# GAM: General Auxetic Metamaterial with Tunable 3D Auxetic Behavior Using the Same Unit Cell Boundary Connectivity

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Description of the Unit Cell

## 3. Examples of Simple Implementations of the Metamaterial

#### 3.1. Shell Generation with Layers of This Metamaterial

#### 3.2. Numerical Testing of the Metamaterial

#### 3.3. Determination of the Mechanical Properties through Inverse Analysis with a Simple ML Model

- Inputs. Geometrical parameters: H, ${H}_{star}$, ${D}_{star}$, ${D}_{joint}$.
- Outputs. Mechanical properties: ${E}_{z}$, ${\nu}_{t}$

`scikit-learn`library from the Python programming language [56]. As a loss function for this model, the mean squared error (MSE) was used, which, denoting the label (output) as y, is expressed as:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Structural Matrix Calculus Model and Equivalent Young’s Modulus and Poisson’s Ratio Calculation

**f**

_{r}at the top face by the cross-sectional area perpendicular to the loading application, the stress ${\sigma}_{zz}$ can be estimated, and therefore the Young’s modulus can be computed as

**Table A1.**Comparison of macroscale variable results between Nastran and the GAM code (code used for the ML examples).

${\mathit{\nu}}_{\mathbf{xz}}$ | ${\mathit{\nu}}_{\mathbf{yz}}$ | ${\mathit{E}}_{\mathit{z}}$ [MPa] | |
---|---|---|---|

Nastran | $-0.84$ | $-0.73$ | 174 |

GAM code | $-0.85$ | $-0.65$ | 186 |

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**Figure 1.**The unit cell in auxetic configuration (

**a**), the non-auxetic configuration (

**b**), the connection between the auxetic unit cell with its neighbors (

**c**), and the connection between the conventional unit cell with its neighbors (

**d**).

**Figure 3.**Auxetic (

**a**) and conventional (

**b**) lateral and top (plan) schematic views of the unit cells in different configurations.

**Figure 5.**Three-dimensional (

**a**) and top (

**b**) views of a shell created from the proposed metamaterial cell simply using $\Delta {z}_{n}=20\phantom{\rule{0.166667em}{0ex}}sin\left(\frac{{y}_{n}}{10}\right)$.

**Figure 6.**Different sections of a shell created from the proposed metamaterial cell using $\Delta {z}_{n}=10\phantom{\rule{0.166667em}{0ex}}sin\left(\frac{{x}_{n}}{10}\right)\phantom{\rule{0.166667em}{0ex}}sin\left(\frac{{y}_{n}}{10}\right)$, where (

**a**) is a slice perpendicular to the x-axis, (

**b**) is a top (plan) view of the previous slice from the z-axis, and (

**c**) represents the whole domain.

**Figure 7.**Typical numerical results for the auxetic (top,

**a**–

**d**), and non-auxetic (bottom,

**e**–

**h**) configurations. In both configurations, an increasing incremental vertical displacement load is applied—from unloaded (left) to fully loaded (right).

**Figure 8.**Range of mechanical properties obtained from the numerical models, considering different geometries, where (

**a**,

**b**) represent the magnitude (size) of these geometrical variables against the equivalent Young’s modulus ${E}_{z}$ and transverse Poisson’s ratio ${\nu}_{t}$, respectively, and (

**c**) displays the relationship between Young’s modulus and Poisson’s ratio in the simulations.

**Figure 9.**The ML model predictions of longitudinal Young’s modulus ${E}_{z}$ (

**a**) and transverse Poisson’s ratio ${\nu}_{t}$ (

**b**), using RF as the regressor. Blue represents the training set, and orange represents the test set. Ground-truth values (from simulations) are on the horizontal axes, while predicted values are on the vertical axes (perfect prediction lies on the bisectrix). A high accuracy is achieved in the test set in terms of the coefficient of determination ${R}^{2}$.

E [GPa] | $\mathit{\nu}$ |
---|---|

200 | $0.33$ |

**Table 2.**Basic statistics of geometrical and mechanical variables of the numerical model, namely the mean, standard deviation (std), minimum, quartiles, and maximum values.

Mean | Std | Min | 25% | 50% | 75% | Max | |
---|---|---|---|---|---|---|---|

${D}_{joint}$ | $-0.38$ | 0.38 | $-1.40$ | $-0.60$ | $-0.36$ | $-0.12$ | 1.2 |

${H}_{star}$ | $-0.09$ | 0.12 | $-0.70$ | $-0.14$ | $-0.06$ | $-0.01$ | 0.4 |

${D}_{star}$ | 0.38 | 0.20 | 0.05 | 0.20 | 0.40 | 0.55 | 0.7 |

H | 2.41 | 0.69 | 1.50 | 2.00 | 2.50 | 3.00 | 3.5 |

${E}_{z}$ [MPa] | 184.78 | 159.64 | 0.00 | 87.78 | 148.62 | 242.30 | 766.2 |

${\nu}_{t}$ | $-0.18$ | 0.28 | $-1.00$ | $-0.33$ | $-0.10$ | 0.00 | 0.5 |

**Table 3.**Cell parameters, Young’s modulus, and schematic representation of three extreme cases for the transverse (to the z-axis) Poisson’s ratio i.e., ${\nu}_{t}\in \{-1,0,0.5\}$.

${\mathit{D}}_{\mathbf{joint}}$ | ${\mathit{H}}_{\mathbf{star}}$ | ${\mathit{D}}_{\mathbf{star}}$ | H | ${\mathit{E}}_{\mathit{z}}$ [MPa] | Representation | |
---|---|---|---|---|---|---|

${\nu}_{t}=-1$ | $-0.8$ | $-0.24$ | 0.25 | 2.0 | 272.03 | |

${\nu}_{t}=0$ | 0.0 | 0.00 | 0.05 | 3.0 | 88.36 | |

${\nu}_{t}=0.5$ | 0.6 | 0.00 | 0.20 | 3.0 | 266.09 |

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**MDPI and ACS Style**

Ben-Yelun, I.; Gómez-Carano, G.; San Millán, F.J.; Sanz, M.Á.; Montáns, F.J.; Saucedo-Mora, L.
GAM: General Auxetic Metamaterial with Tunable 3D Auxetic Behavior Using the Same Unit Cell Boundary Connectivity. *Materials* **2023**, *16*, 3473.
https://doi.org/10.3390/ma16093473

**AMA Style**

Ben-Yelun I, Gómez-Carano G, San Millán FJ, Sanz MÁ, Montáns FJ, Saucedo-Mora L.
GAM: General Auxetic Metamaterial with Tunable 3D Auxetic Behavior Using the Same Unit Cell Boundary Connectivity. *Materials*. 2023; 16(9):3473.
https://doi.org/10.3390/ma16093473

**Chicago/Turabian Style**

Ben-Yelun, Ismael, Guillermo Gómez-Carano, Francisco J. San Millán, Miguel Ángel Sanz, Francisco Javier Montáns, and Luis Saucedo-Mora.
2023. "GAM: General Auxetic Metamaterial with Tunable 3D Auxetic Behavior Using the Same Unit Cell Boundary Connectivity" *Materials* 16, no. 9: 3473.
https://doi.org/10.3390/ma16093473