# Cell-Dependent Mechanical Properties of Asymmetric Crosslinked Metallic Wire Mesh with Hybrid Patterns Based on Arbitrary Poisson’s Ratio

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Representation of Metallic Wire Mesh

_{1}(longer one) and l

_{2}(shorter one), the x-axis projection length of the inclined members l, the cell thickness b, the angles between the inclined members and the horizontal line θ

_{1}and θ

_{2}, and the cross-section diameter of the wire D.

^{∞}) is assumed to impose the PPRC, which corresponds to the vertical force 2P

_{A}= 2P

_{C}= 2lbσ

^{∞}applied at points A and C in Figure 2.

_{A}, M

_{A}, N

_{C}, and M

_{C}. Additionally, vertex B is entirely fixed. The vertices A and C have identical deformation in horizontal displacement but can move freely along vertical displacement. Therefore, considering the special geometric features and deformation compatibility conditions, the following equilibrium equations can be obtained:

_{ij}represents the j-th displacement component due to the force in i direction. The strain field in the x and y direction can be derived as:

_{A}= 2P′

_{C}= 2lbσ

^{∞}. The loading diagram is shown in Figure 3. Similarly, the equivalent Poisson’s ratio of NPRC structure in the y can be expressed as:

## 3. Materials and Methods

#### 3.1. Structural Design of Asymmetric Crosslinked Metallic Wire Mesh

#### 3.2. Experimental Procedures

#### 3.2.1. Fabrication of Asymmetric Crosslinked Metallic Wire Mesh

#### 3.2.2. Uniaxial Tensile Test

#### 3.2.3. Three-Point Bending

#### 3.3. Finite Element Analysis

#### 3.3.1. Numerical Modeling

#### 3.3.2. Preliminary Results of Numerical Simulation

## 4. Results and Discussion

#### 4.1. Influence of Cell Beam Angle on the Poisson’s Ratio

_{2}when θ

_{2}is a constant. It can be observed that when θ

_{2}< 0, the cell is an NPRC model, and the value of v is negative. On the other hand, when θ

_{2}> 0, the cell is a PPRC model, and the value of v is positive. As θ

_{2}decreases, the negative Poisson’s ratio of the structure becomes more pronounced. Figure 11b shows a simulated deformation state of the NPRC model during the uniaxial tensile test. It can be observed that the NPRC model expands transversely from cell 1 to cell 2 under longitudinal stretch, and the negative Poisson’s ratio maintained until it reaches to cell 2. If the longitudinal stretch continues, the angle θ will change from a concave angle to a convex angle and cell 2 will turn into cell 3. It is noteworthy that the Poisson’s ratio of the entire structure changes from negative to positive during this transformation. Additionally, the corrugated wire undergoes a greater deformation, indicating that it bears the maximum force.

_{1}and θ

_{2}. As demonstrated in Equations (6) and (7), the cell beam angle is the primary factor that affects Poisson’s ratio. Therefore, Poisson’s ratio is less correlated with l and b.

_{1}and θ

_{2}are positively correlated with the absolute Poisson’s ratio value. The numerical results presented above regarding the evolution of cell-dependent Poisson’s ratio are consistent with the theoretical analyses.

#### 4.2. Effects of Hybrid Cell Pattern on Macroscopic Poisson’s Ratio

#### 4.3. Influence of Interwoven Joint on Elastic Bending Property

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{A}, M

_{A}, N

_{C}and M

_{C}. Additionally, vertex B is entirely fixed. The vertices A and C have identical deformation in horizontal displacement but can move freely along vertical displacement. A uniform remote stress (σ

^{∞}) is assumed to impose the PPRC, which corresponds to vertical force 2P

_{A}= 2P

_{C}= 2lbσ

^{∞}applied at points A and C in Figure 2. Therefore, considering the special geometric features and the deformation compatibility conditions, the following equilibrium equations can be obtained:

_{ij}represents the j-th displacement component due to the force in i direction, as given by:

_{A}, M

_{A}, N

_{C}and M

_{C}can be obtained from Equations (A1)–(A14) as follows:

_{xy}is defined as the ratio of strains in the transverse and longitudinal directions.

