## 3 Model Validation
All the curve-fitted parameters are shown in Table 2.  
Figure 10 compares the results of the present model and experimental data. The cases of no tension and 2.4 N/mm tension are used to curve-fit the parameters of the present model, \(t\), \(\varphi_{0}\), \(F_{0}^{c}\), and \(E_{com}\). The cases of 0.8 N/mm and 1.6 N/mm are used for validation. Equation (28) provides a method to quantitatively characterize the average error between experimental data and the predicted results. \[\text{Err}=\sum\
olimits_{i=1}^{M}\frac{1}{M}\frac{\sqrt{\left(\tau_{sh}^{pr} (\varphi_{i})-\epsilon_{sh}^{\varpi}(\varphi_{i})\right)^{2}}}{\tau_{sh}^{ \varpi}(\varphi_{i})}, \tag{28}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Loading Cases** & **No Tension** & **0.8 N/mm** & **1.6 N/mm** & **2.4 N/mm** \\ \hline \(\varphi_{0}\) (\({}^{\circ}\)) & 1.5\({}^{\circ}\) & 1.5\({}^{\circ}\) & 1.5\({}^{\circ}\) \\ \hline \(F_{0}^{c}\) (N) & 0.2 & 0.33 & 0.47 & 0.6 \\ \(t\) (mm) & \(t\) & 0.86 \(t\) & 0.73 \(t\) & 0.6 \(t\) \\ \(E_{com}\) (MPa) & 0.25 & 0.83 & 1.41 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Curve-fitting parameters for a carbon fiber plain-woven fabric. Figure 9: Thickness-compressive pressure of a dry carbon yarn [23]. where \(\varphi_{i}\) is the evenly distributed shear angle point with an angle-interval of around \(5^{\circ}\). \(\tau_{sh}^{pr}(\varphi_{i})\) and \(\tau_{sh}^{cx}(\varphi_{i})\) are the corresponding predicted and measured shear stresses for each \(\varphi_{i}\). \(M\) is the number of collected data points of a curve. The average errors for the four load cases are shown in Figure 10. The average errors of the four cases are around 10%, which is good enough considering the scatter of measured data. For the no tension and 2.4 N/mm tension cases, the theoretical curves agree well with experimental results, proving that the present model can effectively describe the shear mechanism and the tension-shear coupling effect of fabrics under biaxial tension. For the cases of 0.8 N/mm and 1.6 N/mm, the deviations between predicted and experimental results are very small, verifying the prediction capability of the present model. Comparing the curves of the four cases, it is found that the shear rigidity of a fabric significantly increases with the applied tension force. Thus, it is evident that the tension-shear coupling must be taken into account in the material forming process of fabrics. As shown in Figure 10, a stress-strain curve can be divided into two parts: the initial linear part and the nonlinear part. The initial part is induced by the static friction between overlapped yarns before a yarn rotates. The maximum value of the static friction force determines the initial shear stress value of the nonlinear part. A higher tension force leads to a larger maximum static friction force, thereby resulting in stiffer shear property of a fabric. Thus, the friction between overlapped yarns is one source of the tension-shear coupling. For the nonlinear part, the stiffness rapidly increases when the tension force varies from zero to 2.4 N/mm. This happens because the fiber density of a yarn becomes larger due to yarn shrinking induced by a tension force. A higher fiber density means a higher transverse compressive Young's modulus. Furthermore, higher transverse stiffness of yarns brings a larger shear resistance for the shear deformation of fabrics. Obviously, the transverse compression of yarns is one main source of the tension-shear coupling. For general fabrics, there should be a flat part between the initial linear part and the nonlinear rising part. The flat part is induced by friction during shear deformation before transverse compression of parallel yarns occurs. However, the fill gap is zero for the case in Figure 10, meaning that parallel yarns contact each other when a yarn starts to rotate. Thus, no obvious flat part is observed in Figure 10.