Numerical analysis of deformation behavior of a 3D textile structure with negative Poisson’s ratio under compression

Abstract

A numerical analysis using the finite element (FE) method was conducted to investigate the deformation behavior of a novel three-dimensional (3D) textile structure made by a special arrangement of the weft and warp yarns in a 3D orthogonal structure to achieve negative Poisson’s ratio (NPR). The minimum number of unit cells required for the FE analysis was first determined by comparing the calculating results of Poisson’s ratio values based on the FE models built with fixed structural parameters and yarn properties. Then Poisson’s ratio values of the 3D textile structure with different structural parameters and yarn properties under compression were calculated using the FE models built with the determined unit cells. The calculating results show that the 3D structure can achieve NPR in one direction if the weft and warp yarns used have different bending stiffness and radii. This numerical analysis can provide useful information for the design and fabrication of this kind of 3D textile structure, which has high potential applications in composite reinforcement and other technical areas.

Keywords

finite element method negative Poisson’s ratio compression behavior 3D textile structure

The finite element (FE) method used in numerical simulations has been considered to be the most widely general-purpose technique in engineering and applied mathematics.¹ In the textile area, the FE method has widely been used to simulate and predict deformation behaviors and mechanical properties of different textile structures and their composites. Thom proposed a FE model for plain weave composites to examine their longitudinal Young’s modulus and peak stresses by considering the effects of mesh methods and size, interface layer thickness, matrix skin thickness, yarn shape, and yarn linear and nonlinear properties.² Based on a micromechanics study, Song et al. presented a FE model to assess the compressive strengths of two-dimensional (2D) tri-axial braided carbon fiber composites and their dependence of microstructural parameters.³ Lim et al. used a three-dimensional (3D) FE code (DYNA3D) to simulate the response of Twaron® fabric under high-speed projectile impact.⁴ Duan et al. created a 3D FE model to simulate impacting behavior of a square patch of plain-woven Kevlar fabric impacted in the transverse direction by a rigid cylinder.^5,6 Based on an accurate 3D parametric geometric model of a jersey structure, Hagège et al. generated a FE model to simulate its biaxial tensile behavior and found that the numerical simulation had a better agreement with experimental results.⁷

The FE method is also effective to analyze and simulate the deformation behaviors and mechanical properties of structures with negative Poisson’s ratio (NPR). Also known as auxetics, these structures becomes expandable when stretched and contractive when compressed,⁸ and compared with conventional materials have demonstrated a series of enhanced properties such as shear modulus,⁹ indentation resistance,^10–12 fracture toughness,^13,14 and acoustic response.¹⁵ Yang et al. derived a 2D triangular FE formulation to investigate the relationship between the Poisson’s ratio of a re-entrant honeycomb structure and structural parameters.¹⁶ Lira et al. used a quarter of a representative volume element as the FE model of auxetic multiple re-entrant honeycomb structures with periodic shear conditions to analyze and evaluate their transverse shear properties.¹⁷ Wright et al.,¹⁸ who investigated the static tensile behavior of a helical auxetic structure by FE analysis, found that the structure exhibited nonlinear tensile behavior due to nonzero component Poisson’s ratios and that the NPR of the structure could be as low as −5. Zhang et al. developed a FE model for a new cellular vehicle body structure with NPR and analyzed the influence of relative density and topological structure on the linear and nonlinear properties of the structure.¹⁹ Hou et al. investigated an isotropic auxetic composite structure by randomly including re-entrant triangles into a matrix and conducted a FE analysis to reveal the deformation mechanism and mechanical properties of the composites by considering geometrical features, stiffness, and boundary conditions of inclusions.^20–22 Zhang et al. conducted a FE analysis of an auxetic honeycomb structure made of tubes and corrugated sheets.²³

