# A Methodology for the Statistical Calibration of Complex Constitutive Material Models: Application to Temperature-Dependent Elasto-Visco-Plastic Materials

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## Abstract

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## 1. Introduction

## 2. Fundamentals

#### 2.1. Global Sensitivity Analysis

#### 2.2. Meta-Models

#### 2.3. Bayesian Inference and Gaussian Processes

#### 2.4. Material Models

#### 2.4.1. Johnson–Cook Constitutive Relations

#### 2.4.2. Zerilli–Armstrong Constitutive Relations

#### 2.4.3. Arrhenius-Type Model Constitutive Relations

## 3. Application

## 4. Results

#### 4.1. Sensitivity Analysis

#### 4.2. Linear Interpolation of Meta-Models

#### 4.3. Bayesian Calibration

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Iooss, B.; Lemaître, P. A review on global sensitivity analysis methods. In Uncertainty Management in Simulation-Optimization of Complex Systems; Springer: Berlin/Heidelberg, Germany, 2015; pp. 101–122. [Google Scholar]
- Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D.; Saisana, M.; Tarantola, S. Global Sensitivity Analysis: The Primer; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Box, G.; Draper, N. Empirical Model-Building and Response Surfaces; John Wiley & Sons: Hoboken, NJ, USA, 1987. [Google Scholar]
- Iooss, B.; Van Dorpe, F.; Devictor, N. Response surfaces and sensitivity analyses for an environmental model of dose calculations. Reliab. Eng. Syst. Saf.
**2006**, 91, 1241–1251. [Google Scholar] [CrossRef] - Rohmer, J. Dynamic sensitivity analysis of long-running landslide models through basis set expansion and meta-modelling. Nat. Hazards
**2014**, 73, 5–22. [Google Scholar] [CrossRef] - Todri, E.; Amenaghawon, A.; Del Val, I.; Leak, D.; Kontoravdi, C.; Kucherenko, S.; Shah, N. Global sensitivity analysis and meta-modeling of an ethanol production process. Chem. Eng. Sci.
**2014**, 114, 114–127. [Google Scholar] [CrossRef] - Welch, W.; Buck, R.; Sacks, J.; Wynn, H.; Mitchell, T.; Morris, M. Screening, predicting, and computer experiments. Technometrics
**1992**, 34, 15–25. [Google Scholar] [CrossRef] - Buhmann, M. Radial Basis Functions: Theory and Implementations; Cambridge University Press: Cambridge, UK, 2003; Volume 12. [Google Scholar]
- MacAllister, A.; Kohl, A.; Winer, E. Using High-Fidelity Meta-Models to Improve Performance of Small Dataset Trained Bayesian Networks. Expert Syst. Appl.
**2019**, 139, 112830. [Google Scholar] [CrossRef] - Guttman, I.; Tiao, G. A Bayesian Approach to Some Best Population Problems; Technical Report; Wisconsin Univ-Madison: Madison, WI, USA, 1963. [Google Scholar]
- Aitchison, J.; Dunsmore, I. Statistical Prediction Analysis; CUP Archive: Cambridge, UK, 1980. [Google Scholar]
- Kennedy, M.; O’Hagan, A. Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B Stat. Methodol.
**2001**, 63, 425–464. [Google Scholar] [CrossRef] - Higdon, D.; Kennedy, M.; Cavendish, J.; Cafeo, J.; Ryne, R. Combining Field Data and Computer Simulations for Calibration and Prediction. SIAM J. Sci. Comput.
**2004**, 26, 448–466. [Google Scholar] [CrossRef][Green Version] - O’Hagan, A. Bayesian analysis of computer code outputs: A tutorial. Reliab. Eng. Syst. Saf.
**2006**, 91, 1290–1300. [Google Scholar] [CrossRef] - Draper, D.; Pereira, A.; Prado, P.; Saltelli, A.; Cheal, R.; Eguilior, S.; Mendes, B.; Tarantola, S. Scenario and parametric uncertainty in GESAMAC: A methodological study in nuclear waste disposal risk assessment. Comput. Phys. Commun.
**1999**, 117, 142–155. [Google Scholar] [CrossRef] - Janse, J.; Scheffer, M.; Lijklema, L.; Van Liere, L.; Sloot, J.; Mooij, W. Estimating the critical phosphorus loading of shallow lakes with the ecosystem model PCLake: Sensitivity, calibration and uncertainty. Ecol. Model.
**2010**, 221, 654–665. [Google Scholar] [CrossRef] - Johnson, G. A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures. In Proceedings of the 7th International Symposium on Ballistics, Hague, The Netherlands, 19–21April 1983; pp. 541–547. [Google Scholar]
- Zerilli, F.; Armstrong, R. Dislocation-mechanics-based constitutive relations for material dynamics calculations. J. Appl. Phys.
