Theoretical Porosity of Elastic Single Jersey Knitted Fabric Based on 3D Geometrical Model of Stitch Overlapping

ABSTRACT To improve the dimensional stability of weft knitted fabrics, spandex yarns are incorporated with the base yarn by the plaiting technique. Adding spandex leads to stitch overlapping, increases the fabric thickness, and affects the pore’s size and distribution. This in turn affects the geometrical and physical properties of weft knitted fabrics such as air permeability. In this paper, the theoretical 3D modeling of stitch overlap, maximum set, and open structures is presented by using AutoCAD software. Fabric thickness was divided into several sections, and the theoretical porosity at each section was analyzed and calculated. Furthermore, single jersey knitted fabrics SJKF with and without spandex were produced to obtain overlapping (by using spandex weight percent, SWP 8%), and open structures. The calculated and theoretical overall porosity were calculated and compared. The study showed the influence of stitch overlapping on fabric porosity. Moreover, the results show that the proposed model captured to some extent the change in fabric porosity as its structure changes.


Introduction
It is known that the fabric porosity influences the physical and thermal properties and end-use of fabric, particularly knitted fabric. The fabric porosity can give us a clue about thermal resistance, air permeability, and water vapor resistance. The heat, liquid sweat generation, and water vapor must be transferred and dissipated from the body to the environment (Kumar and Sood 2020). The water vapor moves primarily through fabric pores by a diffusion process in the air from one fabric side to the other (Mezarciöz and Tuğrul Oğulata 2015). The fabric porosity depends on pore size, volume, and pore distribution. These factors in turn are influenced by fabric construction parameters, such as yarn count, fabric structure, machine setting, and finishing process. Guidoin defined porosity as the ratio of void space within the boundaries of a solid material to the total volume occupied by this material, including void spaces (Guidoin et al. 1987;Siddiqui and Sun 2015).
Mostly, the fabric porosity is investigated using three methods, image processing, air permeability, and geometrical modeling (Benltoufa et al. 2007). A lot of research investigated the pore size and its distribution for woven and knitted fabrics using image processing techniques based on yarn and fiber parameters from fabric cross-section images (Benltoufa et al. 2007;Imrith, Unmar, and Rosunee 2016;Kumar and Sood 2020;Turan et al. 2012). The surface porosity of single jersey knitted fabrics was investigated by many researchers using image analysis techniques with and without yarn hairiness considerations. Inter pores between yarns were measured by computing the equivalent pores diameter, while intra pores inside yarns were calculated from Neckar's theoretical equation (Havlová and Špánková 2017).
Since air permeability has a direct relationship to pore size (Kumar and Sood 2020), some research has linked air permeability and knitted fabric porosity based on the geometrical parameters, such as fiber density, yarn count, stitch length, courses density, wales density, and fabric thickness (Benltoufa et al. 2007;Elnashar 2017;Mezarciöz and Tuğrul Oğulata 2015;Ogulata and Mavruz 2010).
Furthermore, geometrical modeling was conducted using the geometry of the unit cell of a single loop. Fabric porosity was estimated from a 2D knitted fabric model by calculating the area covered by yarn and the total area of one repeat (Fouda and Eldeeb 2020;Fouda, El-Hadidy, and El-Deeb 2015). Few models for 3D porosity are available that investigated weft knitted fabrics, such as Benltoufa and Karaguzel theoretical models and Guidoin empirical model (Abdolmaleki, Ali, and Amani 2012). Benltoufa calculated the knitted fabric porosity from the geometrical representation of the elementary loop geometry, assuming a circular yarn cross-section (Benltoufa et al. 2007). In addition to this method, Benltoufa used the air permeability and image processing method and concluded that geometrical modeling is the most appropriate and the easiest way to evaluate the fabric porosity. Havlová calculated the fabric density by using three theoretical models (based on density, area, and volume) (Havlová 2021) and the relation between fabric porosity and air permeability was found.
Karaguzel's model predicted pore volume in addition to inter-yarn pore size for simple weft knitted fabric structures, from fabric parameters, such as courses and wales density, yarn linear density, and fabric thickness, which characterize the structure (Karaguzel 2004). A plug-in was developed using Python script to predict the plain knitted fabric porosity by using the actual parameters of the fabric. 3D solid and multifilament models of knitted fabric were generated automatically with two alternatives of models (Pierce and parametric). It was assumed that the yarn cross-section was circular that is swept around the yarn's central axis (Siddiqui and Sun 2015). Adam et al. modeled the air permeability of knitted fabric by using three-dimensional models of knitted fabrics and mapping the geometry in the microscale (Puszkarz and Krucińska 2018).
It is known that spandex yarns are incorporated with yarns in knitting machines to enhance the dimensional stability of knitted fabric during usage and after repeated stresses, which is considered the main defect of knitted fabrics, particularly single jersey knitted fabric (Khalil et al. 2021). Adding Spandex turns the knitted structure from an open and normal structure into a very compact structure and causes stitch overlapping where the courses spacing becomes less than 2 p 3d, wales spacing becomes less than 4d, and fabric thickness becomes higher than 3d, where d is yarn diameter (Fouda, El-Hadidy, and El-Deeb 2015;Khalil, Těšinová, and Aboalasaad 2021b;Khalil et al., 2021a). Since the stitch overlapping effect was not studied earlier using the above-mentioned geometrical modeling method, this research presents a 3D modeling of the stitch overlap geometry to calculate the fabric porosity. The research developed a novel method of calculating porosity by making several sections of fabric thickness and calculating the accumulative volume of fibers in a unit cell of single jersey knitted fabric. The results are compared to porosity calculated by using the Guidoin model. The pore size and distribution was measered at each section of fabric thickness.

