# Identification of the LLDPE Constitutive Material Model for Energy Absorption in Impact Applications

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Table 1.**LLDPE properties [22].

Physical Properties | Unit | Tolerance ± | Value | Testing Method |
---|---|---|---|---|

Thickness | µm | 2 | 12 | Thickness gauge |

Width | mm | 5 | 500 | Measuring tape |

Length | - | 5 | High-speed encoder | |

Density | $\mathrm{g}/{\mathrm{cm}}^{3}$ | - | 0.91–0.92 | ASTM D-1505 [23] |

Mechanical Properties | Unit | Tolerance± | Value | Testing Method |

Tensile strength MD | MPa | 10 | 29.2 | ASTM D-882 [23] |

Tensile strength TD | 14.1 | |||

Break elongation MD | % | 245 | ||

Break elongation TD | 540 | |||

Dart drop | g | 40 | ASTM D-1709 [23] | |

Puncture | kg | 1.7 | High-light tester | |

Stretching level | - | - | 110 |

#### 2.1. Quasi-Static Loading

#### 2.2. Dynamic Loading

#### 2.3. Identification of Dynamic Material Parameters

- Update both the MD and TD curves according to Equations (12) and (13):
- (a)
- $\forall \u03f5\in [0,{\u03f5}_{{y}_{1}}]$ update the stiffness by changing the slopes of the curves using $\u03f5:=\frac{1}{{k}_{1}{k}_{e}}\u03f5$;
- (b)
- $\forall \u03f5\in ({\u03f5}_{{y}_{1}},{\u03f5}_{{y}_{2}}]$ update the stiffness by changing the slopes of the curves using $\u03f5:=\frac{1}{{k}_{e}}(\u03f5-{\u03f5}_{{y}_{1}})+{\u03f5}_{{y}_{1}}$;
- (c)
- $\forall \u03f5\in [0,{\u03f5}_{{y}_{2}}]$ update the resultant stress as ${\sigma}_{h}:={k}_{y}{\sigma}_{h}$;
- (d)
- $\forall \u03f5>{\u03f5}_{{y}_{2}}$ connect the parts of the curves in Regions II and III to the second yield point using $\u03f5:=\u03f5+\Delta {\u03f5}_{{y}_{2}}$ and ${\sigma}_{:}=h{\sigma}_{h}+\Delta {\sigma}_{h{y}_{2}}$ where $[\Delta {\u03f5}_{{y}_{2}},\Delta {\sigma}_{{hy}_{2}}]$ is the shift of the second yield point;

- Run the VPS simulation to get ${a}_{s}\left({t}_{i}\right)$ for $i\in \{1,2,\cdots ,m\}$;
- Evaluate the cost function f in Equation (16);
- Repeat the loop from 1 until the cost function f reaches its minimum;
- Return both the MD and TD curves according to Equations (12) and (13) for the optimized coefficients, ${k}_{1}$, ${k}_{e}$, and ${k}_{y}$.

## 3. Results

#### 3.1. Quasi-Static Loading

#### 3.2. Dynamic Loading

#### 3.3. Identification of the Dynamic Material Parameters

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LLDPE | Linear low-density polyethylene |

MD | Machine direction |

TD | Transverse direction |

D3 | Skewed direction by 45${}^{\xb0}$ |

D4 | Skewed direction by −45${}^{\xb0}$ |

VPS | Virtual Performance Solution |

## Appendix A. Iteration Process Flowchart

## Appendix B. List of Equations

## References

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**Figure 2.**Chain structures of HDPE, LLDPE, and LDPE [4].