This paper presents a FE analysis of a novel kind of 3D textile structure made by a special arrangement of the weft and warp yarns in a 3D orthogonal structure. A similar 3D textile structure made by the arrangement of the weft and warp yarns, which was found to have a NPR in the weft yarn direction and zero Poisson’s ratio in the warp yarn direction when compressed in the thickness direction, has been investigated in our previous studies.^24–26 Its deformation behavior and auxetic effect were analyzed using the geometrical,²⁴ elastic,²⁵ and FE methods.²⁶ In this study, a 3D orthogonal textile structure made with a different arrangement of the weft and warp yarns is analyzed by using the FE method. The new arrangement of the weft and warp yarns in the 3D textile structure leads to more complicated deformation behaviors of the structure. In order to better understand deformation behaviors of the structure under compression, FE models will be established with different combinations of geometrical parameters and material properties, and will be used to calculate the Poisson’s ratio values. The objective is to know under what conditions the 3D structure will have NPR. It is expected that such a FE analysis would provide useful information for the design and fabrication of this kind of 3D textile structure which has a high application potential in composite reinforcement and other technical areas.

3D textile structure

The 3D textile structure to be analyzed is shown in Figure 1. It is composed of three yarn systems, i.e. weft yarns, warp yarns, and stitch yarns. In the structure, the straight warp and weft yarns are the basic framework and they are orthogonally arranged in the horizontal plane like a wood pile. The stitch yarns are used to bind all the layers of weft and warp yarns together and provide the stability of the stucture. To obtain a better binding effect, elastic yarns are preferably used as the stitch yarns. Different from the structure proposed in our previous studies,^24–26 the method of yarn arrangement in both the y–x plane and y–z plane is the same in the current structure, i.e. the yarns arranged in the even number layers are always shifted by a half spacing of the yarns arranged in the odd number layers. Because of this special arrangement, the deformation behaviors of the structure become very complicated when it is compressed in the y direction. The warp or weft yarns would be bent due to the pressure exerted by the yarns contacted. Under compression, the elastic stitch yarns would change from the tensioned state to the relaxed state.

Figure 1.

The 3D textile structure.

From Figure 1, it can be seen that the following geometrical parameters are sufficient to describe the structural features of the 3D structure. They are the radius of the warp yarn, r₁, the radius of the weft yarn, r₂, the spacing of two adjacent warp yarns in the x direction, 2l₁, and the spacing of two adjacent weft yarns in the z direction, 2l₂. These parameters directly affect the deformation behavior of the structure and will be used in the subsequent analysis.

FE analysis

FE model of unit cell

Since the elastic yarns are used as the stitch yarns in the structure, they cannot withstand the compression loads and only provide the stability of the stucture.²⁶ Therefore, the stitch yarns are eliminated from the unit cell for the FE analysis. In a unit cell of the structure, which is shown in Figure 2, the warp and weft yarns are alternately arranged.

Figure 2.

The unit cell of the structure.

In this study, the FE method is used to calculate the Poisson’s ratio values of the structure when compressed in the vertical direction. To do that, the unit cell of the structure is first meshed using the solid element SOLID186 of ANSYS software via the mapped mesh method. The meshed unit cell is shown in Figure 3. In order to simulate the compression deformation of the structure, the boundary conditions are determined in such a way that the bottom planes AB and CD are fixed and the compression loads are applied on the top planes A′B′ and C′D′, as shown in Figure 4. Due to the symmetrical feature of the structure, two mutually perpendicular planes, X₀–X₀′ and Z₀–Z₀′, are fixed during the compression process. And some ‘CP’ commands defining coupled degrees of freedom are used in the planes AA′, BB′, CC′, and DD′ to keep them perpendicular to the xz-coordinate plane. The 3D contact elements ‘CONTA174’ and ‘TARGE170’ are applied to simulate the contacts between the warp and weft yarns used in the fabric under compression. To facilitate the analysis, the yarn materials used are assumed to be isotropic linear elastic materials.

Figure 3.

The FE model of the unit cell.

Figure 4.

Boundary conditions of the unit cell: (a) front view; (b) lateral view.

