## 2 Multiscale Modeling of Carbon-Fiber-Reinforced Composite
### Mesoscale Model Simulation Analysis
#### 2.3.1 Mesoscale Model Simulation Analysis Method  
Two-dimensional woven composite materials are created by intertwining fiber yarns using various weaving techniques and combining them with a small amount of resin, resulting in a distinct type of composite material. Depending on the weaving technique of the fiber yarns, these composites can be classified into plain weave, twill weave, and satin weave, as depicted in Figures 5 and 6. The current section delves into the mesoscale cell construction, specifically focusing on plain weave composites. Electron microscope scans of plain weave composite cross-sections show that fiber yarns form a carbon fiber fabric with a wavy pattern through weaving. For the multiscale analysis of mesoscale cells, the mesoscale structure of plain weave composites needs simplification. The division of mesoscale cells for plain weave composites is illustrated in Figure 7.  
Figure 5: The plain weave mesoscale cell structure: (**a**) Plain weave mesoscale cell structure parameters; (**b**) Equivalent stress contour result of plain weave mesoscale cell structure. In developing mesoscale cell models for plain weave composites, [PERSON] and [PERSON] [24] initially introduced the concept of mesoscale cells in fabrics, representing the yarns as zigzag lines. [PERSON] et al. [25] enhanced this model by substituting the zigzag transitions with sine curve transitions, thereby bringing the model closer to actual structures. Further model simplification was proposed by [PERSON] et al. [26], who suggested linear transitions for yarn curves. This approach not only simplified the modeling process but also ensured that the resulting model met design requirements. Consequently, this study adopts the approach proposed by [PERSON] to establish the mesoscale structure of plain weave composite materials, as illustrated in Figure 5a. From Figure 5a, it is evident that the mesoscale cell primarily comprises warp yarns, weft yarns, and resin. The geometric parameters of the mesoscale cell are defined as follows: \(h_{\text{g}}\) is the thickness of resin between adjacent fiber yarns, \(h_{\text{y}}\) denotes the thickness of fiber  
Figure 6: Mesoscale cell structure: (**a**) The full weave mesoscale cell structure; (**b**) The satin weave mesoscale cell structure. Figure 7: Schematic view of plain weave mesoscale cell structure. yarns, \(w_{y}\) represents the width of the yarns, \(u_{y}\) is the distance between adjacent yarns, \(h\) indicates the total thickness of the mesoscale cell, \(g_{\text{w}}\) refers to the width of the curved part of the yarns, and \(w\) is the overall width of the mesoscale cell. Literature studies have shown that the volume fraction of fiber yarns (\(V_{y}\)) is a crucial factor influencing the elastic properties of the mesoscale cell [17; 23]. Therefore, in this study, all parameters except \(w_{y}\) and \(h_{y}\) are kept constant, and the fiber yarn volume fraction is controlled by adjusting the widths \(w_{y}\) and thicknesses \(h_{y}\) of the yarns. That is, \(w_{y}\) and \(h_{y}\), shown in Figure 5a, are selected as the design variables for the mesoscale, denoted as \(x_{2}\) and \(x_{3}\), respectively. In woven composite materials, mesoscale fiber yarns are comprised of numerous filaments and resin. To predict the properties of woven fiber composites using the thermal stress method, it is necessary to input the material parameters of both fiber yarns and resin. In this section, unidirectional fiber yarns with a fiber volume fraction of 51% and EPOLAM 5015 RESIN epoxy are used as mesoscale material inputs, as discussed in Section 2.2, with specific parameters detailed in Table 3. A mesoscale cell model, as depicted in Figure 6a, is established in ABAQUS, and the homogenization method based on thermal stress, introduced in Section 2.1, is applied for analysis. Figure 5b shows the equivalent stress contour map obtained from the finite element analysis.