Computer geometric modeling approach with filament assembly model for 2 × 1 and 3 × 1 twill woven fabric structures

The forces affecting the yarns and the formation of skewness for 2 × 1 twill woven fabrics are shown in Figure 2(a–d), which also shows the 2D geometrical model of linear yarn path skewness segments that illustrates the projection of floating and intersecting yarn segments. Figure 3 illustrates the coordinate system of filaments in yarn cross-section for 2 × 1 and 3 × 1 twill weave fabric structures and the linear yarn path skewness of the unit weave length (pr), respectively. In the case of the 2 × 1 and 3 × 1 twill weaves, the unit weave length was defined in equations (1) and (2) as related to the projection length of floating (l _f) and the intersection region (l _s). The warp yarn float makes an angle with the y axis, which is the skewness angle shown by δ; the intersection line makes an angle β with the y axis and the warp and weft settings of the fabric (S ₁, S ₂), respectively. In the equations, subscripts 1 and 2 refer to parameters in the warp and weft directions, respectively, as shown below.

Figure 2

Figure 3

For 2 × 1 twill weaves (1) pr 1 = 3 S 2 = l f 1 cos δ 1 + l s 1 cos β 1 , {\text{pr}}_{1}=\frac{3}{{S}_{2}}={l}_{\text{f}1}\text{cos}\hspace{0.25em}{\delta }_{1}+{l}_{\text{s}1}\text{cos}\hspace{0.25em}{\beta }_{1},

and for 3 × 1 twill weaves (2) pr 1 = 4 S 2 = l f 1 cos δ 1 + l s 1 cos β 1 . {\text{pr}}_{1}=\frac{4}{{S}_{2}}={l}_{\text{f}1}\text{cos}\hspace{0.25em}{\delta }_{1}+{l}_{\text{s}1}\text{cos}\hspace{0.25em}{\beta }_{1}. In the 2 × 1 and 3 × 1 twill weaves, we used the values of free space (c) between the two yarns over the floating region. The free space makes it possible for the floating yarn to push away the crossing yarn at the interlacing point [4]. In the floating region, the floating length of warp yarn can be estimated as follows: (3) l f2 = p 1 p 1 + c cos δ 2 . {l}_{\text{f2}}=\frac{{p}_{1\text{p}1}+c}{\cos \hspace{.25em}{\delta }_{2}}.

The intersecting length of warp yarn in the intersecting region can be estimated as follows: (4) l s2 = p 1,1 cos β 2 . {l}_{\text{s2}}=\frac{{p}_{\text{1,1}}}{\cos \hspace{.25em}{\beta }_{2}}. The skewness angle (δ, β), the floating length (l _f), and the intersecting length (l _s) were determined in equations (1)–(4). Therefore, the projection of the linear lines of the return yarn movement region (M _fi) of warp or weft yarn in the floating and intersection region on the XY plane is obtained by: (5) M f1 = M f2 = d i / 2 . {M}_{\text{f1}}={M}_{\text{f2}}={d}_{i}/2.

The coordinate system of filaments in twill woven fabrics and the projection of linear yarn path of the floating and intersecting region for 2 × 1 and 3 × 1 twill weaves are shown in Figure 3(a–d), respectively. The origin point of the 2D model (X = 0, Z = 0) of yarn in 2 × 1 twill woven fabric in the XZ axis is shown in Figure 3(a). Some of the twill woven fabric parameters are also shown in the figure. In this study, a geometrical model considering their inherent skewness has been proposed, which can be used in precise geometrical description of twill woven structures. The characteristic property of twill weaves is that the warp/weft floatings and intersectings are not perpendicular, but form an angle. Figure 3(b) shows the projection of linear yarn path of the floating and intersecting region on the XY plane for 2 × 1 twill weave. Figure 3(c) illustrates the origin point of the 2D model (Y = 0, Z = 0) of yarn in 3 × 1 twill woven fabric in the YZ axis and also the projection of linear yarn path of the floating and intersecting region on the XY plane for 3 × 1 twill weave is illustrated in Figure 3(d).

