Geometrical modeling of honeycomb woven fabric architecture

Abstract

Honeycomb woven fabric is considered as a single layer fabric produced only using common weaving looms, but it forms a unique three-dimensional (3D) architecture with inverted pyramidal pits on the fabric surface and repeated tetrahedral-closed space inside the fabric, which is greatly different from the traditional 3D woven fabrics, such as angle-interlock and orthogonal fabrics, showing good prospect for various applications in fields such as geotextiles, medical textiles, air filter, tower packing and underclothing. This paper proposes an analytical model to characterize the geometrical shape and position of each yarn in a honeycomb fabric unit-cell and the volume of the internal space. The model is based on the assumption of fabric thickness in the sum of yarn height and the linear relationship of yarn position and fabric unit-cell dimensions. The model involves geometric parameters, including yarn width, height, spacing, crimp and the number of yarns in a fabric unit-cell. Six honeycomb woven fabrics were manufactured to verify the model. Based on the position and crimp prediction of each yarn node, the architectures of the six fabrics were simulated numerically, which shows close agreement with the observed manufactured fabrics, indicating good accuracy of the geometrical model. A sensitivity study shows that the volume of the internal space decreases with the increase of fabric density, and the application of the elastic yarns to the fabric reduces the volume significantly.

Keywords

geometrical model matrix coding simulation honeycomb fabric

A woven fabric structure has been designed out, called ‘waffle-shape’ or ‘honeycomb’, for various applications in such fields as geotextiles, medical textiles, air filters, tower packing and underclothing.^1–5 Although this fabric is considered as a single layer of woven fabric, owing to the periodical emergence of yarn long floats and short floats gradually in the honeycomb fabric, the single layer of fabric exhibits a special three-dimensional (3D) architecture, which is greatly different compared with the traditional two-dimensional (2D) fabrics, such as plain and twill fabrics and 3D woven fabrics like angle-interlock and orthogonal fabrics with binding yarns in the through-thickness direction.^6,7

An ordinary honeycomb pattern and a modified pattern with their corresponding woven fabrics in a 3D structure are illustrated in Figures 1(a) and (b), respectively. The yarn path in honeycomb fabric contains interwoven curves and straight yarn floats. The length and altitude of yarn float, no matter whether in the warp or weft, are both increasing and then decreasing gradually and simultaneously, showing a periodicity and forming repeats of inverted pyramidal space on both surfaces of the fabric. The pyramidal pits can accommodate steel balls, as shown in Figure 1. In a honeycomb fabric such as that shown in the top schematic of Figure 1, the part of a warp yarn underneath a weft float is a warp float. Two layers of floats form a closed internal space. Owing to the periodical repeats of the inverted pyramidal space and the closed internal space, the honeycomb fabric has been reported experimentally with good sound and moisture absorption, good insulation and a quick dry rate.^8–10 Due to the existence of a yarn straight float, the mechanical properties, such as strength, compression and shear of this fabric, are different from totally interlaced plain woven fabric. The internal closed space formed by the warp and weft floats is also interesting for further investigation, such as accommodation of bio-capsules inside. Owing to these potential interesting performances based on such special fabric architecture, a geometrical model as function of fabric structure is a desirable tool that can be used to further understand the relationship between the honeycomb fabric architecture and property. Moreover, the geometrical model will be very useful in predicting thereafter the honeycomb fabric performance and designing more new honeycomb architectures for the targeted fabric properties.

Figure 1.

Patterns of honeycomb woven fabric and corresponding fabrics with the three-dimensional effect: (a) ordinary honeycomb weave; (b) modified honeycomb weave.

Geometrical studies on woven fabrics can be found in the numerical design for traditional 3D fabric or composites, as well as in the following property analysis.^11–14 As for analysis for the woven fabric structure, whether 2D or 3D, a matrix with 0 and 1 is commonly used to describe the interlacement of warp and weft yarns in a fabric, as shown in Figure 2.¹⁵ This matrix, however, cannot reflect the fabric thickness and other fabric geometrical parameters, such as yarn cross-section and yarn spacing. On the basis of this two-value code, Chen¹⁶ considered the fabric structure factors that include the length and number of yarn floats, step number and warp/weft repeat. A set of algorithms was proposed to describe the 3D woven fabric structure, for instance, orthogonal, angle-interlock, cellular weaves and trapezoid weaves. All 3D structures are the function of the number of weft (warp) layers, the layers interlocked by warp (weft) yarns and weave repeats. Lomov et al.^17,18 proposed a geometrical model of textile structure in which the 3D woven fabric was expressed by the matrix coding based on the yarn placement, crimp and paths as well as yarn layers. Based on this matrix, the fabric structures can be simulated and implemented by WiseTex, which is a commercialized software for describing the internal structure of textiles or textile composites at the unit-cell level.