Determination of unit cell number

In the commonly used method, only one unit cell is required for FE analysis as one unit cell could be sufficient to represent the behavior of the whole structure. However, a problem is found in the present study. Since the yarns in the y–x plane and y–z plane are arranged in the same way, the calculated results of Poisson’s ratio in the x and z directions (ν _yx and ν _yz ) under compression along the y direction should be the same if the weft and warp yarns used are the same. If the weft and warp yarns are different, calculated ν _yx and ν _yz values should also be the same if the weft yarns and the warp yarns are swapped between the y–x plane and y–z plane. However, by comparing the FE model of the unit cell and boundary conditions, as shown in Figures 3 and 4, it can be found that the forms of the unit cell model shown in the y–x plane and y–z plane are different, especially in the y direction. This difference may lead to different calculated results of ν _yx and ν _yz . Although the analysis can be conducted on the full scale structure with all unit cells, this requires much more computing power and much more time for calculation. In order to solve this problem, the first step is to determine the minimum number of unit cells required for the FE analysis. The way used is to increase the unit cell number in one coordinate direction, and keep the unit cell number unchanged in the other two directions and compare the calculated results of ν _yx and ν _yz . The comparison includes two parts. The first part is to compare ν _yx and ν _yz . If ν _yx and ν _yz calculated from two FE models with the same unit cell number, but with swapping of the weft and warp yarns, are different, this means that the FE models with the number of unit cells selected cannot well represent compression behavior of the structure, and an increase of the unit cell number is required. The second part is to compare ν _yx and ν _yz calculated from the new models with increased unit cells in one direction with those calculated from the previous models with less unit cell number. If the differences are very small, this means that the unit cell number increased in this direction does not affect the calculating results and the increase of unit cell number can be changed to other direction. If the differences are evident, the continuous increase of the unit cell number in this direction is required. The process is performed until a suitable unit cell number is found with which ν _yx and ν _yz calculated from two models are very close due to swapping of the weft and warp yarns.

For yarn properties, as they were assumed to be isotropic linear elastic materials, only two parameters, modulus and Poisson’s ratio, are required. For most yarn materials, Poisson’s ratio is between 0.2 and 0.3, so 0.25 was chosen for this study. For modulus, 100 MPa was selected based on the yarns used in our previous study.²⁴ Table 1 shows the structural parameters (r₁, r₂, l₁, and l₂) and number of unit cells in each coordinate direction used for comparison. The previous study has shown that a high difference between the radii of the weft and warp yarns (r₁ and r₂) and low spacing values between two adjacent yarns (l₁ and l₂) can achieve a high NPR of the structure.²⁴ Based on this conclusion, the selected weft and warp yarns have radii that are different by a factor of eight. For the l₁ and l₂, the minimum values were calculated according to the condition in which two adjacent warp or weft yarns in the horizontal distance just touch together using the geometric model proposed previously.^24,26 As shown in Table 1, in order to compare the calculation results, the weft and warp yarns in the FE models of odd number are swapped with those in the FE models of even number. Ten FE models were built for determination of the number of unit cells. Models 1 and 2 have only one unit cell in all three directions. Their weft and warp yarns are just swapped. Then the number of unit cells is increased in x, y, and z directions, respectively. Models 3 and 4 have two unit cells in the x direction, and one unit cell in y and z directions; models 5 and 6 have two unit cells in the z direction and one unit cell in the x and y directions; models 7 and 8 have two unit cells in the y direction and one unit cell in the x and z directions; models 9 and 10 have four unit cells in the y direction and one unit cell in the x and z directions.

Table 1.

The structural parameters and number of unit cells selected

Model number	Poisson’s ratio		Warp yarns		Weft yarns		Number of unit cells in each coordinate direction
Model number	x direction	z direction	r₁ (mm)	l₁ (mm)	r₂ (mm)	l₂ (mm)	x	y	z
1	ν _yx ₋₁	ν _yz ₋₁	0.5	3.01	4	8.84	1	1	1
2	ν _yx ₋₂	ν _yz ₋₂	4	8.84	0.5	3.01	1	1	1
3	ν _yx ₋₃	ν _yz ₋₃	0.5	3.01	4	8.84	2	1	1
4	ν _yx ₋₄	ν _yz ₋₄	4	8.84	0.5	3.01	2	1	1
5	ν _yx ₋₅	ν _yz ₋₅	0.5	3.01	4	8.84	1	1	2
6	ν _yx ₋₆	ν _yz ₋₆	4	8.84	0.5	3.01	1	1	2
7	ν _yx ₋₇	ν _yz ₋₇	0.5	3.01	4	8.84	1	2	1
8	ν _yx ₋₈	ν _yz ₋₈	4	8.84	0.5	3.01	1	2	1
9	ν _yx ₋₉	ν_yz−9	0.5	3.01	4	8.84	1	4	1
10	ν _yx ₋₁₀	ν _yz ₋₁₀	4	8.84	0.5	3.01	1	4	1

