## 2 Tension-Shear Coupling Model
### Incremental Energy Equilibrium Theory  
In forming manufacturing industries, the shape and stress state of a formed fabric depend on the forming process parameters, such as molding shape, forming rate, etc. Numerical simulations are generally used to optimize such forming process parameters. The key to such a simulation is a reliable constitutive relation that describes the mechanical behavior of an RVE of a fabric. In practice, blank-holders are often used to introduce tension forces along yarns so that the wrinkling phenomenon may be postponed. Besides, a fabric is majorly shear deformed in the forming process. Thus, it is critical to establish a constitutive relation for large shear deformation of a fabric RVE with the consideration of tension-shear coupling. Once the constitutive relationship is obtained, the numerical simulation can be realized subsequently. Thus, this work mainly investigates a fabric RVE subjected to a combined tension and shear load, as shown in Figure 3.  
The RVE is applicable for different weave patterns. Thus, no specific weave pattern is illustrated in Figure 3. The RVE is firstly subjected to tension loads along yarns, \(F_{t}\). The tension forces remain constant throughout the load case. Then, the RVE undergoes shear deformation under the force \(F_{nde}\) at the node P of the RVE. The node P moves to P' due to the shear deformation. The shear force \(F_{sh}\) is the projection of \(F_{nde}\) to the boundary of the RVE. The relationship between \(F_{sh}\) and \(F_{nde}\) is given as follows:  
\[F_{sh}=\frac{F_{nde}}{2 cos\left(\frac{\pi}{4}-\frac{\varphi}{2}\right)}. \tag{2}\]  
As mentioned in the above paragraph, the tension forces along yarns are applied before the shear deformation, meaning that a yarn has been stretched before the RVE is sheared. Thus, no tension input energy is involved in the shear deformation. The input energy generated by the node force \(F_{nde}\) will be dissipated by friction, transverse compression, thickness variation, yarn elongation, yarn sliding, and waviness angle variation. Among these dissipation energies, the former two are dominant. The yarn thickness may increase due to in-plane transverse compression, but the variation is small since the increase of yarn thickness will be restricted by overlapped yarns. The yarn elongation is negligible, since the longitudinal modulus of a yarn is much larger than the shear and transverse compressive properties of a fabric. The yarn sliding is also negligible, since a couple of balanced tension forces are applied to the end of a yarn. The waviness angle variation mainly results from yarn sliding and thickness variation, thereby also being insignificant. Thus, only the friction and transverse compression dissipation energies are considered. According to the energy equilibrium theory, the input energy should equal the sum of the dissipation energies. Hence, the following equation is obtained:  
\[\mathrm{d}W_{nde}=\mathrm{d}W_{fri}+\mathrm{d}W_{com}, \tag{3}\]  
Figure 3: Shear deformation of a fabric RVE.  
where \(\mathrm{d}W_{node}\) is the input energy increment generated by the node force, \(F_{node}\). \(\mathrm{d}W_{fri}\) and \(\mathrm{d}W_{com}\) are incremental energy dissipated by the respective friction and transverse compression of yarns.  
Our goal is to express all items in Equation (3) as functions of shear force, shear angle, geometric parameters, and mechanical properties of yarns so that a shear stress-angle curve can be obtained eventually.