# The Three-Level Elastoviscoplastic Model and Its Application to Describing Complex Cyclic Loading of Materials with Different Stacking Fault Energies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Three-Level Model for Describing Complex Cyclic Deformation

**z**is the velocity gradient (strain rate change) at the mesolevel-1.

**z**

^{in}, which is calculated from the dislocation velocities and densities found in the mesolevel-2 submodel; the components of the Cauchy stress tensor are also defined. The stress tensor

**σ**is averaged over the aggregate of crystallites (the macrolevel RV), which yields the Cauchy macrostress tensor

**Σ**. Evaluation of the stresses

**σ**and unit vectors of the normal and the direction of sliding provides the shear stresses ${\tau}^{(k)}$ acting on each k-th SS at the mesolevel-1. The shear rates ${\dot{\gamma}}^{(k)}$ are determined at the mesolevel-2 from dislocation velocities and dislocation current densities. They are used to calculate the inelastic strain rate at the mesolevel-1. To describe the evolution of the internal variables of the mesolevel-1, we apply the following system of equations (the numbers of crystallites are omitted here and below) [15]:

**п**is a 4-valent elastic tensor,

**z**

^{e}and ${\mathbf{z}}^{in}$ are the elastic and inelastic constituents of the strain rate, $o$ is the orientation tensor (orthogonal tensor transforming the basis of the laboratory coordinate system into the basis of the crystallographic coordinate system),

**ω**is the spin of the corotational coordinate system related to the lattice [56],

**b**is the unit vector in the direction of the Burgers edge dislocation vector,

**n**is the unit normal vector to the plane of dislocation sliding, ${\dot{\gamma}}^{(k)}$ is the shear rate along the slip plane, and k is the number of the slip system (SS). The evolution of the dislocation micro (sub)structure is described at the mesolevel-2.

^{k}is the average length of the mean free path of the dislocations on the k-th SS, and $v$ is the Debye frequency.

^{–3}); changes in these densities are expressed by the following relation:

_{ann}. The number of the reacted dislocations per unit time is proportional to the swept volume and the dislocation density on slip systems. In order to describe this annihilation, we use the following relation [60]:

^{ki}for making estimates of the levels of interactions between the k-th and i-th SSs at the intersection of the k-th row and i-th column of the matrix M

^{ki}.

^{ki}, which was constructed like the matrix M

^{ki}. The matrix components were normalized to the initial critical stresses of the corresponding slip systems (Peierls stresses) ${\tau}_{c0\pm}^{(k)}={\tau}_{c\_lat\pm}^{(k)}$ (they are dimensionless quantities). Using the introduced matrices, the evolutionary relations showing how the critical shear stress and its components change can be written as [62]:

## 3. The Results of Application of the Model to Describing Cyclic Deformation

^{9}m

^{–2}, the initial barrier densities are equal to zero.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the model (transfer of influences and determination of response).

**Figure 3.**Relationship between stress and strain intensities obtained in the numerical simulations of simple cyclic loading experiments on the copper macrosample subjected to tension-compression deformation (

**a**) and on the brass polycrystal sample under tension-compression deformation (

**b**); maximum strain intensity −3%, 125 crystallites.

**Figure 4.**Dependence of stress intensity on accumulated strain intensity at ${\mathrm{P}}_{m}$ = 1, maximum strain amplitude—1.5%, (

**a**)—copper, (

**b**)—brass, $\mathsf{\varphi}$ = 60°, 125 crystallites.

**Figure 5.**Dependence of average dislocations density (

**a**), copper (red), brass (blue), and dependence of average barriers density (

**b**), copper (red), brass (blue), at ${\mathrm{P}}_{m}$ = 1, $\mathsf{\varphi}$ = 60°, maximum strain amplitude—1.5%, 125 crystallites.

