Deformation behavior of auxetic laminated fabrics with rotating square geometry

Abstract

Auxetic fabrics are becoming a tempting option for many modern textile products due to their unusual deformation behavior under tensile load. In spite of the developments that have been made, it remains difficult to produce auxetic fabrics more efficiently using existing technologies. In this research, a novel type of auxetic fabrics is proposed based on rotating square geometry and manufactured using commercially available raw materials and lamination technique. Rotating square structures with different parameters are firstly produced through laser cutting a frame fabric, and then bonded to a base fabric using a hot-melt adhesive membrane. Tensile tests are conducted on various two-layer and three-layer laminated fabrics to investigate their deformation behaviors and auxetic effects. Results show that the auxetic behavior of the laminated fabrics mainly depends on initial modulus difference between the base material and the laminated frame material and can be affected by the initial intersection angle and size of square units. It is also found that three-layer laminated fabrics are more desirable than two-layer laminated fabrics when it comes to avoiding out-of-plane deformation and extending tensile strain range of auxeticity.

Keywords

Auxetic material laminated fabric rotating square structure shoe upper

Auxetic materials are known for their unusual deformation behavior under uniaxial tension. Instead of contracting, they will expand in the lateral direction when stretched longitudinally.¹ The Poisson’s ratio is defined as the negative of the ratio of transverse strain to axial strain. It is a measure of deformation of a material in directions perpendicular to the loading direction. Auxetic materials usually have a negative Poisson’s ratio, while most common materials have a positive Poisson’s ratio, for example, nearly 0.5 for rubber, and nearly 0.33 for steel.² Enhanced properties in auxetic materials, especially in auxetic textiles, include variable permeability, synclastic curvature, indentation resistance, energy absorption, etc.^3–7 These properties make auxetic textiles a prospective candidate for applications of smart garments and shoes, among which auxetic footwear upper can be an example. In 2016, Under Armour launched its first-ever 3D printed performance trainer UA Architech, which featured a 3D printed midsole and 3D ClutchFit auxetic upper design, claiming that it could provide athletes with ultimate stability and cushioning on intense workouts. Later, Under Armour Clone Magnetico Pro boots were launched, containing an internal layer of auxetic synthetic material in the upper to provide a precisely customized fit to every foot. In order to develop auxetic fabrics for meeting practical applications, selecting a suitable auxetic structure is very important. According to the monographs,^8–10 typical auxetic structures include reentrant cells, rotating polygons, folded structures, nodule fibril models, missing rib models, chiral and anti-chiral lattices, egg rack structure, etc. Those that have been adopted in knitted or woven fabric designs are mainly folded structures,^11–16 rotating rectangles,^12,17 rotating hexagons,^18–21 reentrant hexagons,^17,22–28 and reentrant triangles.²⁹ Developing auxetic fabrics directly through existing yarn materials and textile formation techniques has always been the end goal of researchers.

A laminated fabric is defined as a material composed of two or more layers of materials that are bonded closely together, at least one of which is a textile fabric. Lamination is deemed a mature technique in the garment industry as a replacement for, or supplement to, sewing, with the benefit of shortening production time, reducing costs, and allowing more consistent quality.³⁰ Another attractive attribute of laminated fabric is that it allows the combination of various types of textiles with different structures and properties to tailor the property of its final product.³¹ Using the lamination technique to fabricate auxetic textiles is, therefore, a feasible and cost-effective option, and has already been reported in some publications. Novak et al.³¹ produced the two-layer laminated fabrics by first cutting ethylene-vinyl acetate (EVA) foam into auxetic cellular structures (reentrant hexagon and chiral structure), and then applying them to weft-knitted fabrics impregnated with adhesive. Although the EVA foams with auxetic structures showed a negative Poisson’s ratio at lower tensile strains, the laminated fabrics showed a positive Poisson’s ratio through the entire range of strains. In another study,³² the combination of three-dimensional (3D) printed auxetic reentrant hexagonal structure and weft knitted fabrics was investigated in terms of permeability and mechanical properties. Results confirmed the possibility of adjusting permeability and mechanical properties of knitted fabrics by 3D printing auxetic structures on top. However, the auxetic effect of this composite fabric system was not mentioned. 3D printed auxetic sinusoidal patterns/nylon plain weave composite fabrics were also investigated, which showed negative Poisson’s ratio values and better mechanical properties (e.g. flex stiffness and bursting property), but the influence factors were not thoroughly studied.³³ Despite these trials, little still remains known about how to obtain and tailor the auxetic effect of composite fabrics through lamination.