Figure 2.

Matrix code for two-dimensional woven fabric.

Dash et al.¹⁹ investigated the geometrical modeling of 3D orthogonal and angle-interlock structures with fabric cross-section and yarn cross-section modeling, which improved the model accuracy significantly. The yarn cross-section was modeled, including elliptical, race-track, lenticular and circular shapes. Endruweit et al.²⁰ and Wong et al.²¹ mentioned an equation to classify the profile ( $ψ$ ) of the yarn cross-section:

ψ = \pm \frac{h}{2} (1 - \frac{x 2}{(w / 2) 2}) ψ

(1)

where h and w are the yarn height and width, respectively, the exponent ψ determines the shape of the yarn cross-section and special cases are ψ = 0.5 for elliptical, ψ = 0 for rectangular and ψ = 1 for lenticular shapes. Considering the yarn cross-section, Dash et al.’s model showed 10% prediction error in structural parameters compared with corresponding experimental measurements. With the accurate yarn cross-sectional simulation, Nottingham University developed an open platform, called TexGen, for simulating all weave geometrical structures based on the accurate set values of yarn cross-sections and nodes for 3D yarn locations.^22–25

As the literature indicates, there is a scarcity of information on the modeling of the honeycomb fabric structure. To address the necessity of such special fabric structure in potential applications, this study aims to establish a geometric model for characterizing the 3D architecture of honeycomb woven fabrics. On the basis of the reviewed matrix expression for 3D woven fabric, this paper takes into account the exact 3D geometry of the ordinary honeycomb fabric and assumes a linear relationship of the yarn through-thickness position with its perpendicular in-plane direction. Two matrices are developed for warp and weft positions in 3D coordinates and employed for honeycomb fabric geometrical simulation. A number of honeycomb fabrics are weaved by experiment to validate the analytical model (matrices) and fabric samples using elastic yarns as weft yarns are also used to assess the sensitivity.

Geometrical modeling

Analysis of honeycomb fabric architecture

In a honeycomb woven fabric, for instance as shown in Figure 3 where the black mark represents the warp float and the white one indicates the weft float, the tight part (weave area ‘’) is made of interlaced yarns in plain weave pattern and the loose parts (floats ‘’ or ‘’), are two orthogonal sets of yarn floats. The fabric is relatively thinner in the weave area owing to the yarn interlacement. The yarn floats are on the fabric surface showing the yarn position is higher with the increase of the length of the yarn float. A fabric unit-cell forms a thick 3D, cuspated or pyramidal fabric profile.

Figure 3.

A honeycomb woven fabric pattern and its simulated fabric.

In the sketch of the fabric pattern, the point ‘A’ is on the top surface, which is caused by the warp floats above and below, and weft floats on the left and right. The long warp and weft floats pull the point ‘A’ on the top of the fabric surface. The point ‘B’ is in the opposite situation. The weft floats above and below and warp floats left and right lead to the point ‘B’ on the top surface of the opposite fabric, that is, at the back side of fabric. The length of the float is decreasing slowly to the plain weave area, resulting in the repeats of the inverted pyramdial profile in a kind of honeycomb appearance.

Figures 3(I), (II), (III) and (IV) around the fabric pattern sketch are the four schematic illustrations of the slice along the longest weft and warp floats in a honeycomb fabric unit-cell, in which Figures 3(I) and (II) show the interlacement of weft yarns ‘a’ and ‘e’ with the warp yarns, respectively, while the sectional diagrams of 3(III) and (IV) reflect how the warp yarns ‘1’ and ‘5’ interlace with the weft yarns, respectively. From the front view of the honeycomb fabric in Figures 3(I) and (IV), the weft yarn ‘a’ is on the highest position, whereas the warp yarn ‘1’ is analyzed in the highest place based on Figures 3(II) and (III). Therefore, the intersection of the first warp and the first weft are in the top position (point ‘A’). However, the fifth warp ‘5’ is at the lowest position from the front face on the basis of Figures 3(I) and (IV), and the same situation is found with the fifth weft ‘e’ based on Figures 3(II) and (III), hence the intersection thereof is at the lowest position (point ‘B’). Figure 3 also shows a simulated honeycomb fabric according to the weave pattern on the left-hand side, and the representative points ‘A’ and ‘B’ the weft yarns ‘a’ and ‘e’ and the warp yarns ‘1’ and ‘5’ are labeled on the simulated fabric.

