## 2 Derivation of Boundary Conditions of 4D C/C Composite
### The Temperature Boundary Conditions of UC2
#### 2.3.1 Boundary Conditions for Calculating \(\lambda_{xx}\) of UC2  
Boundary Conditions of \(((0,l/2),0,z)\) and \(((0,l/2),w/2,z)\)  
By assuming an arbitrary node \(M=(x_{1},0,z_{1})\) situated on the plane \(((0,l/2),0,z)\), the temperature of \(M\) is depicted as \(T_{(x_{1},0,z_{1})}\), and the heat flux of \(M\) is expressed as \(q_{y|(x_{1},0,z_{1})}\). Similarly, the temperature and heat flux of \(M1=(x_{1},w,h-z_{1})\) can be represented by \(T_{(x_{1},w,h-z_{1})}\) and \(q_{y|(x_{1},w,h-z_{1})}\), respectively, which is the corresponding node of \(M\) for the first rotational transformation. Through Equations (3) and (2), the temperature boundary conditions of plane \(((0,l/2),0,z)\) can be expressed as:  
\[T_{(x_{1},0,z_{1})}=T_{(x_{1},w,h-z_{1})}q_{y|(x_{1},0,z_{1})}=q_{y|(x_{1},w,h- z_{1})}. \tag{6}\]  
In the same way, the temperature and the heat flux of \(M=(x_{1},w/2,z_{1})\) should be represented by \(T_{(x_{1},w/2,z_{1})}\) and \(q_{y|(x_{1},w/2,z_{1})}\), when \(M\) is located on the plane \(((0,l/2),w/2,z)\)  
Figure 6: The two rotational transformations of UC2 from UC1. Figure 7: UC1 and UC2 model of four-directional carbon/carbon (4D C/C) composites. \(M1=(x_{1},w/2,h-z_{1})\) should be represented by \(T_{(x_{1},w/2,h-z_{1})}\) and \(q_{y|(x_{1},w/2,h-z_{1})}\), respectively, which is the corresponding node of \(M\) for the first rotational transformation in the plane \(((0,l/2),w/2,z)\). So, the temperature boundary conditions of plane \(((0,l/2),w/2,z)\) can be described as  
\[T_{(x_{1},w/2,z_{1})}=T_{(x_{1},w/2,h-z_{1})}\mathcal{A}_{y|(x_{1},w/2,z_{1})}= q_{y|(x_{1},w/2,h-z_{1})}. \tag{7}\]  
Boundary Conditions of \((0,(0,w/2),z)\) and \((l/2,(0,w/2),z)\)  
The temperature of \(M=(0,y_{1},z_{1})\) can be expressed by \(T_{(0,y_{1},z_{1})}\) and the heat flux of \(M\) can be depicted as \(q_{x|(0,y_{1},z_{1})}\), when the \(M\) is on the plane \((0,(0,w/2),z_{1})\). The temperature of \(M2=(l,y_{1},h-z_{1})\) should be represented by \(T_{(l,y_{1},h-z_{1})}\), and the heat flux of \(M2\) can be expressed as \(q_{x|(l,y_{1},h-z_{1})}\). The \(M2\) is the corresponding node of the \(M=(0,y_{1},z_{1})\) for the second rotational transformation. Since the heat flux is perpendicular to the plane \((0,(0,w/2),z)\), the antisymmetric thermal stimulus is formed. Referring to Equations (1) and (3), the temperature boundary conditions of plane \((0,(0,w/2),z)\) can be expressed as:  
\[T_{(0,y_{1},z_{1})}+T_{(l,y_{1},h-z_{1})}=2T_{(0,0,0)}+\
abla T_{x}q_{x|(0,y_{ 1},z_{1})}=-q_{x|(l,y_{1},h-z_{1})}. \tag{8}\]  
In the same way, the temperature and the heat flux of \(M=(l/2,y_{1},z_{1})\), when located on plane \((l/2,(0,w/2),z)\), should be represented by \(T_{(l/2,y_{1},z_{1})}\) and \(q_{x|(l/2,y_{1},z_{1})}\), respectively. The temperature and the heat flux of \(M2=(l/2,y_{1},h-z_{1})\) can be expressed by \(T_{(l/2,y_{1},h-z_{1})}\) and \(q_{x|(l/2,y_{1},h-z_{1})}\), which is the corresponding node of \(M=(l/2,y_{1},z_{1})\). Thus, the boundary conditions of planes \((l/2,(0,w/2),z)\) can be expressed as:  
\[T_{(l/2,y_{1},z_{1})}+T_{(l/2,y_{1},h-z_{1})}=2T_{(0,0,0)}+\
abla T_{x}q_{x|( l/2,y_{1},z_{1})}=-q_{x|(l/2,y_{1},h-z_{1})}. \tag{9}\]  
Based on Equations (6)-(9), the temperature boundary conditions of UC2 are acquired in x direction when calculating the \(\lambda_{\rm xx}\). Considering the nodes located at the faces, the nodes and the edges [40], the separate equations are summarized as:  
\[ \begin{array}{c}q_{x|(0,y_{1},z_{1})}=-q_{x|(l/2,y_{1},z_{1})},\\ T_{(0,y_{1},z_{1})}-T_{A}=0,\ T_{(l/2,y_{1},z_{1})}+T_{(l/2,y_{1},h-z_{1})}-2T _{A}=\
abla T_{x},\ T_{(l/2,y_{1},z_{1})}-T_{A}=\
abla T_{x}/2\\ q_{z|(x_{1},y_{1},0)}=-q_{z|(x_{1},y_{1},h)}=0,\\ T_{(x_{1},y_{1},0)}=T_{(x_{1},y_{1},h)},\end{array} \]  
\[ \begin{array}{c}T_{A=(0,0,0)}=T_{B=(l/2,0,0)}=T_{C=(l/2,0,h)}=T_{D=(0,0,h)}, \\ T_{A^{\prime}=(0,w/2,0)}=T_{B^{\prime}=(l/2,w/2,0)}=T_{C^{\prime}=(l/2,w/2,h) }=T_{D^{\prime}=(0,w/2,h)}=T_{A=(0,0,0)}+\
abla T_{x}/2,\end{array} \]  
\[