## 2 Modeling Approach
### Multiscale Modelling Procedure of QNCF Composites  
The workflow of multiscale analysis in this paper is given in Figure 4. On the microscale, mechanical properties of the resin and fiber are input to micro-scale analysis model, which are then used to determine mechanical properties of the composites tows. A periodic microstructure three-dimensional (3D) finite element (FE) model is constructed, with the assumption that fibers are uniformly distributed in the matrix, as shown in Figure 4a. In this micromechanics model, the fiber-matrix interface is modeled as perfectly bonded with nodes merged in a conventional mesh. In the failure analysis process, the failure state of the interface is evaluated by the failure of fiber and matrix that is adjacent to the interface. Similar modeling methods are explained in reference [7]. Henceforth, the term \"composite tows\" is refers straight fiber tows impregnated with resin. \"1-direction\" and \"longitudinal direction\" are the direction along the axial of the fiber, \"2-direction\" and \"transverse direction\" are the direction transverse to the axial of the fiber, and \"3-direction\" is in the thickness direction of the fabrics.  
Figure 4: The workflow of multiscale analysis: (**a**) microscale; and (**b**) mesoscale.  
Figure 3: Meso-scale geometrical model of unidirectional (UD) specimens with different angle between fiber direction and process direction: (**a**) \(90^{\circ}\); (**b**) \(0^{\circ}\); and (**c**) \(45^{\circ}\). 1—Process direction; 2—Fiber direction; 3—Face channel; 4—Inner crack; 5—Stitching yarn; 6—Face crack.  
On the mesoscale, three unit cells are used to describe the fiber distortion patterns in the NCF plies (Figure 4b). The in-plane stiffness and strength of the NCF lamina are obtained by numerical analysis. These 3D FE models of single NCF lamina are developed based on the previous geometries (Figure 1c). The meso-scale models are shown in Figure 5, where Cell A is unit cell with a large crack, Cell B is unit cell with a small crack, Cell C is unit cell with a channel, and \(b\) and \(l\) are the width and length of the crack, respectively. In this mechanical model, the local variation of fibers orientation (Area with distorted fiber direction) is also shown in Figure 5.  
The local variation of the fiber orientations can be localized near the crack in ply. It is assumed that the width of the area with distorted fiber direction has the same size as small crack, which is shown in Figure 5b. Based on the experiment data [4], the value of large crack width is 0.480 mm and length is 7.20 mm, and the value of small crack width is 0.352 mm and length is 2.64 mm, and the fiber in-plane crimp angle \(\theta=4^{\circ}\). The fiber crimp angle can be modeled by the rotation of material orientation according to the actual distorted fiber direction [11].  
The volume fraction of composite tows (is also the local fiber volume fraction) is calculated in Reference [4]. The averaged fiber volume fraction of the lamina \(V_{\mathrm{f}}^{\mathrm{I}}\) and the volume fraction of composite tows \(V_{\mathrm{f}}^{\mathrm{I}}\) are related according to the Equation (1).  
\[V_{\mathrm{f}}^{\mathrm{t}}=V_{\mathrm{f}}^{\mathrm{I}}\frac{A^{\mathrm{I}}}{ A^{\mathrm{t}}}=V_{\mathrm{f}}^{\mathrm{I}}\frac{A\times B}{A\times B-S_{ \mathrm{Void}}} \tag{1}\]  
where \(A^{\mathrm{I}}\) is the cross-section of the lamina, \(A^{\mathrm{t}}\) is the cross-section of the tow, and \(S_{\mathrm{Void}}\) is the area of a crack or a channel per one knitting needle. \(A\), \(B\) is the space of stitching loops, \(A\) is perpendicular to the machine direction, \(B\) is in the machine direction (see Figure 1), and assuming \(B=2.74\) mm, and \(A=5.07\) mm in this paper.  
Figure 5: Finite element (FE) model of meso unit cells with different fiber distortion in the individual non-crimp fabric (NCF) lamina: (**a**) Cell A; (**b**) Cell B; and (**c**) Cell C.