# Machine-Learning-Based Model of Elastic—Plastic Deformation of Copper for Application to Shock Wave Problem

^{*}

## Abstract

**:**

## 1. Introduction

^{7}–10

^{9}s

^{–1}[14,15,16,17]. These strain rates can be realized in molecular dynamics (MD) simulations for representative volume elements, but direct MD simulation of the shock-wave processes [18,19,20] is restricted from bellow by much higher strain rates—typically of about 10

^{11}s

^{–1}. Thus, continuum mechanics models are used to investigate the shock-wave problem numerically [21,22,23,24]. It makes essential a transfer of information from the MD, which allows one to study the rich physics of inelastic deformation, to continuum models applicable for realistic spatial and temporal scales.

## 2. Materials and Methods

#### 2.1. MD Simulations of Uniform Axisymmetric Deformation

#### 2.2. Training of ANNs for Tensor Equation of State and Structural Transformations

#### 2.3. Dislocation Plasticity Model and Parameter Fitting by Bayesian Algorithm

^{−4}, the probability value is corrected in accordance with the difference between the model and the MD:

#### 2.4. Application to Plane Shock Wave

## 3. Results

#### 3.1. Results of MD Simulations

#### 3.2. Results of ANN Training

#### 3.3. Results of Bayesian Parameterization of Dislocation Plasticity Model

## 4. Discussion

^{9}s

^{−1}. This coincidence additionally verifies the used interatomic potential [40].

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Parameter | Cu.TEOS1.ANNp | Cu.DISL1.ANNp | Cu.VOID1.ANNp | Cu.BCC1.ANNp |
---|---|---|---|---|

Number of inputs | 3 | |||

Inputs | $\left\{{E}_{\mathrm{V}},{E}_{\mathrm{S}},T\right\}$ | |||

Number of outputs | 6 | 1 | 1 | 1 |

Outputs | $\begin{array}{l}\left\{\left(P-{P}_{\mathrm{i}}\right),\text{\hspace{0.33em}}S,\text{\hspace{0.33em}}\left(U-{U}_{\mathrm{i}}\right),\right.\\ \left.\text{\hspace{0.33em}}\mathrm{ln}\left(K\right),\mathrm{ln}\left(G\right),\rho \right\}\end{array}$ | $\left\{{Q}_{\mathrm{disl}}\right\}$ | $\left\{{Q}_{\mathrm{void}}\right\}$ | $\left\{{Q}_{\mathrm{bcc}}\right\}$ |

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**Figure 1.**Deformation paths at the room temperature in the following coordinates: (

**a**) Green–Lagrange strain tensor components E

_{11}and E

_{22}, and (

**b**) relative volume E

_{V}= V/V

_{0}—shear strain E

_{S}= (E

_{11}− E

_{22})/2. The green line “h/d” shows the case of hydrostatic (volumetric) deformation.

**Figure 3.**Data for training of the EOS ANN at room temperature: (

**a**) shear-strain-dependent component of pressure and (

**b**) stress deviator.

**Figure 4.**Evolution of shear stress, atomic structure and dislocation structure: compression along the path DG4 at 300 K. The stripes of hcp phase show the stacking faults left behind the partial Shockley dislocations.

**Figure 5.**Fcc–bcc phase transition in copper: (

**a**) beginning of this transition under compression along the deformation path DG1 (

**b**) and the pressure of bcc transition as a function of temperature for compression along several deformation paths.

**Figure 6.**Tensile fracture of copper: Evolution of pressure and atomic configurations for the case of tension along the deformation path DG3 at 300 K. The void surface is shown by blue surface mesh obtained by the “Construct surface mesh” algorithm of OVITO.

**Figure 7.**Results of ANN training for the tensor equation of state: MD data (black circles) and ANN predictions (red crosses) for (

**a**) the pressure correction $\left(P-{P}_{\mathrm{i}}\right)$, (

**c**) the stress deviator ($S$ ) and (

**e**) the correction $\left(U-{U}_{\mathrm{i}}\right)$ for the specific internal energy; correlation of the MD and ANN for (

