## 2 Tension-Shear Coupling Model
### Dissipation Energy of Friction  
Figure 4a is a crossover point of two yarns. As shown in Figure 4b, a friction force is generated when a yarn rotates. An equivalent circle is used to approximate the contact area of overlapped yarns. It is reasonable to assume that the friction force is evenly distributed in the contact area. Imagine an infinitesimal sector whose side lengths are \(dr\) and \(rd\phi\), as shown in Figure 4c. The sector moves from A to A' with the yarn rotation. Then, the moment of the friction force to the point O is given by  
\[M_{fri}=\int_{0}^{2\pi}\int_{0}^{R}\frac{F_{fri}}{\pi R^{2}}r\cdot drd\phi= \frac{2}{3}F_{fri}R. \tag{11}\]Furthermore, the friction dissipation energy in an RVE is obtained as  
\[dW_{fri}=N_{y}^{2}M_{fri}\mathrm{d}\varphi=\frac{2}{3}N_{y}^{2}F_{fri}R\mathrm{d }\varphi, \tag{12}\]  
where \(R\) is the equivalent radius. The contact area varies with the shear angle until parallel yarns contact each other in in-plane transverse direction. Thus, \(R\) is expressed as  
\[R= \begin{cases}\frac{w_{0}}{\sqrt{\pi\ cos\varphi}}&\varphi<\varphi_{L}\\ \frac{w_{0}}{\sqrt{\pi\ cos\varphi_{L}}}&\varphi\geq\varphi_{L}\end{cases} , \tag{13}\]  
where \(\varphi_{L}\) is the shear-locking angle, meaning that parallel yarns start to contact each other in in-plane transverse direction. The shear-locking angle can be expressed as  
\[cos\varphi_{L}=\frac{w_{0}}{w_{0}+g_{0}}. \tag{14}\]  
It should be noted that the shear angle keeps increasing after \(\varphi_{L}\) due to in-plane transverse compression of yarns. In other words, the shear angle is actually not locked. However, following the nomenclature of textile forming, the shear-locking angle is still employed. For simplicity, [PERSON]'s friction theory is employed. Before a yarn rotates, the static friction force can be approximately seen as linearly dependent on the shear angle. When a yarn rotates, the friction force \(F_{fri}\) is calculated from the friction factor \(\mu\) and the contact force between overlapped yarns \(F_{c}\). The friction force is expressed as:  
\[F_{fri}= \begin{cases}\frac{\varphi}{\varphi_{0}}\mu F_{c}&\varphi<\varphi_{0} \\ \mu F_{c}&\varphi\geq\varphi_{0}\end{cases} , \tag{15}\]  
where \(\varphi_{0}\) is the critical shear angle when a yarn starts to rotate. Obviously, the friction force depends on the contact force between overlapped yarns, \(F_{c}\). \(F_{c}\) consists of two parts: the one induced by the tension force along a yarn, \(F_{t}^{c}\), and the initial contact force induced by the yarn bending, \(F_{0}^{c}\), as shown in the following equation:  
\[F_{c}=F_{t}^{c}+F_{0}^{c}. \tag{16}\]  
Ideally, \(F_{0}^{c}\) should be calculated from the yarn bending stiffness and the bending angle. However, the measured bending stiffness has large scatter, and its value is sensitive to the load status of a yarn. Thus, it is better to curve-fit \(F_{0}^{c}\) by comparing prediction results to experimental data. Figure 4: Dissipation energy of friction. Figure 5 illustrates the contact force induced by the tension force along a yarn for a plain-woven fabric. \(\theta\) is the waviness angle of a yarn. \(L_{in}\) is the length of the inflected part of a yarn. Then, \(F_{t}^{c}\) is:  
\[F_{t}^{c}=2F_{t}sin\theta=2F_{t}\frac{t}{L_{in}}=\frac{2 tF_{t}}{\sqrt{\left(w_{ 0}+g_{0}\right)^{2}+t^{2}}}. \tag{17}\]  
For fabrics with different weave patterns, the illustration of the contact force is shown in Figure 6.  
Then, the contact force at one cross point is:  
\[F_{t}^{c}=\frac{2\eta F_{t}sin\theta}{N_{y}}=\frac{2\eta_{y}tF_{t}}{N_{y}\sqrt {\left(w_{0}+g_{0}\right)^{2}+t^{2}}}. \tag{18}\]  
The friction force can be calculated by substituting Equation (18) into Equation (16) and then into Equation (15). Finally, the dissipation energy, \(dW_{fri}\), is obtained by substituting Equations (15) and (13) into Equation (12). The expression of \(dW_{fri}\) is obtained:  
\[dW_{fri}=