# Modelling of Auxetic Woven Structures for Composite Reinforcement

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Model

_{l}and E

_{s}, and elasticity of warp yarn be E

_{p}.

_{l}is the elasticity of the weft yarn, $\delta {x}_{2}$ is the final extension in the float region. Now, the total extension would be the sum of extension in the plain weave and float part. The strain in the axial direction is total extension divided by the length of the fabric (in this case unit cell).

#### 2.1. Material

#### 2.2. Test Method

## 3. Computational Model

**Geometry:**The unit cell shown in Figure 8, is created by varying the float length along the warp and weft direction. Here, we use a combination of the plain weave along with the 4 up 1 down twill weave. Here, in each weft, 9 warp yarns are interlaced in plain weave and 10 warp yarns are interlaced in a 4/1 twill pattern, forming a foldable geometry. The twill weave forms the folded part with a long float length, which unfolds when extended in the axial direction. The unit cell created in Texgen, when joined, can produce full-scale fabric. This sort of portrayal is significantly more helpful for computational analysis of material properties.

**FEM:**The prepared unit cell is imported to Ansys Workbench as a Stereo Lithography file to FEM Modeler to generate the initial geometry. Then these data are transferred to the transient structural feature. Then the type of material is assigned to the yarns, that is, the weft yarns were kept elastic and warp yarns are kept non-elastic.

**Contact:**Initially the contact is created automatically on Ansys mechanical, analyzing the face-to-face detection shown in Figure 9 but the contacts are modified according to weave design. The contact method is changed to frictional with the coefficient of friction equal to 0.1. The formulation is set to Augmented Lagrange as the penetration matter in this model. The initial contact condition is analyzed through the contact tool and accordingly pinball radius is adjusted manually.

**Mesh generation:**After the contact point generation, the mesh was generated as shown in Figure 10. The mesh size taken here was medium for easy processing and accuracy and element size was 0.125 mm by default.

**Boundary conditions:**The boundary conditions are imposed, as shown in Figure 11, on the unit cell, keeping in mind that it is a repeating unit and not a full-scale fabric. The following kinematic boundary condition will be applied as in Equations (9) and (10).

_{1}, v

_{2}, v

_{3}, and v

_{4}are displacement vectors of points on the edges of 1, 2, 3, and 4, respectively, and x

_{1}, x

_{2}, x

_{3}and x

_{4}are the absolute value of distances corresponding to displacement vectors v

_{1}, v

_{2}, v

_{3}, and v

_{4}. E stands for macroscopic strain tensor of the fabric [51,52,53,54].

**Simulation:**The left boundary was fixed along the x-direction and a directional displacement was applied along the right boundary, with the displacement rate of 2 mm/min. For the tensile test environment simulation, z and y directional displacements were constrained on both left and right boundaries. For the mechanical analysis of the model, elastic strain in the transverse direction in Figure 12.

## 4. Results and Discussion

#### 4.1. Experimental Results

#### 4.2. Analytical Model Results

#### 4.3. Computational Model Results

#### 4.4. Comparison of the Computational and Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Free body diagram of loading. (T: Outward Tension, T

_{1}: Inward tension, F: Total frictional force, f.p.w: friction on plain weave, f.f: friction on float).

**Figure 4.**Couple formation in auxetic structure. (A, B, C: are nodal points of the zigzag pattern, X, Y: are axes representing warp and weft respectively).

**Figure 5.**Saw tooth model for plain weave. (A

_{1}, A

_{2}, B

_{1}, B

_{2}: are nodal points of the saw tooth model, H

_{1}, H

_{2}: are crimp amplitudes in X and Y axes, h

_{1}/2: crimp height from crossover point, F

_{1}, F

_{2}: Total force along the saw tooth axes, p

_{2}/2: half wavelength, V

_{1}, V

_{2}: inter yarn pressure).

S.No | Yarn Fineness | Tenacity at Breaking Extension (cN/tex) | Breaking Extension (%) |
---|---|---|---|

A | Cotton (30/2 tex) | 13.608 | 6.304 |

B | Cotton (20 tex) | 11.890 | 3.404 |

C | Core spun spandex 38 tex | 5.674 | 63.24 |

Parameters | Warp | Weft |
---|---|---|

Yarn spacing (mm) | 0.635 | 0.635 |

Crimp (%) | 8 | 7.1 |

Bending modulus (MPa) | 0.53 | 0.53 |

Coefficient of friction | 0.1 | 0.1 |

Direction | Plain Weave | Float |
---|---|---|

Weft (Axial strain %) | 1.1660 | 0.5253 |

Warp (Transverse strain %) | 0.5268 | 0.3909 |

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**MDPI and ACS Style**

Shukla, S.; Behera, B.K.; Mishra, R.K.; Tichý, M.; Kolář, V.; Müller, M. Modelling of Auxetic Woven Structures for Composite Reinforcement. *Textiles* **2022**, *2*, 1-15.
https://doi.org/10.3390/textiles2010001

**AMA Style**

Shukla S, Behera BK, Mishra RK, Tichý M, Kolář V, Müller M. Modelling of Auxetic Woven Structures for Composite Reinforcement. *Textiles*. 2022; 2(1):1-15.
https://doi.org/10.3390/textiles2010001

**Chicago/Turabian Style**

Shukla, Shivangi, Bijoya Kumar Behera, Rajesh Kumar Mishra, Martin Tichý, Viktor Kolář, and Miroslav Müller. 2022. "Modelling of Auxetic Woven Structures for Composite Reinforcement" *Textiles* 2, no. 1: 1-15.
https://doi.org/10.3390/textiles2010001