**b**) the pressure correction, (

**d**) the stress deviator and (

**f**) the correction of the specific internal energy, where black line shows the limit of perfect match.

**Figure 8.**Description of the homogeneous nucleation of dislocations in copper on the basis of trained ANNs: (

**a**) the threshold strain states in the coordinates $\left\{{E}_{\mathrm{V}},{E}_{\mathrm{S}}\right\}$; (

**b**) the same threshold in the strain–stress coordinates $\left\{{E}_{\mathrm{V}},S\right\}$.

**Figure 9.**Description of the void nucleation in copper on the basis of trained ANNs: (

**a**) the threshold strain states in the coordinates $\left\{{E}_{\mathrm{V}},{E}_{\mathrm{S}}\right\}$; (

**b**) the same threshold in the strain–stress coordinates $\left\{{E}_{\mathrm{S}},P\right\}$.

**Figure 10.**Description of the beginning of bcc transformation in copper on the basis of trained ANNs: (

**a**) the strain states of the beginning of bcc transformation in the coordinates $\left\{{E}_{\mathrm{V}},{E}_{\mathrm{S}}\right\}$; (

**b**) the same states in the strain–stress coordinates $\left\{{E}_{\mathrm{V}},S\right\}$.

**Figure 11.**Results of Bayesian algorithm of parameter identification: contour maps of probability of model parameters: (

**a**–

**c**) show different pairs of parameters to be identified.

**Figure 12.**Comparison of the MD with the dislocation plasticity model with optimal parameters determined by means of the Bayesian algorithm: strain dependencies of stress deviator for different strain paths at 300 K. Panels (

**a**–

**f**) correspond to different deformation trajectories and modes as indicated near the curves.

**Figure 13.**Comparison of the MD with the dislocation plasticity model with optimal parameters determined by means of the Bayesian algorithm: strain dependencies of stress deviator for different strain paths at 800 K. Panels (

**a**–

**f**) correspond to different deformation trajectories and modes as indicated near the curves.

**Figure 14.**Comparison of the MD with the dislocation plasticity model with optimal parameters determined by means of the Bayesian algorithm: evolution of dislocation density with applied strain at 300 K. Panels (

**a**–

**f**) correspond to different deformation trajectories and modes as indicated near the curves.

**Table 1.**Deformation paths studied in MD for copper single crystal with the axis of symmetry coinciding with the lattice direction [100].

Identifier | Axial Strain Rate (ns^{−1}) | Transverse Strain Rate (ns^{−1}) |
---|---|---|

DG0 | 0.4 | 0.3 |

DG1 | 0.6 | 0.2 |

DG2 | 0.2 | 0.4 |

DG3 | 0 | 0.5 |

DG4 | 1 | 0 |

DG5 | 0.8 | 0.1 |

DG6 | 0.6 | −0.2 |

DG7 | 0.2 | −0.4 |

DG8 | 0.4 | −0.3 |

DG9 | 0.8 | −0.1 |

**Table 2.**Parameters of the dislocation plasticity model determined by fitting with MD by means of the Bayesian algorithm: $B$ is the friction coefficient of gliding dislocation, $\alpha $ is the hardening coefficient, ${k}_{\mathrm{m}}$ is the coefficient of efficient energy of dislocation multiplication, ${k}_{\mathrm{n}}$ is the coefficient of nucleation barrier and ${k}_{\mathrm{a}}$ is the annihilation coefficient.

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

$B$ | 1.88 × 10^{−5} Pa × s |

$\alpha $ | 1.56 |

${k}_{\mathrm{m}}$ | 1.4 |

${k}_{\mathrm{n}}$ | 0.3 |

${k}_{\mathrm{a}}$ | 50 |

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**MDPI and ACS Style**

Mayer, A.E.; Lekanov, M.V.; Grachyova, N.A.; Fomin, E.V.
Machine-Learning-Based Model of Elastic—Plastic Deformation of Copper for Application to Shock Wave Problem. *Metals* **2022**, *12*, 402.
https://doi.org/10.3390/met12030402

**AMA Style**

Mayer AE, Lekanov MV, Grachyova NA, Fomin EV.
Machine-Learning-Based Model of Elastic—Plastic Deformation of Copper for Application to Shock Wave Problem. *Metals*. 2022; 12(3):402.
https://doi.org/10.3390/met12030402

**Chicago/Turabian Style**

Mayer, Alexander E., Mikhail V. Lekanov, Natalya A. Grachyova, and Eugeniy V. Fomin.
2022. "Machine-Learning-Based Model of Elastic—Plastic Deformation of Copper for Application to Shock Wave Problem" *Metals* 12, no. 3: 402.
https://doi.org/10.3390/met12030402