# Multiaxial Deformations of Elastomeric Skins for Morphing Wing Applications: Theoretical Modeling and Experimental Investigations

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## Abstract

**:**

## 1. Introduction

## 2. Experimental

#### 2.1. Experimental Set Up

#### 2.2. Specimen Geometry, Elastomer Synthesis, and Experimental Conditions

#### 2.3. Various Mechanical Tests under Multiaxial Modes of Deformation

#### 2.4. Strain Measurement under Various Modes of Deformations

## 3. Material Modeling

#### 3.1. Kinematics of Hyperelastic Deformation

#### 3.2. Governing Equations of Hyperelastic Deformation

#### 3.2.1. Uniaxial Mode of Deformation

#### 3.2.2. Pure Shear Mode of Deformation

#### 3.2.3. Biaxial Mode of Deformation

#### 3.2.4. Equibiaxial Mode of Deformation

#### 3.3. A Newly Proposed Material Model

## 4. Results and Discussions

#### 4.1. Identification of Material Parameters

#### 4.2. Mechanical Comparison of Latex, Oppo, and Ecoflex Elastomers

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Representation of (

**a**) Biaxial testing device and (

**b**) enlarged view of the specimen holder for conducting uniaxial, pure shear, biaxial and equibiaxial test at a fixed strain rate [40].

**Figure 2.**Dimension of specimens used for biaxial testing of three different elastomers (

**a**) Latex (

**b**) Oppo and (

**c**) Ecoflex [40].

**Figure 3.**Representation of multi axial deformation modes at the (

**i**) initial and (

**ii**) final position of the specimen. The various deformation modes are shown in (

**a**) UX (

**b**) PS (

**c**) BX and (

**d**) EB.

**Figure 4.**Strain maps developed under all possible modes of deformation are represented for (i) UX (ii) PS (iii) BX (iv) EB in the biaxial testing device. The strain maps for X direction are shown in (

**a**,

**c**,

**e**,

**g**) and corresponding strain maps for Y direction are shown in (

**b**,

**d**,

**f**,

**h**), respectively for each deformation modes.

**Figure 5.**Experimental validation of the proposed model with Latex test data compared to the existing material models under all possible deformation modes, (

**a**) uniaxial, (

**b**) pure shear, (

**c**) biaxial, and (

**d**) equibiaxial for a single set of material constants.

**Figure 6.**Experimental validation of the proposed model with Oppo test data compared to the existing material models under all possible deformation modes, (

**a**) uniaxial, (

**b**) pure shear, (

**c**) biaxial, and (

**d**) equibiaxial for a single set of material constants.

**Figure 7.**Experimental validation of the proposed model with Ecoflex test data compared to the existing material models under all possible deformation modes, (

**a**) uniaxial, (

**b**) pure shear, (

**c**) biaxial, and (

**d**) equibiaxial for a single set of material constants.

Constitutive Model | Material Constants |
---|---|

Proposed model | $\mu =0.30$ MPa, $a=0.08$, $b=0.28$, $c=0.15$ |

Mooney Rivlin model | ${C}_{1}=0.14$ MPa, ${C}_{2}=0.11$ MPa |

Gent model | $\mu =0.39$ MPa, ${J}_{m}=6$ |

Constitutive Model | Material Constants |
---|---|

Proposed model | $\mu =0.39$ MPa, $a=0.1$, $b=0.0018$, $c=0.001$ |

Mooney Rivlin model | ${C}_{1}=0.21$ MPa, ${C}_{2}=0.19$ MPa |

Gent model | $\mu =0.6$ MPa, ${J}_{m}=6$ |

Constitutive Model | Material Constants |
---|---|

Proposed model | $\mu =0.012$ MPa, $a=0.1$, $b=0.0018$, $c=0.001$ |

Mooney Rivlin model | ${C}_{1}=0.009$ MPa, ${C}_{2}=0.007$ MPa |

Gent model | $\mu =0.023$ MPa, ${J}_{m}=6$ |

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**MDPI and ACS Style**

Ahmad, D.; Kumar, D.; Ajaj, R.M.
Multiaxial Deformations of Elastomeric Skins for Morphing Wing Applications: Theoretical Modeling and Experimental Investigations. *Polymers* **2022**, *14*, 4891.
https://doi.org/10.3390/polym14224891

**AMA Style**

Ahmad D, Kumar D, Ajaj RM.
Multiaxial Deformations of Elastomeric Skins for Morphing Wing Applications: Theoretical Modeling and Experimental Investigations. *Polymers*. 2022; 14(22):4891.
https://doi.org/10.3390/polym14224891

**Chicago/Turabian Style**

Ahmad, Dilshad, Deepak Kumar, and Rafic M. Ajaj.
2022. "Multiaxial Deformations of Elastomeric Skins for Morphing Wing Applications: Theoretical Modeling and Experimental Investigations" *Polymers* 14, no. 22: 4891.
https://doi.org/10.3390/polym14224891