Hardness obtained from conical indentations with various cone angles
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The relationship between hardness and cone angle of conical indenters was studied using finite element analysis for elastic–plastic solids with work-hardening. Comparisons were made between the present simulation results, slip line theory, and experimental results. Tabor’s concept of representative strain based on indentation experiments in metals (The Hardness of Metals, Oxford, 1951) was shown to be applicable to a wide range of materials. The relative size of plastic zone with respect to the contact radius was found to influence the variation of hardness with indenter cone angle. The method proposed by Atkins and Tabor [J. Mech. Phys. Solids, 13, 149 (1965)] for constructing stress-strain curves using representative strains was also examined, and the conditions under which the method is valid were obtained. I. INTRODUCTION
Indentation experiments have been performed for over one hundred years to measure the hardness of materials.1 In the last decade, major improvements in nanoindentation techniques have made them applicable for studying the material properties in the respective nanometer and micro-Newton ranges.2,3 Using these techniques, the hardness and Young’s modulus may be obtained from the peak load and the initial slope of the unloading curves2,3 or from the work of indentation.4,5 The Meyer hardness1,6 is frequently used in the indentation literature. It is defined as the ratio of indentation load F to the projected contact area A of indentation impression; i.e., H ⳱ F/A. Early work showed that the Meyer hardness can be expressed as CYo for metals,1 where C is the constraint factor and is approximately 37,8 and Y is Tabor’s representative yield stress at about 8% to 10% strain. The physical meaning of the Meyer hardness and the effects of indenter geometry on hardness have been studied by a number of researchers.1–16 Atkins and Tabor10 have measured hardness of annealed and work-hardened Cu and steel samples using conical indenters of various cone angles. They showed that, with increasing cone angle, the annealed materials exhibit a continually decreasing hardness; fully workhardened materials show an initial decrease of hardness followed by an increase in hardness with increasing cone angle. For some intermediate hardening conditions, hardness decreases initially and then retains a constant value. Analytical theories exist only for conical indentation in rigid-plastic solids11 or elastic solids.12 Elastic–plastic
finite element calculations have been reported for a few indenter angles,13–15 and the elastic finite element analyses have also been made for the semi cone angles of 42.28°, 60°, 70.32°, and 80°.16 Although significant progress has been made, fundamental questions remain, including how does hardness vary with indenter geometry according to continuum mechanics? Is Tabor’s concept of representative strain applicable to materials other than metals? Is the method proposed by Atkins and Tabor for obtaining stress–strain curves using various cone angles valid for all materials? In this paper, we a
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