On determination of the constitutive behavior of tempered martensitic steels from micro-indentations: Application to Eurofer97 steel

Abstract

This paper proposes a simple but powerful approach to determine the constitutive equation of tempered martensitic steels from micro-hardness tests. Finite element simulations were used to investigate the plastic flow in the contact region between the tip and the specimen. The simulations were validated by experimental tests carried out on the tempered martensitic steel Eurofer97 using a simple constitutive Ludwik-type equation. A series of simulations using different constitutive behaviors representative of possible irradiation-induced changes were run. In all cases, a pile-up of material against the indenter tip was observed that is strongly dependent on the constitutive law. Considering the real contact height of the indenter with the material, it was shown that the hardness scales with an averaged value of the flow stress over 30% of plastic strain. In addition, the parameters of Ludwik equation were shown to be determinable from the two experimental quantities that are the hardness and the ratio between the contact height and the penetration depth of the tip.

Introduction

Among the small specimen test techniques used to determine mechanical properties, micro-hardness and nano-hardness experiments have been extensively used to extract parameters of the plastic flow [1], [2]. Hardness measurement requires a very small amount of material, which makes it an attractive method to follow the evolution of mechanical properties with irradiation, providing that a reliable method exists to correlate hardness data to other standard data. However, hardness is not a fundamental property of materials, so relating hardness to the flow stress (taken at a given strain or averaged over a strain range) is not straightforward [3]. Based upon both theoretical and experimental considerations, various analytical expressions between hardness and yield stress or ultimate tensile strength have been proposed [4], [5], [6]. These empirical calibrations are not universal in the sense that the parameters of the expressions are material dependent. Therefore, there is no guarantee that a calibration well-established for an unirradiated material remains the same for the same material after irradiation. A close look at the residual profile of the indentation and analysis of this pile-up with finite element simulations yields additional useful information on the strain-hardening of the materials as it was demonstrated in Santos et al. [7]. When hardness experiments are performed on materials that develop a pile-up or sink-in, the hardness, defined by the load divided by the projected contact area, can be determined from the experimental load-penetration depth curves only if one correctly accounts for the pile-up or sink-in geometry. Indeed, the contact height between the indenter tip and the materials is a function of the pile-up or sink-in geometry, which in turn depends on the overall constitutive behavior. Therefore, the hardness can be properly calculated only if a reliable model is available to relate the penetration depth to the contact height. This study was undertaken to search for an appropriate description of the contact area during the indent experiment on tempered martensitic steel. Using the knowledge of the experimental values of hardness and pile-up geometry, a simple method is proposed to determine the true-stress–strain curve.