# Inverse Finite Element Approach to Identify the Post-Necking Hardening Behavior of Polyamide 12 under Uniaxial Tension

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{2}and a total length of 170 mm and were subjected to a tensile load with a constant cross-head velocity of 50 mm/min until fracture. The tests were performed at 23 °C, 37 °C, 50 °C, 80 °C, and 100 °C. This temperature range was adopted for the present work since it covers the temperatures used during the manufacturing steps of PTCA balloons [1] and shafts.

- Necking followed by stable neck propagation until fracture (23 °C ambient, conditioned) (Group I);
- Necking followed by strain hardening (37 °C ambient and conditioned, 50 °C ambient) (Group II);
- Without necking (50 °C conditioned, 80 °C ambient and conditioned, 100 °C) (Group III).

#### 2.1. Stress and Strain Conversion

- ${R}^{2}$ = 0: no fit
- ${R}^{2}$ = 1: perfect fit
- 0 < ${R}^{2}$ < 1: partial correct fit

#### 2.2. Elastoplastic Constitutive Model and Inverse Identification Approach

#### 2.2.1. Hardening Model until Necking

#### 2.2.2. Hardening Model Post Necking

## 3. Results

#### 3.1. Linear Elastic Values

#### 3.2. Curve Fitting until Necking

#### 3.3. Curve Fitting Post-Necking

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Comparison of the force–displacement curves of uniaxial tensile tests at 100 °C for a mesh size of 0.5 mm, 1 mm, and 2 mm.

## Appendix B

**Table A1.**Variables of fit to the Ludwik, Swift, and Ghosh models until necking for 23 °C (Ambient and Conditioned), 37 °C (Ambient and Conditioned), and 50 °C (Ambient) including the resulting ${R}^{2}$.

Temp. (°C) | Variable | ${\mathit{\sigma}}_{\mathit{L}\mathit{u}\mathit{d}\mathit{w}\mathit{i}\mathit{k}}$ | ${\mathit{\sigma}}_{\mathit{S}\mathit{w}\mathit{i}\mathit{f}\mathit{t}}$ | ${\mathit{\sigma}}_{\mathit{G}\mathit{h}\mathit{o}\mathit{s}\mathit{h}}$ | ${\mathit{\sigma}}_{\mathit{V}\mathit{o}\mathit{c}\mathit{e}}$ | ||||
---|---|---|---|---|---|---|---|---|---|

Amb | Cond | Amb | Cond | Amb | Cond | Amb | Cond | ||

$23(\mathrm{valid}\mathrm{up}\mathrm{to}{\sigma}_{VM,amb}=47.5\mathrm{MPa},$${\sigma}_{VM,cond}=43\mathrm{MPa}$) | ${\epsilon}_{0}$ | − | − | 1.155 × 10^{−4} | 2.92 × 10^{−5} | 2.34 × 10^{−14} | 2.34 × 10^{−14} | − | − |

$k$ | 159.2 | 49.21 | − | − | 159.2 | 49.21 | 902.9 | 69.34 | |

$n$ | 0.426 | 0.2328 | 0.1149 | 0.1494 | 0.426 | 0.2328 | − | − | |

${\sigma}_{\infty}$ | − | − | − | − | − | − | 15.46 | 29.3 | |

${R}^{2}$ | 0.9629 | 0.9760 | 0.992 | 0.9923 | 0.9629 | 0.9760 | 0.9881 | 0.8632 | |

$37(\mathrm{valid}\mathrm{up}\mathrm{to}{\sigma}_{VM,amb}=41\mathrm{MPa},$${\sigma}_{VM,cond}=35\mathrm{MPa}$) | ${\sigma}_{e}$ | − | − | 4.23 × 10^{−5} | 1.193 × 10^{−4} | 2.34 × 10^{−14} | 2.33 × 10^{−14} | - | − |

$k$ | 47.82 | 49.94 | − | − | 47.82 | 49.94 | 60.91 | 34.98 | |

$n$ | 0.2467 | 0.3017 | 0.1515 | 0.2156 | 0.2467 | 0.3017 | - | - | |

${\sigma}_{\infty}$ | − | − | − | − | − | − | 27.36 | 27.31 | |

${R}^{2}$ | 0.9852 | 0.9853 | 0.9959 | 0.9947 | 0.9852 | 0.9853 | 0.8319 | 0.9395 | |

$50(\mathrm{valid}\mathrm{up}\mathrm{to}{\sigma}_{VM,amb}=35\mathrm{MPa}$) | ${\sigma}_{e}$ | − | − | 1.059 × 10^{-4} | − | 2.34 × 10^{-14} | − | − | − |