Figure 4.

Simulation of yarns in honeycomb fabric: (a) weave; (b) yarn path.

Input parameters for geometrical modeling

Characterization of yarns in honeycomb fabric

When a honeycomb woven fabric is in a relaxed state, the unit-cell contains a pyramidal pit, an internal closed-tetrahedral space and an interwoven-yarn layer. We assume the unit-cell has $2 m \times 2 m$ warp and weft yarns, where the m is a natural number. The cuspate point of the pyramidal pit is at the plain weave interlacement of the $(m + 1) th$ warp and the $(m + 1) th$ weft. The float length of the warp and weft yarns decreases until their $(m - 1) th$ yarn, as shown by the example (8 × 8 warps × wefts) in Figure 4. In a fabric unit-cell, the yarns at both fabric surfaces have one yarn interlacement, as shown the dashed lines in Figure 4(a), while the remainder have two or three yarn interlacements. The yarn interlacement causes the yarn path to have straight floats and curved intersections, as shown in Figure 4(b).

Figure 5 shows the dimension of the yarn crossover and schematic of the yarn cross-section. D is the height of the yarn crossover, which equals the sum of warp height (h_j) and weft height (h_w), l_j and l_w represent the warp yarn length and weft yarn length of the crimped part, respectively, W_j and W_w are the warp yarn width and weft yarn width, respectively, and s is the yarn spacing, which equals the distance between two neighboring yarn centerlines where the subscripts j and w represent the warp and weft symbols, respectively. Because the yarns used in the experiment are plied strands, their cross-sections are assumed as an elliptical shape with the height of one strand diameter and the width of two times the diameter. The yarn crimp (c, fractional) has the definition for the plain weave fabric as follows:²⁶

c j = \frac{l j}{s w} - 1 and c w = \frac{l w}{s j} - 1

(2)

Figure 5.

Schematic representation of yarn crossover and yarn cross-section.

The ratio of crimps, the balance between warp and weft crimps, in the two directions of fabric, is governed by the given yarn spacing and the yarn linear density. However, as for the yarn crimps in a honeycomb woven fabric as shown in Figure 4(a), the weft yarns normally have one interlacement (j₁,w₁) at the edge of the unit-cell and two (j₂,w₂) or three (j_3,w₃) interlacements in the rest of the unit-cell, and the same situation is practicable for the warps in the unit-cell. It is supposed that the unit-cell has 2 m warps and 2 m wefts; the crimp in a honeycomb woven fabric is thereafter calculated as

c j 1 = \frac{l w - s j}{2 ms j} and c w 1 = \frac{l j - s w}{2 ms w}

(3a)

c j 2 = \frac{l w - s j}{ms j} and c w 2 = \frac{l j - s w}{ms w}

(3b)

c j 3 = \frac{3 l w - 3 s j}{2 ms j} and c w 3 = \frac{3 l j - 3 s w}{2 ms w}

(3c)

In Equation (3), the length of crimp parts Lj and Lw can be obtained by integrating the l expression if the crimp (ζ) profile was provided with a cosine function:

ζ = \frac{D}{4} \cos (\frac{π}{s} x)

(4)

l = \int_{0}^{s} \sqrt{1 + (ζ') 2} d x

(5)

In Equation (4), the parameters D and s can be measured experimentally, then ζ and l are thereafter calculated, and $(ζ')$ is the derivative of the yarn crimp expression $(ζ)$ to x. Substituting Equation (4) into Equation (5), and combining with Equation (3), the length of each yarn crimp in the honeycomb fabric can be obtained.

Apart from the length of yarn crimp, the yarn width and height, as well as the cross-section shape are important input parameters in terms of yarn characteristics in the development of the geometrical model. These yarn factors can be obtained by experimental measurement before fabric manufacturing.

Fabric structural parameters

Yarn spacing (s) in Figure 5 is determined by warp density ( $ρ j$ ) and weft density ( $ρ w$ ), which means the amount of aligned yarns across the fabric unit length. Both densities can be set on the weaving loom as input fabric parameters. On a weaving loom, the $ρ j$ value is usually constant due to the fixed reed number and the number of yarns through the dent of the reed. On occasions, the $ρ j$ value is affected by the weft materials, such as application of elastic weft.

The number of yarns in a honeycomb unit-cell also affects the honeycomb appearance. Together with $ρ j$ and $ρ w$ , it determines the unit-cell size. In the development of the geometrical model, the distribution of aligned yarns in the unit-cell is assumed to be uniform.

