# Optimization of Johnson–Cook Constitutive Model Parameters Using the Nesterov Gradient-Descent Method

^{*}

## Abstract

**:**

## 1. Introduction

^{4}–10

^{5}s

^{−1}. Originally, the Taylor test was designed to calculate the impact yield stress of a cylindrical specimen, including its residual length, after impact with a non-deformable target (rigid wall) using a simple model of a rigid plastic material [15]. This approach has often been used to determine the dynamic yield strength [16,17,18,19,20] and to select constitutive relations and constants in numerical simulations [21,22,23,24,25,26,27].

^{6}s

^{−1}). Armstrong R.W. [33] presented the constitutive relations of metals, including α-titanium, copper, α-iron, and tantalum for a wide range of strain rates. The focus was on the Taylor high-velocity impact tests (solid cylinder) and the simulation of the strain characteristics. Gao C. et al. [34] developed a modified Taylor test using a split Hopkinson pressure bar and high-speed imaging to obtain stress–strain curves. The proposed method has been verified by experiments and finite element method simulations at different impact velocities. Jia B. et al. [35] used a single shear specimen to investigate the thermo-viscoplastic behavior of aluminum alloy (2024-T351) subjected to simple shear stress. A hybrid material model was developed and verified via numerical simulation of the Taylor test. Li J-C. et al. [36] performed the Taylor tests for projectiles with four types of nose shapes (blunt, hemispherical, truncated ogive, and truncated conical) and studied the characteristics of loading conditions depending on the nose shape and impact velocity. In [37], Li J-C. et al. applied the results of [36] to a numerical and experimental study of the high-velocity loading of a missile-borne recorder at different velocities using the Taylor impact test. Selyutina N.S. et al. [38] determined the dynamic yield strength of metals using the structural–temporal approach, but the range of interaction velocities was limited, which reduced the role of the shock waves. Pantalé O. et al. [39] simulated the dynamic tensile properties of materials using a specially designed target in the Taylor impact test to generate tensile deformation in its central area. Rodionov E.S. et al. [40] experimentally and numerically investigated the dynamic plasticity of oxygen-free high-conductivity copper (OFHC) at strain values of 0.3 and strain rates up to 1.7 × 10

^{4}s

^{−1}. Experimental data for OFHC copper at higher impact velocities of 150–450 m/s are presented in [18,41].

## 2. Formulation of the Problem

_{Z}—1 layers of the same height, where N

_{Z}is the number of nodes along the vertical Z axis. Each layer was divided into N

_{Y}—1 equal sectors, where N

_{Y}is the number of rays, and into N

_{R}—1 rings of equal thickness, where N

_{R}is the number of nodes along the ray. As a result, the cylinder was divided into quadrangular and triangular (near the axis) prisms. The quadrangular prisms consisted of 6 tetrahedral elements, while the triangular prisms consisted of 3. Figure 1 shows the finite element model of a cylinder. This problem is symmetric, so 1/2 of the cylinder is modelled. The number of elements in the finite element model was 1314, and the number of nodes was 6120 (N

_{R}= 9, N

_{Y}= 9, N

_{Z}= 18).

^{3}), bulk sound velocity (3940 m/s), shear modulus (41 GPa), yield strength (90 MPa), Grüneisen parameter (2.04), specific heat capacity (392.4 J/kg·K), Hugoniot adiabat coefficients (3940 m/s and 1.49) [16].

## 3. Modification of the JC Constitutive Model

^{3}s

^{−1}). However, the strain rates reached values of 10

^{6}s

^{−1}and more at high-velocity impacts. Model constants determined for numerical simulation of high-velocity processes at low strain-rates led to discrepancies between numerical and experimental results. This raised the question of determining the parameters of the JC constitutive model for high strain-rates.

**B**:

_{max}**σ**is the quasi-static yield strength (denoted as

_{0}**A**in the original model [1]);

**ε**is the equivalent plastic strain; ${\dot{\epsilon}}^{\ast}={\dot{\epsilon}}_{pl}/{\dot{\epsilon}}_{0}$ is the dimensionless plastic strain rate; ${\dot{\epsilon}}_{pl}$ is the plastic strain rate; ${\dot{\epsilon}}_{0}$ is the initial strain rate (${\dot{\epsilon}}_{0}$ = 1 s

_{pl}^{−1});

**T**,

_{0}**T**, and

**T**are the initial, current, and melting temperatures, respectively; and

_{m}**B**,

**n**,

**C**, and

**m**are the material model constants.

## 4. Solution-Quality Function

_{f}(Figure 2). A less-reliable measured parameter is the maximum cylinder radius, R

_{f}, due to the possible fracture of the cylinder in contact with the rigid wall. It is worth noting that the specimens were partially broken along the external radius at impact velocities of 316 m/s and above [41].

