Parametric modeling based on the real geometry of glass fiber unidirectional non-crimp fabric

Abstract

Meso-structure modeling of fiber reinforcement is a critical step for predictions of the manufacturing process and mechanical properties of composites. In the work reported in this paper, a meso-scale model based on the real geometry of glass fiber non-crimp fabric (NCF) was developed, and an attempt for the parametric modeling has been made. First, the buckling of warp yarns due to the tension of stitching yarn has been considered, and the buckling can be described by a curve equation. In addition, cross-sectional characterizations of warp and weft yarns were studied via metallographic observations, and 3D models have been constructed. Second, the path of the polyester stitching yarn was extracted by the tracer fiber technique and fitted by the B-spline spline curve; this trace can be can be simplified to several equations in Matlab. Finally, the unit cell model has been assembled and the penetration has been checked for each component. In this study, a meso-scale model of NCF was accurately established which truly reflects the relationship among geometry and position of yarns, providing a precise geometric model for further study on compressive deformation and permeability prediction of NCF in the future.

Keywords

unidirectional non-crimp fabric meso-scale model real geometry tracer fiber technique 3D stitching path

In uniaxial or multi-axial warp-knitted non-crimp fabrics (NCFs), reinforced yarns are parallel and straight aligned at an angle without interlacing, which fully utilizes the tensile properties of reinforced yarns. In recent years, NCFs have been widely applied in the field of composites due to their excellent tensile properties, shear resistance, tear resistance, good drapability and designability.^1,2 A meso-scale model therefore needs to be established which accurately reflects the geometric relationship among yarns in NCF and the compression behavior of a reinforcement and the infusion process, and which determines the fiber volume fraction and mechanical properties of composites.^3,4 Zhuo et al. proposed a warp-knitted model in the early 1990s that assumed that the transversal cross-section of circular yarns was incompressible and inextensible, and the thread tension in each part was equal; the amount of delivery yarn per unit was calculated.⁵ In 2001, scholars from the University of Leeds put forward a 3D stitching loop model; the Non-Uniform Rational B-Spline (NURBS) interpolation algorithm was applied to develop a simple geometry model, and 3D simulation was achieved via computer.⁶ Adanur and Liao assumed the geometry was highly related to the cross-section of the yarn and its axis; where the central axis was continuously differentiable, the computer-aided geometric design (CAGD) technique was used to construct an ideal 3D stitching yarn.⁷ A multi-axial unit cell model has been established based on the NURBS curve which considered reinforcing yarns arranged in parallel, and their cross-section exhibited a racetrack shape. Since geometry of the stitching yarn was derived from the pure knitted fabric, the binding effect of the stitching yarn to reinforced yarns was not considered.⁸ The NCF model proposed by Uhlig et al. considered the warp yarn deformation caused by the stitching compaction and local fiber volume fraction, and stress–strain distribution was predicted under the tensile load via finite element method (FEM).⁹ Lomov et al. employed the textile modeling software WiseTex to establish a multi-axial NCF model, various thickness of several segments in a stitching yarn were predicted, and experimental results were consistent with theoretical ones based on the Kawabata Evaluation System (KES) and metallographic method.¹⁰ In other research, steps for constructing a multi-axial NCF model were stated; it was believed stitching yarns resulted in various yarn orientations in adjacent layers within a fabric, which also contributed to a slight out-of-plane buckling of reinforcing yarns.¹¹ In addition, this buckling could be expressed by a cosine function.¹²

However, some simplifications and assumptions exist in previous models. In this study, meso-scale structure has been investigated based on the real geometry of NCF, and an attempt for parametric modeling has been made. Geometric parameters of warp and weft yarns were obtained by means of metallographic microscope, and the stitching trace in 3D configuration inside the NCF was extracted by the tracer fiber technique according to differences in refractive index of resin and polyester fiber; the trace can be simplified and expressed by curve equations. Finally, the geometric model has been constructed.