**Figure 3.**Typical stress-strain curve of LLDPE with different types of loading [21].

**Figure 12.**Material response in all directions: (

**a**) Force versus displacement in MD. (

**b**) Force versus displacement in TD. (

**c**) Force versus displacement in D3. (

**d**) Force versus displacement in D4.

**Figure 13.**Averaged constitutive material model curves: (

**a**) Force versus displacement averaged per direction in all stretching velocities. (

**b**) Stress versus strain averaged per direction.

**Figure 14.**Performance of a single element model: (

**a**) Force versus displacement of a single element in the MD. (

**b**) Force versus displacement of a single element in TD.

**Figure 16.**Optimization iterations for the drop height $H=10$ dm and $n=8$ layers: (

**a**) Impactor displacement. (

**b**) Impactor acceleration.

**Figure 17.**Optimization iterations for the drop height $H=10$ dm and $n=9$ layers: (

**a**) Impactor displacement. (

**b**) Impactor acceleration.

**Figure 18.**Optimization iterations for the drop height $H=10$ dm and $n=10$ layers: (

**a**) Impactor displacement. (

**b**) Impactor acceleration.

**Figure 19.**Optimization iterations for the drop height $H=15$ dm and $n=9$ layers: (

**a**) Impactor displacement. (

**b**) Impactor acceleration.

**Figure 20.**Optimization iterations for the drop height $H=15$ dm and $n=10$ layers: (

**a**) Impactor displacement. (

**b**) Impactor acceleration.

**Figure 21.**Strain rate-dependent constitutive material model curves: (

**a**) Particular drop test response. (

**b**) Particular drop test response per layer.

**Figure 22.**Averaged strain rate-dependent constitutive material model curves: (

**a**) Resultant stress versus strain in MD/TD. (

**b**) Resultant shear stress versus shear strain.

**Figure 23.**Comparison of the drop test simulation with the experiment for the drop height $H=10$ dm and $n=8$ layers: (

**a**) Impactor displacement (

**b**) Impactor acceleration. (

**c**) Impactor energy loss. (

**d**) Impactor total energy.

**Figure 24.**Comparison of the drop test simulation with the experiment for the drop height $H=10$ dm and $n=9$ layers: (

**a**) Impactor displacement (

**b**) Impactor acceleration. (

**c**) Impactor energy loss. (

**d**) Impactor total energy.

**Figure 25.**Comparison of the drop test simulation with the experiment for the drop height $H=10$ dm and $n=10$ layers: (

**a**) Impactor displacement (

**b**) Impactor acceleration. (

**c**) Impactor energy loss. (

**d**) Impactor total energy.

**Figure 26.**Comparison of the drop test simulation with the experiment for the drop height $H=15$ dm and $n=9$ layers: (

**a**) Impactor displacement (

**b**) Impactor acceleration. (

**c**) Impactor energy loss. (

**d**) Impactor total energy.

**Figure 27.**Comparison of the drop test simulation with the experiment for the drop height $H=15$ dm and $n=10$ layers: (

**a**) Impactor displacement (

**b**) Impactor acceleration. (

**c**) Impactor energy loss. (

**d**) Impactor total energy.

Stretching Velocity v (m/s) | Direction | Number of Samples N |
---|---|---|

$0.0002$ | MD | 6 |

TD | 6 | |

D3 | 6 | |

D4 | 6 | |

$0.02$ | MD | 6 |

TD | 6 | |

D3 | 6 | |

D4 | 6 | |

$0.2$ | MD | 6 |

TD | 6 | |

D3 | 6 |

Drop Height H (dm) | Number of Layers n | Optimization | Designation |
---|---|---|---|

10 | 8 | √ | 1008 |

9 | √ | 1009 | |

10 | √ | 1010 | |

15 | 8 | × (material ruptured) | |

9 | √ | 1509 | |

10 | √ | 1510 |

Direction | MD | TD | ||
---|---|---|---|---|

Variable | Tensile Stress (MPa) | Break Elongation (%) | Tensile Stress (MPa) | Break Elongation (%) |

Data sheet [22] | 29.2 | 245 | 14.1 | 540 |

Experiment | 29.3 | 139 | 16.2 | 701 |

Error (%) | 0.5 | −43 | 15 | 30 |

Yield Point | MD | TD | ||||
---|---|---|---|---|---|---|

${\mathit{\u03f5}}_{\mathit{y}}$ (-) | ${\mathit{\sigma}}_{\mathit{y}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{hy}}$ (N/mm) | ${\mathit{\u03f5}}_{\mathit{y}}$ (-) | ${\mathit{\sigma}}_{\mathit{y}}$ (MPa) | ${\mathit{\sigma}}_{\mathit{hy}}$ (N/mm) | |