The calculated results of ν _yx and ν _yz as a function of compression strain are shown in Figures 5 to 7, respectively. Figure 5 shows the results calculated from models 1 to 4. Comparing the results calculated from model 1 and model 2, it can be found that although the curves of ν _yx ₋₁ and ν _yz ₋₂ are close, the curves of ν _yx ₋₂ and ν _yz ₋₁ are very different. This implies that only one unit cell is not enough for the FE analysis. After the unit cell number is increased to two in the x direction, comparing the results calculated from models 1 and 3 and models 2 and 4, it can be found that the curves of ν _yx ₋₁ and ν _yx ₋₃, curves of ν _yz ₋₁ and ν _yz ₋₃, curves of ν _yx ₋₂ and ν _yx ₋₄, and curves of ν _yx ₋₂ and ν _yx ₋₄ are very close. This means that the unit cell number increased in the x direction does not affect the calculated results. However, comparing the results calculated from models 3 and 4, it can be found that although the curves of ν _yx ₋₃ and ν _yz ₋₄ are close, the curves of ν _yx ₋₄ and ν _yz ₋₃ are very different. Therefore, an increase of the unit cell number in the other two coordinate directions is necessary.

Figure 5.

Effect of unit cell number in x direction.

Figure 6.

Effect of unit cell number in z direction.

Figure 7.

Effect of unit cell number in y direction.

Figure 6 shows the results calculated from models 1, 2, 5, and 6. After the unit cell number is increased to two in the z direction, comparing the results calculated from models 1 and 5 and models 2 and 6, it can be found that the curves of ν _yx ₋₁ and ν _yx ₋₅, curves of ν _yz ₋₁ and ν _yz ₋₅, curves of ν _yx ₋₂ and ν _yx ₋₆, and curves of ν _yx ₋₂ and ν _yx ₋₆ are very close. This means that the unit cell number increase in the y direction also does not affect the calculated results. However, comparing the results calculated from models 5 and 6, it can be found that although the curves of ν _yx ₋₅ and ν _yz ₋₆ are very close, the curves of ν _yx ₋₆ and ν _yz ₋₅ are very different. Therefore, an increase of the unit cell number in the y direction is necessary.

Figure 7 shows the results calculated from models 1, 2, and 7–10. First, comparing the results calculated from models 1 and 7 and models 2 and 8, it can be found that the curves of ν _yx ₋₁ and ν _yx ₋₇ and curves of ν _yz ₋₁ and ν _yz ₋₇ are different, although curves of ν _yx ₋₂ and ν _yx ₋₈ and curves of ν _yx ₋₂ and ν _yx ₋₆ are very close. This means that the unit cell number increase in the z direction affects the calculated results. Comparing the results calculated from models 7 and 8, it can be found that the curves of ν _yx ₋₇ and ν _yz ₋₈ are very close, but the curves of ν _yx ₋₈ and ν _yz ₋₇ are still very different. So it is necessary to continue to increase the unit cell number in the y direction. Comparing the results calculated from models 9 and 10, it can be found that the curves of ν _yx ₋₉ and ν _yz ₋₁₀ and the curves of ν _yx ₋₁₀ and ν _yz ₋₉ become very close after the unit cell number is increased to four. This means that the FE model with four unit cells in the y direction can well represent the deformation behavior of the whole fabric structure under compression. This number of unit cells will be used for building the FE models to predict deformation behaviors of the structure with different structural parameters and yarn properties.