Materials

In this paper, glass fiber unidirectional NCFs, consisting of warp yarns, weft yarns and stitching yarns, were used as shown in Figure 1. Warp yarns account for 94% of total weight of the fabric, mainly aligned in 0-degree direction for assuming the force. The small content of weft and stitching yarn contributes little to the mechanical property; their main role is to maintain a stable fabric structure. Table 1 reports the parameters of NCF, while a and b are the width and length of the unit cell, respectively.

Figure 1.

Photos of NCF of (a) the front surface and (b) the back surface.

Table 1.

Fabric parameters

Name	Taishan UD
Surficial density (g/m²)	1200
Glass fiber density (g/cm³)	2.56
Stitching pattern	Tricot
Linear density (tex)
Warp	4000
Weft	360
Stitching	8
Density (number/10 mm)
Warp	3.5
Weft	5
Unit cell size (mm)
a	3.5
b	10.0

Geometric modeling

NCF geometric model consists of the modeling of warp, weft and stitching yarn. Each component includes the longitudinal modeling (along the yarn path) and cross-section modeling.

Warp modeling

As is well known, warp yarns are laid at 0 degrees, which should be straight for an ideal state. However, in actual production, stitching yarns are unwound from the beam, bringing about a certain tension which is helpful for fixing warp and weft yarns and beneficial for the overall fabric structure. Due to the binding effect of stitching yarns, warp yarns are deformed to a certain degree, which is depicted in Figure 2.

Figure 2.

Knitting process and warp deformation: (a) knitting process; (b) Tricot knitting; (c) warp buckling.

Placing the fabric under a microscope, a regular buckling deformation of the warp yarn was observed owing to the stitching tension, as shown in Figure 3(a).

Figure 3.

Warp modeling along the longitudinal direction: (a) warp buckling deformation; (b) warp modeling.

Figure 3(b) extracts the longitudinal profile of a warp yarn, which can be expressed by Equation (1).

y = b_{R} \cdot cos [(\frac{π}{b} \cdot (x - \frac{b}{2}))], x \in (0, 10)

(1)

where, b_R is the offset distance in Y direction under a stitching tension; b_R measured is 0.25 mm. Therefore, the contour length of the crimp warp L in a unit cell can be calculated by Equation (2).

L = 2 * \int_{0}^{5} \sqrt{1 + \frac{1}{16} * cos 2 (\frac{π}{10} * (x - 5))} d x

(2)

Solving the arithmetic solution via Matlab, L = 10.155 mm.

Currently, there are several assumptions about cross-section shapes of warp yarns, including the circular,^13–16 flat^17,18 and elliptic.¹⁹ The cross-sections of elliptic¹⁹ and biconvex^20–22 shapes have been long accepted; recently, scholars proposed the rectangular and runway section,^23,24 as shown in Figure 4.

Figure 4.

Common cross-sections of yarns: (a) circular; (b) flat; (c) elliptical; (d) runway; (e) biconvex; (f) rectangular.

In our work, the cross-section of warp yarns was observed by metallographic microscopy and proved to be approximately a half elliptic, as shown in Figure 5(a). The commercial software Catia was employed to sweep the cross-section of warp yarns along the path which is from Figure 3(b), and a 3D geometry of a unit cell warp is shown in Figure 5(b).

Figure 5.

Warp modeling: (a) warp cross-section and (b) warp 3D model.

Further, the cross-section of a warp yarn can be described by the elliptic Equation (3),

\frac{y^{2}}{1 . 625^{2}} + \frac{z^{2}}{0 . 65^{2}} = 1, z \in (- 0.3, 0.65)

(3)

in which the warp height is 0.95 mm, and the width is 3.25 mm; as a result, the cross-section area is 2.448 mm² according to the elliptical formula. Combined with Equation (2) to solve, the warp volume in a unit cell is 24.86 mm³.