Rotating square geometry, firstly proposed by Grima and Evans,³⁴ is one of the most typical auxetic structures that have been widely studied. It is a lattice formed with rigid squares that are connected together at their vertices by hinges. When the whole structure is under loading, the rigid square units can rotate, leading to an auxetic behavior and a constant value of Poisson’s ratio = –1 in the ideal case.^8–10 Efforts have been made to reconstruct the rotating squares with fabric structures. A weft knitted fabric design was firstly proposed based on the rotating square geometry, but the produced fabric had only auxetic effect when stretched in the course direction.¹² Also adopting the rotating square geometry, auxetic non-woven fabrics were fabricated by laser cutting a highly ordered pattern of slits on commercial needle-punched non-woven fabrics. The modified samples proved to be auxetic, while their breaking force dropped massively.³⁵ The rotating square structure was applied to warp knitted fabrics through Jacquard technology lately, but the auxetic behavior was detected only when samples were stretched in the wale direction.¹⁷ From the application perspective, the rotating square geometry is quite popular when developing metamaterials for applications like auxetic esophageal stents,^36,37 novel scaffolds for tissue engineering,³⁸ and reconfigurable nano-filters.³⁹ The rotating square geometry is thus selected as the auxetic structure to be imbedded into the design of laminated fabrics in this study. The laminated fabrics are formed by attaching rotating square frames cut from a very stable fabric to a base fabric with easy deformation and high extensibility. The in-plane auxetic properties of different laminated fabric samples are tested using uniaxial tensile method, and the effects of different parameters on the auxetic effect are analyzed. The aim is to provide a basic understanding on the design principles and deformation mechanisms of auxetic laminated fabrics with rotating square geometry. It is expected that this type of auxetic laminated fabrics could be an advanced option for footwear upper material.

Design, material and fabrication

Design

The deformation of an ideal rotating square structure is shown in Figure 1(a). The square units should be perfectly rigid, and the hinge connections (indicated in black dots in the sketch) should allow the rigid square units to rotate freely. The edge length of the square units is a, and the acute angle between two adjacent squares is β. When the structure is stretched uniaxially, the square units will rotate around the hinge joints until β turns to 90°, leading to an ideal auxetic behavior. It should be noted that the ideal hinge joints between square units are almost impossible to realize in fabrics. In order to apply this auxetic geometry to the fabric design, a revised rotating square structure was adopted. As shown in Figure 1(b), the hinge joints between square units are replaced by connections of 1 mm. To avoid possible influence of connections on the auxetic property of the revised structure, the connection width was kept the same for all samples. The width of 1 mm was determined on two considerations: (a) a larger connection width may largely hinder the rotation of square units; (b) a connection width smaller than 1 mm is easily broken during the laser-cutting procedure. The edge length of the approximate squares and intersection angle between two adjacent squares of the revised geometry are also represented by a and β.

Figure 1.

Design philosophy: (a) deformation of an ideal rotating square structure and (b) revised rotating square structure.

Though the Poisson’s ratio of an ideal rotating square geometry is constantly –1 regardless of the geometric parameters such as initial angle and square size, the actual deformation of the laminated fabric with rotating square geometry is largely restricted by the base fabric. The deformation of the laminated fabrics should thus be a lot different from the ideal deformation of the rotating square geometry. There is no doubt that geometrical parameters of rotating square frames including initial intersection angle β₀ and edge length a will affect the Poisson’s ratio of laminated fabrics. In order to investigate their influences on the auxetic effect of laminated fabrics, five different rotating square frames were designed, as listed in Table 1, numbered as RS1, RS2, RS3, RS4, and RS5 respectively.

Table 1.

Design of rotating square frames with different parameters

No.	RS1	RS2	RS3	RS4	RS5
Initial angle (β₀)	30°	45°	60°	45°	45°
Edge length (a)	10 mm	10 mm	10 mm	6 mm	14 mm

Materials

To minimize the misshaping of square units and the restriction of the base to the rotating of the square units during deformation, stiff fabric with high shape stability and elastic fabric with high extensibility and low stiffness should be selected as the frame fabric and base fabric, respectively. In this regard, a twill fabric (100% polyester, black color, 210 g/m²) was selected as the frame fabric because of its structural stability, and a warp-knitted tricot fabric (75% polyamide, 25% spandex, white color, weight 160 g/m²) was adopted as the base fabric due to its easy deformation. For the coupling of the two types of fabrics, the hot-melt adhesive membrane made of thermoplastic polyurethane (TPU) was also used. It is a commercial product (type XJU120) from Shanghai Xingxia Polymer Products Company with a thickness of 0.1 mm. The temperature range required to process the TPU membrane is 140–165°C.