_{f}is adequate to approximate the shape of the external surface with high accuracy.

**σ**,

_{0}**B**,

**n**,

**B**,

_{max}**C**,

**m**), gradient-descent methods were effective for solving this problem. However, it should be noted that calculating the ${Q}_{f}$ value for a single numerical experiment takes a long time (up to several hours, Intel Xeon 3.2 GHz). Therefore, the number of calculation steps should be minimized. For this purpose, the Nesterov gradient-descent method was chosen to determine the ${Q}_{f}$ minimum [42].

_{0}is a quasi-static yield strength and cannot change at high strain-rates from a physical point of view. Thus, three parameters of the modified model remained under consideration:

**B**,

**B**, and

_{max}**C**.

## 5. Numerical Results and Discussion

- Optimization of parameters for each of tests 1, 2, 3, 4, 5, and 6;
- Optimization of parameters for tests 1 and 2:

- 3.
- Optimization of parameters for tests 3–6:

- 4.
- Optimization of parameters for tests 1–6:

**B**to a value of ~2.8, which corresponds to the microhardness experiments [41]. However, in the experiments [41], the increase in maximum microhardness at individual points reached 3.5 times, and the average microhardness increase at the front end of the cylinder increased by 1.8–2.3 times.

_{max}**B/B**and

_{0}**C/C**are the ratios between optimal and original parameters, Q

_{0}_{f0}is the solution quality before the optimization of parameters, and Q

_{f}is the solution quality after the optimization of parameters. Good agreement between the calculated and experimental data was observed at

**Q**< 0.07 (tests 1, 2, 5, 1 + 2, 3 + 4 + 5 + 6).

_{f}**B**-decrease by 30%,

**C**-decrease by 10%). At impact velocities of 162 m/s and 167 m/s [41], the experiments are described exactly by the same set of parameters. However, with an increase in the impact velocity, the deviation of the calculated values from the experimental values increases. The set of experimental data [41] is better described under the assumption that hardening parameter