Weft modeling

Perpendicular to the knitting direction, weft yarns consist of small content. With a small constraint applied by stitching yarns, weft yarns were regarded as straight along the Y direction. Figure 6(a) shows the cross-section of a weft yarn, which could also be described by the shape of a half elliptic. The weft geometry is shown in Figure 6(b), and the cross-section is expressed by Equation (4).

\frac{x^{2}}{0 . 8^{2}} + \frac{z^{2}}{0 . 15^{2}} = 1, z \in (- 0.15, 0)

(4)

where 0.8 and 0.15 are the long and short semi-axis of the ellipse, respectively. The height of a weft yarn is 0.15 mm, the width is 1.6 mm, and the length in a unit cell is 3.5 mm; thus, the weft volume in a unit cell is 0.66 mm³.

Figure 6.

Weft modeling: (a) weft cross-section and (b) weft modeling.

Stitching modeling

Stitching yarn was employed to constrain the reinforced yarns in NCF. Many scholars have already investigated traditional warp-knitted fabric. The geometric properties of warp-knitted fabric were first studied by Fletcher and Roberts.²⁵ The stitching model was first built by Allison; the model consists of a semicircle and two tangent lines, and the third straight line is considered as the loop extension line.²⁶ Later, Grosberg established the warp-knitted loops and adopted elastica curves to model the loop faces and loop connecting parts.^27,28 In these models, curvatures at the points between the loop faces and connecting parts are not equalized; later, Kurbak attempted to model the loop connection via continuous curves for warp-knitted fabrics.²⁹

For NCFs, the stitching modeling is a problem since its geometric information is hidden in warp and weft yarns. Currently, the modeling usually adopts some assumptions. There are few models on the trace of stitching yarn in NCF; some work is summarized in Figure 7. Du and Ko established the multi-axial NCF model and assumed the bend segments of stitching yarns as 90 degrees.³⁰ Later, this modeling method was adopted by Gu to calculate the relationship between the geometric model and process parameters as well as the fiber volume fraction.³ In addition, a multi-axial NCF model proposed by Lomov et al. considered the diameter variations of stitching yarns at various segments; the corners of stitching yarns were still regarded as 90 degrees.¹⁰ Gu established the multi-axial NCF by combining the stretched reinforced yarns with the stitching yarns obtained from the pure knitted fabric, which has not considered the real geometry of stitching yarns.³¹

Figure 7.

Modeling of stitching and NCFs: (a) Allison; (b) Du; (c) Gu; (d) Lomov; (e) Gu.

In this work, the tracer fiber technique was introduced to track the stitching path. The tracer fiber technique has been applied in fields of scientific research and testing. As early as 1952, Yan pioneered this method by immersing the yarn sliver in a liquor. Their refractive indexes were close, making the arrangement of tracer fiber in the sliver clearly visible. This work has made a significant contribution to the textile field and opened access to applications of the tracer fiber technique.^32,33

The tracer fiber technique is characterized by the principle of light refraction and reflection—when a beam of light goes through one material from another, if the refractive index of the two materials is close, refraction and reflection hardly occur at the interface, and the light passes through smoothly. If the two materials have different refractive indexes, refraction and reflection at the interface lead to the optical path altering direction.

Table 2 reports the refractive index of materials. The refractive index of glass fiber is 1.50, while the polyester filament is 1.62. An epoxy resin with a refractive index close to glass fiber was chosen to prepare a composite to highlight the stitching yarn; a schematic diagram of the optical path in a composite is shown in Figure 8.

Figure 8.

Fiber refraction method.

Table 2.

