in Equation (1):  
\[\frac{1}{c^{2}}\bigg{(}\frac{\partial^{2}u}{\partial t^{2}}\bigg{)}=\ \frac{1}{r}\bigg{(}\frac{\partial u}{\partial r}\bigg{)}+\bigg{(}\frac{ \partial^{2}u}{\partial r^{2}}\bigg{)}+\frac{1}{r^{2}}\bigg{(}\frac{\partial^ {2}u}{\partial\theta^{2}}\bigg{)} \tag{1}\]  
where \(u\) is the strain, \(r\) is the radius, \(\theta\) is the angle in degrees, and \(c\) is the speed of a sound in a medium as shown in Equation (2) [42].  
\[c=\ frequency*wavelength \tag{2}\]  
The two-dimensional discretized wave equation in the cylindrical coordinate system is given in Equation (3).  
\[\frac{1}{c^{2}}\bigg{(}\frac{(u_{i,j}^{t-1}{-}2u_{i,j}^{t}{+}u_{i,j}^{t+1})} {(\Delta t)^{2}}\bigg{)}{=\ \frac{1}{r}\bigg{(}\frac{u_{i+1,j}^{t}{-}u_{i-1,j}^{t}{ -}}{2\Delta r}\bigg{)}+\ \bigg{(}\frac{(u_{i-1,j}^{t}{-}2u_{i,j}^{t}{+}u_{i+1,j}^{t})}{( \Delta r)^{2}}\bigg{)}+\ \frac{1}{r^{2}}\bigg{(}\frac{(u_{i,j-1}^{t}{-}2u_{i,j}^{t}{+}u_{i,j+1}^{t})} {(\Delta\theta)^{2}}\bigg{)} \tag{3}\]  
Figure 6: Diagrammatic representation of the experiment.  
The two-dimensional heat conduction equation in a cylindrical coordinate system with no energy-generation term is given in Equation (4).  
\[\alpha\frac{\partial T}{\partial t}\ =\ \frac{1}{r}\left(\frac{\partial}{ \partial r}\right)\left(r\frac{\partial T}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial}{\partial\theta}\left(r\frac{\partial T}{\partial\theta}\right) \tag{4}\]  
The two-dimensional discretized heat equation in a cylindrical coordinate system is shown in Equation (5):  
\[\frac{T_{i,j}^{t+1}T_{i,j}^{t}}{\Delta t} =\alpha\left[\frac{1}{r}\left(\frac{(T_{i+1,j}^{t}-T_{i,j}^{t})}{ (\Delta r)}\right)+r\frac{(T_{i-1,j}^{t}-2T_{i,j}^{t}+T_{i+1,j}^{t})}{(\Delta r )^{2}}+\frac{1}{r^{2}}\left(\frac{(T_{i,j-1}^{t}-2T_{i,j}^{t}+T_{i,j+1}^{t})}{ (\Delta\theta)^{2}}\right)\right] \tag{5}\]  
where \(T\) is the temperature; \(t\), \(r\), and \(\theta\) refer to time and positions; \(\Delta t\) is the timestep size in (s); and \(\Delta r\) and \(\Delta\theta\) are the differences in the spatial positions of the temperature nodes. \(\alpha\) = 0.12 mm\({}^{2}\)/s in-plane thermal conductivity of CFRPs [43; 44].  
Overall temperature changes in the CFRP plate can be obtained by solving the coupled heat and wave equation [45; 46; 47] as given in Equation (6):  
\[\Delta T_{total}\ =\ \Delta T_{conduction}+\Delta T_{thernelastic} \tag{6}\]  
where \(\Delta T_{conduction}=\ \frac{T_{i,j}^{t+1}T_{i,j}^{t}}{\Delta t}\), which can be obtained from the heat equation in a cylindrical coordinate system as given in Equation (5). \(\Delta T_{thernelastic}=\frac{\epsilon}{\text{CTE}}\), \(\epsilon\) is the thermoelastic strain obtained from wave equation in a cylindrical coordinate system as given in Equation (3), and \(\text{CTE}=-4.5\cdot 10^{-5}\ ^{\circ}\ \text{C}^{-1}\)[33; 42] is the coefficient of thermal expansion of the CFRP.