**B**of the used copper increases by a factor of 1.9, while parameter

**C**decreases slightly.

## 6. Conclusions

- The JC constitutive model was modified by introducing a material-hardening limit for plastic deformation,
**B**, at high strain-rates._{max} - A solution quality function,
**Q**, was proposed to estimate the deviation of calculations from the experimental data. The final length of the cylinder, the radius of the lateral surface of the cylinder at five points, and the maximum radius of the cylinder were taken as the function parameters, with weighting factors of 20, 2, and 1 according to the effect on the final quality of the solution and reliability of the parameter measurement._{f} - An optimization algorithm for selecting parameters
**B**and**C**of the JC constitutive model and the limiter**B**was developed to find the best agreement between the calculated and experimental data for the Taylor impact test using the Nesterov gradient-descent method._{max} - The optimal parameters, namely,
**B**,**B**, and_{max}**C**, of the modified JC constitutive model were calculated for nine sets of experimental data. The solution quality in some experiments increased by several times when using optimal parameters. For all experiments, the solution quality improved by 10% after optimization. - The developed method for optimizing the selection of the constitutive model constants can be adapted for a wide range of problems (arbitrary set of optimized parameters, arbitrary material models, and software codes, including ANSYS/LS Dyna).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Johnson, G.R.; Cook, W.H. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In Proceedings of the 7th International Symposium on Ballistics, The Hague, The Netherlands, 19–21 April 1983; American Defense Preparedness Association: Arlington, VA, USA, 1983; pp. 541–547. [Google Scholar]
- Johnson, G.R.; Cook, W.H. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech.
**1985**, 21, 31–48. [Google Scholar] [CrossRef] - Zhang, C.; Li, Y.; Wu, J. Mechanical Properties of Fiber-Reinforced Polymer (FRP) Composites at Elevated Temperatures. Buildings
**2023**, 13, 67. [Google Scholar] [CrossRef] - Xie, H.; Zhang, X.; Miao, F.; Jiang, T.; Zhu, Y.; Wu, X.; Zhou, L. Separate Calibration of Johnson–Cook Model for Static and Dynamic Compression of a DNAN-Based Melt-Cast Explosive. Materials
**2022**, 15, 5931. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Fu, X.; Xu, N.; Pan, Y.; Zhang, Y. Spatial Constitutive Modeling of AA7050-T7451 with Anisotropic Stress Transformation. Materials
**2022**, 15, 5998. [Google Scholar] [CrossRef] - Zhang, F.; He, K.; Li, Z.; Huang, B. Strain-Rate Effect on Anisotropic Deformation Characterization and Material Modeling of High-Strength Aluminum Alloy Sheet. Metals
**2022**, 12, 1430. [Google Scholar] [CrossRef] - Sun, X. Uncertainty Quantification of Material Properties in Ballistic Impact of Magnesium Alloys. Materials
**2022**, 15, 6961. [Google Scholar] [CrossRef] - Yang, S.; Liang, P.; Gao, F.; Song, D.; Jiang, P.; Zhao, M.; Kong, N. The Comparation of Arrhenius-Type and Modified Johnson–Cook Constitutive Models at Elevated Temperature for Annealed TA31 Titanium Alloy. Materials
**2023**, 16, 280. [Google Scholar] [CrossRef] - Yin, W.; Liu, Y.; He, X.; Li, H. Effects of Different Materials on Residual Stress Fields of Blade Damaged by Foreign Objects. Materials
**2023**, 16, 3662. [Google Scholar] [CrossRef] - Liu, L.; Wu, W.; Zhao, Y.; Cheng, Y. Subroutine Embedding and Finite Element Simulation of the Improved Constitutive Equation for Ti6Al4V during High-Speed Machining. Materials
**2023**, 16, 3344. [Google Scholar] [CrossRef] - Rodríguez Prieto, J.M.; Larsson, S.; Afrasiabi, M. Thermomechanical Simulation of Orthogonal Metal Cutting with PFEM and SPH Using a Temperature-Dependent Friction Coefficient: A Comparative Study. Materials
**2023**, 16, 3702. [Google Scholar] [CrossRef] - Niu, W.; Wang, Y.; Li, X.; Guo, R. A Joint Johnson–Cook-TANH Constitutive Law for Modeling Saw-Tooth Chip Formation of