# Large-Strain Hyperelastic Constitutive Model of Envelope Material under Biaxial Tension with Different Stress Ratios

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material

^{2}was used. The stratospheric airship envelope material was composed of five functional layers. Functional layers were composed of a wearable layer, ultraviolet layer, gas retention layer, sealing layer, and a woven fabrics layer.

#### 2.2. Biaxial Tensile Tests

#### 2.3. Dimension of Cruciform Specimen

^{2}× 160 mm

^{2}and the shape of the cross corner was rounded with a radius of 25 mm. The effective length of the arm was 160 mm. The double welded zone was the clamping area and was 110 mm in length. The end of the clamping region had a ring with a radius of 20 mm.

#### 2.4. Result of Biaxial Tensile Test

## 3. A New Constitutive Model for Envelope Materials under Large Strains

- Bonding between the substrate fabric and any other functional layers was assumed to be perfect.
- The envelope material was made up of matrix and fabric components. In addition to the fabric layer, other functional layers are regarded as the matrix.
- The shear stress mainly contributed to the matrix stress. In this work, the modulus of shear was kept constant.
- The failure mode of the envelope material during biaxial tension was a brittle fracture. The failure of the matrix and fiber stress occurred at the same time.

#### 3.1. Constitutive Model Theory with Equal Stress Ratios

^{T}F, and fiber directional vector a

_{0}. Here, F is the deformation gradient tensor. The large strain response of the envelope was assumed to originate from the resistance of the matrix, fibers, fiber interactions and fiber–matrix interaction. Therefore, the strain energy can be divided into four parts, i.e.:

^{M}is the strain energy contribution from matrix resistance, W

^{F}is the strain energy contribution from the fiber stretch, W

^{FF}is the strain energy contribution from the fiber–fiber interaction due to the weaving, and W

^{FM}is the strain contribution from the fiber-matrix shear interaction.

_{1}, I

_{2}, I

_{3}are invariants of the strain tensor and I

_{4a}is the squares of the stretches along the fiber directions [19]. I

_{5a}is the fourth power of the stretching along the fiber directions.

^{M}:

^{FM}equal zero when biaxial tensile tests in warp and weft directions for envelope material.

_{0i}is defined as follows:

_{i}is defined as follows:

#### 3.2. Constitutive Model Theory with Different Stress Ratios

_{1}:σ

_{2}= C (C ≥ 1). In this section, the warp loading speed is constant. The weft fiber–fiber strain energy is assumed to be as follows:

#### 3.3. Identification of the New Constitutive Model Parameters

- (1)
- k
_{11}(warp fiber initial stiffness) and k_{12}(weft fiber initial stiffness) represent the initial fiber stiffness. The two parameters can be obtained by the initial slope of the biaxial tensile tests. - (2)
- c
_{1}(matrix shear modulus) and k (fiber-fiber interaction parameter in equal stress ratio) can be solved by the small strain in the biaxial tensile tests. The two parameters can be obtained by the following equation [21]:$$\begin{array}{l}{\frac{\partial {\mathit{\sigma}}_{11}}{\partial {\mathit{\lambda}}_{1}}|}_{\begin{array}{l}{\mathit{\lambda}}_{1}=1\\ {\mathit{\lambda}}_{2}=1\end{array}}=4\mathrm{c}+4{\mathrm{k}}_{11}\\ {\frac{\partial {\mathit{\sigma}}_{22}}{\partial {\mathit{\lambda}}_{1}}|}_{\begin{array}{l}{\lambda}_{1}=1\\ {\lambda}_{2}=1\end{array}}=2\mathrm{c}+4\mathrm{k}\\ {\frac{\partial {\mathit{\sigma}}_{11}}{\partial {\mathit{\lambda}}_{2}}|}_{\begin{array}{l}{\mathit{\lambda}}_{1}=1\\ {\mathit{\lambda}}_{2}=1\end{array}}=2\mathrm{c}+4\mathrm{k}\\ {\frac{\partial {\mathit{\sigma}}_{22}}{\partial {\mathit{\lambda}}_{2}}|}_{\begin{array}{l}{\mathit{\lambda}}_{1}=1\\ {\mathit{\lambda}}_{2}=1\end{array}}=4\mathrm{c}+4{\mathrm{k}}_{12}\end{array}$$ - (3)
- Parameters k
_{21}(warp fiber large strain control parameter) and k_{22}(weft fiber large strain control parameter) are related to the slope of stress–strain curves under large strains. The two parameters can be obtained by biaxial tensile tests with large strain slopes.