Prediction of deformation behaviors with different structural parameters and yarn properties

Poisson’s ratio values of the 3D textile structure with different structural parameters and yarn properties under compression will be calculated using the FE models built with four unit cells in the y direction. The structural parameters include radii of warp and weft yarns (r₁ and r₂) and spacing between two adjacent yarns in the y–x plane and y–z plane (2l₁ and 2l₂). In order to obtain a large variation of the structure, four values from 0.5 mm to 4 mm were first selected for r₁ and r₂. After r₁ and r₂ were determined, l₁ and l₂ were then calculated based on the geometrical model proposed previously under the condition that the spacing between two adjacent yarns should be as close as possible,^24,26 as the low spacing makes the structure have high NPR. For yarn properties, they were also assumed to be isotropic linear elastic materials with a Poisson’s ratio of 0.25. For the modulus, three values from 100/16 to 100*16 MPa were selected. It can be found that the change of one modulus to the next involves a factor of 16 times as bending stiffness is proportional to the fourth power of diameter of the yarn and the change of the yarn diameter is a factor of two in the current study. Based on these structural parameters and yarn moduli, 12 structural variations were suggested based on the orthogonal experimental method and are listed in Table 2.

Table 2.

Combinations of structural parameters and material properties

Structure Variation	Poisson’s ratio		Warp yarns			Weft yarns			Ratios of warp yarn to weft yarn
Structure Variation	x direction	z direction	Modulus E₁ (MPa)	r₁ (mm)	l₁ (mm)	Modulus E₂ (MPa)	r₂ (mm)	l₂ (mm)	Radius	Bending stiffness
I	ν _yx _−I	ν _yz _−I	100/16	0.5	1.44	100/16	0.5	1.44	1	1
II	ν _yx _−II	ν _yz _−II	100/16	1	2.89	100	1	2.89	1	1/16
III	ν _yx _−III	ν _yz _−III	100/16	2	5.78	100*16	2	5.78	1	1/(16*16)
IV	ν _yx _−IV	ν _yz _−IV	100/16	4	11.56	100/16	4	11.56	1	1
V	ν _yx _−V	ν _yz _−V	100	1	3.51	100	2	5.05	1/2	1/16
VI	ν _yx _−VI	ν _yz _−VI	100	0.5	3.01	100*16	4	8.84	1/8	1/(161616*16)
VII	ν _yx _−VII	ν _yz _−VII	100	4	8.84	100/16	0.5	3.01	8	161616*16
VIII	ν _yx _−VIII	ν _yz _−VIII	100	2	5.05	100	1	3.51	2	16
IX	ν _yx _−IX	ν _yz _−IX	100*16	2	7.03	100*16	4	10.09	1/2	1/16
X	ν _yx _−X	ν _yz _−X	100*16	4	10.09	100/16	2	7.03	2	161616
XI	ν _yx _−XI	ν _yz _−XI	100*16	0.5	1.76	100	1	2.52	1/2	1
XII	ν _yx _−XII	ν _yz _−XII	100*16	1	2.52	100*16	0.5	1.76	2	16

To understand the deformation behavior of the 3D structure, the structural variation VII is first chosen as an example to demonstrate how yarns are deformed under compression. The deformation process of the structure at different compression strains is shown in Figure 8. It can be found that the structure contracts in the x direction (front view) due to bending of the finer weft yarns and almost maintains unchanged in the z direction (lateral view) under compression. This indicates that the structure has NPR in the x direction and zero Poisson’s ratio in the z direction. Figure 8 also shows that the bending deformation is the main deformation mode of yarns in the structure.

Figure 8.

Deformation process of structural variation VII at different compression strains: (a) front view; (b) lateral view.

