# On the Derivation of Boundary Conditions for Continuum Dislocation Dynamics

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Notation

**u**denote the displacement field. We employ the additive decomposition of the displacement gradient in the elastic and plastic part as

#### 2.2. Kinematics of Single Crystals and CDD

## 3. Principle of Virtual Power

#### 3.1. Internal and External Expenditure of Power

**b**within the volume and due to defect flow across the surface. To this end, we introduce chemical potential-like quantities $\left({\Lambda}^{\alpha},{\Xi}_{i}^{\alpha},{X}^{\alpha},{X}_{j}^{\alpha}\right)$ conjugate, respectively, to the normal flows $\left({j}_{i}^{\alpha}{n}_{i},{J}_{ij}^{\alpha}{n}_{j},{k}_{i}^{\alpha}{n}_{i},{k}_{ji}{n}_{i}\right)$ through a surface with outer normal

**n**. In line with Gurtin, we additionally introduce micro tractions ${\Pi}^{\alpha}\left(\mathit{n}\right)$ conjugate to the shear rates ${\dot{\gamma}}^{\alpha}$ at the surface. That is, we have the external power $\mathcal{W}\left(\Omega \right)$ expended on $\Omega $ defined as

#### 3.2. Principle of Virtual Power

**Theorem**

**1.**

**n**. Furthermore, $\partial S$ denotes the boundary curve with normal vector field

**ν**(which is tangential to the surface). Moreover, let

**X**be a vector field $\mathit{X}:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ which is defined in a surrounding of S. The normal vector field

**n**is likewise supposed to be extended to a unit vector field in a neighborhood of S. Moreover, let κ be the mean curvature function on the surface S. With these notations, the following equality holds:

**Proof.**

**n**(cf. [11]) on slip system $\alpha $. This seems to be an unreasonable requirement, and we conclude that $\Delta {X}^{\alpha}=0$ should be enforced. Alternatively, one might restrict variations to dislocation speeds with vanishing normal derivative ${n}_{k}{\partial}_{k}{\tilde{v}}^{\alpha}=0$. We note that when transferred to a boundary condition for the actual dislocation velocity, setting the normal derivative of the velocity to zero becomes another way to model open boundary conditions. This has been employed for CDD in that the computational domain contains a layer around the actual crystal where the dislocation flux quantities are duplicated from the last layer of the bulk [22].

## 4. Constitutive Theory

#### 4.1. Energy Imbalance

#### 4.2. Dissipation Inequality

#### 4.3. Surface Constitutive Theory

## 5. Discussion

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDD | Continuum dislocation dynamics |

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**MDPI and ACS Style**

Hochrainer, T. On the Derivation of Boundary Conditions for Continuum Dislocation Dynamics. *Crystals* **2017**, *7*, 235.
https://doi.org/10.3390/cryst7080235

**AMA Style**

Hochrainer T. On the Derivation of Boundary Conditions for Continuum Dislocation Dynamics. *Crystals*. 2017; 7(8):235.
https://doi.org/10.3390/cryst7080235

**Chicago/Turabian Style**

Hochrainer, Thomas. 2017. "On the Derivation of Boundary Conditions for Continuum Dislocation Dynamics" *Crystals* 7, no. 8: 235.
https://doi.org/10.3390/cryst7080235