A discrete dislocation analysis of size-dependent plasticity in torsion

TL;DR

The paper tackles size-dependent plasticity in micron-scale crystals undergoing torsion by introducing a monopole-based discrete dislocation framework coupled with image-field finite-element calculations to enforce boundary conditions. Dislocations are represented as monopoles with positions, Burgers vectors, and line elements, enabling a density-accurate yet entanglement-free description of 3D dislocation dynamics, including topological updates and nucleation. The study reveals a bilinear size dependence of the torque–torsion response and a transition from nucleation-controlled to interaction-controlled plasticity as wire diameter increases, with Geometrically Necessary Dislocations and dislocation pile-ups playing key roles. The approach provides deep insight into boundary effects on plasticity at small scales and offers a path toward more efficient, boundary-aware dislocation simulations with potential for extended physics such as dislocation reactions and improved numerical acceleration.

Abstract

A method for solving three dimensional discrete dislocation plasticity boundary-value problems using a monopole representation of the dislocations is presented. At each time step, the displacement, strain and stress fields in a finite body are obtained by superposition of infinite body dislocation fields and an image field that enforces the boundary conditions. The three dimensional infinite body fields are obtained by representing dislocations as being comprised of points, termed monopoles, that carry dislocation line and Burgers vector information. The image fields are obtained from a three dimensional linear elastic finite element calculation. The implementation of the coupling of the monopole representation with the finite element method, including the interaction of curved dislocations with free surfaces, is presented in some detail because it differs significantly from an implementation with a line based dislocation representation. Numerical convergence and the modeling of dislocation loop nucleation for large scale computations are investigated. The monopole discrete dislocation plasticity framework is used to investigate the effect of size and initial dislocation density on the torsion of wires with diameters varying over three orders of magnitude. Depending on the initial dislocation source density and the wire diameter, three regimes of torsion-twist response are obtained: (i) for wires with a sufficiently small diameter, plastic deformation is nucleation controlled and is strongly size dependent; (ii) for wires with larger diameters dislocation plasticity is dislocation interaction controlled, with the emergence of geometrically necessary dislocations and dislocation pile-ups playing a key role, and is strongly size dependent; and (iii) for wires with sufficiently large diameters plastic deformation becomes less heterogeneous and the dependence on size is greatly diminished.

Figures (12)

• Figure 1: Decomposition of the problem into the superposition of interacting dislocations in a homogeneous infinite solid, the ( $\tilde{}$ ) fields, and the image problem that enforces the boundary conditions, the ( $\hat{}$ ) fields.
• Figure 2: Schematic of a dislocation loop exiting the free surface and the introduction of virtual monopoles in three steps: (a) Step 1: check for monopoles that are outside the body $\Omega$ (black) and remove them. (b) Step 2: identify the two monopoles that are closest to the surface. (c) Step3: create virtual monopoles (red) and the corresponding element of line $\xi$ that closes the loop.
• Figure 3: (a) Boundary-value problem for torsion of an fcc single crystal wire subjected to a prescribed twist angle $\theta$. In the calculations, the wire is taken to have a [001] orientation. (b) Example finite element meshes for wire diameters of $D=150$nm, 500nm, and 1000nm, nested into each other.
• Figure 4: Illustration of the exclusion distance $e$. (a) Sketch of a dislocation loop in a transverse plane of the wire; (b) Sketch of a dislocation loop on a slip plane inclined relative to the wire axis.
• Figure 5: Plots of resolved shear stress, $\tau$, for a general dislocation loop, defined by \ref{['eq:tau']} versus loop radius $r$. (a) Comparison of the monopole discretization results with the analytical solution of HirthBook82 for current line element values $\xi=0.86$b, $\xi=27.52$b and $\xi=137.6$b. The value of the regularization parameter $\epsilon$ is fixed at $2$b. (b) The effect of the choice of initial values of line element $\xi_0$ and choice of the value of the regularization parameter $\epsilon$ for dislocation loops with various values of initial radius $r_0$.