An Expanding Cavity Model for Indentation Analysis of Shape Memory Alloys

Abstract

The mechanical and functional responses of shape memory alloys (SMAs), which are often used in small volume applications, can be evaluated using instrumented indentation tests. However, deciphering the indentation test results in SMAs can be complicated due to the combined effects of the non-uniform state of stress underneath the indenter and stress-induced phase transformation. To address this issue, an expanding cavity model (ECM) applicable to spherical indentation of SMAs is developed in this work based on an analytical solution for an internally pressurized hollow sphere. Analytical expressions for key indentation parameters such as the mean contact pressure and size of the transforming zone are obtained, whose validity is evaluated by recourse to finite element simulations and published experimental data for a Ni–Ti alloy. It is shown that the ECM predicts the above parameters reasonably well for indentation strains varying from 0.01 to 0.04. Also, a method is proposed to determine the critical stress required to initiate phase transformation under uniaxial compression based on the application of the ECM to interpret the indentation stress–strain response.

Indentation of glasses

“…Lawn et al [72] suggested that the widely-used expanding cavity models for describing the deformation fields underneath indenter are then no longer valid when a material undergoes densification by compaction within the immediate contact zone of a sharp indenter. (This is also true when phase transformations occur underneath the indenter [73,74].) This is because of the intense hydrostatic compressive stresses that render the volume of the indentation to be more readily accommodated within the compaction zone, diminishing the intensity of any residual stresses.…”

Effect of pressure sensitivity on spherical indentation response of shape memory alloys

“…In this section, approximate equations representing the mean contact pressure and transformed zone size evolution are developed, based on ECM, following the approach proposed by Johnson [42] and Narasimhan [43] for elastic-plastic solids and Anuja et al [38] for pressure insensitive SMAs. As a first step towards this end, a pressure sensitive SMA sphere, initially in the A phase, of radius b having a cavity with radius a (a < b) subjected to pressure p c is analyzed and expressions for stresses and displacements are obtained.…”

“…However, owing to the complex stress state present beneath the indenter and the concurrent occurence of plastic yielding and SIMT, interpretation of the results is not straightforward. Hence, the spherical indentation response of SMAs has been studied using a combination of experiments and simulations [24][25][26][29][30][31][32][33][34][35][36][37][38]. One of the most important challenges involved with indentation is to develop correlations between bulk properties like elastic modulus, transformation stress and strain, yield strength, etc and nano-indentation test data such as load-depth curves, hardness and depth recovery ratio recorded using small samples.…”

“…In a very recent study, Anuja et al [38] developed an expanding cavity model (ECM) to analyze spherical indentation of pressure insensitive SMAs, following the procedure outlined by Johnson [42] and Narasimhan [43] for elastic-plastic solids. In particular, analytical expressions were derived from ECM to determine the size of the transformed zone as well as the mean contact pressure.…”

“…• An ECM incorporating pressure sensitivity of phase transformation needs to be developed to analyse the indentation response of SMAs which display tension-compression asymmetry. This is because ECM can characterize indentation stress-strain curves through simple analytical expressions [42,43], which are readily amenable for inverse analysis to determine material parameters from experimental results (see, for example, [38,45]).…”