Model building
For the current study, building a model based on the actual fabric parameters necessitates the production of single jersey knitted fabric.

Material and method
Giza 86 Egyptian cotton fibers (172 ± 3.1 mtex, 32 ± 0.45 mm) were used to spin a 35 Ne ring-spun yarn (English twist factor, α e = 3.6). To obtain the overlapping structure, full plaited SJKF were produced using this yarn at a loop length of 2.9 mm and using SWP 8% and 30 dtex spandex yarn.
To obtain an open structure, 100% cotton SJKF (without spandex) was produced using the same yarn count and loop length to obtain the open structure. Samples were produced on the VIGNONI SJ-B weft knitting machine (machine gauge: 24 needles/inch, number of feeders: 57, diameter: 19 inches). Fabrics were dyed at 95°C and compacted at 90°C, however, the elastic knitted fabric was first heat set at 185°C before the dyeing process. Moreover, one more fabric structure called "maximum set" was not produced, but instead, it was drawn since the maximum set is the transitional case between open and overlapping structures.
The samples were conditioned in a standard atmosphere (65%±5 relative humidity and 20°C ± 2 for 24 h) before testing. Yarn diameter was measured by capturing images of the yarns using a (ProgRes-CT3) camera attached to a microscope. The captured images were analyzed by using NIS-elements software, and yarn diameter was calculated by computing the average of 600 readings. The air permeability of fabric samples was measured according to EN ISO 9237. Fabric thickness was measured according to EN (1996;International Standard IS 1996). Fabric weight was measured using a high-precision electronic balance (±.01 g) according to the standard test method ASTM D3776. Five readings were recorded for fabric thickness, fabric weight, fabric air permeability, and wales and courses densities, and averages along with the coefficient of variation were calculated. The fabric density (ρ fabric ) and calculated overall porosity (ε) were obtained from equations (1) and (2) (Eltahan, Sultan, and Mito 2016;Salama, El-Deeb, and El-Shahat 2015): where W is fabric weight per unit area (g.m −2 ), h is fabric thickness (mm), and ρ f is fiber density (kg. m −3 ). In the case of 100% cotton SJKF, fiber density is taken as ρ c . While in the case of elastic knitted fabric, fiber density is taken as ρ e and calculated according to equation (3).

Model design
A 3-dimensional multifiber model (rather than a solid cylindrical model) was developed taking into consideration the following parameters: loop length, equivalent yarn diameter since real yarns do not have a circular cross-section, fiber length in a single loop, fiber diameter, fiber cross-section shape, total number of fibers in yarn cross-section, and yarn twist. The model was constructed to simulate fabric structures.