Fabrication

As shown in Figure 2, the manufacture process of laminated fabrics with the rotating square geometry can be divided into three steps:

Figure 2.

Fabrication process of laminated fabrics.

Step 1: the frame fabric is bonded with the adhesive membrane under the temperature of 150°C;

Step 2: the frame fabric with adhesive is laser-cut into the designed rotating square geometries using the machine Han’s Yueming Laser CMA1309-B-A;

Step 3: attach the auxetic frame(s) with adhesive to one side of the elastic base fabric to form a two-layer laminated fabric, or to both sides of the elastic base fabric to form a three-layer laminated fabric.

Altogether eight series of laminated fabrics were produced, with their designations listed in Table 2. The two-layer and three-layer laminated fabrics are coded as L2 and L3 respectively. Considering that the tensile properties of the base fabric in two principal directions are different, samples with different orientations were also produced. As shown in Figure 3, the laminated samples with the warp direction of the frame fabric along the wale direction of the base fabric are coded as PW, while the samples with the warp of the frame along the course of the base are coded as PC.

Table 2.

Designation of the laminated fabric samples

Type	Orientation	Frame no.	Code
Two-layer laminates	Warp-Wale	RS1	L2-PW-RS1
		RS2	L2-PW-RS2
		RS3	L2-PW-RS3
		RS4	L2-PW-RS4
		RS5	L2-PW-RS5
	Warp-Course	RS2	L2-PC-RS2
Three-layer laminates	Warp-Wale	RS2	L3-PW-RS2
Three-layer laminates	Warp-Course	RS2	L3-PC-RS2

Figure 3.

Different orientations: (a) warp direction of the frame fabric along the wale direction of the base fabric (PW) orientation and (b) warp direction of the frame fabric along the course of the base fabric (PC) orientation.

Tensile tests

Material property

In the fabric plane, the laminated fabric with auxetic geometry can be divided into the base area of the knitted fabric only and the frame area of the woven fabric bonded to the knitted fabric. It should be noted that the knitted base fabric without woven frame will undergo the heat process during the fabrication process so that its tensile property should be different from that of the knitted base fabric without the heat process. In order to compare the tensile properties of the base section and the frame section in laminated fabrics, tensile tests were performed on the heat-processed base fabric and uncut laminated fabrics with PW orientation using an Instron 4411 Universal Testing Machine. For the heat-processed base fabric, sample strips were prepared according to the standard ISO 13934-1, with a size of 150 mm × 50 mm. Gauge length was set as 100 mm, and the rate of extension was 100 mm/min. For the uncut laminated fabrics, sample strips were prepared with a size of 250 mm × 50 mm. Gauge length was set as 200 mm, and the rate of extension was 100 mm/min. Three specimens were tested for each type of specimen. To obtain the initial modulus of different materials, the true stress and strain were calculated, respectively, using

σ_{t} = \frac{F}{A_{0}} \times (1 + \frac{Δ L}{L_{0}})

(1)

ε_{t} = \ln (1 + \frac{Δ L}{L_{0}})

(2)

where σ_t is true stress (MPa), ε_t is true strain, F is tensile force (N), A₀ is original cross-sectional area (mm²), ΔL is extension value (mm), and L₀ is original length (mm).

Poisson’s ratio

To characterize the auxetic property, uniaxial tensile tests were conducted on the laminated fabrics with rotating square frame(s). The set-up of the test is shown in Figure 4. All of the laminated fabric samples were prepared with a length of 200 mm and a width of two repeating units of the rotating squares. Gauge length was set as 150 mm, and the rate of extension was 30 mm/min. A digital camera Canon EOS 800D was installed right ahead of the sample. Three specimens were tested for each type of laminated fabric. During the tests, the upper clamp kept moving until the sample fractured with the camera video-recording of the fabric deformations at the same time. Photos were extracted from the videos at an interval of 3 s. The width of one repeating unit of the rotating square geometry at the center of the sample is measured from each photo. The transverse and axial true strains were then calculated according to Equation (2). The Poisson’s ratio is obtained through Equation (3).

