## 4 Parametric Study  
It is essential to understand the effect of geometry and mechanical properties of yarns on the large shear behavior. In this work, a kind of carbon fiber woven fabric is taken as an example. It should  
Figure 10: Comparison between the analytical model and experimental data of a carbon plain-woven fabric under bi-axial tension and shear load. be noted in Figure 10 that the unit of tension force along a yarn is N/mm, meaning that the force is homogenized by the side length of an RVE. Thus, the force along a yarn is dependent on the thickness of a fabric. Conversely, the parametric study requires that the tensile load is independent of thickness variation. Thus, a tensile stress \(\sigma_{y}\) with unit of MPa is used. As a reference, some basic parameters are reasonably set in Table 3. The sensitivity study is performed by varying one of these parameters while keeping the others constant. Figure 11a shows the shear deformation curves of fabrics with different wave patterns. The weave pattern is characterized by \(\eta_{y}/N_{y}\), the ratio between the inflection points of a yarn and the number of yarns in an RVE. The ratios are 1, 1/2, and 1/4 for plain woven, 2*2 will, and 8 harness satin, respectively. As shown in Figure 11a, a higher ratio corresponds to a higher shear stiffness. The result is reasonable because a higher ratio means more interaction between fill and warp yarns. Figure 11b is the parametric study of the fabric geometry. It is found that the shear stiffness increases with the increase of yarn thickness but decreases with the increase of yarn gap. This is because the waviness angle becomes larger when a yarn becomes thicker. Thus, the contact force between overlapped yarns increases, though the tension force along the yarn remains unchanged. However, the increase of yarn gap reduces the waviness angle, thereby weakening the effect of the tension force on shear deformation. Besides, the shear-locking phenomenon is postponed with the increase of gap size, \(25^{\circ}\) for \(g_{0}=0.5\), \(34^{\circ}\) for \(g_{0}=1\), and \(44^{\circ}\) for \(g_{0}=2\). The transverse compression of parallel yarns is one main source of the shear resistance. Thus, the rising shear-locking angle significantly reduces the shear rigidity of a fabric. Figure 11c illustrates the effect of the yarn property and the tensile load on the shear behavior of a fabric. It should be noted that in real cases, the transverse compressive modulus \(E_{com}\) is sensitive to the tensile load \(F_{t}\). However, in the sensitivity study, the two parameters are assumed to be independent so as to investigate the effect of each parameter on the shear behavior. It can be seen that the increase of tensile force along yarns results in a larger shear rigidity of a fabric. Moreover, each curve in Figure 11 can be divided into three parts, the initial linear part, the flat part, and the nonlinear rising part. Curves in Figure 11a-c indicate that the weave pattern, the yarn geometry, and the tensile load along yarns can only influence the maximum stress of the first part and the shear stress of the flat part but have no effect on the third part. However, as shown in Figure 11c, the growth of \(E_{com}\) significantly increases the slope of the third part of a curve, thereby enhancing the shear rigidity of a fabric. This happens because the effects of the weave pattern, the geometry, and the tensile load on shear behavior are characterized by the friction force which controls the linear and the flat part. However, the third nonlinear rising part is mainly controlled by the transverse compression of yarns.