Matrix code for the unit-cell of honeycomb fabric

The honeycomb woven fabric cross-section is illustrated in Figure 6. Owing to the particular yarn interlacement, the yarn position (z_j/w) in the through-thickness varies regularly and periodically depending on the yarn count (tex), $ρ j$ and $ρ w$ (number of threads per centimeter), where the profile of yarn positions is assumed linearly:

z j = \frac{2 T}{U j} x (nU j \leq x < nU j + \frac{U j}{2}) and z j = 2 T (1 - \frac{x}{U j}) (nU j + \frac{U j}{2} \leq x < (n + 1) U j)

(6a)

z w = \frac{2 T}{U w} y (nU w \leq y < nU w + \frac{U w}{2}) and z w = 2 T (1 - \frac{y}{U w}) (nU w + \frac{U w}{2} \leq y < (n + 1) U w)

(6b)

where T is the fabric thickness, U_j and U_w are the fabric unit-cell width and length and n represents the natural numbers, that is, 0, 1, 2, 3…. In Figure 6, ϕ is the angle of one weft/warp with a set of perpendicular warps/wefts, which theoretically equals

\tan - 1 \frac{2 T}{U}

. The fabric thickness is actually the accumulation of the yarn height. It is found that the (m + 1)^th yarn is at the top or bottom position in a fabric unit-cell (2 m × 2 m yarns). Owing to the pyramidal profile in the honeycomb fabric, an expression (Equation (7)) is assumed for simplifying the fabric thickness:

T = (m + 1) \cdot (\frac{h j + h w}{2})

(7)

Figure 6.

Honeycomb woven fabric cross-section and fabric weave pattern.

In a unit-cell of honeycomb fabric (2 m × 2 m yarns) along the x-axis and y-axis, as shown in Figure 6, the position of an arbitrary $(i th)$ yarn in (x,y) is

x i = is j (0 \leq i \leq 2 m)

(8a)

y i = is w (0 \leq i \leq 2 m)

(8b)

m = \frac{U j}{2 s j} = \frac{U w}{2 s w}

(8c)

The z values of the $(i th)$ yarn in the warp and weft directions are

z j = \frac{(m + 1) \cdot (h j + h w)}{2 m} i (\frac{n U j}{s j} \leq i < \frac{n U j}{s j} + \frac{U j}{2 s j});

z j = (h j + h w) (m + 1) (1 - \frac{i}{2 m}) (\frac{nU j}{s j} + \frac{U j}{2 s j} \leq i < (n + 1) \frac{U j}{s j})

(9a)

z w = \frac{(m + 1) \cdot (h j + h w)}{2 m} i (\frac{nU w}{s w} \leq i < \frac{nU w}{s w} + \frac{U w}{2 s w});

z w = (h j + h w) (m + 1) (1 - \frac{i}{2 m}) (\frac{nU w}{s w} + \frac{U w}{2 s w} \leq i < (n + 1) \frac{U w}{s w})

(9b)

Herein, z_j and z_w are the transverse locations of the warp and weft yarns. When the yarns are in interlacement at the unit-cell diagonal, z_j and z_w should be swapped over within half of the yarn height according to the yarn crimp calculation. Then the matrix for describing the 3D coordinates of warp yarns in a honeycomb unit-cell with $2 m \times 2 m$ nodes is

The matrix for recording the 3D coordinates of weft yarns in a honeycomb unit-cell with $2 m \times 2 m$ nodes is

(x j, y j, z j) = [((0, (U w - s w), \frac{- h j}{2}) (s j, (U w - s w), \frac{(m + 1) \cdot (h j + h w)}{2 m} + \frac{h j}{2}) \dots ((U j - s j), (U w - s w), (\frac{(m + 1) \cdot (h j + h w)}{2 m} + \frac{h j}{2})) (0, (U w - 2 s w), 0) (s j, (U w - 2 s w), \frac{(m + 1) \cdot (h j + h w)}{2 m} - \frac{h j}{2}) \dots ((U j - s j), (U w - 2 s w), \frac{(m + 1) \cdot (h j + h w)}{2 m}) (0, (U w - 3 s w), 0) : (s j, (U w - 3 s w), \frac{(m + 1) \cdot (h j + h w)}{2 m}) : \dots ((U j - s j), (U w - 3 s w), \frac{(m + 1) \cdot (h j + h w)}{2 m}) (0, is w, 0) : (s j, is w, \frac{(m + 1) \cdot (h j + h w)}{2 m}) : \dots ((U j - s j), is w, \frac{(m + 1) \cdot (h j + h w)}{2 m}) (0, 2 s w, 0) (s j, 2 s w, \frac{(m + 1) \cdot (h j + h w)}{2 m} - \frac{h j}{2}) \dots ((U j - s j), 2 s w, \frac{(m + 1) \cdot (h j + h w)}{2 m}) (0, s w, \frac{- h j}{2}) (s j, s w, \frac{(m + 1) \cdot (h j + h w)}{2 m} + \frac{h j}{2}) \dots ((U j - s j), s w, (\frac{(m + 1) \cdot (h j + h w)}{2 m} + \frac{h j}{2})) (0, 0, \frac{h j}{2}) (s j, 0, \frac{(m + 1) \cdot (h j + h w)}{2 m}) \dots ((U j - s j), 0, (\frac{(m + 1) \cdot (h j + h w)}{2 m} - \frac{h j}{2}))]

