# Planar Mechanical Metamaterials with Embedded Permanent Magnets

^{1}

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**Mat**S @ FIT—Freiburg Center for Interactive Materials and Bioinspired Technologies, University of Freiburg, 79110 Freiburg im Breisgau, Germany

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

#### 3.1. Validation of the Dipole–Dipole Interaction

#### 3.2. Numerical Results for Periodic Metamaterials

#### 3.3. Experimental Results

## 4. Conclusions

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Design of studied mechanical metamaterial. (

**a**) elementary cell in the open state; (

**b**) elementary cell in the closed state; (

**c**) modified (repulsive) elementary cell with co-directional magnets; (

**d**) modified (attractive) elementary cell with counter-directional magnets; and (

**e**) unit cell with embedded modified cell (shown by blue color) under biaxial compression.

**Figure 2.**Dependencies of the force between two identical magnets with out-of-plane magnetic moments. Solid lines correspond to experimental measurements, solid symbols represent calculations according to Equation (2) for several values of the magnetic moment.

**Figure 3.**Numerically obtained force–strain curves for the metamaterial with unit cell (Figure 1e) modified by the embedment of the magnets in repulsive and attractive configurations. Solid lines correspond to the results for the magnets with $m=0.105{\text{}\mathrm{Am}}^{2}$, dashed lines for the magnets with $m=0.075{\text{}\mathrm{Am}}^{2}$. Blue circles on the horizontal axis denote the strain ${\epsilon}_{st}$ corresponding to the snap-through phenomena. The numbers near the snapshots represent applied strain $\epsilon $.

**Figure 4.**Numerically obtained dependencies of the energy of the unit cell on the applied strain $\epsilon $. (

**a**) Total energy vs. $\epsilon $ for various configurations of magnets. Solid lines correspond to the magnets with $m=0.105{\text{}\mathrm{Am}}^{2}$, dashed lines to the magnets with $m=0.075{\text{}\mathrm{Am}}^{2}$. (

**b**) Magnetic energy for repulsive configuration (red), attractive configuration (blue), and elastic energy for neutral configuration (dashed). Note that all energy curves are shifted vertically to have a (0, 0) origin.

**Figure 5.**Arrangements of the magnets in 16 × 16 metamaterial used for experiments and simulations. (

**a**) Blue squares mark the positions of the magnets for the arrangement I, white circles mark the positions for the arrangement II. (

**b**) Example of 3D printed metamaterial with embedded magnets (arrangement II). Four magnets in the corners are used for the mounting in the fixtures and do not affect the behavior of the metamaterial during tests.

**Figure 6.**(

**a**) Force–displacement curves obtained experimentally (solid lines) and numerically (dashed lines) for finite 16 × 16 metamaterials with arrangements I and II. Solid lines show the means for three consecutive loadings of the metamaterial, while gray areas correspond to the standard deviation. (

**b**,

**c**) Two stable states of the metamaterial (arrangement I), observed experimentally. The snap-through occurs at the displacement marked by the blue circle (Figure 6a).

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**MDPI and ACS Style**

Slesarenko, V. Planar Mechanical Metamaterials with Embedded Permanent Magnets. *Materials* **2020**, *13*, 1313.
https://doi.org/10.3390/ma13061313

**AMA Style**

Slesarenko V. Planar Mechanical Metamaterials with Embedded Permanent Magnets. *Materials*. 2020; 13(6):1313.
https://doi.org/10.3390/ma13061313

**Chicago/Turabian Style**

Slesarenko, Viacheslav. 2020. "Planar Mechanical Metamaterials with Embedded Permanent Magnets" *Materials* 13, no. 6: 1313.
https://doi.org/10.3390/ma13061313