Ti-6AL-4V Based on an Improved Smoothed Particle Hydrodynamics Method. Materials
**2023**, 16, 4465. [Google Scholar] [CrossRef] - Ben Said, L.; Wali, M. Accuracy of Variational Formulation to Model the Thermomechanical Problem and to Predict Failure in Metallic Materials. Mathematics
**2022**, 10, 3555. [Google Scholar] [CrossRef] - Wang, Z.; Cao, Y.; Gorbachev, S.; Kuzin, V.; He, W.; Guo, J. Research on Conventional and High-Speed Machining Cutting Force of 7075-T6 Aluminum Alloy Based on Finite Element Modeling and Simulation. Metals
**2022**, 12, 1395. [Google Scholar] [CrossRef] - Taylor, G.I. The use of flat ended projectiles for determining yield stress. 1: Theoretical considerations. Proc. R. Soc. Lond. A
**1948**, 194, 289–299. [Google Scholar] [CrossRef] - Gust, W.H. High impact deformation of metal cylinders at elevated temperatures. J. Appl. Phys.
**1982**, 53, 3566–3575. [Google Scholar] [CrossRef] - Bogomolov, A.N.; Gorel’skii, V.A.; Zelepugin, S.A.; Khorev, I.E. Behavior of bodies of revolution in dynamic contact with a rigid wall. J. Appl. Mech. Tech. Phys.
**1986**, 27, 149–152. [Google Scholar] [CrossRef] - Pakhnutova, N.V.; Boyangin, E.N.; Shkoda, O.A.; Zelepugin, S.A. Microhardness and Dynamic Yield Strength of Copper Samples upon Impact on a Rigid Wall. Adv. Eng. Res.
**2022**, 22, 224–231. [Google Scholar] [CrossRef] - Włodarczyk, E.; Sarzynski, M. Strain energy method for determining dynamic yield stress in Taylor’s test. Eng. Trans.
**2017**, 65, 499–511. [Google Scholar] - Scott, N.R.; Nelms, M.D.; Barton, N.R. Assessment of reverse gun taylor cylinder experimental configuration. Int. J. Impact Eng.
**2021**, 149, 103772. [Google Scholar] [CrossRef] - Bayandin, Y.V.; Ledon, D.R.; Uvarov, S.V. Verification of Wide-Range Constitutive Relations for Elastic-Viscoplastic Materials Using the Taylor–Hopkinson Test. J. Appl. Mech. Tech. Phys.
**2021**, 62, 1267–1276. [Google Scholar] [CrossRef] - Volkov, G.A.; Bratov, V.A.; Borodin, E.N.; Evstifeev, A.D.; Mikhailova, N.V. Numerical simulations of impact Taylor tests. J. Phys. Conf. Ser.
**2020**, 1556, 012059. [Google Scholar] [CrossRef] - Acosta, C.A.; Hernandez, C.; Maranon, A.; Casas-Rodriguez, J.P. Validation of material constitutive parameters for the AISI 1010 steel from Taylor impact tests. Mater. Des.
**2016**, 110, 324–331. [Google Scholar] [CrossRef] - Revil-Baudard, B.; Cazacu, O.; Flater, P.; Kleiser, G. Plastic deformation of high-purity α-titanium: Model development and validation using the Taylor cylinder impact test. Mech. Mater.
**2015**, 80, 264–275. [Google Scholar] [CrossRef] - Nguyen, T.; Fensin, S.J.; Luscher, D.J. Dynamic crystal plasticity modeling of single crystal tantalum and validation using Taylor cylinder impact tests. Int. J. Plast.
**2021**, 139, 102940. [Google Scholar] [CrossRef] - Takagi, S.; Yoshida, S. Development of estimation method for material property under high strain rate condition utilizing experiment and analysis. Int. J. Press. Vessel. Pip.
**2022**, 199, 104771. [Google Scholar] [CrossRef] - Ho, C.S.; Mohd Nor, M.K. An Experimental Investigation on the Deformation Behaviour of Recycled Aluminium Alloy AA6061 Undergoing Finite Strain Deformation. Met. Mater. Int.
**2020**, 27, 4967–4983. [Google Scholar] [CrossRef] - Sen, S.; Banerjee, B.; Shaw, A. Taylor Impact Test Revisited: Determination of Plasticity Parameters for Metals at High Strain Rate. Int. J. Solids Struct.
**2020**, 193–194, 357–374. [Google Scholar] [CrossRef] - Zerilli, F.J.; Armstrong, R.W. Dislocation-mechanics-based constitutive relations for materials dynamics calculations. J. Appl. Phys.
**1987**, 61, 1816–1825. [Google Scholar] [CrossRef] [Green Version] - Steinberg, D.J.; Cochran, S.G.; Guinan, M.W. A constitutive model for metals applicable at high-strain rate. J. Appl. Phys.
**1980**, 51, 1498–1504. [Google Scholar] [CrossRef] [Green Version] - Preston, D.L.; Tonks, D.L.; Wallace, D.C. Model of plastic deformation for extreme loading conditions. J. Appl. Phys.
