The Role of Geometrically Necessary Dislocations in Cantilever Beam Bending Experiments of Single Crystals.

Edgar Husser,Swantje Bargmann

doi:10.3390/ma10030289

Abstract

The mechanical behavior of single crystalline, micro-sized copper is investigated in the context of cantilever beam bending experiments. Particular focus is on the role of geometrically necessary dislocations (GNDs) during bending-dominated load conditions and their impact on the characteristic bending size effect. Three different sample sizes are considered in this work with main variation in thickness. A gradient extended crystal plasticity model is presented and applied in a three-dimensional finite-element (FE) framework considering slip system-based edge and screw components of the dislocation density vector. The underlying mathematical model contains non-standard evolution equations for GNDs, crystal-specific interaction relations, and higher-order boundary conditions. Moreover, two element formulations are examined and compared with respect to size-independent as well as size-dependent bending behavior. The first formulation is based on a linear interpolation of the displacement and the GND density field together with a full integration scheme whereas the second is based on a mixed interpolation scheme. While the GND density fields are treated equivalently, the displacement field is interpolated quadratically in combination with a reduced integration scheme. Computational results indicate that GND storage in small cantilever beams strongly influences the evolution of statistically stored dislocations (SSDs) and, hence, the distribution of the total dislocation density. As a particular example, the mechanical bending behavior in the case of a physically motivated limitation of GND storage is studied. The resulting impact on the mechanical bending response as well as on the predicted size effect is analyzed. Obtained results are discussed and related to experimental findings from the literature.

Highlights

Micromechanical testing of small-scaled single crystals has been excessively practiced in the last two decades to study the mechanical size-dependence of diverse materials [1,2,3,4]
A sufficiently low initial dislocation density provokes a rapid starvation of available dislocations [7]
With a sufficiently small number of obstacles in the single crystalline sample, the size-dependent flow stress is governed by the applied deformation rate relative to the dislocation nucleation rate which determines the required stress level for continuing operation of individual dislocation sources, see Balint et al [45]

Summary

Introduction

Micromechanical testing of small-scaled single crystals has been excessively practiced in the last two decades to study the mechanical size-dependence of diverse materials [1,2,3,4]. Different intrinsic (microstructural) effects have been found to be triggered by the interplay of physical size limitation such as free surfaces and the underlying microstructure in which the initial density of dislocations plays a crucial role. Stable plastic deformation for crystals of „360 nm size was achieved and no strengthening effect in the range of 360 nm–1500 nm was observed for Mo alloy microcrystals [6]. In both cases, the crystals were machined from relatively strong pre-strained bulk. A sufficiently low initial dislocation density provokes a rapid starvation of available dislocations [7]

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Conclusion