Poisson’s ratios ν _yx and ν _yz as a function of compression strain calculated according to the values listed in Table 2 are shown in Figures 9 and 10, respectively. To ontain a better demonstration of the results, the scale above zero for the coordinate of Poisson’s ratio is enlarged. It can be found that Poisson’s ratio values vary from quasi-zero to negative for all the structural variations. Since the bending deformation is the main deformation mode of yarns in the structure under compression, the deformation behavior of the structure will mainly depend on the bending stiffness of the yarns. Two cases can be considered. In the first case, the weft yarn and warp yarn used have the same bending stiffness, such as structural variations I, IV, and XI. Comparing all the ν _yx and ν _yz curves of these structural variations, it can be found that all the ν _yx and ν _yz curves fluctuate around zero, except that the ν _yz curve of XI negatively increases with compression strain. These results imply that when the weft and warp yarns have the same bending stiffness due to use of the yarns with the same radii and moduli, the 3D structure has quasi-zero Poisson’s ratios; when the weft and warp yarns have the same bending stiffness but have different radii and moduli, the NPR can be achieved in one direction in which the yarns perpendicular to this direction are bigger than the yarns parallel to this direction.

Figure 9.

ν _yx as a function of compression strain curves.

Figure 10.

ν _yz as a function of compression strain curves.

In the second case, the weft yarn and warp yarn used have different bending stiffness, such as structural variations II, III, V, VI, VII, VIII, IX, X, and XII. Comparing the ν _yx and ν _yz curves of structural variations V, VIII, IX, and XII, in which the weft yarn and warp yarn have the same modulus but have different radii, it can be found that all these structural variations have NPR in one direction where the yarns perpendicular to this direction have bigger radii than the yarns parallel to this direction, and quasi-zero Poisson’s ratio in other direction where the yarns perpendicular to this direction have smaller radii than the yarns parallel to this direction. Comparing the ν _yx and ν _yz curves of structural variations II and III in which the weft yarn and warp yarn have the same radius but have different moduli, it can be found that the ν _yx and ν _yz curves fluctuate around zero. This means that although the yarns used have different bending stiffness, the NPR cannot be achieved when the radii of the yarns used are the same. Comparing the ν _yx and ν _yz curves of structural variations VI, VII, and X in which both the radii and moduli of the weft yarn and warp yarn are different, it can be found that all these structural variations have NPR in one direction where the bending stiffness of the yarns perpendicular to this direction is higher than that of the yarns parallel to this direction, and quasi-zero Poisson’s ratio in other direction where the bending stiffness of the yarns perpendicular to this direction is lower than that of the yarns parallel to this direction. Based on the above comparisons, it can be confirmed that the 3D textile structure can only achieve NPR when weft and warp yarns with different bending stiffness and radii are used. If the weft and warp yarns used are the same, it is impossible to achieve NPR.

Conclusions

A numerical analysis was conducted using the FE method on a novel kind of 3D textile structure made by a special arrangement of the weft and warp yarns in a 3D orthogonal structure to achieve NPR. The minimum number of unit cells required for the FE analysis was first determined by comparing the calculating results of Poisson’s ratio values by increasing the unit cell number in each coordinate direction. Then Poisson’s ratio values of the 3D textile structure with different structural parameters and yarn properties under compression were calculated using the FE models built with the determined unit cells. Based on the calculation results, the following conclusions could be obtained.

The FE method is efficient for analyzing deformation behavior of 3D textile structures. The FE model built with four unit cells in the thickness direction of the structure can represent the deformation behavior of the structure under compression.

Bending stiffness and yarn radius are the main factors affecting compression deformation behavior of the 3D structure.

When the weft and warp yarns have the same radius, no matter whether they have the same bending stiffness or not, the 3D structure has quasi-zero Poisson’s ratio.

When the weft and warp yarns have the same bending stiffness with the use of different radii and moduli, the structure 3D has NPR in one direction where the yarns perpendicular to this direction have bigger radius than the yarns parallel to this direction.

When the weft and warp yarns have different bending stiffness with the use of different radii, the 3D structure has NPR in one direction where the yarns perpendicular to this direction have bigger bending stiffness than the yarns parallel to this direction, and quasi-zero Poisson's ratio in other direction where the yarns perpendicular to this direction have lower bending stiffness than the yarns parallel to this direction.

Footnotes

Funding

This work was supported by the Research Grants Council of Hong Kong Special Administrative Region Government in the form of a GRF project (grant number 516510).

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