**2003**, 93, 211–220. [Google Scholar] [CrossRef] - Lee, S.; Yu, K.; Huh, H.; Kolman, R.; Arnoult, X. Dynamic Hardening of AISI 304 Steel at a Wide Range of Strain Rates and Its Application to Shot Peening Simulation. Metals
**2022**, 12, 403. [Google Scholar] [CrossRef] - Armstrong, R.W. Constitutive Relations for Slip and Twinning in High Rate Deformations: A Review and Update. J. Appl. Phys.
**2021**, 130, 245103. [Google Scholar] [CrossRef] - Gao, C.; Iwamoto, T. Instrumented Taylor Impact Test for Measuring Stress-Strain Curve through Single Trial. Int. J. Impact Eng.
**2021**, 157, 103980. [Google Scholar] [CrossRef] - Jia, B.; Rusinek, A.; Xiao, X.; Wood, P. Simple Shear Behavior of 2024-T351 Aluminum Alloy over a Wide Range of Strain Rates and Temperatures: Experiments and Constitutive Modeling. Int. J. Impact Eng.
**2021**, 156, 103972. [Google Scholar] [CrossRef] - Li, J.-C.; Chen, G.; Huang, F.-L.; Lu, Y.-G. Load Characteristics in Taylor Impact Test on Projectiles with Various Nose Shapes. Metals
**2021**, 11, 713. [Google Scholar] [CrossRef] - Li, J.-C.; Chen, G.; Lu, Y.-G.; Huang, F.-L. Investigation on the Application of Taylor Impact Test to High-G Loading. Front. Mater.
**2021**, 8, 717122. [Google Scholar] [CrossRef] - Selyutina, N.S.; Petrov, Y.V. Prediction of the dynamic yield strength of metals using two structural-temporal parameters. Phys. Solid State
**2018**, 60, 244–249. [Google Scholar] [CrossRef] - Pantalé, O.; Ming, L. An Optimized Dynamic Tensile Impact Test for Characterizing the Behavior of Materials. Appl. Mech.
**2022**, 3, 1107–1122. [Google Scholar] [CrossRef] - Rodionov, E.S.; Lupanov, V.G.; Gracheva, N.A.; Mayer, P.N.; Mayer, A.E. Taylor Impact Tests with Copper Cylinders: Experiments, Microstructural Analysis and 3D SPH Modeling with Dislocation Plasticity and MD-Informed Artificial Neural Network as Equation of State. Metals
**2022**, 12, 264. [Google Scholar] [CrossRef] - Zelepugin, S.A.; Pakhnutova, N.V.; Shkoda, O.A.; Boyangin, E.N. Experimental study of the microhardness and microstructure of a copper specimen using the Taylor impact test. Metals
**2022**, 12, 2186. [Google Scholar] [CrossRef] - Nesterov, Y.E. A method of solving a convex programming problem with convergence rate O(1/k
^{2}). Soviet Math. Dokl.**1983**, 27, 372–376. [Google Scholar] - Wilkins, M.L.; Guinan, M.W. Impact of cylinders on a rigid boundary. J. Appl. Phys.
**1973**, 44, 1200–1206. [Google Scholar] [CrossRef] - Johnson, G.R. Numerical algorithms and material models for high-velocity impact computations. Int. J. Impact Eng.
**2011**, 38, 456–472. [Google Scholar] [CrossRef] - Gorel’skii, V.A.; Zelepugin, S.A.; Sidorov, V.N. Numerical solution of three-dimensional problem of high-speed interaction of a cylinder with a rigid barrier, taking into account thermal effects. Int. Appl. Mech.
**1994**, 30, 193–198. [Google Scholar] [CrossRef] - Gorelski, V.A.; Zelepugin, S.A.; Smolin, A.Y. Effect of Discretization in Calculating Three-Dimensional Problems of High-Velocity Impact by the Finite-Element Method. Comput. Math. Math. Phys.
**1997**, 37, 722–730. [Google Scholar] - Zelepugin, S.A.; Nikulichev, V.B. Numerical modeling of sulfur–aluminum interaction under shock-wave loading. Combust. Explos. Shock. Waves
**2000**, 36, 845–850. [Google Scholar] [CrossRef] - Pashkov, S.V.; Zelepugin, S.A. Probabilistic approach in modelling dynamic fracture problems. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2022**, 236, 10681–10689. [Google Scholar] [CrossRef] - Tolkachev, V.F.; Ivanova, O.V.; Zelepugin, S.A. Initiation and development of exothermic reactions during solid-phase synthesis under explosive loading. Therm. Sci.
**2019**, 23, S505–S511. [Google Scholar] [CrossRef] - Banerjee, B. Validation of the material point method and plasticity with Taylor impact tests. Report no. C-SAFE-CD-IR-04-004. arXiv
**2012**, arXiv:1201.2476. [Google Scholar] - Ojoc, G.G.; Totolici Rusu, V.; Pîrvu, C.; Deleanu, L. How friction could influence the shape and failure mechanism in impact, with the help of a finite element model. UPB Sci. Bull. Ser. D Mech. Eng.
**2021**, 83, 185–192. [Google Scholar]