υ_{i j} = - \frac{ε_{j}}{ε_{i}}

(3)

where i represents the axial direction, j represents the transverse direction, ε_i is the axial strain, and ε_j is the transverse strain.

Figure 4.

Set-up of tensile tests for Poisson’s ratio.

Tensile properties of materials

Heat-processed base fabric

The tensile property of the heat-processed base fabric in the course and wale directions are shown in Figure 5 and Table 3. From the force-extension curves in Figure 5(a), it can be seen that the extensibility of the processed base fabric in either direction is excellent, and the maximum extension in the course direction almost doubles that in the wale direction. The increase rate of tensile force is much higher in the wale direction than in the course direction. The force at break is over 600 N when stretched in the wale direction, almost 1.5 times that in the course direction. The true stress of the processed base fabric within the first 40% strains is plotted, as shown in Figure 5(b). Data within the first 10% strains are subjected to linear fitting, the slope of which can be used as the initial modulus, around 0.52 MPa in the course direction and 1.5 MPa in the wale direction, respectively. It means that it is easier for the processed base fabric to be extended in the course direction than in the wale direction.

Figure 5.

Tensile property of heat-processed base fabric in two principal directions: (a) force-extension curves and (b) stress-strain curves.

Table 3.

Tensile properties of processed base fabric and uncut laminates

Type	Thickness/mm	Direction	Strain at break/%	SD	Force at break/N	SD	Initial modulus/MPa
Processed base fabric	0.328	Course	163.63	2.00	405.19	6.94	0.52
Processed base fabric	0.328	Wale	119.11	1.93	608.14	3.34	1.5
Two-layer laminates (PW orientation)	0.580	Weft	41.26	0.70	632.48	6.58	12.21
Two-layer laminates (PW orientation)	0.580	Warp	34.35	2.37	1361.74	154.62	87.06
Three-layer laminates (PW orientation)	1.100	Weft	45.41	0.19	1377.84	8.57	11.17
Three-layer laminates (PW orientation)	1.100	Warp	37.19	0.71	2551.67	44.18	76.49

SD: standard deviation.

Uncut laminated fabrics

Two-layer laminates with PW orientation

The tensile property of the uncut two-layer laminated fabric with PW orientation in two principal directions is shown in Figure 6 and Table 3. From the force-extension curves in Figure 6(a), It can be seen that maximum extension in the weft direction is slightly larger than in the warp direction. The force at break in the warp direction is more than double than that in the weft direction. And the increase rate of tensile force is also faster in the warp direction than in the weft direction. It implicates that the uncut two-layer laminates with PW orientation are more reluctant to extend in the warp direction than in the weft direction. The true stress of the sample within the first 30% strains is shown in Figure 6(b). From linear fitting the data within the first 10% strains, we know that the initial modulus in the warp direction is around 87.06 MPa, much higher than that in the weft direction, around 12.21 MPa. When compared with the processed base fabric, the initial modulus of the two-layer laminated material in the weft and warp direction is over 20 times and 50 times that of the base fabric in the course and wale direction, respectively. It means that the two-layer laminated fabric is much more stable and harder to deform than the base fabric in either direction.

Figure 6.

Tensile property of uncut two-layer laminates (warp direction of the frame fabric along the wale direction of the base fabric (PW) orientation) in two principal directions: (a) force-extension curves and (b) stress-strain curves.

Three-layer laminates with PW orientation

The tensile property of the uncut three-layer laminated fabric with PW orientation in two principal directions are shown in Figure 7 and Table 3. It can be seen from Figure 7(a) that the force variation trend, as well as the maximum elongation, of the three-layer laminated fabrics is close to that of the two-layer ones, while the force at break of the three-layers is almost doubled compared with the two-layer samples either in the weft direction or in the warp direction. True stress within the first 30% tensile strains is also plotted in Figure 7(b). From linear fitting the data in the first 10% strain range, the initial modulus of 11.17 MPa and 76.49 MPa is obtained, respectively, for the weft and warp direction. The result in either direction is slightly smaller than the modulus of the two-layer material, mainly due to the increase in the cross-sectional area. Compared with the processed base fabric, the modulus of the three-layer laminated fabric in the weft and warp direction is, respectively, 20 times and 50 times of that of the base fabric in the course and wale direction. It means that a much better structural stability is also shown in the three-layer laminated fabric in comparison with the processed base fabric. The difference in initial modulus between the three-layer laminated fabric and the processed base fabric is close to that between the two-layer material and the base fabric.

