# Symmetric-in-Plane Compression of Polyamide Pantographic Fabrics—Modelling, Experiments and Numerical Exploration

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## Abstract

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## 1. Introduction

#### 1.1. Generalised Continua and Their Use to Model Pantographic Structures

#### 1.2. Model-Driven Design and Synthesis of Pantographic Structures

#### 1.3. Out-of-Plane and In-Plane Compression of Pantographic Fabrics

#### 1.4. Plan of the Present Work

## 2. Mathematical and Numerical Modelling of Pantographic Structures

#### 2.1. Geometrically Non Linear Second Gradient Elasticity Model

#### 2.1.1. Kinematics

#### 2.1.2. Potential Energy

#### External Work

#### Strain Energy—Linear Elasticity

#### Strain Energy—Higher Deformations

#### 2.1.3. Variational Approach

- prescribed null displacement at the lower edge: ${\underline{u}}_{|\mathrm{lower}}=\underline{0}$
- no applied load on either side: ${\partial}_{n}{\underline{u}}_{|\mathrm{sides}}=\underline{0}$
- prescribed displacement ${\underline{u}}_{0}$ on the upper edge: ${\underline{u}}_{|\mathrm{upper}}={\underline{u}}_{0}$

#### 2.2. Finite Elements Numerical Model

#### Function Space

#### Modelling with Software

## 3. Experimental Specimen

#### 3.1. Studied Class of Pantographic Structure

#### 3.2. Modelled Specimen: Production and Characterisation

- Width of beam: 0.9 mm
- Height of beam: 4 mm
- Width of pivot: 0.9 mm
- Height of pivot: 1 mm

#### 3.3. Experimental Procedure

#### 3.4. Outputs of the Experiments

#### Measurement Uncertainty

## 4. Model Fitting and Numerical Validation

#### 4.1. Numerical Parameters

#### 4.1.1. Fixed Parameters

#### 4.1.2. Parameters to Be Fitted

- begin with the linear case values: ${c}_{num}={c}_{lin}$; ${a}_{num}={a}_{lin}$ and ${c}_{33}=0$
- fit ${c}_{33}$ to experimental results (force-displacement graph)

#### 4.2. First Experiment—Single Swelling

#### 4.2.1. Without Torsional Energy

#### Parameter Values

#### Force-Displacement Graph Comparison

#### 4.2.2. Fitting the Torsional Parameter

#### 4.3. Second Experiment—Two Swellings

#### Parameter Values

#### Qualitative Comparison

## 5. Conclusion and Possible Future Work

#### 5.1. Work Done

#### 5.2. Results

#### 5.3. Possible Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Numerical-experimental comparison—Specimen 1, not fitted (${c}_{num}={c}_{lin}$, ${a}_{num}={a}_{lin}$, ${c}_{33}=0$ N/m). The grey area represents the error-type.

**Figure 5.**Numerical-experimental comparison—Specimen 1, fitted (${c}_{num}={c}_{lin}$, ${a}_{num}={a}_{lin}$, ${c}_{33}=140$ N/m). The grey area represents the error-type.

**Figure 11.**Fibers, ${u}_{0}=5$ cm—arbitrary parameter values: ${a}_{num}=0.00123$ J, ${c}_{num}=12310$ N/m, ${c}_{33}=140$ N/m.

Young Modulus | Size of Cross-Section | Cross-Section Area | Second Moment of Area |
---|---|---|---|

${\mathit{E}}_{\mathit{m}}$ (Pa) | ${\mathit{l}}_{\mathit{m}}$ (m) | ${\mathit{A}}_{\mathit{m}}$ (m^{2}) | ${\mathit{I}}_{\mathit{m}}$ (m^{4}) |

$2.5\times {10}^{7}$ | $1.5\times {10}^{-3}$ | $2.3\times {10}^{-6}$ | $4.2\times {10}^{-13}$ |

Specimen | 1 | 2 |
---|---|---|

${d}_{m}$ (m) | $7.1\times {10}^{-3}$ | $3.5\times {10}^{-3}$ |

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**MDPI and ACS Style**

Tran, C.A.; Gołaszewski, M.; Barchiesi, E.
Symmetric-in-Plane Compression of Polyamide Pantographic Fabrics—Modelling, Experiments and Numerical Exploration. *Symmetry* **2020**, *12*, 693.
https://doi.org/10.3390/sym12050693

**AMA Style**

Tran CA, Gołaszewski M, Barchiesi E.
Symmetric-in-Plane Compression of Polyamide Pantographic Fabrics—Modelling, Experiments and Numerical Exploration. *Symmetry*. 2020; 12(5):693.
https://doi.org/10.3390/sym12050693

**Chicago/Turabian Style**

Tran, Chuong Anthony, Maciej Gołaszewski, and Emilio Barchiesi.
2020. "Symmetric-in-Plane Compression of Polyamide Pantographic Fabrics—Modelling, Experiments and Numerical Exploration" *Symmetry* 12, no. 5: 693.
https://doi.org/10.3390/sym12050693