The concept of virtual location is used to simulate the fiber distributions in yarn cross-section of 2 × 1 and 3 × 1 twill woven fabrics. The Cartesian coordinates from polar coordinates of the center of yarn to coordinate system x′z′ are as follows: (6) f N j x ′ = R j c cos γ , {f}_{N\hspace{.03em}j}^{x^{\prime} }={R}_{j\text{c}}\text{cos}\hspace{0.25em}\gamma , (7) f N j z ′ = R j c sin γ . {f}_{N\hspace{.03em}j}^{z^{\prime} }={R}_{j\text{c}}\text{sin}\hspace{0.25em}\gamma .

The coordinate system of yarn cross-section in y′z′ is as follows: (8) f N j x ′ = R j c cos γ , \hspace{.25em}{f}_{Nj}^{x^{\prime} }=\text{}{R}_{j\text{c}}\text{cos}\hspace{0.25em}\gamma , (9) f N j z ′ = R j c sin γ . \hspace{.25em}{f}_{Nj}^{z^{\prime} }=\text{}{R}_{j\text{c}}\text{sin}\hspace{0.25em}\gamma .

In the 2 × 1 and 3 × 1 twill weave fabric structures, the position of the center fabric structure is characterized by Cartesian coordinates (X, Z) and (Y, Z) of the center of twill woven fabric structure (X = 0, Z = 0) and (Y = 0, Z = 0) that are shown in Figure 3(a) and (c), respectively. In 2D twill weave, the number of warp/weft yarns were defined as i _d1 (1, 2,…, N ₁) or i _d2 (1, 2,…, N ₂). The coordinates for warp yarn of the center of yarn at i _d1 in the Y axis for 2 × 1 and 3 × 1 for one unit cell twill weave fabric are defined as follows:

For 2 × 1 twill weave fabric, the equation is as follows: (10) d i Y = ( p 1,1 ) 2 d 1 + ( p 1 P 1 + c ) 3 d 1 + ( p 1 , 2 ) 4 d 1 . {{d}_{i}}^{Y}=\text{}({p}_{\text{1,1}}){2}_{d1}+\text{}({p}_{1P1}+c){3}_{d1}+\text{}({p}_{1,2}){4}_{d1}.

For 3 × 1 twill weave fabric, the equation is as follows: (11) d i Y = ( p 1,1 ) 2 d 1 + ( p 1 P 1 + c ) 3 d 1 + ( p 1 P 2 + c ) 4 d 1 + ( p 1 , 2 ) 5 d 1 . {{d}_{i}}^{Y}=\text{}({p}_{\text{1,1}}){2}_{d1}+\text{}({p}_{1P1}+c){3}_{d1}+\text{}({p}_{1P2}+c){4}_{d1}+\text{}({p}_{1,2}){5}_{d1}.

The coordinates of weft yarn in the X axis and the coordinate of center of yarn at i _d2 are defined in the same way as in equations (10) and (11). The position of the center fabrics structure is characterized by Cartesian coordinates (X, Z) of the center of the one unit cell twill woven fabric structure (X = 0, Z = 0).

For 2 × 1 twill weave fabric, the equation is as follows: (12) d i X = ( p 2,1 ) 2 d 2 + ( p 2 P 1 + c ) 3 d 2 + ( p 2 , 2 ) 4 d 2 . {{d}_{i}}^{X}=\text{}({p}_{\text{2,1}}){2}_{d2}+\text{}({p}_{2P1}+c){3}_{d2}+\text{}({p}_{2,2}){4}_{d2}.

For 3 × 1 twill weave fabric, the equation is as follows: (13) d i X = ( p 2,1 ) 2 d 2 + ( p 2 P 1 + c ) 3 d 2 + ( p 2 P 2 + c ) 4 d 2 + ( p 2 , 2 ) 5 d 2 . {{d}_{i}}^{X}=({p}_{\text{2,1}}){2}_{d2}+({p}_{2P1}+c){3}_{d2}+({p}_{2P2}+c){4}_{d2}+({p}_{2,2}){5}_{d2}.