Figure 7.

Tensile property of uncut three-layer laminates (warp direction of the frame fabric along the wale direction of the base fabric (PW) orientation) in two principal directions: (a) force-extension curves and (b) stress-strain curves.

Deformation behavior of laminated fabrics

Deformation behavior of a typic laminated fabric with rotating square geometry

Taking the two-layer laminated fabric (type L2-PW-RS2) as an example, the deformation behaviors of a typic laminated fabric with rotating square geometry under warp-directional and weft-directional tensile strains are analyzed respectively, based on the Poisson’s ratio results as shown in Figure 8.

Figure 8.

Poisson’s ratio of a type of laminated fabric (L2-PW-RS2) under warpwise and weftwise tensile strains.

Under warp-directional tensile strains

It can be seen that the maximum negative Poisson’s ratio of the fabric is approaching –0.8 under warp-directional tensile strains. With the increase of tensile strains, the Poisson’s ratio rapidly increases at first, in an almost linear way, and then slows down, gradually turning to positive at around 7% strain. This is rather different from the theoretical Poisson’s ratio value of –1 in an ideal rotating square structure, which means that the lateral width of the laminated fabric does not increase proportionally with the increase of tensile strain as presumed. Due to the constraint of the base fabric, the auxetic effect in the laminated fabric diminishes as the tensile strain increases, and finally disappears.

The deformation process of the fabric with different tensile strains up to 14% is shown in Figure 9, in which one repeating unit in each picture is enlarged for better illustration, as outlined by a blue box. To represent the change in intersection angle, β with different subscripts are used for better indication. At the initial stage (0% tensile strain), the squares in the repeating unit are neat and regular, as indicated in the red box, and the original intersection angle between two rotated squares is denoted as β₀. At 2.8% tensile strain, the square units rotate with their shape well retained, leading to an increase in the intersection angle, denoted as β₁. As tensile strain increases to 5.6%, the square units keep rotating with minor distortion, and the angle continues to increase, denoted as β₂. With the tensile strain further increasing to 14%, the square unit becomes increasingly misshaped, which can be directly observed from Figure 9(f). Meanwhile the intersection angle is kept almost unchanged. It should be noted that the out-of-plane deformation of the sample is also significantly increased with the increase of tensile strains, which might be difficult to be noticed from the photos.

Figure 9.

Deformation process of L2-PW-RS2 under different warpwise tensile strains: (a) 0%; (b) 2.8%; (c) 5.6%; (d) 8.4%; (e) 11.2% and (f) 14%.

To sum up, the laminated fabric mainly undergoes three stages of deformation when stretched along the warp direction. In the first stage, the square units in the frame would rotate with barely no distortion, leading to an increase in intersection angle and generating a good auxetic effect. In the second stage, the square units continue to rotate, pushing the intersection angle increase to its maximum. In this stage, distortion of the square units and the out-of-plane deformation starts to be evident. In the last stage, the distortion of the square units, as well as the out-of-plane deformation, escalates as the tensile strain increases. However, the intersection angle is kept almost unchanged because the rotating of square units is totally constrained by the base fabric.

Under weft-directional tensile strains

It can be seen from Figure 8 that the Poisson’s ratio turns from zero to positive as tensile strain increases from 1% to 14% in the weft direction, which means that the auxetic effect almost does not exist under weft-directional tensile strains. The deformation process of the laminated fabric is shown in Figure 10. The lateral contraction in the sample is rather clear compared with the deformations under warpwise tensile strains. There is also an increase in the intersection angle at the initial stage, as shown in Figure 10(b), but the angle stops increasing afterwards. At the same time, the distortion of the square unit becomes increasingly observable as the tensile strain increases to 14%.

Figure 10.

Deformation process of L2-PW-RS2 under different weftwise tensile strains: (a) 0%; (b) 2.8%; (c) 5.6%; (d) 8.4%; (e) 11.2% and (f)14%.