Refractive index of materials

Materials	Polyester filament	E-glass fiber	Epoxy resin
Refractive index	1.62³⁴	1.50³⁵	1.50–1.56³⁶

In Figure 8, as natural light enters the composite, because the refractive index of resin differs from that of the polyester fiber, the light is therefore refracted while passing through the polyester fiber and reflected at the surface of the polyester fiber. As the refractive index of glass fiber and resin are close, the light enters glass fiber smoothly from the resin without refraction and reflection, there is no an obvious boundary between glass fiber and resin. Glass fiber and resin are transparent and the light at the surface of polyester fiber is reflected into the air, leading to the profile of polyester stitching yarn being distinct and clear.^35,37

The composite was prepared via the resin transfer process, and the cavity thickness was set sufficiently high to avoid contact of the upper plate with the fabric. After the resin curing, the sample was observed by optical microscopy in two views; the results are shown in Figure 9.

Figure 9.

The stitching profile highlights in a composite.

Figure 9 shows the X–Y and X–Z plane of a single stitching loop; the origin coordinate is (0,0,0). Interpolated points a, b, c … m along the stitching path in the X–Z plane correspond to points a', b', c' … m' in the X–Y plane. Coordinates were calibrated to confirm that the x-coordinates in the X–Y plane and X–Z plane are consistent. In the X–Y plane, (x, y) coordinates of interpolated points were measured; meanwhile, measurements were made on (x, z) coordinates of corresponding points in the X–Z plane. As a result, (x, y, z) coordinates of each interpolated point are determined.

In order to ensure continuous and smooth interpolation points along the yarn path, a series of discrete points are used to fit the 3D space trace of the stitching yarn, thus, the B-spline curve is introduced.³⁸ Due to the compaction of warp yarns from both sides, the loop part of the stitching almost overlaps and compacts into a straight line. More control points need to be inserted at the loop inflection due to a large bending curvature. Finally, these points are fitted via a smooth B-spline curve to obtain the stitching space trace, and the result is shown in Figure 10.

Figure 10.

3D path of the stitching yarn.

Figure 10 shows the complete unit cell of a stitching yarn, in which the loop segment is almost a straight line, which varies a lot from the stitching yarn from a pure warp-knitted fabric. An attempt to describe the continuous path of a stitching yarn by curve equations has been made; here, several segments are divided for the sake of simplicity. In the X–Y plane in Figure 9, supposed a'b'g' and g'h'i' are straight lines, traces of c'd'e' and e'f'c', i'j'k' and k'l'i’ could be considered as a cosine function depicted by Equation (1). In the X–Z plane, segments of abg–cde and ghi–ijk are approximated to an elliptic equation. Equations for each segment are as follows:

For segments of a’b’g’ and abg, curve functions are shown in Equation (5),

\begin{matrix} {\begin{matrix} y = - 0.615 x + 3.5 \\ \frac{(x - 2.642)^{2}}{2 . 642^{2}} + \frac{(z - 0.103)^{2}}{0 . 575^{2}} = 1 \end{matrix} \\ x \in (0, 5.284), z \in (0.103, 0.678) \end{matrix}

(5)

For c’d’e’ and cde,

\begin{matrix} {\begin{matrix} y = 3.75 - 0.25 cos (\frac{π}{10} x) \\ \frac{(x - 2.642)^{2}}{2 . 642^{2}} + \frac{(z - 0.103)^{2}}{0 . 575^{2}} = 1 \end{matrix} \\ y \in (3.5, 3.75), z \in (- 0.472, 0.103) \end{matrix}

(6)

The distance between c'd'e' and e'f'c' is short, and a straight line was adopted.

\begin{matrix} {\begin{matrix} x = 5.284 \\ z = 0.103 \end{matrix} y \in (3.75, 3.85) \end{matrix}

(7)

For e’f’c’ and efc,

\begin{matrix} {\begin{matrix} y = 3.85 - 0.25 cos (\frac{π}{10} x) \\ \frac{(x - 2.642)^{2}}{2 . 642^{2}} + \frac{(z - 0.103)^{2}}{0 . 575^{2}} = 1 \end{matrix} \\ y \in (3.6, 3.85), z \in (- 0.472, 0.103) \end{matrix}

(8)