The 2 × 1 twill weave was formed of asymmetric floatings indicated in Figure 3(a and b), which shows the center line of the yarn path at the point A, B, C,…, G, A along the yarn length on the XZ plane. The main dimensionless parameters are p _2,1 (g _p2,1, s _p2,1, k _p2,1), p _2,2 = (k _p2,2, s _p2,2, g _p2,2) and p _2p1 + c is the spacing of center line of yarn at the point on the XZ plane. For the 3 × 1 twill weave, the unit length of yarn path based on center line of yarn point A, B,…, H, A on the YZ plane is shown in Figure 3(c and d). The main dimensionless parameters are p _1,1 (g _p1,1, s _p1,1, k _p1,1), p _1,2 = (k _p1,2, s _p1,2, g _p1,2), p _1p1 + c and p _1p2 + c is the spacing of the center line of the yarn at the point on the YZ plane. For 2 × 1 and 3 × 1 twill weaves, the center line of the yarn path made an angle by skewing in the floating region (δ _i ), and then it made an angle in the intersecting region (β _i ) of the linear part of yarn segment. θ _i is the crimp angle of the yarn.

The crimp value in the Z axis is accepted as the fabric thickness direction that is derived from the linear segmented model of yarn path of 2 × 1 and 3 × 1 twill structure. The coordinate system of (X, Z) or (Y, Z) of the twill woven fabric structure can be separated into two parts: the first part was defined as positive (+) h _i /2 and the second part was defined as negative (−) h _i /2. In the Z axis, the coordinate of h _i /2 is defined as: (14) d i Z = ± h i / 2 h 2 2 , where the coordinate of yarn in the upper quadrant regions , − h 2 2 , where the coordinate of yarn in the lower quadrant regions . {d}_{i}^{Z}\hspace{.25em}=\pm {h}_{i}/2\left\{\begin{array}{cc}\frac{\hspace{.25em}{h}_{2}}{2},& \text{where}\hspace{.5em}\text{the}\hspace{.5em}\text{coordinate}\hspace{.5em}\text{of}\hspace{.5em}\text{yarn}\hspace{.5em}\text{in}\hspace{.5em}\text{the}\hspace{.5em}\text{upper}\hspace{.5em}\text{quadrant}\hspace{.5em}\text{regions},\\ -\frac{{h}_{2}}{2},& \text{where}\hspace{.5em}\text{the}\hspace{.5em}\text{coordinate}\hspace{.5em}\text{of}\hspace{.5em}\text{yarn}\hspace{.5em}\text{in}\hspace{.5em}\text{the}\hspace{.5em}\text{lower}\hspace{.5em}\text{quadrant}\hspace{.5em}\text{regions}.\end{array}\right.

Equations (6)–(9) are defined as the coordinate system (x′, z′) or (y′, z′) of filament in yarn cross section and also equations (10)–(14) are calculated for the coordinate system (X, Z) or (Y, Z) of yarn in twill woven fabric structures. The coordinates of the Nth filament of jth layers on the X axis ( f N j X d i ) ({f}_{Nj}^{X}{d}_{i}) , on the Y axis ( f N j Y d i ) ({f}_{Nj}^{Y}{d}_{i}) , and on the Z axis ( f N j Z d i ) ({f}_{Nj}^{Z}{d}_{i}) can be defined. The positions of the filaments in yarn cross section of i _d2 in the X axis are as follows: (15) f N j X d i = d i X + R j c cos γ or = d i X + f N j x ′ . {f}_{Nj}^{X}{d}_{i}={d}_{i}^{X}+{R}_{j\text{c}}\text{cos}\hspace{0.25em}\gamma \hspace{1em}\text{or}={d}_{i}^{X}+{f}_{Nj}^{x^{\prime} }.

In yarn cross section of i _d1 in the Y axis, the positions of the filaments are defined as: (16) f N j Y d i = d i Y + R j c cos γ or = d i Y + f N j y ′ . {f}_{Nj}^{Y}{d}_{i}={d}_{i}^{Y}+{R}_{j\text{c}}\text{cos}\hspace{0.25em}\gamma \hspace{1em}\text{or}={d}_{i}^{Y}+{f}_{Nj}^{y^{\prime} }.