It is also observed from experiments that the two-layer laminated fabric is easily deformed out of the fabric plane under weft-directional tensile strains. Considering the difficulty to observe the out-of-plane deformation from the photos taken from the front, pictures were taken from different angles, as shown in Figure 11(a). It can be seen that the fabric surface is rather humpy and hillocky, with some of the convex and concave areas encircled in blue and red, respectively. It is noticed that the out-of-plane deformation appears in a regular pattern when the sample is stretched along the weft direction. The rhombi in the base part of the laminated fabric can be segmented into two types, as shown in Figure 11(b), one arranged with their short diagonal line aligned with the weft direction, and the other with their long diagonal aligned with the weft direction. When stretched, the former type tends to bulge out with their short diagonal (indicated in blue dotted line) forming into a protruded ridge, while the latter indents with their short diagonal (indicated in red dotted line) sagging into a groove. The out-of-plane deformation largely explains the results of an increasing positive Poisson’s ratio obtained through in-plane measurement.

Figure 11.

Out-of-plane deformation of L2-PW-RS2 when stretched in the weft direction: (a) photos from different angles and (b) sketch of rhombi arranged in different orientations.

It is clear that the auxetic behavior of the laminated fabric is more obvious when stretched in the warp direction than in the weft direction. When tensile force is applied in the warp direction, the auxetic behavior of the laminated sample mainly depends on the course-directional extensibility of the base fabric, and vice versa. From the tensile property analysis, we know that the initial modulus of the heat-processed base fabric is much lower in its course direction than in its wale direction. The course-directional constraint of the base fabric to the rotating of the laminated square units is, thus, weaker. It allows the better auxetic behavior under the warp-directional stretch. On the other hand, the limitation of the base fabric to the laminated frame is too high in the wale direction, which may lead to structural imbalance in the composite system. The out-of-plane deformation, therefore, is more significant under weft-directional tensile strains.

Effect of different parameters

Effect of initial intersection angle (β₀) of the frame

The Poisson’s ratios of the two-layer laminated fabrics with different initial intersection angles under warp-directional tensile strains are shown in Figure 12(a). It can be seen that the Poisson’s ratio of L2-PW-RS1 with β₀ of 30° is close to that of L2-PW-RS2 with β₀ of 45°, with the highest negative Poisson’s ratio approaching –0.8 at 1% tensile strain. With the increase of tensile strains, the auxetic effect of L2-PW-RS1 also drops at a decreasing rate, and disappears at tensile strains around 8%. Comparatively, the sample L2-PW-RS3 with β₀ of 60° shows a weaker auxetic effect. Its highest auxetic effect is also obtained at the initial stage, with a negative Poisson’s ratio of –0.33. The value of Poisson’s ratio becomes positive after tensile strain exceeds 4%. From observations, it is also found that the intersection angle stops increasing and the square units start to distort at a much earlier stage for the laminated sample with β₀ of 60°, which directly leads to the faster fading away of its auxetic effect. In this case, the room left for the square units to rotate to their fullest extent, that should be 90°, is only 30° which is smaller than the other two types of samples. After the squares rotate to their extremity, the whole material tends to distort leading to a non-auxetic behavior. Therefore, the tensile strain range for auxetic effect is narrower in the sample with β₀ of 60°.

Figure 12.

Effect of geometrical parameters of the frame on the auxetic effect of laminated fabrics: (a) different β₀ and (b) different a.

Effect of square size (edge length a) of the frame

The Poisson’s ratios of the two-layer laminated fabrics with different square sizes under warp-directional tensile strains are shown in Figure 12(b). It can be found that all three types of fabric samples are auxetic to a certain extent, and their auxetic effect decreases with the increase of tensile strain. However, the auxetic effect of both L2-PW-RS4 (6 mm) and L2-PW-RS5 (14 mm), with a maximum negative Poisson’s ratio of –0.3 at 1% tensile strain, is lower than that of L2-PW-RS2 (10 mm), with a maximum negative Poisson’s ratio of –0.8, also at 1% tensile strain. Although the Poisson’s ratio variation trend of L2-PW-RS4 is the mildest, its tensile strain range with negative Poisson’s ratio is the widest, around 0∼8%. Comparatively, the tensile strain range having negative Poisson’s ratio in L2-PW-RS5 approximates to 3% only. It is also observed from the experiments that the sample L2-PW-RS5 tends to deform out of the fabric plane more than the other two.

In general, the laminated fabrics with smaller or larger squares size cannot achieve a higher auxetic effect. The laminated fabric with a = 10 mm is most preferable when considering both auxetic effect and tensile range having an auxetic effect. The underlying mechanisms to explain the effects of square size of the frame on the auxetic effect of laminated fabrics are quite complicated. For one thing, the oversized square units of the frame may contribute to a more severe structural instability in the composite system, thus leading to an undesired out-of-plane deformation. For the other, if square units are tiny the area of base fabric in the laminated system may be too small, further limiting the rotating of squares. Therefore, an appropriate size of the square units has to be selected to obtain a desired auxetic effect in the laminated fabrics.

