Extracting elastic-plastic properties from experimental loading-unloading indentation curves using different optimization techniques

Highlights

• Elastic-plastic material properties are extracted from indentation tests.

• Three numerical optimisation techniques are applied with Finite Element Analysis (FEA).

• Optimised solutions are compared to the corresponding experimental test data.

• Applicability and limitations of Finite Element (FE) and optimisation techniques are evaluated.

Abstract

This work is focused on the determination of elastic-plastic material properties from indentation loading-unloading curves using optimisation techniques and experimental data from instrumented indentation tests. Three different numerical optimisation methods (namely, FE analysis, dimensional mathematical functions and simplified mathematical equations approaches) have been used to determine three material properties; Young's modulus, yield stress and work-hardening exponent. The predictions of the material properties from the three approaches have been validated against the values obtained from uniaxial tensile tests and compared to the experimental loading-unloading curves. In general, the elastic-plastic material properties predicted from these three proposed optimisation methods estimate the Young's modulus to within 6% and the yield stress and work-hardening exponent to within 12%, compared to the values obtained from the uniaxial tensile tests.

Introduction

Indentation techniques have been used for mechanical characterisation of materials for decades due to their non-destructive nature and applicability to small sized samples. Fig. 1 shows a schematic illustration of an indentation testing system [1] where a downward load is applied to the indenter to penetrate the test sample, and the reaction force and the displacement at the indenter tip are recorded during the test. Different approaches have been proposed to obtain the mechanical material properties, such as Young's modulus (E), yield stress (σ_y) and work-hardening exponent (n), from the indentation data, see e.g. [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In many studies, there is only one interpreting method involved and it is usually performed using numerical simulations, see e.g. [4], [8]. Experimental indentation tests have been carried out for different materials using different indenter geometries and compared to the corresponding numerical simulations, e.g. [6], [7], [12].

To analyse the material response of an indented specimen, the effects of the indenter geometry on the prediction of the material properties have been investigated by Kang et al [13] using the commercial FE software ABAQUS with new optimisation approaches combining three different methods: (i) Combined FE Simulation and optimisation [14] (ii) Combined dimensional analysis and optimisation [15] and (iii) Optimisation using simplified equations [16]. However, the previous optimisation techniques [14], [15], [16] have been mainly based on simulated target FE loading-unloading curves, rather than curves obtained from experimental tests. It has been found in a previous study [14] that determining elastic-plastic properties from indentation data using only FE simulation and optimisation is less accurate when it is based on experimental indentation data with random errors. Therefore, it is worth extending the investigation to the other two developed optimisation approaches to evaluate their feasibility and robustness.

This study highlights the extraction of elastic-plastic properties from experimental instrumented indentation loading-unloading curves, using the three developed optimization techniques. The general performance and the applicability of these techniques are evaluated and some limitations and areas that need to be explored in the future are addressed. In this study, the experimental loading-unloading curves are obtained using a single Berkovich indenter under different indentation loads [14].

To investigate the mechanical properties of materials that exhibit a power law hardening, which is generally assumed to characterise the work-hardening plasticity behaviour of metals including steels, the stress-strain relationship is given as follows:

$$\sigma =\{\begin{array}{c}E\varepsilon ,\varepsilon \le \frac{{\sigma }_y}{E}\hfill \\ K{\varepsilon }^n,\varepsilon >\frac{{\sigma }_y}{E}\hfill \end{array}$$

where the coefficient K is given by:

$$\mathrm{K}=E^n{{\sigma }_y}^{1-n}$$