(10a)

(x w, y w, z w) = [(0, (U w - s w), \frac{h w}{z}) (s j, (U w - s w), \frac{(m + 1) \cdot (h j + h w)}{2 m} - \frac{h w}{2}) \dots ((U j - s j), (U w - s w), \frac{(m + 1) \cdot (h j + h w)}{2 m} - \frac{h w}{2}) (0, (U w - 2 s w), 0) (s j, (U w - 2 s w), \frac{(m + 1) \cdot (h j + h w)}{2 m} + \frac{h w}{2}) \dots ((U j - s j), (U w - 2 s w), \frac{(m + 1) h w}{m}) (0, (U w - 3 s w), 0) : (s j, (U w - 3 s w), \frac{(m + 1) \cdot (h j + h w)}{2 m}) : \dots : ((U j - s j), (U w - 3 s w), \frac{(m + 1) h w}{m}) : (0, is w, 0) : (s j, is w, \frac{(m + 1) \cdot (h j + h w)}{2 m}) : \dots : ((U j - s j), is w, \frac{(m + 1) h w}{m}) (0, 2 s w, 0) (s j, 2 s w, \frac{(m + 1) \cdot (h j + h w)}{2 m} + \frac{h w}{2}) \dots ((U j - s j), 2 s w, \frac{(m + 1) h w}{m}) (0, s w, \frac{h w}{2}) (s j, s w, \frac{(m + 1) \cdot (h j + h w)}{2 m} - \frac{h w}{2}) \dots ((U j - s j), s w, \frac{(m + 1) \cdot (h j + h w)}{2 m} - \frac{h w}{2}) (0, 0, \frac{- h w}{2}) (s j, 0, \frac{(m + 1) \cdot (h j + h w)}{2 m}) \dots ((U j - s j), 0, \frac{(m + 1) \cdot (h j + h w)}{2 m} + \frac{h w}{2})]

(10b)

Internal space and pyramidal architecture

Owing to the π phase of amplitude difference of Equation (6) between the warp and weft directions, the top warps (wefts) and the back-side wefts (warps) form a closed internal space like a tetrahedral shape, as shown in the schematic representation in Figure 7.

Figure 7.

Internal closed space in tetrahedral shape.

The distance between the maximum warp float and the weft float is the fabric thickness subtracted by the yarn height. The depth of the one-side inverted pyramidal space is

T' = (m + 1) \cdot (\frac{h j + h w}{2}) - h j - h w = (m - 1) (\frac{h j + h w}{2})

(11)

The volume of the one-side inverted pyramidal space is

V p = \frac{1}{3} U j U w T' = \frac{2}{3} m 2 (m - 1) s j s w (h j + h w)

(12)

The other side of the inverted pyramidal space has the same volume as the opposite space, therefore the volume of yarns and internal space has the same volume of Equation (12). Apart from the inverted pyramidal and internal space, the volume of the yarns in a unit-cell is

V y = \frac{4}{3} m 2 s j s w (h j + h w)

(13)

Therefore, the volume of the internal tetrahedral space (Vt) equals the value of Equation (12) subtracted by Equation (13):

Vt = \frac{2 (m - 3)}{3} m 2 s j s w (h j + h w)

(14)

On the basis of Equation (14), it is found that the m value should be larger than three, which assures that the internal volume positive. Equation (14) indicates that the fabric unit-cell at least has 6 × 6 warps × wefts to form the honeycomb woven fabric architecture.