**1987**, 61, 1816–1825. [Google Scholar] [CrossRef][Green Version] - Samantaray, D.; Mandal, S.; Bhaduri, A. A comparative study on Johnson–Cook, modified Zerilli–Armstrong and Arrhenius-type constitutive models to predict elevated temperature flow behaviour in modified 9Cr–1Mo steel. Comput. Mater. Sci.
**2009**, 47, 568–576. [Google Scholar] [CrossRef] - Portillo, D.; del Pozo, D.; Rodríguez-Galán, D.; Segurado, J.; Romero, I. MUESLI—A Material UnivErSal LIbrary. Adv. Eng. Softw.
**2017**, 105, 1–8. [Google Scholar] [CrossRef] - Taylor, G. The use of flat-ended projectiles for determining dynamic yield stress I. Theoretical considerations. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1948**, 194, 289–299. [Google Scholar] - Whiffin, A. The use of flat-ended projectiles for determining dynamic yield stress-II. Tests on various metallic materials. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1948**, 194, 300–322. [Google Scholar] - Menga, E.; Sánchez, M.; Romero, I. Anisotropic meta-models for computationally expensive simulations in nonlinear mechanics. Int. J. Numer. Methods Eng.
**2019**. [Google Scholar] [CrossRef] - Hoff, P. A First Course in Bayesian Statistical Methods; Springer: Berlin/Heidelberg, Germany, 2009; Volume 580. [Google Scholar]
- Wernick, M.; Yang, Y.; Brankov, J.; Yourganov, G.; Strother, S. Machine learning in medical imaging. IEEE Signal Process. Mag.
**2010**, 27, 25–38. [Google Scholar] [CrossRef][Green Version] - Koenig, N.; Matarić, M. Robot life-long task learning from human demonstrations: A Bayesian approach. Auton. Robot.
**2017**, 41, 1173–1188. [Google Scholar] [CrossRef] - Cofino, A.; Cano Trueba, R.; Sordo, C.; Gutiérrez Llorente, J. Bayesian networks for probabilistic weather prediction. In Proceedings of the 15th Eureopean Conference on Artificial Intelligence, ECAI’2002, Lyon, France, 21–26 July 2002. [Google Scholar]
- Rasmussen, C.; Williams, C.K. Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Carmassi, M.; Barbillon, P.; Keller, M.; Parent, E.; Chiodetti, M. Bayesian calibration of a numerical code for prediction. arXiv
**2018**, arXiv:1801.01810. [Google Scholar] - Shrot, A.; Bäker, M. Determination of Johnson–Cook parameters from machining simulations. Comput. Mater. Sci.
**2012**, 52, 298–304. [Google Scholar] [CrossRef] - Li, H.; He, L.; Zhao, G.; Zhang, L. Constitutive relationships of hot stamping boron steel B1500HS based on the modified Arrhenius and Johnson–Cook model. Mater. Sci. Eng. A
**2013**, 580, 330–348. [Google Scholar] [CrossRef] - Banerjee, A.; Dhar, S.; Acharyya, S.; Datta, D.; Nayak, N. Determination of Johnson Cook material and failure model constants and numerical modelling of Charpy impact test of armour steel. Mater. Sci. Eng. A
**2015**, 640, 200–209. [Google Scholar] [CrossRef] - Valentin, T.; Magain, P.; Quik, M.; Labibes, K.; Albertini, C. Validation of constitutive equations for steel. J. Phys. IV
**1997**, 7, C3-611. [Google Scholar] [CrossRef] - Sobol’, I. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki
**1967**, 7, 784–802. [Google Scholar] [CrossRef] - Carmassi, M.; Barbillon, P.; Chiodetti, M.; Keller, M.; Parent, E. CaliCo: An R package for Bayesian calibration. arXiv
**2018**, arXiv:1808.01932. [Google Scholar] - Craig, P.; Goldstein, M.; Rougier, J.; Seheult, A. Bayesian forecasting for complex systems using computer simulators. J. Am. Stat. Assoc.
**2001**, 96, 717–729. [Google Scholar] [CrossRef] - Riddle, M.; Muehleisen, R. A guide to Bayesian calibration of building energy models. In Proceedings of the Building Simulation Conference, Atlanta, GA, USA, 10–12 September 2014. [Google Scholar]

**Figure 3.**Iterative process for a two-stage approach of screening and calibration of model parameters.

**Figure 4.**Linear interpolation of meta-model predictions of $\Delta R$ for the Johnson–Cook constitutive relation. Each piecewise linear interpolation connects predictions with the same model parameters.