$k$ | 48.86 | − | − | − | 48.85 | − | 34.9 | − | |

$n$ | 0.2987 | − | 0.2163 | − | − | − | − | − | |

${\sigma}_{\infty}$ | − | − | − | − | − | − | 27.04 | − | |

${R}^{2}$ | 0.9844 | − | 0.9941 | − | 0.9844 | − | 0.9396 | − |

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**Figure 1.**Exemplary engineering stress–strain and true stress–strain curve of a semi-crystalline thermoplastic material. The point of necking is defined as the highest engineering stress followed by a stress decline. ${\epsilon}_{e}$ marks the end of the elasticity and the true strain ${\epsilon}_{n,true}$ the beginning of the necking.

**Figure 2.**Optimization routine in LS-OPT 6.0.0 using an FE simulation in LS-DYNA of the dog bone tensile specimen and a user-defined HC ${\sigma}_{h}$ to minimize the difference between the simulated and the experimental force–displacement curve.

**Figure 3.**A 3D FE model of the dog bone specimen (ISO 527-2, Type 1A with a total length of 170 mm and a cross section of 10 × 4 mm

^{2}) using symmetry in the x-direction. The left-hand side of the specimen (blue) is fixed, and the right-hand side (black) is subjected to a displacement with kinematic coupling functionality. The red point, which is 35 mm from the symmetry plane, indicates the point where local displacement is measured. (

**a**) MS = 1 mm (1960 elements); (

**b**) MS = 2 mm (250 elements).

**Figure 4.**Comparison of the model to the ambient and conditioned experimental linear elastic values in a temperature range of 23–100 °C (296–373 K). (

**a**) Temperature-dependent model of Mahieux et al. and experimental values of the experimental Young’s modulus E. (

**b**) Temperature-dependent model of Mahieux et al. and values of the experimental proportionality limit ${\sigma}_{p}$.

**Figure 5.**Comparison of the experimental data to the simulation of Group III for two mesh sizes. The shaded area indicates a ±10% force region around the experimental curve. Green: MS = 1 mm, Blue: MS = 2 mm. (

**a**) Experimental and simulated force–displacement curves at 50 °C cond and 80°C cond; (

**b**) experimental and simulated force–displacement curves at 80 °C ambient and 100 °C; (

**c**) HCs for 50 °C (conditioned), 80 °C (ambient/conditioned), and 100 °C for an MS of 1 and 2 mm.

**Figure 6.**Comparison of the experimental data to the simulation of Group I for two mesh sizes (MS = 1 mm and MS = 2 mm). HC = 1 mm: HC identified with MS = 1 mm, HC = 2 mm: HC identified with MS = 2 mm. The shaded area indicates a ±10% force region around the experimental curve. (

**a**) Experimental and simulated force–displacement curves at 23 °C (ambient) including the resulting ${R}^{2}$. (

**b**) Experimental and simulated force–displacement curves at 23 °C (ambient) including the resulting ${R}^{2}$. (

**c**) HCs for 23 °C (ambient/conditioned) for an MS of 1 and 2 mm. (

**d**) Neck forming and propagation in the FE model.

**Figure 7.**Comparison of the experimental data to the simulation of Group II for two mesh sizes (MS = 1 mm and MS = 2 mm). HC = 1 mm: HC identified with MS = 1 mm, HC = 2 mm: HC identified with MS = 2 mm. The shaded area indicates a ±10% force region around the experimental curve. (

**a**) Experimental and simulated force–displacement curves at 37 °C (ambient) with the successive fitting of the HC including the resulting ${R}^{2}$. (

**b**) Experimental and simulated force–displacement curves at 37 °C (ambient) with the extended Hockett–Sherby model including the resulting ${R}^{2}$. (

**c**) HCs for the extended Hockett–Sherby model and the successive fit for an MS of 1 and 2 mm.

**Figure 8.**Comparison of the experimental data to the simulation of Group II for two mesh sizes (MS = 1 mm and MS = 2 mm). HC = 1 mm: HC identified with MS = 1 mm, HC = 2 mm: HC identified with MS = 2 mm. The shaded area indicates a ±10% force region around the experimental curve. (

**a**) Experimental and simulated force–displacement curves at 37 °C (conditioned) with the extended Hockett–Sherby model including the resulting ${R}^{2}$. (

**b**) Experimental and simulated force–displacement curves at 50 °C (ambient) with the extended Hockett–Sherby model including the resulting ${R}^{2}$. (

**c**) HCs for the extended Hockett–Sherby model for an MS of 1 and 2 mm.

**Table 1.**Linear elastic values. $E$ and $\nu $ were taken from [18]. ${\sigma}_{e}$ represents the first value of the HC.