Experimental and numerical demonstration

Two patterns (unit-cell 6 × 6 and 18 × 18) for the honeycomb woven fabric were attempted in the experimental demonstration. Twelve fabrics (six for the 6 × 6 pattern and six for the 18 × 18 pattern) were manufactured by an SL8900S rapier loom (CCI TECH INC.). The yarn specifications for the 12 honeycomb woven fabrics are listed in Table 1. The designed fabric specifications and setting parameters on the loom are listed in Table 2.

Table 1.

Specifications of yarn for honeycomb fabrics (±standard deviation)

Yarn samples	Yarn 1: 100% cotton weft yarns 20 Ne (40 ^s/2)	Yarn 2: 91.4% cotton/ 8.6% spandex weft yarns 20 Ne (40 ^s/2)	Yarn 3: 100% cotton warp yarns 16 Ne (32 ^s/2)
Yarn count (tex)	28.6 (±0.47)	31.5 (±0.82)	35.1 (±0.19)
Yarn width (mm)	0.31 (±0.01)	0.47 (±0.03)	0.35 (±0.01)
Yarn height (mm)	0.18 (±0.01)	0.25 (±0.02)	0.22 (±0.02)
Yarn breaking elongation (%)	6.27 (±0.36)	19.10 (±1.41)	5.54 (±0.38)
Yarn picture

Table 2.

Fabric specifications

Fabric samples	Fabric content		Fabric density (No. threads/cm)		No. yarns in unit-cell
Fabric samples	Warp	Weft	Warp	Weft	Warp	Weft
1c	Yarn 3	Yarn 1	21	17	6	6
2c	Yarn 3	Yarn 1	21	23	6	6
3c	Yarn 3	Yarn 1	21	29	6	6
4c	Yarn 3	Yarn 1	21	17	18	18
5c	Yarn 3	Yarn 1	21	23	18	18
6c	Yarn 3	Yarn 1	21	29	18	18
7 s	Yarn 3	Yarn 2	21	17	6	6
8 s	Yarn 3	Yarn 2	21	23	6	6
9 s	Yarn 3	Yarn 2	21	29	6	6
10 s	Yarn 3	Yarn 2	21	17	18	18
11 s	Yarn 3	Yarn 2	21	23	18	18
12 s	Yarn 3	Yarn 2	21	29	18	18

There are 20 harness frames on the rapier loom. Two harness frames were used for forming fabric selvage and the remaining 18 harness frames were used for weaving the honeycomb structure. Therefore, the unit-cell can only have 18 or 6 warps owing to the even number of warp/weft yarns in a unit-cell of the honeycomb fabric and common divisors of 18 harness frames according to the design rule and consistency of the honeycomb structure. The value (warps/cm) was constant (21 ends/cm) as the number of yarns wound on the warp beam, the number of yarns in each dent of the reed and the reed number are all predetermined on the loom in this study. The warp and weft were all plied yarns, which can provide higher strength during the weaving procedure.

In the weaving process, as shown in Table 2, the weft density was set as 17, 23 and 29 wefts/cm, respectively. The fabric specifications in Table 2 were obtained after 24 hours of condition balance. Yarn cross-section, yarn width, yarn spacing and fabric cross-section were examined using a microscope (Leica M165C). Fabric thickness was measured using the Kawabata Evaluation System (KES-FB3) compression tester at the starting point without load. Six specimens were examined for their yarn cross-section, yarn width, yarn spacing and fabric thickness and cross-section. Based on the specifications, the fabric structure was matrix coded according to Equation (10) for the sake of geometrical modeling. The geometries of six honeycomb fabrics without elastic yarns (Fabrics 1c–6c) were simulated by TexGen²⁵ based on the matrix codes, as shown in the simulation procedures in Figure 8.

Figure 8.

Procedure of geometrical simulation in TexGen: (a) input 8 m² nodes as two layers for warps and wefts; (b) define three coordinates for each node according to Equation (10); (c) set up yarn cross-sections.

Figure 8(b) shows the most important step in the simulation, which determines the accuracy of the modeling. In TexGen, yarns were simulated as solid and flexible bars with elliptical cross-section (Figure 8(c)) on the basis of the measured yarn width and height. The yarn crossover was simulated based on the yarn crimp calculation (Equations (3)–(5)). The simulated fabric can be used for the future study of fabric structures versus properties, including the prediction of fabric mechanical properties, permeability and heat transfer properties.