**Figure 5.**Linear interpolation of meta-model predictions of $\Delta R$ for the Zerilli–Armstrong constitutive relation.

**Figure 6.**Linear interpolation of meta-model predictions of $\Delta R$ for the Arrhenius-type constitutive relation.

**Figure 7.**Global Sensitivity Analysis (GSA) results for the Johnson–Cook (JC) model considering $\Delta R$ and $\Delta L$ at 200 and $320\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$.

**Figure 8.**GSA results for the Zerilli–Armstrong (ZA) model considering $\Delta R$ and $\Delta L$ at 200 and $320\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$.

**Figure 9.**GSA results for the Arrhenius-type model considering $\Delta R$ and $\Delta L$ at 200 and $320\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}$.

**Figure 10.**Comparison of meta-models interpolated results and FEM simulations, considering $\Delta R$ for the Johnson–Cook model.

**Figure 11.**Comparison of meta-models interpolated results and FEM simulation results, considering $\Delta R$ for the Zerilli–Armstrong model.

**Figure 12.**Comparison of meta-models interpolated results and FEM simulation results, considering $\Delta R$ for the Arrhenius-type model.

**Figure 13.**Prior vs. posterior probability distribution functions of parameters $A\sim {\theta}_{1},B\sim {\theta}_{2},C\sim {\theta}_{3},$ in the JC model considering $\Delta R$ as the QoI.

0.48 | 0.48 | 0.48 |

(a) | (b) | (c) |

**Figure 14.**Prior vs. posterior probability distribution functions of parameters $A\sim {\theta}_{1},B\sim {\theta}_{2},C\sim {\theta}_{3},$ in the JC model considering $\Delta L$ as the QoI.

0.48 | 0.48 | 0.58 |

(a) | (b) | (c) |

**Figure 15.**Prior vs. posterior parameter probability distributions for ${C}_{0}\sim {\theta}_{1},{C}_{3}\sim {\theta}_{2},{C}_{5}\sim {\theta}_{3},n\sim {\theta}_{4}$, of the ZA model considering $\Delta R$ as the QoI.

0.48 | 0.48 | 0.48 | 0.48 |

(a) | (b) | (c) | (d) |

**Figure 16.**Prior vs. posterior parameter probability distributions for ${C}_{0}\sim {\theta}_{1},{C}_{3}\sim {\theta}_{2},{C}_{5}\sim {\theta}_{3},n\sim {\theta}_{4}$, of the ZA model considering $\Delta L$ as the QoI.

0.48 | 0.48 | 0.48 | 0.48 |

(a) h | (b) h | (c) h | (d) h |

**Figure 17.**Prior vs. posterior parameter probability distributions for parameters ${A}_{2}\sim {\theta}_{1},{A}_{3}\sim {\theta}_{2},{\alpha}_{3}\sim {\theta}_{3},n\sim {\theta}_{4}$ of the Arrhenius-type model considering $\Delta R$ as the QoI.

0.48 | 0.48 | 0.48 | 0.48 |

(a) | (b) | (c) | (d) |

**Figure 18.**Prior vs. posterior parameter probability distributions for parameters ${A}_{2}\sim {\theta}_{1},{A}_{3}\sim {\theta}_{2},{\alpha}_{3}\sim {\theta}_{3},n\sim {\theta}_{4}$ of the Arrhenius-type model considering $\Delta L$ as the QoI.

0.48 | 0.48 | 0.48 | 0.48 |

(a) h | (b) h | (c) h | (d) h |

Term/Parameter | Probability Distribution Function |
---|---|

$\theta $ | $\mathcal{N}(\mu ,1)$ JC/ARR or $\mathcal{N}(\mu ,10)$ ZA (see Table 2) |

${\lambda}^{2}$ | ${\Gamma}(1,0.1)$ |

${\sigma}_{\delta}^{2}$ | ${\Gamma}(1,0.1)$ |

${\psi}_{\delta}$ | $U(0,1)$ |

Material Model | Mean Values |
---|---|

Johnson–Cook | $A=113\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}B=211\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}C=0.073,\phantom{\rule{0.277778em}{0ex}}n=0.218,\phantom{\rule{0.277778em}{0ex}}m=0.818$ |

Zerilli–Armstrong | ${C}_{0}=707.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}{C}_{1}=575\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}{C}_{3}=0.00698\phantom{\rule{3.33333pt}{0ex}}{\mathrm{K}}^{-1},\phantom{\rule{0.277778em}{0ex}}$ |

${C}_{4}=0.00032\phantom{\rule{3.33333pt}{0ex}}{\mathrm{K}}^{-1},\phantom{\rule{0.277778em}{0ex}}{C}_{5}=637.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}n=0.41$ | |