Temp (°C) | $\mathit{E}$ (MPa) | ${\mathit{\sigma}}_{\mathit{e}}\left(\mathbf{MPa}\right)$ | $\mathit{\nu}$ (-) | $\mathit{E}$ (MPa) | ${\mathit{\sigma}}_{\mathit{e}}\left(\mathbf{MPa}\right)$ | $\mathit{\nu}$ (-) |
---|---|---|---|---|---|---|

Ambient | Conditioned | |||||

23 | 1553.5 | 29.76 | 0.37 | 882.0 | 11.38 | 0.31 |

37 | 1036.3 | 11.86 | 0.44 | 641.5 | 7.67 | 0.46 |

50 | 606.2 | 7.26 | 0.46 | 446.4 | 5.32 | 0.46 |

80 | 315.2 | 4.69 | 0.47 | 297.6 | 4.42 | 0.47 |

100 | 272.1 | 4.04 | 0.47 | - | - | - |

**Table 2.**Variables of the mathematical model to fit the linear elastic values in a temperature range of 23–100 °C or 296–373 K.

Variable | $\mathbf{Fit}\mathbf{of}\mathit{E}$ (MPa) | $\mathbf{Fit}\mathbf{of}{\mathit{\sigma}}_{\mathit{e}}\left(\mathbf{MPa}\right)$ | ||
---|---|---|---|---|

Ambient | Conditioned | Ambient | Conditioned | |

${P}_{g}$ | 1952 | 1103 | 5857 | 18.57 |

${P}_{r}$ | 395 | 310.0 | 4.722 | 4.575 |

${T}_{d}$ | 401.3 | 411 | 408 | 408.8 |

${T}_{g}$ | 312.8 | 312.7 | 200 | 302.1 |

$m$ | 21.17 | 19.13 | 4.328 | 15.96 |

$n$ | 12.8 | 18.16 | 21.2 | 22.84 |

${R}^{2}$ | 0.9994 | 0.9995 | 0.99958 | 1 |

**Table 3.**Fitted variables of the Hockett–Sherby model until necking for 23 °C (Ambient and Conditioned), 37 °C (Ambient and Conditioned), and 50 °C (Ambient) ${R}^{2}$.

Temp. (°C) | Variable | ${\mathit{\sigma}}_{\mathit{H}\mathit{o}\mathit{c}\mathit{k}\mathit{e}\mathit{t}\mathit{t}-\mathit{S}\mathit{h}\mathit{e}\mathit{r}\mathit{b}\mathit{y}}$ | |
---|---|---|---|

Amb | Cond | ||

23 (valid up to ${\sigma}_{VM,amb}=47.5\mathrm{MPa}$, ${\sigma}_{VM,cond}=43\mathrm{MPa}$) | $k$ | 195.6 | 4.149 |

$n$ | 0.8 | 0.4159 | |

${\sigma}_{\infty}$ | 46.46 | 48.8 | |

${R}^{2}$ | 0.996 | 0.9946 | |

37 (valid up to ${\sigma}_{VM,amb}=41\mathrm{MPa}$, ${\sigma}_{VM,cond}=35\mathrm{MPa}$) | $k$ | 2.539 | 4.56 |

$n$ | 0.3681 | 0.519 | |

${\sigma}_{\infty}$ | 53.43 | 41.74 | |

${R}^{2}$ | 0.9925 | 0.9997 | |

50 (valid up to ${\sigma}_{VM,amb}=35\mathrm{MPa}$) | $k$ | 4.646 | - |

$n$ | 0.5207 | - | |

${\sigma}_{\infty}$ | 40.67 | - | |

${R}^{2}$ | 0.9997 | - |

**Table 4.**Variables of the extended Hockett–Sherby model for Group III (50 °C Cond, 80 °C Amb/Cond, and 100 °C). ${R}^{2}$–HC is the goodness of the extended Hockett–Sherby model to the experimental HC. ${R}^{2}$–FD, MS = 1 mm is the goodness of the resulting force–displacement curve using a mesh size of 1 mm; and ${R}^{2}$–FD, MS = 2 mm is the quality of the resulting force–displacement curve using a mesh size of 2 mm.