Results and discussion

Validation of the analytical model by experimental measurement

Table 3 lists the measured dimension of average value for each fabric, including the width and length of the fabric unit-cell as well as the fabric thickness. It is noted that the U_j values of fabric samples 4c, 5c and 6c are much higher than that of fabric samples 1c, 2c and 3c because the unit-cell size of fabric samples 4c, 5c and 6c is three times larger than that of fabric samples 1c, 2c and 3c. Meanwhile, the U_w value decreases as $ρ w$ increases for the fabric samples in the same unit-cell size. As shown in Table 3, the fabric thickness increases with the application of elastic yarn in the same fabric pattern. This is because the yarn height of elastic yarn is slight larger than that of the pure cotton weft yarn. Meanwhile, the shrinkage of elastic yarn pushes the inverted pyramidal space into an extension in the z direction. Figure 9 shows the effects of weft density and elastic yarn on the size of the fabric unit-cell. It is evident that the size is decreasing with the increase of weft density and the use of elastic yarn for both honeycomb patterns.

Figure 9.

Effects of elastic yarn and weft density on the area of the fabric unit-cell.

Table 3.

Measured dimension of the fabric unit-cell (±standard deviation)

Fabric samples	U_j (mm)	Uw (mm)	T (mm)
1c	2.84 (±0.06)	3.60 (±0.02)	0.88 (±0.02)
2c	2.87 (±0.04)	2.81 (±0.03)	0.79 (±0.01)
3c	2.67 (±0.029)	2.20 (±0.01)	0.85 (±0.02)
4c	8.65 (±0.01)	10.21 (±0.08)	1.63 (±0.03)
5c	8.60 (±0.02)	7.57 (±0.08)	1.94 (±0.02)
6c	8.51 (±0.04)	5.98 (±0.07)	2.04 (±0.02)
7 s	2.02 (±0.03)	3.65 (±0.05)	1.03 (±0.03)
8 s	2.02 (±0.05)	2.78 (±0.10)	1.08 (±0.03)
9 s	2.04 (±0.07)	2.24 (±0.06)	1.11 (±0.04)
10 s	3.76 (±0.17)	10.75 (±0.07)	5.43 (±0.05)
11 s	4.04 (±0.02)	7.96 (±0.08)	5.73 (±0.02)
12 s	3.85 (±0.011)	6.05 (±0.25)	5.48 (±0.07)

Table 4 compares three input geometrical parameters between prediction from the analytical model (setting

ρ j

and

ρ w

and Equation (7)) and corresponding measurement from the experiment. Warp and weft spacing (s_j and s_w) determines the x and y values for each warp and weft yarn. It can be seen from Table 4 that the percentage errors between the predicted values and experimental values of two parameters s_j and s_w are less than 11%, indicating that the model generated has sufficient accuracy in predicting yarn spacing of the honeycomb fabric. In addition, Table 4 shows that the error of fabric thickness in comparison is also less than 11%, except for fabric sample 4c. This implies that the model gives satisfactory accuracy for the prediction of fabric thickness.

Table 4.

Comparison of yarn spacing and fabric thickness between analytical prediction (P) and experimental measurement (E)

Fabric samples	s_j (mm)			s_w (mm)			T (mm)
Fabric samples	P	E	*Error (%)	P	E	*Error (%)	P	E	*Error (%)
1c	0.471	0.473	0.42	0.564	0.600	6.38	0.88	0.88	0
2c	0.471	0.478	1.49	0.423	0.468	10.64	0.88	0.79	10.23
3c	0.471	0.445	5.52	0.339	0.367	8.26	0.88	0.85	3.41
4c	0.471	0.481	2.12	0.564	0.567	0.53	2.00	1.63	22.70
5c	0.471	0.478	1.49	0.423	0.421	0.47	2.00	1.94	3.09
6c	0.471	0.473	0.42	0.339	0.332	2.06	2.00	2.04	1.96

Note: * $Error = \frac{P - E}{E} \times 100$

Comparison of characterized and simulated fabric structures

The numerical simulation was implemented according to the matrices (Equation (10)) and suggested simulation procedures (Figure 8). The matrices are based on the measured yarn geometry and setting parameters on the loom, including the number of yarns in a fabric unit-cell and weft/warp density. Three views of the fabric were characterized for the 3D fabric unit-cell architecture: top, warp and weft laterals. Six fabrics (fabrics 1c–6c) are compared for their unit-cells between the experimental characterization and numerical simulation.