For g’h’i’ and ghi,

\begin{matrix} {\begin{matrix} y = 0.689 x + 3.6 \\ \frac{(x + 2.358)^{2}}{2 . 358^{2}} + \frac{(z - 0.103)^{2}}{0 . 575^{2}} = 1 \end{matrix} \\ x \in (- 4.716, 0), z \in (0.103, 0.678) \end{matrix}

(9)

For i’j’k’ and ijk,

\begin{matrix} {\begin{matrix} y = 0.1 + 0.25 cos (\frac{π}{10} (x + 5)) \\ \frac{(x + 2.358)^{2}}{2 . 358^{2}} + \frac{(z - 0.103)^{2}}{0 . 575^{2}} = 1 \end{matrix} \\ y \in (0.1, 0.35), z \in (- 0.472, 0.103) \end{matrix}

(10)

For segments of i’j’k’ and k’l’i’, a straight line was used,

\begin{matrix} {\begin{matrix} x = 0 \\ z = 0.103 \end{matrix} y \in (0, 0.1) \end{matrix}

(11)

For k’l’i’ and kli,

\begin{matrix} {\begin{matrix} y = 0.25 cos (\frac{π}{10} (x + 5)) \\ \frac{(x + 2.358)^{2}}{2 . 358^{2}} + \frac{(z - 0.103)^{2}}{0 . 575^{2}} = 1 \end{matrix} \\ y \in (0, 0.25), z \in (- 0.472, 0.103) \end{matrix}

(12)

Matlab was used to draw the simplified knitted loop; here, Equations (5)–(10) need to be converted into functions of t.

For segments of a’b’g’ and abg,

\begin{matrix} {\begin{matrix} x = 2.642 cos (t) + 2.642 \\ y = - 1.625 cos (t) + 1.875 \\ z = 0.575 sin (t) + 0.103 \end{matrix} t \in (0, π) \end{matrix}

(13)

For c’d’e’ and cde,

\begin{matrix} {\begin{matrix} x = 2.642 cos (t) + 2.642 \\ y = 3.75 - 0.25 cos (\frac{π}{4} cos (t) + \frac{π}{4}) \\ z = 0.575 sin (t) + 0.103 \end{matrix} t \in (- π, 0) \end{matrix}

(14)

For c’d’e’and e’f’c’,

\begin{matrix} {\begin{matrix} x = 5.284 + 0 * t \\ y = t \\ z = 0.103 + 0 * t \end{matrix} t \in (3.75, 3.85) \end{matrix}

(15)

For e’f’c’ and efc,

\begin{matrix} {\begin{matrix} x = 2.642 cos (t) + 2.642 \\ y = 3.85 - 0.25 cos (\frac{π}{4} cos (t) + \frac{π}{4}) \\ z = 0.575 sin (t) + 0.103 \end{matrix} t \in (- π, 0) \end{matrix}

(16)

For g’h’i’ and ghi,

\begin{matrix} {\begin{matrix} x = 2.358 cos (t) - 2.358 \\ y = 1.625 cos (t) + 1.975 \\ z = 0.575 sin (t) + 0.103 \end{matrix} t \in (0, π) \end{matrix}

(17)

For i’j’k’ and ijk,

\begin{matrix} {\begin{matrix} x = 2.358 cos (t) - 2.358 \\ y = 0.1 + 0.25 cos (\frac{π}{4} cos (t) + \frac{π}{4}) \\ z = 0.575 sin (t) + 0.103 \end{matrix} t \in (- π, 0) \end{matrix}

(18)

For i’j’k’ and k’l’i’,

\begin{matrix} {\begin{matrix} x = 0 * t \\ y = t \\ z = 0.103 + 0 * t \end{matrix} t \in (0, π) \end{matrix}

(19)

For k’l’i’ and kli,

\begin{matrix} {\begin{matrix} x = 2.358 cos (t) - 2.358 \\ y = 0.25 cos (\frac{π}{4} cos (t) + \frac{π}{4}) \\ z = 0.575 sin (t) + 0.103 \end{matrix} t \in (- π, 0) \end{matrix}

(20)

Figure 11 shows the knitted loop fitted by equations via Matlab in various views, where Figure 11(a), (b), (c), and (d) are from the X–Y plane, X–Z plane and Y–Z plane, and (−1,1,1) view, respectively. Among these, the hollow points are from the experimental results shown in Figure 9. Comparison of the real configuration of the stitching yarn from experimental results with the fitted equations confirms that the trace of a stitching yarn can be expressed using the above Equations (5)–(12).

