# Composites with Re-Entrant Lattice: Effect of Filler on Auxetic Behaviour

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Design of Auxetic Structures

#### 2.2. Additive Manufacturing and Mechanical Testing of Auxetic Structures

#### 2.3. Digital Image Correlation

#### 2.4. Finite-Element Analysis

^{3}.

#### 2.5. Statistical Analysis

## 3. Results

#### 3.1. FE Simulations of Porous Lattice Structure

#### 3.2. Effect of Filler on NPR

_{1}for the axially and transversely oriented two-phase structures with filler’s modulus starting from 600 MPa to 0.2 MPa (ratio of filler modulus to modulus of lattice ${E}_{filler}/{E}_{lattice}$ varies from 0.3 to 10

^{−4}) and porous structure. The gradient of displacements fields reflects the behaviour of the structure: with the X axis directed to the right, negative values (blue colour) on the right and positive values (red colour) on the left correspond to lateral contraction of the structure, i.e., positive Poisson’s ratios. This behaviour is observed for the structures with elastic modulus equal to 600 MPa, 200 MPa and 60 MPa $({E}_{filler}/{E}_{lattice}$ equal to 0.3, 0.1 and 0.03). On the contrary, positive values on the right and negative values on the left indicate lateral expansion (negative Poisson’s ratio), i.e., auxetic behaviour. The structures from the considered cases with elastic modulus of 2 MPa, 0.2 MPa and 0 MPa (porous lattice) ${(E}_{filler}/{E}_{lattice}\text{}$ equal to ${10}^{-3}$, ${10}^{-4}$ and 0) fall in that category. The chosen tensile displacement values allowed us to see this tendency for the two-phase structure with both axial and transverse orientations of unit-cells (see Table 2). When a filler’s elastic modulus is 20 MPa ${(E}_{filler}/{E}_{lattice}=0.01)$, both axially and transversely oriented structures demonstrate a close to zero value of Poisson’s ratio.

_{filler}/E

_{lattice}= 0.1, 0.01, 10

^{−4}). All the measured values and the mean value of the Poisson’s ratio are confidently in the positive zone for the filler with elastic modulus of 200 Mpa (E

_{filler}/E

_{lattice}= 0.1). When the modulus of the filler decreased 10-fold, down to 20 Mpa (E

_{filler}/E

_{lattice}= 0.01), the Poisson’s ratio for one of the measured pair of points (line 6) falls below 0, while other points—as well as the globally measured value—were still positive. This marks the threshold for the elastic properties of the filler corresponding to the transition from the negative to positive values of the overall Poisson’s ratio, i.e., loss of the auxetic behaviour. The filler with the elastic modulus of 0.2 Mpa (10,000 times lower than the modulus of the lattice, E

_{filler}/E

_{lattice}= 10

^{−4}) did not affect the designed auxetic properties of the lattice.

_{filler}/E

_{lattice}= 0.1). For the case of 2 Mpa (E

_{filler}/E

_{lattice}= 10

^{−3}), the average value of the Poisson’s ratio is negative, while there are some locally measured values demonstrating the opposite behaviour. This can be explained by proximity of the measured points to the constrained surfaces. The average values of the Poisson’s ratio also indicated the transition point for the elastic properties of the filler that divides the auxetic and the non-auxetic behaviours of the filled lattice. The filler with the lowest elastic modulus of 0.2 Mpa (E

_{filler}/E

_{lattice}= 10

^{−4}) embedded into the transversally oriented structure had a value much closer to the zero value of the global Poisson’s ratio than the axially oriented structure.

#### 3.3. Analysis of Stress Distributions in Lattice

_{filler}/E

_{lattice}= 0.1, 0.01 and 10

^{−4}). The tensile strength of the lattice material (HIPS) was measured with in-house experiments and was taken as 30 Mpa.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Model of axially oriented re-entrant unit-cell; (

**b**) axially oriented porous auxetic-lattice structure; (

**c**) model of transversely oriented re-entrant unit-cell; (

**d**) transversely oriented porous auxetic-lattice structure; (

**e**) lattice of two-phase auxetic structure.

**Figure 4.**(

**a**) Axially oriented unit-cell of AM sample with applied speckles; (

**b**) transversely oriented unit-cell of AM sample with applied speckles; (

**c**) tensile test of sample with Vic-3D Micro-DIC system.

**Figure 7.**Tensile force vs. displacement diagram (loading curve) recorded in tensile tests for axially (

**a**) and transversally (

**b**) oriented samples.

**Figure 8.**(

**a**) Field of ${\epsilon}_{22}$ strain for axial direction auxetic-lattice. Data for cell obtained numerically (

**b**) and experimentally (

**c**).

**Figure 9.**(

**a**) Field of ${\epsilon}_{22}$ strain for transversal direction auxetic-lattice. Data for cell obtained numerically (

**b**) and experimentally (

**c**).

**Figure 10.**Dependence of Poisson’s ratio on lg[${E}_{filler}/{E}_{lattice}$] for axially (

**a**) and transversally (

**b**) oriented re-entrant auxetic structures.

**Figure 11.**Comparison of distribution of normalised maximum principal stress (with tensile strength) fields for axial (