Also, the positions of filaments in twill wave fabric of i _d1 or i _d2 in the Z axis are as follows: (17) f N j Z d i = ± h i / 2 + R j c sin γ or = d i Z + f N j z ′ . {f}_{Nj}^{Z}{d}_{i}=\pm {h}_{i}/2+{R}_{j\text{c}}\text{sin}\hspace{0.25em}\gamma \hspace{1em}\text{or}={d}_{i}^{Z}+{f}_{Nj}^{z^{\prime} }.

The fabric pattern and setting were designed. The automatic generation of 2D and 3D weaves represented by 2D binary matrices to a general vector matrix was developed by Jiang and Chen [14] and was extended by Patumchat and Sriprateep [16]. The structure of the initial vector matrix H₀ of the deflection of the yarns can be obtained. In this work, we used the element vectors for 2 × 1 and 3 × 1 twill weaves that are shown in Figure 3 that connected points (A, B, C,…, A) of the center line of the yarn path along the yarn length. The structure of one unit cell for 2 × 1 twill weave, the number of connected points of the piecewise for the arc abscissa models in warp and weft directions is 8 points, and also one unit cell for 3 × 1 twill weave in warp and weft directions is 9 points, respectively. The fabric pattern and setting of twill weave were designed, so the structure of the initial vector matrix H ₀ in the floating region in warp or weft of 2 × 1 twill weave (A _1 or 2, D _1 or 2, E _1 or 2, A _1 or 2) and 3 × 1 twill weave (A _1 or 2, D _1 or 2, E _1 or 2, F _1 or 2, A _1 or 2) of the yarns can be obtained as follows: (18) H 0 ( 2 × 1 ) = T → A 1 , A 2 T → D 1 , D 2 T → E 1 , E 2 T → A 1 , A 2 T → E 1 , E 2 T → A 1 , A 2 T → D 1 , D 2 T → E 1 , E 2 T → D 1 , D 2 T → E 1 , E 2 T → A 1 , A 2 T → D 1 , D 2 T → A 1 , A 2 T → D 1 , D 2 T → E 1 , E 2 T → A 1 , A 2 , {H}_{0(2\times 1)}=\left[\begin{array}{cccc}{\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}\\ {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}\\ {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}\\ {\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}\end{array}\right], (19) H 0 ( 3 × 1 ) = T → A 1 , A 2 T → D 1 , D 2 T → E 1 , E 2 T → F 1 , F 2 T → A 1 , A 2 T → F 1 , F 2 T → A 1 , A 2 T → D 1 , D 2 T → E 1 , E 2 T → F 1 , F 2 T → E 1 , E 2 T → F 1 , F 2 T → A 1 , A 2 T → D 1 , D 2 T → E 1 , E 2 T → D 1 , D 2 T → E 1 , E 2 T → F 1 , F 2 T → A 1 , A 2 T → D 1 , D 2 T → A 1 , A 2 T → D 1 , D 2 T → E 1 , E 2 T → F 1 , F 2 T → A 1 , A 2 . {H}_{0(3\times 1)}=\left[\begin{array}{lllll}{\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{F}_{1},{F}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}\\ {\overrightarrow{T}}_{{F}_{1},{F}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{F}_{1},{F}_{2}}\\ {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{F}_{1},{F}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}\\ {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{F}_{1},{F}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}\\ {\overrightarrow{T}}_{{A}_{1},{A}_{2}}& {\overrightarrow{T}}_{{D}_{1},{D}_{2}}& {\overrightarrow{T}}_{{E}_{1},{E}_{2}}& {\overrightarrow{T}}_{{F}_{1},{F}_{2}}& {\overrightarrow{T}}_{{A}_{1},{A}_{2}}\end{array}\right]. T → A 1 , A 2 {\overrightarrow{T}}_{{A}_{1},{A}_{2}} is the vector of the distance between the yarn centerline and center plane of the twill fabrics at the original point (X = 0, Y = 0, Z = 0) relative to the connected points of the linear segment of yarn path in warp (A1) and weft (A2) directions, respectively (Figure 4).

