# The $\Lambda$-invariant and topological pathways to influence sub-micron strength and crystal plasticity

## TL;DR

This paper introduces the $\Lambda$-invariant as a new measure of dislocation topology, demonstrating how it can be used to control and understand the influence of sample dimensions on the strength and plasticity of micro- and nanoscale materials.

## Contribution

It presents a novel observable for dislocation ensembles, the $\Lambda$-invariant, enabling control over dislocation microstructures independent of sample size in simulations.

## Key findings

- Dislocation microstructures can be engineered to be size-independent.
- The $\Lambda$-invariant correlates with dislocation linking and vortex character.
- Sample dimension effects can be mitigated through specific dislocation configurations.

## Abstract

In small volumes, sample dimensions are known to strongly influence mechanical behavior, especially strength and crystal plasticity. This correlation fades away at the so-called mesoscale, loosely defined at several micrometers in both experiments and simulations. However, this picture depends on the entanglement of the initial defect configuration. In this paper, we study the effect of sample dimensions with a full control on dislocation topology, through the use of a novel observable for dislocation ensembles, the $\Lambda$-invariant, that depends only on mutual dislocation linking: It is built on the natural vortex character of dislocations and it has a continuum/discrete correspondence that may assist multiscale modeling descriptions. We investigate arbitrarily complex initial dislocation microstructures in sub-micron-sized pillars, using three-dimensional discrete dislocation dynamics simulations for finite volumes. We demonstrate how to engineer nanoscale dislocation ensembles that appear virtually independent from sample dimensions, either by biased-random dislocation loop deposition or by sequential mechanical loads of compression and torsion.