Effect of orientation of the base fabric

The Poisson’s ratio variation of L2-PC-RS2 (PC orientation) and L2-PW-RS2 (PW orientation) under warp-directional tensile strains are compared in Figure 13(a). From the curves, it can be seen that the increasing trends of Poisson’s ratio of the two samples are similar, both at a faster rate at first and then slower, but their auxetic effects are quite different. For the sample L2-PC-RS2, there barely exists an auxetic effect under warp-directional tension. The minimum value of its Poisson’s ratio is nearly zero at the tensile strain of 1%. Then, the Poisson’ ratio keeps growing until the tensile strain reaches 9% and levels off at around one. Comparatively, the L2-PW-RS2 shows a much better auxetic effect, with a maximum negative Poisson’s ratio of –0.78 and the auxetic behavior keeps for over 6% strains. This difference is mainly due to the different properties of the base fabric in its course and wale direction. A larger initial modulus is shown in the wale direction, which is exactly the direction perpendicular to the warp-directional tension in L2-PC-RS2. More resistance is thus encountered when the laminated fabric intends to expand laterally. A distinct out-of-plane deformation was also observed in L2-PC-RS2 when stretched along the warp direction, which may take the major accountability for its undesirable auxetic effect.

Figure 13.

Effect of orientation on the auxetic effect of laminated fabrics: (a) under warp-directional tensile strains and (b) under weft-directional tensile strains.

The Poisson’s ratio variations of L2-PC-RS2 and L2-PW-RS2 under weft-directional tensile strains are shown in Figure 13(b). It is found that the auxetic effect of L2-PC-RS2 is better than L2-PW-RS2 in overall terms. A zero Poisson’s ratio is kept for around 5% tensile strains in L2-PC-RS2, and then the value of Poisson’s ratio is still within the range of 0∼0.2 as tensile strain increases to the tested maximum. In contrast, an almost non-auxetic effect is shown in L2-PW-RS2, with a zero Poisson’s ratio at 1% tensile strain and an ever-increasing Poisson’s ratio afterwards. When stretched along the weft direction, the course direction of the base fabric, with a lower initial modulus, is oriented perpendicular to tension in L2-PC-RS2, while the wale direction of the base fabric, with a higher initial modulus, is oriented perpendicular to tension in L2-PW-RS2. Therefore, the lateral expansion is less limited in L2-PC-RS2 under this condition, leading to relatively better auxetic effect.

It should be noted that the auxetic effect of L2-PC-RS2 under weft-directional tension is worse than that of L2-PW-RS2 under warp-directional tension, even though the course direction of the base fabric is oriented perpendicular to tension in either case or, to put it another way, equivalent restrictions are imposed on the rotating square frame by the base fabric in both cases. It indicates that the rotating square frame areas with different laminating orientations have different mechanical properties. The stiffness of the laminated frame area with PC orientation may be smaller than that with PW orientation, thus imposing a weaker rotating tendency.

Effect of number of plies

The Poisson’s ratio results of L3-PW-RS2 (three-layer laminated fabric) and L2-PW-RS2 (two-layer laminated fabric) under warp-directional tensile strains are shown in Figure 14(a). It can be seen that the L3-PW-RS2 exhibits an improved auxetic property compared with its two-layer counterpart. An auxetic effect is exhibited across nearly 22% tensile strains, which is tripled than L2-PW-RS2 and is the largest strain range endowing negative Poisson’s ratio among all samples tested in this research. A maximum negative Poisson’s ratio approximating –0.75 is obtained in L3-PW-RS2, close to that acquired in L2-PW-RS2. Likewise, as tensile strain increases, the growth rate of Poisson’s ratio is fast followed by a slow-down in L3-PW-RS2, but the overall trend is much milder than L2-PW-RS2.

Figure 14.

Effect of number of plies on the auxetic effect of laminated fabrics under warp-directional tensile strains: (a) warp direction of the frame fabric along the wale direction of the base fabric (PW) orientation and (b) warp direction of the frame fabric along the course of the base fabric (PC) orientation.