Figure 4

The center line of yarn in twill woven fabric structure was explained in the previous section. The circular geometry for yarn cross-section is perpendicular to the centerline yarn path along the yarn length. The data points of coordinates x _i , y _i , z _i of the crimp weave in each cross section can be obtain in the vector matric H ₀(2 x 1) and H ₀(3 x 1). The projected radius of the crimp weave at the first layer (R _jc = 0) and the filaments in outer layers are arranged with open packing form using the virtual location concept. Each cross-section is rotated along the length of yarn path by a pre-determined amount in respect of the previous one allowing for the yarn twist. In the 2 × 1 twill weave, the projection of outer layers (x′, z′) of filament in the yarn cross section is defined in equations (6) and (7) and the coordinate system (X, Y) of yarn in twill woven fabrics is calculated by equations (10), (12), and (14). Therefore, the positions of filaments in woven fabrics in the X and Y axes are defined by equations (15) and (16). The positions of filaments in the Z axis are defined by equation (17). For the coordinate system (y′, z′) of filament in yarn cross-section and coordinate system (Y, Z) of yarn in twill woven fabrics is calculated in the same way, but using different axes. In the 3 × 1 twill weave, the projection of outer layers (x′, z′) or (y′, z′) of filament in yarn cross section, the coordinate system (X, Y) of yarn in twill woven fabrics and the positions of filaments in woven fabrics are calculated in the same way. In 2 × 1 and 3 × 1 twill weaves, the center line of the yarn path makes an angle by skewing in the floating region (δ _i ), and then it makes an angle in the intersecting region (β _i ) of the linear part. The coordinates of filaments in each cross section on the normal plane of centerline of yarn, therefore, in the warp direction along the yarn length (x _i ) will be at (y _i , z _i ) and the filament in the weft direction along the yarn length (y _i ) will be at (x _i , z _i ). The projection of the filaments with their inherent skewness in outer layers of warp/weft directions can be calculated by: (20) For warp : x ′ = x i − R j c sin ( γ ) sin θ i y ′ = y i + [ R j c cos ( γ + ψ ) cos θ i ] / cos ( δ i ) cos ( β i ) z ′ = z i + R j c sin ( γ + ψ ) cos θ i , \text{For}\hspace{.5em}\text{warp}:\hspace{.5em}\left\{\begin{array}{l}x^{\prime} ={x}_{i}-{R}_{j\text{c}}\text{sin}(\gamma )\text{sin}\hspace{0.25em}{\theta }_{i}\\ y^{\prime} ={y}_{i}+{[}{R}_{j\text{c}}\text{cos}(\gamma +\psi )\text{cos}\hspace{0.25em}{\theta }_{i}]/\text{cos}({\delta }_{i})\text{cos}({\beta }_{i})\\ z^{\prime} ={z}_{i}+{R}_{j\text{c}}\text{sin}(\gamma +\psi )\text{cos}\hspace{0.25em}{\theta }_{i},\end{array}\right. (21) For weft : x ′ = x i + [ R j c cos ( γ + ψ ) cos θ i ] / cos ( δ i ) cos ( β i ) y ′ = y i − R j c sin ( γ ) sin θ i z ′ = z i + R j c sin ( γ + ψ ) cos θ i . \text{For}\hspace{.5em}\text{weft}:\hspace{.5em}\left\{\begin{array}{l}x^{\prime} ={x}_{i}+{[}{R}_{j\text{c}}\text{cos}(\gamma +\psi )\text{cos}\hspace{0.25em}{\theta }_{i}]/\text{cos}({\delta }_{i})\text{cos}({\beta }_{i})\\ y^{\prime} ={y}_{i}-{R}_{j\text{c}}\text{sin}(\gamma )\text{sin}\hspace{0.25em}{\theta }_{i}\\ z^{\prime} ={z}_{i}+{R}_{j\text{c}}\text{sin}(\gamma +\psi )\text{cos}\hspace{0.25em}{\theta }_{i}.\end{array}\right. In equations (20) and (21), if the angle is formed by skewing in the intersection region (β _i ), then the angle in the floating region (δ _i ) is zero degrees. Also if an angle is formed by skewing in the floating region (δ _i ), then the angle in the intersecting region (β _i ) is zero degrees.