## 2 Multiscale Modeling of Carbon-Fiber-Reinforced Composite  
### An Overview of Multiscale Modeling for Carbon-Fiber-Reinforced Composites  
The multiscale design process of woven-fabric-reinforced composites encompasses three scales: microscale, mesoscale, and macroscale, with the structural representation of each scale illustrated in Figure 1. At the microscale, the microscale cell is composed of fiber filaments and resin, approximated from the homogenization of fiber yarns. At the mesoscale, the mesoscale cell consists of fiber yarns with a small amount of resin. Depending on the weaving pattern of the fiber yarns, mesoscale cells can be categorized into plain, twill, and satin weaves. Notably, Figure 1 shows the mesoscale cells without resin, only displaying the woven fiber yarns. At the macroscale, the laminate scale is formed by stacking multiple single-ply woven composite materials bonded together with resin. In this paper, the layup design of the car door pertains to the macroscale.  
To predict the elastic properties of the representative volume element (RVE), this paper adopts a two-scale progressive homogenization method, which has been verified as feasible in the literature and is suitable for multiscale analysis of carbon fiber composites [18; 21]. In conjunction with the finite element analysis software ABAQUS, thermal stress is applied to predict the elastic properties of the RVE. The formula for calculating the equivalent stiffness coefficients of the RVE is as follows [21; 22]:  
\[\mathbf{D}_{ijmn}^{H}=\frac{1}{|Y|}\int\limits_{Y}\mathbf{D}_{ijkl}\bigg{(}\delta_{km }\delta_{ln}+\frac{\partial\psi_{k}^{mn}}{\partial y_{l}}\bigg{)}dY \tag{1}\]  
where \(\mathbf{D}_{ijmn}^{H}\) represents the equivalent stiffness matrix of the composite material, \(\mathbf{D}_{ijkl}\) is the fourth-order stiffness matrix of the RVE, and the subscripts \(i,j\), \(m\), and \(n\) each represent different directions; \(Y\) denotes the region of the RVE; \(\delta_{km}\delta_{ln}\) is the product of the Kronecker tensor, and \(\psi_{k}^{mn}\) is the displacement function.  
To solve for the equivalent stiffness coefficients \(\mathbf{D}_{ijmn}^{H}\) of the RVE in Equation (1), it is first necessary to determine the displacement function \(\psi_{k}^{mn}\). The solution for the displacement function can be achieved in ABAQUS by applying thermal stress, and the functional relationship between thermal strain and temperature is as follows:  
\[\epsilon_{mn}^{c}=-k_{mn}.\Delta T \tag{2}\]  
where \(k_{mn}\) represents the thermal expansion coefficient, \(\Delta T\) is the loading temperature, and \(\epsilon_{mn}^{c}\) denotes the thermal strain.  
Through theoretical derivation, the displacement function in Equation (1) can be converted into the form of thermal expansion coefficient, as follows:  
\[\mathbf{D}_{ijmn}^{H}=\frac{1}{|Y|}\int\limits_{Y}\mathbf{D}_{ijkl}\bigg{[}\frac{1}{2} \bigg{(}\frac{\partial\psi_{k}^{mn}}{\partial y_{l}}+\frac{\partial\psi_{l}^{ mn}}{\partial y_{k}}\bigg{)}+k_{mn}.\Delta T\bigg{]}dY \tag{3}\]  
In ABAQUS, Multi-Point Constraints (MPC) are applied, and the binding constraint is used to implement periodic boundary conditions. A Fortran subroutine is written to apply thermal stress, and a Python script is developed to read stress-strain information, ultimately solving for the homogenization coefficients of the RVE. Due to space limitations, the two-scale progressive homogenization method is not described in detail; interested readers may refer to the literature for more information [17; 18; 21].  
Figure 1: Flow chart of multiscale design for CFRP car door.