# A New Material Model for Agglomerated Cork

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Successive Linear Approximation Method

#### 2.1. Relative Motion Description

#### 2.2. Linearized Constitutive Equation

## 3. Hyperelastic Material Model

**An extended Mooney–Rivlin material.**The constitutive equation

**Remark on $\beta $ value for cork:**Since, in this paper, we are considering the cork agglomerated as a material, from the definition of $\beta $ and the compressibility of cork, it follows that $\beta \approx 0$.

**Remarks on linear model:**Let $\mathbf{B}=\mathbf{I}+2\mathbf{E}$, hence ${\mathbf{B}}^{-1}=\mathbf{I}-2\mathbf{E}$ and $\mathrm{tr}\mathbf{H}=\mathrm{tr}\mathbf{E}$, for small linear strain $\mathbf{E}$, then

## 4. Parameter Fitting for Cork Agglomerates

#### 4.1. Uniaxial Compression

#### 4.2. Equibiaxial Compression

#### 4.3. Experimental Tests

#### 4.4. Curve Fitting

## 5. Linearized Partial Differential Equations

## 6. Variational Formulation

**A**and

**B**are second-order tensors, $\mathit{A}.\mathit{B}=\mathrm{tr}\left(\mathit{A}{\mathit{B}}^{T}\right)$), we have, $\forall v\in V$,

## 7. Numerical Validation

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Agglomerated cork sample positioned for uniaxial compression testing at quasi-static strain rates.

**Figure 4.**Dimensions of the octagon-shaped sample and the experimental setup for the equibiaxial compression tests.

**Figure 5.**Comparison between analytical and experimental results for uniaxial and equibiaxial tests, respectively.

**Figure 6.**Finite element mesh and boundary condition for both tests, uniaxial (

**left**) and equibiaxial (

**right**).

**Figure 7.**Comparison between numerical, analytical and experimental solutions for uniaxial and equibiaxial compression tests, respectively.

Uniaxial Compression | Equibiaxial Compression | ||
---|---|---|---|

Stretch | Stress [MPa] | Stretch | Stress [MPa] |

0.99990 | −0.00267 | 0.99983 | −0.00154 |

0.98716 | −0.08793 | 0.96683 | −0.18431 |

0.97331 | −0.15326 | 0.93350 | −0.31465 |

0.91794 | −0.25836 | 0.90016 | −0.38643 |

0.86256 | −0.32187 | 0.86683 | −0.45608 |

0.80718 | −0.38418 | 0.83350 | −0.52095 |

0.77949 | −0.41763 | 0.81683 | −0.54994 |

0.72411 | −0.48701 | 0.78350 | −0.61324 |

0.69642 | −0.52242 | 0.76683 | −0.65323 |

0.64104 | −0.59524 | 0.73350 | −0.73154 |

0.61335 | −0.63417 | 0.71683 | −0.77041 |

0.55797 | −0.72245 | 0.68350 | −0.86132 |

0.53028 | −0.77436 | 0.66683 | −0.92065 |

0.47490 | −0.90147 | 0.63350 | −1.06085 |

0.44721 | −0.98206 | 0.61683 | −1.14073 |

0.39183 | −1.19875 | 0.58350 | −1.38921 |

0.36414 | −1.34949 | 0.56683 | −1.59243 |

0.33645 | −1.54421 | ||

0.30876 | −1.80366 | ||

0.25339 | −2.65663 |

Parameter | Value |
---|---|

${s}_{1}$ | −25.860100 |

${s}_{2}$ | 2.7356100 |

${s}_{3}$ | 19.317400 |

${s}_{4}$ | 0.0407647 |

${s}_{5}$ | −8.3524100 |

${s}_{6}$ | −2.5714300 |

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**MDPI and ACS Style**

Pereira, G.T.d.A.; Sousa, R.J.A.d.; Liu, I.-S.; Teixeira, M.G.; Fernandes, F.A.O.
A New Material Model for Agglomerated Cork. *Math. Comput. Appl.* **2022**, *27*, 92.
https://doi.org/10.3390/mca27060092

**AMA Style**

Pereira GTdA, Sousa RJAd, Liu I-S, Teixeira MG, Fernandes FAO.
A New Material Model for Agglomerated Cork. *Mathematical and Computational Applications*. 2022; 27(6):92.
https://doi.org/10.3390/mca27060092

**Chicago/Turabian Style**

Pereira, Gabriel Thomaz de Aquino, Ricardo J. Alves de Sousa, I-Shih Liu, Marcello Goulart Teixeira, and Fábio A. O. Fernandes.
2022. "A New Material Model for Agglomerated Cork" *Mathematical and Computational Applications* 27, no. 6: 92.
https://doi.org/10.3390/mca27060092