Assumptions
The following assumptions were used to describe the fabric structure; (1) fibers have kidney shape and their cross-section areas are equal, (2) yarn cross-section is circular, (3) fibers are evenly distributed along yarn cross-section, (4) yarn hairiness is neglected, (5) individual fibers are continuous, (6) theoretical fabric thickness of 100% cotton, maximum set, and overlapping is 2d, 2d, and 3d respectively, and (7) wales spacing is constant and equal to 4d.

Yarn structure
The total number of fibers in yarn cross-section, N was calculated according to equation (4).
where ρ f is fiber density (kg/m 3 ). The cotton fiber cross-section area, A was obtained according to equation (6).
The circular cross-section that had an area A shown in Figure (1-a) was converted into a kidney shape with the same area to imitate the cotton fiber as shown in Figure (1-b). The number of twists per inch of that yarn, TPI was calculated according to equation (7).
Based on loop length, the number of twists in one loop was calculated. The spandex yarn diameter was calculated according to equation (5).  Figure 2 and Figure 3 show the steps conducted to develop the SJKF 3D model using AutoCAD software. Firstly, a two-dimensional sketch of a single loop was carried out based on the Peirce model (Peirce 1947) and actual yarn diameter, wales, and courses density as shown in Figure 2. The actual loop parameters are shown in Table 1.

Fabric structure
Secondly, since the inclusion of inter-fiber spacing is one of the important factors that determine the utility of the model for theoretical predictions of the porosity of textiles, a cross-sectional sketch of the yarn based on individual fibers was drawn, as shown in Figure (1-b). Thirdly, as shown in Figure 3,  a sketch describing the profile of the loop in a 3D view was created consisting of a continuous spline. For spandex yarn, it follows a shorter trajectory than in the case of 100% cotton yarn since more tension is applied while feeding as shown in Figure 3. Therefore, the loop length in the 3D model varied slightly from the actual loop length, where the loop length of open, maximum set, and overlapping structures were 3.07 mm, 2.9 mm, and 2.85 mm, respectively. Finally, the 3D loop was obtained using the sweep operation of the fiber cross-sections.
Afterward, the loop was repeated to demonstrate the final fabric appearance and Figure 4 shows the different views for each structure. Figures (4-a), (4-e), and (4-h) show the 3D isometric projection of open SJKF, maximum set, and overlapping structures, respectively. Figures (4-b), (4-f), and (4-i) show the side view, and Figures (4-c), (4-g) and (4-j) show the front views. Finally, Figures (4-d) and (4-k) show the optical microscopic images of the produced open and overlapping SJKF structures, respectively.