**Figure 3.**Fields and isolines of (

**a**) specific shear strain energy (GJ/m

^{3}) and (

**b**) temperature (K) in a copper cylinder at 30 and 87 µs upon impact with a rigid wall at an initial velocity of 316 m/s.

**Figure 5.**Calculated (red) and experimental (black) profiles of the external surface of the cylinders (optimal Johnson–Cook model parameters from Table 4) for tests 1–6.

Test | Material | L_{0} (mm) | D_{0} (mm) | ʋ_{0} (m/s) | T_{0} (K) | Reference |
---|---|---|---|---|---|---|

1 | OFHC Cu | 23.47 | 7.62 | 210 | 298 | [43] |

2 | ETP Cu | 30 | 6.0 | 188 | 718 | [16] |

3 | OFHC Cu M1 | 34.5 | 7.8 | 162 | 298 | [41] |

4 | OFHC Cu M1 | 34.5 | 7.8 | 167 | 298 | [41] |

5 | OFHC Cu M1 | 34.5 | 7.8 | 225 | 298 | [41] |

6 | OFHC Cu M1 | 34.5 | 7.8 | 316 | 298 | [41] |

**Table 2.**Original Johnson–Cooke model parameters [2].

σ_{0} (MPa) | B (MPa) | C | n | m | T_{m} (K) |
---|---|---|---|---|---|

89 | 292 | 0.025 | 0.31 | 1.09 | 1356 |

Test | 1 | 2 | 3 | 4 | 5 | 6 | Average | Standard Deviation |
---|---|---|---|---|---|---|---|---|

${Q}_{f}$ | 0.149 | 0.292 | 0.099 | 0.143 | 0.162 | 0.311 | 0.193 | 0.07 |

Test | B (GPa) | C | B_{max} |
---|---|---|---|

1 | 0.202 | 0.023 | 3.509 |

2 | 0.205 | 0.023 | 3.503 |

3 | 0.309 | 0.024 | 3.387 |

4 | 0.286 | 0.022 | 3.623 |

5 | 0.539 | 0.014 | 2.783 |

6 | 0.486 | 0.018 | 2.881 |

1 + 2 | 0.204 | 0.023 | 3.493 |

3 + 4 + 5 + 6 | 0.565 | 0.020 | 2.558 |

1 + 2 + 3 + 4 + 5 + 6 | 0.265 | 0.024 | 3.330 |

Test | B/B_{0} | C/C_{0} | B_{max} | Q_{f} | Q_{f0} | Q_{f0}/Q_{f} |
---|---|---|---|---|---|---|

1 | 0.692 | 0.919 | 3.509 | 0.011 | 0.149 | 13.6 |

2 | 0.702 | 0.907 | 3.503 | 0.046 | 0.292 | 6.4 |

3 | 1.059 | 0.948 | 3.387 | 0.098 | 0.099 | 1.0 |

4 | 0.981 | 0.865 | 3.623 | 0.143 | 0.143 | 1.0 |

5 | 1.846 | 0.566 | 2.783 | 0.041 | 0.162 | 3.9 |

6 | 1.663 | 0.725 | 2.881 | 0.088 | 0.311 | 3.5 |

1 + 2 | 0.697 | 0.910 | 3.493 | 0.028 | 0.221 | 7.8 |

3 + 4 + 5 + 6 | 1.936 | 0.820 | 2.558 | 0.063 | 0.179 | 2.8 |

1 + 2 + 3 + 4 + 5 + 6 | 0.908 | 0.965 | 3.330 | 0.177 | 0.193 | 1.1 |

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**MDPI and ACS Style**

Zelepugin, S.A.; Cherepanov, R.O.; Pakhnutova, N.V.
Optimization of Johnson–Cook Constitutive Model Parameters Using the Nesterov Gradient-Descent Method. *Materials* **2023**, *16*, 5452.
https://doi.org/10.3390/ma16155452

**AMA Style**

Zelepugin SA, Cherepanov RO, Pakhnutova NV.
Optimization of Johnson–Cook Constitutive Model Parameters Using the Nesterov Gradient-Descent Method. *Materials*. 2023; 16(15):5452.
https://doi.org/10.3390/ma16155452

**Chicago/Turabian Style**

Zelepugin, Sergey A., Roman O. Cherepanov, and Nadezhda V. Pakhnutova.
2023. "Optimization of Johnson–Cook Constitutive Model Parameters Using the Nesterov Gradient-Descent Method" *Materials* 16, no. 15: 5452.
https://doi.org/10.3390/ma16155452