[文献] 纤维束中心线与方向的处理

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1. TexGen

Lin H, Zeng X, Sherburn M, Long AC, Clifford MJ. Automated geometric modelling of textile structures. Text Res J. 2012;82(16):1689-702.

In TexGen, the yarn path is represented by a spline S(u) (Bezier, natural or periodic cubic):

\begin{aligned}
& S(u)=S_0(u), \quad u_0 \leq u<u_1 \newline \\
& S(u)=S_1(u), \quad u_1 \leq u<u_2 \newline\\
& \text {........ } \newline \\
& S(u)=S_{k-2}(u) \quad u_{k-2} \leq u \leq u_{k-1} \newline \\
&
\end{aligned}

The given k points u_i are called knots or control nodes. The vector u=\left(u_0, \ldots, u_{k-1}\right) is called a knot vector for the spline, as shown in Figure 1. Yarn cross-sectional geometries will be defined locally at each control node.

In TexGen geometry modeling, the yarn path is usually determined by other parameters, such as fabric structure, yarn cross section, yarn spacing and fabric thickness. It is not necessary to define the yarn mid-line based on direct experimental measurement. In practice, the yarns are initially modeled as a symmetrical and constant cross section with a well-defined central line. Then the yarn cross section is modified locally based on experimental images. The central line is retained as a convenience reference line.

2. X-ray CT based multi-layer unit cell modeling of carbon fiber-reinforced textile composites

Sinchuk Y, Shishkina O, Gueguen M, et al. X-ray CT based multi-layer unit cell modeling of carbon fiber-reinforced textile composites: Segmentation, meshing and elastic property homogenization[J]. Composite Structures, 2022, 298: 116003.

The material properties of a tow are strongly anisotropic and defined by the local fiber orientation field. The following algorithm was implemented to compute tow orientation vectors required for FE modeling:

For each tow:

• 1. Find centerline projection using “family” orientation angle (Fig. 6a);
     
     Fig. 6. A tow (a), the tow projection with the centerline (red) and mean point (green) (b), spline interpolation of the centerline (c).
• 2. Find the tow mean surface (voxel representation);
• 3. Find tow centerline voxels (based on step 1 and 2 results);
• 4. Get tow centerline curve by a spline fitting to the centerline voxels (Fig. 6b);
• 5. Find fiber orientation for centerline points as *tangent vectors**;*
• 6. Get orientation for each tow voxel as the orientation of the closest centerline point.

In step 1, the 2D in-plane projection of the tow centerline was found as the straight line (red line Fig. 6b) through the tow projection mean point (green point Fig. 6b) with the “family” orientation slope (34° in Fig. 6b). This approach provides the same slope for each tow in a “family” (including small tow pieces at image corners, where difficult to determine the orientation accurately). The “family” slopes extracted from the known lay-up (0°, 30°, 60°, 30°, 0°) can be inaccurate as the manual lay-up can result in deviations from the ideal lay-up. Therefore, the new values were computed by averaging tow centerline slopes with length weights. The averaging with the length weight means that each tow contributes its slope into the average values proportional to its length (the longer tow, the more accurate its slope approximation). For example, the found “family” slopes which correspond to the plies lay-up were [1.01°, 33.84°, 63.77°, 34.06°, 1.05°], which shows more than 4° difference compared to the ideal value for 4th ply, due to the physical deviation of the ply lay-up.

In step 3, the voxels of the 2D centerline projections were mapped back to 3D as a corresponding voxel subset of the tow mean surface. The centerline voxels were fitted by the 2nd order spline curve using SciPy spline fitting functionality [53]. In order to simplify the spline fitting procedure, only 1% of the evenly spaced samples of the centerline voxels array were used (green points in Fig. 6b). The curve tangents (for discrete points set) were computed and then used for the orientation interpolation in tow voxels (steps 5 and 6 of the algorithm). Fig. 6b plots the resulting centerline curve (red), the points used for fitting (green) and the tow boundary profile.

In this way, the orientation for all tow voxels was extracted. The statistical analysis of the obtained orientation distribution showed that the fibers had a small out-of-plane inclination angle (less than 10° for most of the voxels).