# Development of the Concurrent Multiscale Discrete-Continuum Model and Its Application in Plasticity Size Effect

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of DCM Coupling Framework

#### 2.1. Two-Dimensional Dislocation Dynamics

#### 2.2. Calculation in Finite Element Module

## 3. Plastic Strain–Stress Transfer in Coupling Scheme

#### 3.1. Plastic Strain and Stress Transfer Scheme

#### 3.2. Coordinate System Conversion Involved in Coupling Scheme

## 4. Subroutines for ABAQUS

#### 4.1. User Subroutines in the Multiscale Framework

#### 4.2. Data Structure for the Coupling Framework

## 5. Uniaxial Compression Simulation for Single Crystal Micropillar

#### 5.1. Computational Model of Micropillars

#### 5.2. Effect of Sample Size on Crystal Strength

## 6. 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 of the coupling scheme of DCM (Yellow block represents the single element in FEM module.).

**Figure 2.**Schematic of distributing plastic strain to integration points of the FEM. The black points represent integration points. The red areas and the shaded areas are the core domains and secondary domains, respectively.

**Figure 3.**The numbering of integration points and nodes in 2D quadrilateral parent elements: (

**a**) 4-node element, (

**b**) 4-node reduced integration element, (

**c**) 8-node element, and (

**d**) 8-node reduced integration element. The dashed lines divide the element into integration point domains. The red dot and black cross represent the node and integration point, respectively.

**Figure 4.**Schematic of two-dimensional coordinate system conversion. (

**a**) (x, y) and (x′, y′) represent the position of point A in the coordinate system OXY and O′X′Y′, respectively. (

**b**) The transformation matrix R can be obtained from the relationship of two coordinate systems.

**Figure 6.**The storage format of arrays in Fortran (

**a**), and the plastic strain tensor re-written into the vector to store in an array (

**b**).

**Figure 8.**The data structures of dislocations (

**a**), obstacles (

**b**), and dislocation sources (

**c**), and the relative user subroutines.

**Figure 9.**Sketch of micropillar compression. The black T-signs represent dislocations. The red triangle and green circle represent obstacle and source, respectively.

**Figure 11.**The yield stress at 0.2% plastic strain offsets, ${\sigma}_{0.2}$, (

**a**) and average secant hardening modulus, ${H}_{ave}$, (

**b**) versus the width of micropillars. The inset in (

**a**) shows the capture of the yield stress from the one stress–strain curve, and the inset in (

**b**) is the ratio of surface area to volume for each micropillar.

Number of Integration Points (n) | $\mathbf{Integration}\mathbf{Point}\mathbf{Location}{\mathit{x}}_{\mathit{i}}$ |
---|---|

1 | ${x}_{1}=0$ |

2 | ${x}_{1}=1/\sqrt{3}$$,{x}_{2}=-1/\sqrt{3}$ |

3 | ${x}_{1}=\sqrt{0.6}$$,{x}_{2}=0$$,{x}_{3}=-\sqrt{0.6}$ |

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**MDPI and ACS Style**

Zhang, Z.; Tong, Z.; Jiang, X.
Development of the Concurrent Multiscale Discrete-Continuum Model and Its Application in Plasticity Size Effect. *Crystals* **2022**, *12*, 329.
https://doi.org/10.3390/cryst12030329

**AMA Style**

Zhang Z, Tong Z, Jiang X.
Development of the Concurrent Multiscale Discrete-Continuum Model and Its Application in Plasticity Size Effect. *Crystals*. 2022; 12(3):329.
https://doi.org/10.3390/cryst12030329

**Chicago/Turabian Style**

Zhang, Zhenting, Zhen Tong, and Xiangqian Jiang.
2022. "Development of the Concurrent Multiscale Discrete-Continuum Model and Its Application in Plasticity Size Effect" *Crystals* 12, no. 3: 329.
https://doi.org/10.3390/cryst12030329