Tables 5 and 6 compare the architecture under three views between the manufactured honeycomb woven fabrics and the corresponding simulations. The simulation based on the analytical model (Equation (10)) shows excellent agreement with the experimental characterization on the basis of unit-cell length, width and height, as shown in the quantity comparison in Table 4. Fabrics 1c–3c do not show 3D architecture as evident as fabrics 4c–6c, because the former does not have internal space according to Equation (14), and the latter displays evident internal space in the 3D architecture. Tables 5 and 6 also show that the honeycomb fabric becomes denser with the increase of weft density, as shown in the top view of the fabrics in Tables 5 and 6.

Table 5.

Comparison of simulated and real fabrics for the unit-cell with 6 × 6 yarns

Table 6.

Comparison of simulated and real fabrics for the unit-cell with 18 × 18 yarns

Effect of elastic yarn on the honeycomb weave structure

The elastic yarn with the spandex filament can shrink the fabric unit-cell length or width of the fabric, causing the angle ϕ to greatly increase and the fabric to have a more prominent honeycomb structure. Herein, the honeycomb weave structure is compared for the factor of the weft yarns Yarn 1 and Yarn 2. One kind of fabric unit-cell and three values of $ρ w$ are considered in the comparison list.

As shown in Table 7, the size of the unit-cell becomes much smaller owing to the existence of elastic yarn than that of the unit-cell with pure cotton wefts under the same

ρ w

value. With the same weft yarn, the size of the unit-cell also decreases with the increase of

ρ w

. From the lateral view of 18 × 18 warps × wefts, the ϕ angle becomes larger owing to the shrinkage of elastic yarn in the weft direction. The honeycomb fabric with Yarn 1 (pure cotton) exhibits a clearer internal space than that of the honeycomb fabric with Yarn 2 (spandex/cotton), and the internal volume of space decreases from the honeycomb fabric with Yarn 1 to the honeycomb fabric with Yarn 2.

Table 7.

Comparison of honeycomb fabrics (unit-cell of 18 × 18, warps × wefts) with weft Yarn 1 and weft Yarn 2.

Figure 10 shows the internal volume of a unit-cell (V_t) for the six honeycomb fabrics in Table 7 on the basis of the specifications listed in Table 3 and the volume calculation according to Equation (14). With the increase of $ρ w$ , the V_t value decreases. This is mainly owing to the decrease of the unit-cell area. Under the same $ρ w$ value (17 wefts/cm and 23 wefts/cm), the elastic yarn causes V_t to decrease significantly. However, for the $ρ w$ of 29 wefts/cm, the V_t value shows a slight variation. This is probably due to the decreased area of the unit-cell and the increased height of the internal tetrahedral space. Therefore, the application of elastic yarn has little effect on V_t for the tight honeycomb woven fabric.

Figure 10.

Comparison of the volume of internal space among six honeycomb woven fabrics.

Conclusions

A 3D woven fabric is analyzed for its structure, which has waffle or honeycomb geometrical features. The honeycomb fabric structure is characterized for its length and width of the unit-cell, warp and weft yarn densities and yarn width, height, spacing and crimp. In the geometrical modeling, the fabric thickness is assumed as the sum of warp and weft height in half the number of yarns in a fabric unit-cell. A linear function is used for predicting the interwoven positions along the warp and weft directions. Based on the linear assumption, two matrices are developed for characterizing the 3D location of each node in warps and wefts. Using the developed matrices, the numerical simulation in describing the 3D honeycomb weave architecture is carried out based on the platform TexGen, which can define the yarn cross-section, yarn path and yarn density.

Six honeycomb fabrics without elastic yarns were manufactured for validating the developed model. Three of them are honeycomb fabrics in the unit-cell of 6×6 warps×wefts, while the other three samples are honeycomb fabrics with 18×18 warps×wefts in the unit-cell. The comparison between the measured values and the simulated values based on the geometrical model shows excellent agreement (average error <5%) with each other in terms of fabric thickness, yarn cross-section, yarn width and spacing, unit-cell dimension and volume of internal tetrahedral space. This indicates that the simulation based on the developed matrices is very effective for simulating the honeycomb fabric architecture.

Six more honeycomb fabrics with elastic wefts were manufactured for the sensitivity study. The experimental results of fabrics show that the honeycomb weave effect becomes more prominent when the elastic yarns are used as weft yarns for fabrics, especially for fabrics with low fabric density. This is also reflected by the decrease of the volume of internal space in honeycomb fabric with the increase of fabric density and the use of elastic yarns. In the view of significance, applied with yarn specific geometries and properties, this geometrical model will thereafter assist with the fundamental study on the relationship of honeycomb fabric architecture and its mechanical, permeability, wettability and heat transfer properties. Based on this, more honeycomb structures will be developed for the desired fabric performance for various applications.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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