# Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations

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## Abstract

**:**

## 1. Introduction

## 2. Pantographic Sheets with Initially Cycloidal Fibers

## 3. Numerical Examples of Equilibrium Shapes

#### 3.1. Bias Extension Test

#### 3.2. Bias Compression Test

#### 3.3. Shear Test

#### 3.4. Torsion Test

## 4. Comparison among the Pantographic Sheets with Orthogonal Lattices

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Distribution of the strain–energy density for the bias extension test: (

**a**) ${W}^{I}$ contribution; and (

**b**) ${W}^{II}$ contribution.

**Figure 3.**Strain–energy contributions for the bias extension test: extensional energy, ${\mathsf{\Psi}}_{e}$; shear energy, ${\mathsf{\Psi}}_{s}$; bending energy, ${\mathsf{\Psi}}_{b}$; and total energy ${\mathsf{\Psi}}_{tot}$.

**Figure 4.**Distribution of the deformation measures of first gradient for the bias extension test: (

**a**) ${\epsilon}_{L}$; (

**b**) ${\epsilon}_{M}$; and (

**c**) $\gamma $.

**Figure 5.**Bias compression test: (

**a**) buckled shape, colors indicate the out-of-plane displacement; and (

**b**) second gradient energy contribution, ${W}^{II}$.

**Figure 6.**Distribution of the strain–energy density for the shear test: (

**a**) ${W}^{I}$ contribution; and (

**b**) ${W}^{II}$ contribution.

**Figure 7.**Strain–energy contributions for the shear test: extensional energy, ${\mathsf{\Psi}}_{e}$; shear energy, ${\mathsf{\Psi}}_{s}$; bending energy, ${\mathsf{\Psi}}_{b}$; and total energy ${\mathsf{\Psi}}_{tot}$.

**Figure 8.**Distribution of the deformation measures of first gradient for the shear test: (

**a**) ${\epsilon}_{L}$; (

**b**) ${\epsilon}_{M}$; and (

**c**) $\gamma $.

**Figure 9.**Torsion test: (

**a**) equilibrium shape without buckling, where colors indicate the out-of-plane displacement; (

**b**) buckled shape, where colors indicate the out-of-plane displacement; and (

**c**) buckled shape, where colors indicate the second gradient energy ${W}^{II}$.

**Figure 11.**Strain–energy contributions for the torsion test: extensional energy, ${\mathsf{\Psi}}_{e}$; shear energy, ${\mathsf{\Psi}}_{s}$; flexural/twisting energy, ${\mathsf{\Psi}}_{b}$; and total energy ${\mathsf{\Psi}}_{tot}$.

**Figure 13.**Bias extension test: (

**a**) equilibrium shape without buckling (imposed displacement ${\tilde{u}}_{1}=1.424$); (

**b**) in-plane buckled shape (${\tilde{u}}_{1}=1.424$); and (

**c**) out-of-plane buckled shape (${\tilde{u}}_{1}=1.017$).

**Figure 14.**Scheme of an oscillatory orthogonal network of fibers. Solid blue lines are the graphs of Equation (32), while solid red lines refer to Equation (33).

**Figure 15.**Strain–energy contributions for the bias extension test for sample with oscillatory fibers: extensional energy, ${\mathsf{\Psi}}_{e}$; shear energy, ${\mathsf{\Psi}}_{s}$; bending energy, ${\mathsf{\Psi}}_{b}$; and total energy ${\mathsf{\Psi}}_{tot}$.

**Figure 16.**Distribution of the strain–energy density for the bias extension test for sample with oscillatory fibers: (

**a**) stretching contribution; (

**b**) shear contribution; and (

**c**) ${W}^{II}$ contribution.

**Figure 17.**Longitudinal reaction ${R}_{{X}_{1}}$ for the bias extension test: comparison among different cases of fibers.

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**MDPI and ACS Style**

Scerrato, D.; Giorgio, I.
Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations. *Symmetry* **2019**, *11*, 1523.
https://doi.org/10.3390/sym11121523

**AMA Style**

Scerrato D, Giorgio I.
Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations. *Symmetry*. 2019; 11(12):1523.
https://doi.org/10.3390/sym11121523

**Chicago/Turabian Style**

Scerrato, Daria, and Ivan Giorgio.
2019. "Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations" *Symmetry* 11, no. 12: 1523.
https://doi.org/10.3390/sym11121523