# Mechanisms-Based Transitional Viscoplasticity

## Abstract

**:**

## 1. Introduction

## 2. Mechanism of Plastic Flow

## 3. Thermal Activation

## 4. Rerouting of Plastic Flow and Consequences

## 5. Dynamic Overstress

_{2}) and (7

_{2}) $\left(C\xb7{\dot{H}}^{pe}={R}_{k}^{-2}\delta \sigma /\tau \right)$ confirms that the viscous overstress in Equation (6) is acting on the plastic strain. Relaxation of the viscous overstress is best described by Maxwell’s process${\dot{S}}^{a}+{S}^{a}/2\tau =C\xb7{\dot{H}}^{pe}$, where${S}^{a}$ is the active overstress (Figure 2). Rerouting of plastic flow is enabled by the excess of energy ${\dot{W}}_{l}={S}^{a}:{\dot{S}}^{a}/2\mu $, where$tr{S}^{a}=0$. During slow processes, a large portion of ${W}_{l}$ is dissipated. At high strain rates, ${W}_{l}$ explicitly affects the plastic flow, increases storage of energy, intensifies plasticity-induced heating and influences the damage mechanism.

## 6. Plasticity-Induced Heating

## 7. Hall–Petch Relation

#### 7.1. Energy-Based Hall–Petch Relation

#### 7.2. Kinematics-Based Construction of Hall–Petch Relation

## 8. Transitional Viscoplasticity

## 9. OFHC Copper

## 10. Plate Impact Problem

## 11. Conclusions

- The macroscopic plastic flow results from plastic slippages and slip reorganizations. The description is constructed on the basis of the tensor representation concept. It is my conviction that tensor representations derived for generic dyads represent useful tools in the hands of a modeler.
- In the proposed model, thermally activated processes are considered stochastic. The concept explains the transition of flow mechanisms from power-law creep to high strain rate dislocation glide.
- The proposed description of plasticity-induced heating is based on the hypothesis that plasticity-induced heating quantifies the efficiency of the plastic flow process, while plastic work aids in configurational entropy (suppleness) of the material.
- Drag on dislocations is activated by dynamic excitations. As shown, the excitations result from the kinematically-necessary readjustments of flow pathways.
- The stress–strain relations are constructed in the framework of transitional viscoplasticity. The power-law relations enable a smooth elastic-plastic transition during loading and unloading processes.
- I developed an energy-based Hall–Petch relation, where the commonly known stress-based relation is replaced by its kinematics-based counterpart. The proposed Hall–Petch concept was born out of extensive discussions with Ron Armstrong, who walked me through the sixty years of Hall–Petch interpretations, for which I am grateful.
- The model is calibrated for OFHC copper, implemented to a deformable discrete element code (my Ph.D. thesis) and validated in simulations of a plate impact problem. The method itself describes a semi-Cosserat medium, where grain translations and rotations are accounted for.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. A Short Note on Deformable Discrete Element Method

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**Figure 2.**Path rerouting activates dynamic excitations. Note that overstress is only partly stored in newly created dislocation structures.

**Figure 4.**Contours of temperature rise are plotted as a function of grain size. (

**a**) Temperature rise is calculated at strain rate 10

^{4}/s and in a broad range of initial temperatures T

_{0}. (

**b**) Temperature rise is plotted at room temperature and for strain rates between 10

^{3}/s and 10

^{6}/s. The coefficient${\xi}_{ef}$ is estimated based on experimental measurements in [46].

**Figure 5.**Grain size strengthening depicted at strains approximately one percent. Experimental data (blue points) gathered from [52]. The red line represents the model predictions. The middle part of the log–log plot complies with the Hall–Petch relation.

**Figure 6.**Deformation maps for copper presented in terms of stress at 20% strain. The blue mesh represents the model predictions taken in a broad range of temperatures and strain rates. (

**a**) Experimental data collected from References [5,35,36,37,38,39,40,41,42,43,60,61,62] and the points are marked in red. (

**b**) Microcrystalline copper (grains size 1 μm) does not exhibit the stress upturn.

**Figure 8.**Copper target is struck by a copper flyer with velocity 308 m/s. The entire system is constructed with the use of 7200 triangle particles, where each particle is composed of three elements. Artificial viscosity is not used in this simulation.

**Figure 9.**VISAR measurement (thick gray line) reported in [64] is compared with results of the numerical simulation (red line). Damage levels from zero (no damage) to one (fully developed crack) are shown at VISAR points A, B, C and D. The contours are rescaled to fit the window.

Bulk Modulus | Shear Modulus | Mass Density | Yield Stress, 298 K | Burgers Vector | Melting Point | Specific Heat, 298 K |
---|---|---|---|---|---|---|

$B$ | $\mu $ | $\rho $ | ${\sigma}_{y0.2\%}$ | $b$ | ${T}_{m}$ | ${C}_{p}$ |

$GPa$ | $GPa$ | $kg/{m}^{3}$ | $MPa$ | $nm$ | $K$ | $J/\left(kgK\right)$ |

$138$ | $56$ | $8930$ | $84$ | $0.2555$ | $1356$ | 385 |

Strain Rate Exponent | Stress Exponent | Heat Coefficient | Crystallographic Constant | Schmid Factor | Transition Temperature | Overstress Exponent |
---|---|---|---|---|---|---|

${\omega}_{p}$ | ${n}_{p}$ | ${\xi}_{ef}$ | ${\alpha}_{yS}$ | ${r}_{0}$ | ${T}_{c}$ | ${n}_{r}$ |

$-$ | $-$ | $-$ | $-$ | $-$ | $K$ | $-$ |

$0.99$ | $0.56$ | $0.031$ | $0.044$ | $0.3$ | $596.6$ | 0.5 |

Activation Energy Factor | Thermal Activation | Stress Pre-Factor | Critical Energy | Schmid Factor | Ductility | Damage Strain |
---|---|---|---|---|---|---|

${g}_{a}$ | ${k}_{a}$ | ${\mathsf{\Lambda}}_{0}$ | ${\mathrm{G}}_{l}$ | ${e}_{r}^{p}$ | ${e}_{d}$ | ${H}_{kk}^{0}$ |

$-$ | $-$ | $-$ | $MJ/{m}^{3}$ | $-$ | $-$ | $-$ |

$1.13$ | $0.087$ | $6.8$ | $5.0$ | $0.016$ | $0.148$ | $0.0296$ |

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Zubelewicz, A.
Mechanisms-Based Transitional Viscoplasticity. *Crystals* **2020**, *10*, 212.
https://doi.org/10.3390/cryst10030212

**AMA Style**

Zubelewicz A.
Mechanisms-Based Transitional Viscoplasticity. *Crystals*. 2020; 10(3):212.
https://doi.org/10.3390/cryst10030212

**Chicago/Turabian Style**

Zubelewicz, Aleksander.
2020. "Mechanisms-Based Transitional Viscoplasticity" *Crystals* 10, no. 3: 212.
https://doi.org/10.3390/cryst10030212