_{A}= 2P′

_{C}= 2lbσ

^{∞}, the following equation of equilibrium is obtained:

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**Figure 1.**Simplified unit cell structures of RVE and the corresponding geometrical parameters: (

**a**) PPRC and (

**b**) NPRC; (

**c**) cell thickness b.

**Figure 5.**Four different types of asymmetric crosslinked metallic wire mesh structures associated with hybrid patterns based on arbitrary Poisson’s ratio.

**Figure 7.**Interwoven joining technologies of asymmetric crosslinked metallic wire mesh: (

**a**) resistance spot welding and (

**b**) adhesive bonding.

**Figure 10.**Comparison of load–displacement curves obtained from the finite element analysis and the three-point bending experiment.

**Figure 11.**(

**a**) Theoretical evolution curve of Poisson’s ratio with θ

_{2}and (

**b**) deformation process of single cell based on FEA.

**Figure 12.**Simulation results of macroscopic Poisson’s ratio variation curves with displacement by using different models.

**Figure 18.**In-plane distortion of asymmetric crosslinked metallic wire mesh with different connection methods after the bending experiment: (

**a**) glue adhesion (

**b**) resistance spot welding.

**Figure 19.**Elastic bending recovery angles of metallic wire mesh with different deflection: (

**a**) bending angle θ

_{B}(θ

_{B}= 180° − θ

_{R1}); (

**b**) instantaneous recovery angle θ

_{IR}(θ

_{IR}= 180° − θ

_{R2}); (

**c**) one-day recovery angle θ

_{DR}(θ

_{DR}= 180° − θ

_{3}).

Elastic Modulus (MPa) | Density (g/mm^{3}) | Poisson’s Ratio | Strain Hardening Exponent (n) | Strength Coefficient (K) |
---|---|---|---|---|

1.9 × 10^{5} | 7.8 × 10^{−3} | 0.3 | 0.45 | 1400 |

Geometric Parameters | l/mm | b/mm | θ_{1} | θ_{2} |
---|---|---|---|---|

Value | 6 | 5.8 | 60° | 30° |

Sample Number | l/mm | b/mm | |θ_{1}| | |θ_{2}| |
---|---|---|---|---|

Model 1 | 6 | 5.8 | 60° | 20° |

Model 2 | 6 | 5.8 | 60° | 30° |

Model 3 | 6 | 5.8 | 60° | 40° |

Model 4 | 6 | 5.8 | 50° | 30° |

Model 5 | 6 | 5.8 | 70° | 30° |

l/mm | b/mm | |θ_{1}| | |θ_{2}| |
---|---|---|---|

12 | 1.5 | 60° | 30° |

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**MDPI and ACS Style**

Wu, F.; Li, Z.; Lin, C.; Ge, S.; Xue, X.
Cell-Dependent Mechanical Properties of Asymmetric Crosslinked Metallic Wire Mesh with Hybrid Patterns Based on Arbitrary Poisson’s Ratio. *Symmetry* **2023**, *15*, 941.
https://doi.org/10.3390/sym15040941

**AMA Style**

Wu F, Li Z, Lin C, Ge S, Xue X.
Cell-Dependent Mechanical Properties of Asymmetric Crosslinked Metallic Wire Mesh with Hybrid Patterns Based on Arbitrary Poisson’s Ratio. *Symmetry*. 2023; 15(4):941.
https://doi.org/10.3390/sym15040941

**Chicago/Turabian Style**

Wu, Fang, Zeyu Li, Congcong Lin, Shaoxiang Ge, and Xin Xue.
2023. "Cell-Dependent Mechanical Properties of Asymmetric Crosslinked Metallic Wire Mesh with Hybrid Patterns Based on Arbitrary Poisson’s Ratio" *Symmetry* 15, no. 4: 941.
https://doi.org/10.3390/sym15040941