Geometrical analysis of 3D dislocation dynamics simulations of FCC micro-pillar plasticity

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Abstract

A geometrical analysis of 3D dislocation dynamics simulations of FCC micro-pillar plasticity shows that the insertion of jogged dislocations before relaxation or enabling cross-slip during plastic flow are prerequisites for the formation of potentially strong natural pinning points and single arm dislocation sources. Absent these conditions, we argue that mobile dislocation starvation will occur naturally in the course of plastic flow and lead to a state in which plastic flow is controlled by the nucleation of dislocations, most likely at the free surfaces.

Introduction

Uniaxial compression testing of micro-pillars, introduced first by Uchic et al., ignited a new wave of studies of small scale plasticity leading to the observation of ‘Smaller is stronger’ even without significant strain gradients [1], [2], [3], [4]. Two main lines of thought have been developed as possible explanations of this size effect: (1) mobile dislocation starvation followed by dislocation nucleation as the controlling mechanism of plastic flow and (2) single arm dislocation sources with lengths that scale with the diameter of the micro-pillar. It is often assumed that for micro-pillars in the deep sub-micrometer size range with very few dislocations (based on the dislocation density of a well-annealed crystal (ρ  1012 m−2)), the dislocation starvation/surface nucleation model might be applicable [5], whereas the single arm dislocation source model might be acceptable for micro-pillars with diameters of several micrometers containing a much larger number of dislocations [6]. However, these two approaches are still being debated, especially with respect to a possible transition size between these two models. Identifying the initial dislocation network structures and understanding their evolution pathways in the course of deformation for a given volume and dislocation density are necessary to clarify this transition.

Transmission electron microscopy (TEM) is the most powerful technique for observing dislocation structures; the dynamic evolution of dislocation structures in micro-pillars has been studied recently through in situ mechanical testing in the TEM [7], [8]. However, the TEM technique requires an electron transparent sample with a thickness usually less than ∼200 nm, and only dislocations that satisfy a diffraction condition are visible. These limitations prevent us from observing the movements of all dislocations in micron-sized pillars. Also, TEM does not permit the study of very fast dynamic processes. Therefore, in order to overcome these limitations, in both bulk materials and micro-pillars, 3D dislocation dynamics (DD) simulations have been developed [9], [10], [11]. In 3D DD simulations, dislocations are usually represented as discrete lines, and their motions and reactions are controlled mainly by mobility laws and rules pertaining to topological changes [11]. The DD simulations allow us to observe the evolution of dislocation structures in the reasonable spatial and temporal scale, compared to atomistic simulations and have become useful tools to understand complex dislocation mechanisms in bulk deformation or, in some cases, to find completely new strengthening mechanisms [12], [13]. For micro-pillars, which typically contain only a few dislocations, the computational cost is much less than that of a bulk simulation. Thus, 3D DD simulation has been widely used to interpret micro-pillar plasticity [14], [15], [16], [17].

In bulk metals, the average or collective behavior of a large number of dislocations typically determines the mechanical properties. For example, with only fundamental material constants and the dislocation density, a flow stress can be estimated using the Taylor equation, without detailed information about the dislocation structures. However, micro-pillars typically do not have a sufficient number of dislocations to show this statistically average effect, so the location and motion of each dislocation is important for determining mechanical properties at the small scale. Furthermore, whereas atomistic simulations render dislocation structures to evolve in a natural way, with the atoms responding to the forces on them, the evolution of dislocation structures in 3D DD follows strictly the implemented rules of dislocation motion. Therefore, the mechanical properties of micro-pillars in 3D DD simulations are expected to be strongly affected by the choice of the initial dislocation structure and by how the rules of dislocation motion are implemented. In this paper we wish to highlight the significance of the choice of the initial dislocation structure and the enabling of cross-slip for FCC pillar plasticity in 3D DD simulations. Then, the potential consequences caused by different initial simulation conditions will be examined in detail using a geometrical analysis and compared with recently reported DD simulations.

Section snippets

Problem statement

The possible transition between the dislocation starvation/nucleation model and the single arm source model is closely related to the sustainability of dislocation networks. If the network has strong pinning points which are connected to single arm dislocation sources, plastic deformation can be controlled by their repeated operation, with the consequence that some mobile dislocations always remain in the micro-pillars. If not, dislocation networks can be expected to unravel completely, with

Possible formation mechanisms of stable single arm dislocation sources

The Lomer–Cottrell (LC) junction is the most plausible structure to supply a strong pinning point in FCC crystals [16]. Because a LC junction is thought to be almost completely immobile, a single arm dislocation connected to a LC junction has the potential to produce a fixed, immortal dislocation source. There might be other possible modes by which strong pinning points are created. In this paper, however, we examine only the mechanisms by which LC junctions connected to single arm dislocation

Concluding remarks

3D DD simulations are powerful tools to study the time evolution of dislocations in pillars. However, the initial conditions should be carefully selected to achieve more realistic results. In FCC pillar plasticity, the formation of strong pinning points with single arm dislocation sources is crucial to keep mobile dislocations in the pillars. Based on our analysis, this structure can be created most likely by the interaction between jogged and straight dislocations or the by LC junction

Acknowledgements

This work was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-FG02-04ER46163. Additional support through an NSF-EU COLLABORATIVE RESEARCH PROGRAM: NANOMESO under NSF Grant No. 0502208 is gratefully acknowledged. We would like to thank Prof. W. Cai and Drs. S. Aubry and C.R. Weinberger for useful discussions.

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