**a**,

**b**) and transversal (

**c**,

**d**) two-phase structures with filler elastic modulus of 200, 20 and 0.2 MPa (${E}_{filler}/{E}_{lattice}=0.1,\text{}0.01\text{}and\text{}{10}^{-4}$): for various applied strains: (

**a**,

**c**) 0.25%; (

**b**,

**d**) 1%.

Elastic Modulus of Auxetic-Lattice, ${\mathit{E}}_{\mathit{l}\mathit{a}\mathit{t}\mathit{t}\mathit{i}\mathit{c}\mathit{e}}$, Mpa | Elastic Modulus of Filler, ${\mathit{E}}_{\mathit{f}\mathit{i}\mathit{l}\mathit{l}\mathit{e}\mathit{r}}$, MPa | Ratio between Elastic Moduli, ${\mathit{E}}_{\mathit{f}\mathit{i}\mathit{l}\mathit{l}\mathit{e}\mathit{r}}/{\mathit{E}}_{\mathit{l}\mathit{a}\mathit{t}\mathit{t}\mathit{i}\mathit{c}\mathit{e}}$ |
---|---|---|

2000 | 600 | 0.3 |

200 | 0.1 | |

60 | 0.03 | |

20 | 0.01 | |

2 | 0.001 | |

0.2 | 0.0001 |

Elastic modulus of filler ${E}_{filler}$, MPa | 600 | 200 | 60 | 20 | 2 | 0.2 | 0 |

Relation between elastic moduli $({E}_{filler}/{E}_{lattice}=0.01)$ | 0.3 | 0.1 | 0.03 | 0.01 | 0.001 | 0.0001 | 0 |

Axial orientation of structure | |||||||

Transversal orientation of structure | |||||||

Non-auxetic | $\nu \approx 0$ | Auxetic |

Modulus of Filler, MPa ${\mathit{E}}_{\mathit{f}\mathit{i}\mathit{l}\mathit{l}\mathit{e}\mathit{r}}/{\mathit{E}}_{\mathit{l}\mathit{a}\mathit{t}\mathit{t}\mathit{i}\mathit{c}\mathit{e}}$ | $\mathbf{Strain}\text{}{\mathit{\epsilon}}_{11}$ | Poisson’s Ratios | Global Poisson’s Ratio | |
---|---|---|---|---|

200 (0.1) | 0.27 | |||

20 (0.01) | 0.09 | |||

0.2 (10 ^{−4}) | −0.27 |

Modulus of Filler, MPa ${\mathit{E}}_{\mathit{f}\mathit{i}\mathit{l}\mathit{l}\mathit{e}\mathit{r}}/{\mathit{E}}_{\mathit{l}\mathit{a}\mathit{t}\mathit{t}\mathit{i}\mathit{c}\mathit{e}}$ | $\mathbf{Strain}\text{}{\mathit{\epsilon}}_{11}$ | Poisson’s Ratio | Global Poisson’s Ratio | |
---|---|---|---|---|

200 (0.1) | 0.21 | |||

2 (10 ^{−3}) | −0.01 | |||

0.2 (10 ^{−4}) | −0.02 |

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**MDPI and ACS Style**

Tashkinov, M.; Tarasova, A.; Vindokurov, I.; Silberschmidt, V.V.
Composites with Re-Entrant Lattice: Effect of Filler on Auxetic Behaviour. *Polymers* **2023**, *15*, 4076.
https://doi.org/10.3390/polym15204076

**AMA Style**

Tashkinov M, Tarasova A, Vindokurov I, Silberschmidt VV.
Composites with Re-Entrant Lattice: Effect of Filler on Auxetic Behaviour. *Polymers*. 2023; 15(20):4076.
https://doi.org/10.3390/polym15204076

**Chicago/Turabian Style**

Tashkinov, Mikhail, Anastasia Tarasova, Ilia Vindokurov, and Vadim V. Silberschmidt.
2023. "Composites with Re-Entrant Lattice: Effect of Filler on Auxetic Behaviour" *Polymers* 15, no. 20: 4076.
https://doi.org/10.3390/polym15204076