Toy nanoindentation model and incipient plasticity

TL;DR

This paper introduces a simplified two-dimensional nanoindentation model using discrete elasticity, revealing dislocation nucleation, hysteresis, and multistability, aligning with experimental observations.

Contribution

It presents a novel toy model for nanoindentation that captures dislocation dynamics and hysteresis phenomena in finite crystals.

Findings

Discontinuities in load vs penetration depth correspond to dislocation loop creation.

Multiple stable solutions exist, showing multistability in the system.

Hysteresis observed during load decrease after dislocation formation.

Abstract

A toy model of two dimensional nanoindentation in finite crystals is proposed. The crystal is described by periodized discrete elasticity whereas the indenter is a rigid strain field of triangular shape representing a hard knife-like indenter. Analysis of the model shows that there are a number of discontinuities in the load vs penetration depth plot which correspond to the creation of dislocation loops. The stress vs depth bifurcation diagram of the model reveals multistable stationary solutions that appear as the dislocation-free branch of solutions develops turning points for increasing stress. Dynamical simulations show that an increment of the applied load leads to nucleation of dislocation loops below the nanoindenter tip. Such dislocations travel inside the bulk of the crystal and accommodate at a certain depth in the sample. In agreement with experiments, hysteresis is observed if the stress is decreased after the first dislocation loop is created. Critical stress values for loop creation and their final location at equilibrium are calculated.