Fabric porosity calculations
An enclosure was drawn around the repeat of the knitted fabric structure to simulate the surrounding air as shown in Figure 5. The fabric porosity was obtained by calculating the porosity of progressive growing sections of the fabric (top to bottom), using a 0.033 mm increment value, as shown in Figure 6. The open structure and maximum set fabrics were divided into 10 sections, while overlapping was divided into 15 sections because its thickness is greater. The pore volume at i th section, V pi can be calculated according to equation (8).
where V xi is enclosure volume of the i th section, and V fi is fiber volume of the i th section. The theoretical porosity in the i th section, ε i can be calculated according to equation (9) The porosity at each section was plotted against its corresponding location. Moreover, the theoretical overall porosity was calculated using the yarn volume and enclosure volume in the last section and compared to the calculated porosity using equations (2). Table 2 shows the calculated overall porosity and air permeability values of produced 100% cotton and overlapping SJKF. Air permeability is the rate of airflow through the fabric when it is subjected to air pressure difference on either surface. Air permeability is mostly influenced by both the fabric porosity and thickness (Atalay 2018;Song 2011). It can be seen that the calculated fabric porosity of 100% cotton is higher than the overlapping, by 6.4%. Since 100% cotton fabric is an open structure as shown in Figures 2 and 4, the courses density is low, and the volume of air pores is evidently more. This results in the presence of free spaces between loops. Adding spandex leads to the courses density increasing, stitch overlapping; therefore, adding an extra layer of fibers as shown in Figure 4, thus the volume of air pores becomes less, and fabric porosity decreases. These results can be verified by the air permeability value where the air permeability of 100% cotton is found to be about 7 times that for overlapping. Table 3 shows the theoretical porosity values at different sections for open structure, maximum set, and overlapping using the proposed 3D model and equations (4-9). The theoretical porosity was calculated at each fabric section. The total pores volume at each section of fabric thickness is shown in  . It is obvious that the open structure has the highest pore volume at all sections followed by the maximum set followed by overlapping structure. At the fifth section where the fabric thickness is equal to d, the total pores volume of the overlapping structure is less than open and maximum set structures by 72% and 68% respectively. Also, at the tenth section where the fabric thickness is equal to 2d, the total pores volume of overlapping structure was less than open and maximum set structures by 74% and 70% respectively. So, adding spandex has a great effect in pore size and distribution of SJKF.  . This trend was almost the same for all structures. Based on the results of the 3D model, in all fabric sections, the fabric porosity of 100% cotton is the highest followed by the maximum set, followed by overlapping. Figure 9 shows the comparison between theoretical and calculated overall fabric porosity. It is reported that the porosity of knitted jersey ranges from 60% to 80% which agrees with the current  finding (Benltoufa et al. 2007). The results show that the proposed model captured to some extent the change in fabric porosity as its structure changes where the prediction difference was −10.4% and −12% for 100% cotton and overlapping, respectively. The difference among the results can be ascribed to the simplifying assumptions that yarn cross-section is circular, which in turn affects the air pores volume and distribution inside the fabric structure. Furthermore, the model assumed that the fabric thickness of 100% cotton and overlapping to be 2d, 3d, but the measured fabric thickness as shown in Table 2 was 2:2d, and 3:2d, respectively, which obviously increases the air pores volume and overall porosity. Moreover, wales spacing was assumed to be 4d which differs from the real wales spacing values shown in Table 1.

Conclusion
The research achieved its objective and can be summarized in the following points: (1) A 3D geometrical model of single jersey knitted fabrics was presented by AutoCAD software for different loop structures, namely open structure (100% cotton), maximum set, and overlap structure.
(2) In the theoretical simulation, the yarn was represented as continuous multifibers, not a solid cylinder, and the fiber's cross-section was represented as a kidney shape to simulate the cotton's fiber cross-section shape. The yarn twist was also taken into consideration and was represented by theoretical simulations. (3) The porosity of single jersey plain knitted fabrics was analyzed at different sections of the fabric thickness, based on the visible pore volume between yarns and within fibers and the total unit volume. The fabric's overall porosity was calculated for each structure. (4) Adding spandex has a significant effect on the pore size and distribution of SJKF. (5) It was found that the theoretical results were less than the calculated results of the porosity by no more than 12%. However, both calculated and theoretical overall porosity ranged from 60% to 80%, and that is consistent with the results of the previous research. (6) From the experimental results, it was found that the thickness of 100% cotton SJKF reached 2:2d while the thickness of elastic SJKF reached 3:2d because of stitch overlapping. (7) The model can be developed by considering the real fabric thickness. Furthermore, it can be useful in analyzing the thermo-physiological properties of knitted fabrics.

Highlights
(1) the theoretical 3D modeling of stitch overlap, maximum set, and open structures is presented by using AutoCAD software based on actual parameter of loop geometry. (2) Fabric thickness was divided into several sections, and the theoretical porosity at each section of fabric thickness was analyzed and calculated. (3) Single jersey knitted fabric with and without spandex were produced to obtain overlapping (by using spandex weight percent, SWP 8%), and open structures. (4) the results show that the proposed model captured to some extent the change in fabric porosity as its structure changes and, adding spandex has a significant effect on pore size and distribution of single jersey knitted fabric. (5) The model can be developed by considering the real fabric thickness. Furthermore, it can be useful in analyzing the thermo-physiological properties of knitted fabrics.