Arrhenius-type | ${Q}_{0}=412.31,\phantom{\rule{0.277778em}{0ex}}{Q}_{1}=-510.82,\phantom{\rule{0.277778em}{0ex}}{Q}_{2}=1873.4,\phantom{\rule{0.277778em}{0ex}}{Q}_{3}=-1872.4$ |

${A}_{0}=36.402,\phantom{\rule{0.277778em}{0ex}}{A}_{1}=-68.301,\phantom{\rule{0.277778em}{0ex}}{A}_{2}=254.32,\phantom{\rule{0.277778em}{0ex}}{A}_{3}=-255.57$ | |

${\alpha}_{0}=0.009481,\phantom{\rule{0.277778em}{0ex}}{\alpha}_{1}=-0.003841,\phantom{\rule{0.277778em}{0ex}}{\alpha}_{2}=-0.012971,\phantom{\rule{0.277778em}{0ex}}{\alpha}_{3}=0.025892,\phantom{\rule{0.277778em}{0ex}}$ | |

$n=5.2248$ |

Material Model | Significant Parameters |
---|---|

Johnson–Cook | $A,B,C$ |

Zerilli–Armstrong | ${C}_{0},{C}_{3},{C}_{5},n$ |

Arrhenius-type | ${A}_{2},{A}_{3},{\alpha}_{3},n$ |

Model | Mean Relative Error | Maximum Relative Error |
---|---|---|

Jonhson–Cook | $6.2\times {10}^{-3}$ | $1.2\times {10}^{-2}$ |

Zerilli–Armstrong | $8.1\times {10}^{-3}$ | $3.2\times {10}^{-2}$ |

Arrhenius-type | $1.1\times {10}^{-2}$ | $6.7\times {10}^{-2}$ |

Model | Mean Relative Error | Maximum Relative Error |
---|---|---|

Jonhson–Cook | $6.9\times {10}^{-3}$ | $2.3\times {10}^{-2}$ |

Zerilli–Armstrong | $9.4\times {10}^{-3}$ | $3.6\times {10}^{-2}$ |

Arrhenius-type | $7.8\times {10}^{-3}$ | $3.0\times {10}^{-2}$ |

**Table 6.**A posteriori mean values obtained for each parameter, considering both QoIs, $\Delta R$ and $\Delta L$, for the calibration of the models.

Material Model | Mean Values $\mathit{\Delta}\mathit{R}$ | Mean Values $\mathit{\Delta}\mathit{L}$ |
---|---|---|

Johnson–Cook | $A=112.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}B=211.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}C=0.065$ | $A=112.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}B=211.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}C=0.073$ |

Zerilli–Armstrong | ${C}_{0}=704.6\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}{C}_{3}=-0.00392\phantom{\rule{3.33333pt}{0ex}}{\mathrm{K}}^{-1},\phantom{\rule{0.277778em}{0ex}}$ | ${C}_{0}=702.8\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}{C}_{3}=1.65\phantom{\rule{3.33333pt}{0ex}}{\mathrm{K}}^{-1},\phantom{\rule{0.277778em}{0ex}}$ |

${C}_{5}=638.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}n=-1.65$ | ${C}_{5}=636.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa},\phantom{\rule{0.277778em}{0ex}}n=1.02$ | |

Arrhenius-type | ${A}_{2}=255.0,\phantom{\rule{0.277778em}{0ex}}{A}_{3}=-254.8$ | ${A}_{2}=254.0,\phantom{\rule{0.277778em}{0ex}}{A}_{3}=-255.9$ |

${\alpha}_{3}=0.077,\phantom{\rule{0.277778em}{0ex}}n=4.9$ | ${\alpha}_{3}=0.59,\phantom{\rule{0.277778em}{0ex}}n=5.54$ |

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**MDPI and ACS Style**

de Pablos, J.L.; Menga, E.; Romero, I. A Methodology for the Statistical Calibration of Complex Constitutive Material Models: Application to Temperature-Dependent Elasto-Visco-Plastic Materials. *Materials* **2020**, *13*, 4402.
https://doi.org/10.3390/ma13194402

**AMA Style**

de Pablos JL, Menga E, Romero I. A Methodology for the Statistical Calibration of Complex Constitutive Material Models: Application to Temperature-Dependent Elasto-Visco-Plastic Materials. *Materials*. 2020; 13(19):4402.
https://doi.org/10.3390/ma13194402

**Chicago/Turabian Style**

de Pablos, Juan Luis, Edoardo Menga, and Ignacio Romero. 2020. "A Methodology for the Statistical Calibration of Complex Constitutive Material Models: Application to Temperature-Dependent Elasto-Visco-Plastic Materials" *Materials* 13, no. 19: 4402.
https://doi.org/10.3390/ma13194402