**Figure 6.**Dependence of the amount of additional hardening on the degree of non-proportionality of cyclic loading for the angle lag (

**a**), and the change in the ratio of shear amplitude to tension (

**b**) (brass—blue graph, copper—red graph).

**Figure 7.**Dependence of maximum effective stresses on accumulated strain intensity in the complex cyclic loading experiments on the polycrystalline bras sample loaded along two trajectories (

**a**)—black, (

**b**)—red graph.

Variable | Magnitudes | No. Equation | Source |
---|---|---|---|

${\mathsf{\rho}}_{+}^{(k)},{\mathsf{\rho}}_{-}^{(k)}$ at t = 0 | 10^{9} m^{−2} | 4 | [59] |

${\mathsf{\rho}}_{\mathrm{b}\mathrm{a}\mathrm{r}}^{(k)}$ at t = 0 | 0 m^{−2} | 10 | hypothesis |

b (for copper and brass similarly) | 0.2556 × 10^{−9} m | 4 | [59] |

$v$ | 7.11 × 10^{12} s^{−1} | 6 | [61] |

$\mathrm{\Delta}{G}_{*}^{k}$ | 0.206 eV | 6 | [19] |

L | 5 b | 7 | numerical identification |

${r}_{av}$ | 15 b | 8 | hypothesis |

${r}_{0}$ | 3 b | 8 | numerical identification |

A | 139 | 8 | [59] |

B | 5 | 8 | [59] |

${h}_{ann}^{}$ | 5 b | 9 | hypothesis |

${\mathrm{R}}_{\mathrm{bar}}^{kl}$ | Matrix, 24 × 24 elements, dimensionless | 10 | lattice’s characteristic |

${\gamma}_{SFE}$ copper | 75 erg/cm^{3} | 10 | [61] |

${\gamma}_{SFE}$ brass | 20 erg/cm^{3} | 10 | [61] |

${M}^{ki}$ | Matrix, 24 × 24 elements, dimensionless | 11 | Calculating using elastic dislocations model |

${B}^{ki}$ | Matrix, 24 × 24 elements, dimensionless | 11 | Calculating using elastic dislocations model |

${\tau}_{c\_lat}^{(k)}$ copper | 17.5 MPa | 11 | [21] |

${\tau}_{c\_lat}^{(k)}$ brass | 22.3 MPa | 11 | [39] |

**Table 2.**Additional hardening ${\mathrm{\Delta}}_{ad}$ vs. φ and ${\mathrm{P}}_{m}$, for Cu and Brass polycrystals (125 elements), equivalent strain amplitude = 1.5%.

Material | P_{m} | φ, Degree | Δ_{ad}, % | φ, Degree | Δ_{ad}, % |

Copper | 1 | 15 | 5.1 | 60 | 6.4 |

30 | 5.2 | 75 | 7.0 | ||

45 | 6.1 | 90 | 7.1 | ||

Brass | 1 | 15 | 11.1 | 60 | 13.1 |

30 | 12.3 | 75 | 16.1 | ||

45 | 12.8 | 90 | 23.1 | ||

Material | φ, Degree | P_{m} | Δ_{ad}, % | P_{m} | Δ_{ad}, % |

Copper | 33 | 0.25 | 1.1 | 0.75 | 4.1 |

0.5 | 1.2 | 1 | 5.2 | ||

Brass | 33 | 0.25 | 6.1 | 0.75 | 11.1 |

0.5 | 10.3 | 1 | 12.3 |

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**MDPI and ACS Style**

Trusov, P.V.; Gribov, D.S. The Three-Level Elastoviscoplastic Model and Its Application to Describing Complex Cyclic Loading of Materials with Different Stacking Fault Energies. *Materials* **2022**, *15*, 760.
https://doi.org/10.3390/ma15030760

**AMA Style**

Trusov PV, Gribov DS. The Three-Level Elastoviscoplastic Model and Its Application to Describing Complex Cyclic Loading of Materials with Different Stacking Fault Energies. *Materials*. 2022; 15(3):760.
https://doi.org/10.3390/ma15030760

**Chicago/Turabian Style**

Trusov, Peter Valentinovich, and Dmitriy Sergeevich Gribov. 2022. "The Three-Level Elastoviscoplastic Model and Its Application to Describing Complex Cyclic Loading of Materials with Different Stacking Fault Energies" *Materials* 15, no. 3: 760.
https://doi.org/10.3390/ma15030760