#### 3.4. Validation of New Constitutive Model

## 4. Results and Discussion

^{2}between the experimental data and the new constitutive model was 0.98. Therefore, the new constitutive model is an acceptable model in predicting envelope material stress–strain behavior.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Stress–strain curves of the biaxial tensile tests under equal loading ratio in warp and weft directions.

**Figure 6.**Stress–strain curves of the biaxial tensile tests under different loading ratios in warp and weft directions.

**Figure 7.**Comparison between the experimental [23] and predicted models for coated fabrics.

**Figure 9.**Relationship between the each constituent of stress and experimental stress with displacement under a stress ratio of 1:1 (warp:weft).

**Figure 11.**Relationship between each constituent of stress and experimental stress with strain under a stress ratio of 1.5:1 (warp:weft).

**Figure 12.**Relationship between each constituent of stress and experimental stress with strain under a stress ratio of 2:1 (warp:weft).

Loading Ratios | Warp (N·mm^{−1}/min) | Weft (N·mm^{−1}/min) | Number of Test Specimens |
---|---|---|---|

1:1 | 40 | 40 | 3 |

1.5:1 | 40 | 27 | 3 |

2:1 | 40 | 20 | 3 |

c_{1} | k_{11} | k_{21} | k_{12} | k_{22} | k | k_{c} (c = 0.5) |
---|---|---|---|---|---|---|

(MPa) | (Mpa) | (1) | (Mpa) | (1) | (Mpa) | (1) |

11 | 32 | 10 | 28 | −20 | 36 | 7 |

_{c}= fiber-fiber interaction parameter in different stress ratios.

Loading Ratio | c_{1} | k_{11} | k_{21} | k_{12} | k_{22} | k | R^{2} |
---|---|---|---|---|---|---|---|

(MPa) | (MPa) | (1) | (MPa) | (1) | (MPa) | (1) | |

1:1 | 146.48 ± 18.62 | 25.57 ± 4.86 | 17.14 ± 1.23 | 40.53 ± 12.52 | 24.78 ± 2.32 | 76.00 ± 9.71 | 0.98 |

Loading Ratios | k_{c} (MPa) | |
---|---|---|

Tests | Predicted | |

1.5:1 | 47.77 ± 4.86 | 50.72 ± 6.94 |

2:1 | 37.86 ± 2.98 | 38.05 ± 4.03 |

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**MDPI and ACS Style**

Qu, Z.; He, W.; Lv, M.; Xiao, H.
Large-Strain Hyperelastic Constitutive Model of Envelope Material under Biaxial Tension with Different Stress Ratios. *Materials* **2018**, *11*, 1780.
https://doi.org/10.3390/ma11091780

**AMA Style**

Qu Z, He W, Lv M, Xiao H.
Large-Strain Hyperelastic Constitutive Model of Envelope Material under Biaxial Tension with Different Stress Ratios. *Materials*. 2018; 11(9):1780.
https://doi.org/10.3390/ma11091780

**Chicago/Turabian Style**

Qu, Zhipeng, Wei He, Mingyun Lv, and Houdi Xiao.
2018. "Large-Strain Hyperelastic Constitutive Model of Envelope Material under Biaxial Tension with Different Stress Ratios" *Materials* 11, no. 9: 1780.
https://doi.org/10.3390/ma11091780