Figure 11.

The trace of a unit stitching yarn fitted by curve equations in various views: (a) the X–Y plane; (b) the X–Z plane; (c) the Y–Z plane; (d) the (–1,1,1) view.

Model assembly

The unit cell model

The unit cell model is the minimum representative volume element, which is a basic structure in a fabric.³⁹ After completing the modeling of each yarn, the unit cell model has been assembled according to the position of yarns. In the fabric, a unit cell consists of a warp yarn and two weft yarns plus a stitching yarn; the geometric relationship among these three yarns was determined with reference to Table 1. Two unit cells are shown in Figure 12.

Figure 12.

The unit cell of NCF.

Penetration check

After completing the assembly, the interference among yarns should be examined to see if a penetration occurs. No penetration between the warp and weft has been found. For the stitching yarn, due to its complex trace, it is necessary to check the cycling features as well as the interference with the warp and weft yarns.

The stitching yarn arrays along the X and Y direction to obtain a larger model, as shown in Figure 13. S represents the stitching yarn, S1 arrays along the X and Y axis and obtains S2 and S3, S4, respectively. Here, it is checked whether the connection at the loop joint is smooth. Figure 13 shows the detail of joints of stitching yarns. In the lower frame, S2 forms the bending loop and bounds S1. In the upper frame, S2 and S3 are connected and wrapped around by the loop of S4. It is found that no penetration occurs in these two regions; in addition, the cycling has no problem.

Figure 13.

Penetration check for stitching yarns.

Excluding the interference of the knitting yarn itself, it is necessary to check the penetration problem between the stitching yarn and other components, including the warp and weft. Here, two regions in which the penetration tends to occur have been examined. Figure 14(a) and (b) show the front and back surface of the model. It is found that no penetration occurs between the stitching yarns and warp yarns in Figure 14(a), and the same result is found for Figure 14(b): no interference exists between the stitching yarns and weft yarns.

Figure 14.

Penetration check for yarns. (a) Interaction among warps and stitching yarns. (b) Interaction among wefts and stitching yarns.

Conclusions

In this paper, an attempt has been made to establish a 3D geometrical model of glass fiber unidirectional NCF. First, the buckling of warp yarns due to the tension of stitching yarn has been considered, and the buckling profile has been extracted and fitted by an equation. Second, cross-section parameters of warp and weft yarns were obtained accurately via metallographic microscopy; the cross-sections can be described by the elliptical shape, and 3D models were established.

With regards to the stitching yarn, since the geometric information is hidden in the fabric, previous researchers have proposed many stitching models which have not been verified. The tracer fiber technique was introduced to track the stitching path; the differences of refractive index among glass fiber, the resin and polyester stitching highlight the profile of the stitching yarn. The 3D configuration of a stitching yarn was extracted and fitted by the B-spline curve; in addition, the trace can be expressed by a series of equations through dividing the yarn into several segments. Finally, via the location and relationship of each yarn, a more realistic meso-scale unit cell was established. Using the penetration check for yarns, no penetration was found to occur among the yarns.

In this study, a meso-scale model of NCF was developed to truly reflect the relationship between the yarn geometrical state and the location, which improves the earlier geometric model. In addition, it lays a good foundation for further process simulation and mechanical prediction of composites.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

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, et al. A meso-scale unit-cell based material model for the single-ply flexible-fabric armor. Mater Des 2009; 30: 3690–3704.