1 | 0.26 | 8.4 | 0.1 | 0.33 | 8 | 0.1 |

2 | 0.84 | 20 | 0.24 | 0.69 | 10 | 0.12 |

Direction | MD | TD | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Stretching Velocity v (m/s) | 0.0002 | 0.02 | 0.2 | 0.0002 | 0.02 | 0.2 | ||||||

Young | 44 | 63 | 63 | 67 | 36 | 42 | 54 | 76 | 55 | 41 | 35 | 26 |

modulus | 57 | 76 | 63 | 47 | 37 | 50 | 39 | 53 | 64 | 47 | 31 | 30 |

E (MPa) | 63 | 41 | 81 | 70 | 31 | 30 | 55 | 44 | 49 | 58 | 27 | 38 |

Young | 57 | 65 | 38 | 53 | 52 | 31 | ||||||

modulus | 53 | 46 | ||||||||||

E (MPa) |

Drop Height H (dm) | 10 | 15 | |||
---|---|---|---|---|---|

Number of layers n | 8 | 9 | 10 | 9 | 10 |

Impact velocity v (m/s) | 4.16 | 4.14 | 4.18 | 4.94 | 5.01 |

Energy absorption D (%) | 90.03 | 88.26 | 87.61 | 89.13 | 89.43 |

Energy absorption $\overline{D}$ (%) | 88.63 | 89.28 | |||

Drop Height H (dm) | 10 | 15 | |||
---|---|---|---|---|---|

Number of layers n | 8 | 9 | 10 | 9 | 10 |

Number of iterations | 278 | 152 | 205 | 234 | 221 |

First part stiffness multiplier ${k}_{1}$ (-) | 2.75 | 2.89 | 2.99 | 3.14 | 3.07 |

Stiffness multiplier ${k}_{e}$ (-) | 3.41 | 3.47 | 3.55 | 1.69 | 2.29 |

Yield stress multiplier ${k}_{y}$ (-) | 1.00 | 0.91 | 0.88 | 1.16 | 1.04 |

Acceleration error ${E}_{s}$ (%) | 3 | 3 | 2 | 2 | 3 |

Displacement error ${E}_{d}$ (%) | 1 | 1 | 1 | 0 | 1 |

Drop Height H (dm) | 10 | 15 | |||
---|---|---|---|---|---|

Number of layers n | 8 | 9 | 10 | 9 | 10 |

Acceleration error ${E}_{s}$ (%) | 6 | 8 | 10 | 6 | 3 |

Displacement error ${E}_{d}$ (%) | 5 | 2 | 2 | 3 | 2 |

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**MDPI and ACS Style**

Hynčík, L.; Kochová, P.; Špička, J.; Bońkowski, T.; Cimrman, R.; Kaňáková, S.; Kottner, R.; Pašek, M.
Identification of the LLDPE Constitutive Material Model for Energy Absorption in Impact Applications. *Polymers* **2021**, *13*, 1537.
https://doi.org/10.3390/polym13101537

**AMA Style**

Hynčík L, Kochová P, Špička J, Bońkowski T, Cimrman R, Kaňáková S, Kottner R, Pašek M.
Identification of the LLDPE Constitutive Material Model for Energy Absorption in Impact Applications. *Polymers*. 2021; 13(10):1537.
https://doi.org/10.3390/polym13101537

**Chicago/Turabian Style**

Hynčík, Luděk, Petra Kochová, Jan Špička, Tomasz Bońkowski, Robert Cimrman, Sandra Kaňáková, Radek Kottner, and Miloslav Pašek.
2021. "Identification of the LLDPE Constitutive Material Model for Energy Absorption in Impact Applications" *Polymers* 13, no. 10: 1537.
https://doi.org/10.3390/polym13101537