Energy dissipated during spherical indentation
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relationship is derived for spherical indentation relating the dissipated energy to the ratio of hardness to elastic modulus and the ratio of indentation depth to radius. The result agrees with recent findings obtained on the basis of scaling relationships in combination with finite element simulations. Furthermore, relationships are given for hardness, elastic modulus and contact area, which permit a determination of these properties independent of the strain hardening characteristics and independent of pileup and sink-in.
Based on the methodology developed by Oliver and Pharr1 indentation testing has become a standard technique to assess elastic modulus and hardness of materials. A problem remains the consideration of pileup and sink-in.2 Relationships between the dissipated energy have been derived.3–7 which in combination with the unloading slope appear to offer a method to determine hardness, elastic modulus and contact area independent pileup and sink-in.2,3,7,8 For indentation using conical indenters, the relationship between dissipated energy, the ratio of hardness to reduced elastic modulus, and the included cone angle have been established on the basis of finite element simulations5,6 and analytically.3 In this paper, a relationship is derived for spherical indentation that relates the dissipated energy to the ratio of hardness to elastic modulus and the ratio of indentation depth to indenter radius. The first suggestion to relate the dissipated energy to the ratio of hardness and elastic modulus has been made by Shorshorov et al.9 Using scaling relationships in combination with finite element simulations, Cheng et al. obtained for a conical indenter with an included halfangle of 70.3° for the ratio of irreversible Wir to total work Wt (Berkovich indenter)6 H Wir We hf =1− = = 1 − 5.33 Wt Wt h Er
,
(1)
where We is the reversible elastic work, hf the residual depth, h maximum indention depth, H the hardness, which is proportional to Wir,10 and Er the reduced elastic modulus. It has been proven that the relationship Wir/W ⳱ hf/h is valid for conical and spherical indenters a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2004.0228 J. Mater. Res., Vol. 19, No. 6, Jun 2004
independent of pileup and sink-in.5,6,8 Malzbender and de With3 derived on the basis of the work by Oliver and Pharr1
冉
 Er Wir ⑀ We hf + =1− = =1− Wt Wt h 2 tan ␥ H
冊
−1
,
(2)
where ␥ is the included half-angle of the indenter,  is a correction factor to consider that the boundary conditions used to derive elastic contact models used in indentation allow for inward displacement of the surface, and ⑀ is a geometric constant, which takes a value of 0.72 for conical indenters and 0.75 for paraboloid indenters. Note that, since conical indenters behave similar to spherical indenter during unloading 0.75 (paraboloid) has commonly been accepted for Berkovich indenters.2 Equation (2) was derived on the basis of the relationships given by Oliver and Pharr;1 however, it appeared that
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