The Poisson’s ratio results of two-layer and three-layer laminated fabrics with PC orientation under warp-directional tensile strains are compared in Figure 4(b). A more distinct auxetic effect is also shown in the three-layer samples. The maximum negative Poisson’s ratio of L3-PC-RS2 is nearly –0.68 at 1% tensile strain. With the increase of tensile strains to 5%, the value of Poisson’s ratio firstly increases, and then tends to level off at around –0.4. As the strain keeps increasing, the value of Poisson’s ratio continues to rise, turning positive at a tensile strain of nearly 8.5%, and finally becomes stationary at around 0.5.

It is known from Table 3 that the initial modulus of the uncut three-layer laminated fabrics is close to, or is slightly smaller than, the initial modulus of the uncut two-layer laminated fabrics either in the warp or weft direction. Since the same base fabric and the same orientation are used for two-layer and three-layer samples, the initial modulus differences between the laminated material and the base material are also similar. This may account for the result that a similar maximum value of negative Poisson’s ratio is obtained in L2-PW-RS2 and L3-PW-RS2, but is not enough to explain a wider tensile strain range showing a negative Poisson’s ratio in L3-PW-RS2 and difference in the auxetic effect between L2-PC-RS2 and L3-PC-RS2. From experimental observations, it is found that the deformation of three-layer laminated fabrics under either warp- or weft-directional tensile strains is always kept in the fabric plane, until the break of the frame fabric. It is mainly because the three-layer laminated fabrics have the rotating square frame attached to both sides of the base fabric, forming a symmetric and balanced structure. It ensures that square units can rotate to their fullest, thus largely extending the tensile strain range showing auxetic effect in L3-PW-RS2. It also helps to avoid the biased results measured from the fabric plane due to severe out-of-plane deformations occurring in L2-PC-RS2.

Conclusions

In this article, a novel type of auxetic laminated fabric is proposed based on rotating square geometry. Fabric samples were produced by laser-cutting rotating square frames with different parameters out of a stiff fabric and attaching them to an elastic base fabric using hot-melt adhesive membrane. Tensile tests were conducted to investigate the deformation behavior of the laminated fabrics under tensile strains along two principal directions. The effects of the initial intersection angle of squares, size of squares, orientation of base fabric, and the number of plies on the auxetic effect of laminated fabrics were also analyzed. Detailed conclusions are as follows:

Based on the deformation behavior analysis, it is known that the auxetic effect of the laminated fabrics usually occurs at the initial stage of stretching, and decreases as the tensile strain increases. The distortion of square units and the out-of-plane deformation are the major signs showing that the in-plane auxetic behavior is about to disappear.

The maximum negative Poisson’s ratio mainly depends on the initial modulus difference between the base material and the laminated frame material, regardless of the layer numbers, but can be diminished by a large initial intersection angle and inappropriate size of squares.

Out-of-plane deformation is one of the major problems when developing two-layer laminated fabrics with in-plane auxeticity, but can be effectively avoided in the three-layer samples. The auxetic behavior of three-layer laminated fabric turns out better than its two-layer counterpart in terms of tensile strain range of auxeticity.

In this study, different auxetic properties have been achieved in various laminated fabrics with rotating square geometry. It proves to be practicable and maneuverable to produce auxetic fabrics using commercial materials and lamination technique. Factors that influence the auxetic properties of the laminated fabrics include the geometric parameters of the rotating squares, the tensile properties of the base fabric, and the number of plies in the laminate. But little is still known about the force distribution of the laminated fabrics under tensile stresses, and the design principles to tailor the auxetic effect and to control their in-plane and out-of-plane deformations. More efforts will be made in the future to obtain a better understanding understanding of the deformation of the auxetic laminated fabrics through simulation.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Research Institute for Intelligent Wearable Systems of The Hong Kong Polytechnic University in form of an internal project (No. P0039471).

ORCID iDs

Yanping Liu

Hu Hong

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Deformation behavior of auxetic laminated fabrics with rotating square geometry

Abstract

Keywords

Design, material and fabrication

Design

Materials

Fabrication

Tensile tests

Material property

Poisson’s ratio

Tensile properties of materials

Heat-processed base fabric

Uncut laminated fabrics

Two-layer laminates with PW orientation

Three-layer laminates with PW orientation

Deformation behavior of laminated fabrics

Deformation behavior of a typic laminated fabric with rotating square geometry

Under warp-directional tensile strains

Under weft-directional tensile strains

Effect of different parameters

Effect of initial intersection angle (β0) of the frame

Effect of square size (edge length a) of the frame

Effect of orientation of the base fabric

Effect of number of plies

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Effect of initial intersection angle (β₀) of the frame