Variable | 50 °C Cond | 80 °C Amb | 80 °C Cond | 100 °C |
---|---|---|---|---|

${\sigma}_{\infty}$ | 30.98 | 24.52 | 25.08 | 20.66 |

$m$ | 7.389 | 6.309 | 4.429 | 6.088 |

$n$ | 0.5794 | 0.5784 | 0.5298 | 0.563 |

$o$ | 48.01 | 62.16 | 76.34 | 63.99 |

$p$ | 7.512 | 8.015 | 2.803 | 8.499 |

$q$ | 79.43 | 70.78 | 79.21 | 61.12 |

$r$ | 2.296 | 2.379 | 12.04 | 2.457 |

${R}^{2}$–HC | 1 | 1 | 1 | 1 |

${R}^{2}$–FD, MS = 1 mm | 0.994 | 0.9971 | 0.9961 | 0.9976 |

${R}^{2}$–FD, MS = 2 mm | 0.9958 | 0.9929 | 0.9937 | 0.9927 |

**Table 5.**Variables of the extend Hockett–Sherby model for 23 °C (conditioned), 37 °C (ambient/conditioned), and 50 °C (ambient).

Variable | 23 °C Cond | 37 °C Amb | 37 °C Cond | 50 °C Amb | ||||||
---|---|---|---|---|---|---|---|---|---|---|

MS (mm) | 1 | 2 | $1({\mathit{\sigma}}_{\mathit{V}\mathit{M},}=144\mathbf{MPa})$ | 1 | $2({\mathit{\sigma}}_{\mathit{V}\mathit{M}}=144\mathbf{MPa})$ | 2 | 1 | 2 | 1 | 2 |

${\sigma}_{\infty}$ | 34.96 | 39.9 | 43.42 | 37.88 | 41.72 | 32.2 | 39.40 | 36.26 | 36.06 | 37.57 |

$m$ | 23.36 | 6.967 | 4.961 | 3.395 | 5.908 | 14.5 | 5.94 | 14.45 | 8.047 | 6.316 |

$n$ | 3.494 | 0.4335 | 0.4241 | 4.579 | 0.4479 | 0.7373 | 0.57 | 0.7913 | 0.6062 | 0.5632 |

$o$ | 10.16 | 3.494 | 52.94 | 30.76 | 2.184 | 8.742 | 36.98 | 38.38 | 0.8628 | 90.57 |

$p$ | 4.373 | 0.7256 | 1.719 | 4.015 | 6.024 | 0.031366 | 2.79 | 2.385 | 4.171 | 2.661 |

$q$ | 41.96 | 88 | 0.4477 | 44.26 | 51.04 | 71.01 | 49.30 | 51.83 | 84.64 | 0.8859 |

$r$ | 1.232 | 2.596 | 8.613 | 0.2106 | 1.588 | 2.172 | 2.79 | 2.694 | 2.469 | 15.07 |

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**MDPI and ACS Style**

Amstutz, C.; Weisse, B.; Haeberlin, A.; Burger, J.; Zurbuchen, A. Inverse Finite Element Approach to Identify the Post-Necking Hardening Behavior of Polyamide 12 under Uniaxial Tension. *Polymers* **2022**, *14*, 3476.
https://doi.org/10.3390/polym14173476

**AMA Style**

Amstutz C, Weisse B, Haeberlin A, Burger J, Zurbuchen A. Inverse Finite Element Approach to Identify the Post-Necking Hardening Behavior of Polyamide 12 under Uniaxial Tension. *Polymers*. 2022; 14(17):3476.
https://doi.org/10.3390/polym14173476

**Chicago/Turabian Style**

Amstutz, Cornelia, Bernhard Weisse, Andreas Haeberlin, Jürgen Burger, and Adrian Zurbuchen. 2022. "Inverse Finite Element Approach to Identify the Post-Necking Hardening Behavior of Polyamide 12 under Uniaxial Tension" *Polymers* 14, no. 17: 3476.
https://doi.org/10.3390/polym14173476