A general relation for contact stiffness including adhesion in indentation analysis
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The Maugis–Barquins (MB) solutions for the adhesive contact between an axisymmetric indenter and an elastic half-space are modified by incorporating the interfacial energy defined by the real area of contact. With the modified MB solutions, general relations for contact stiffness including adhesive effects in indentation analysis are derived. Numerical calculations showed that the difference in expected stiffness for the modified MB model compared to the standard MB results can be significant at low loads of interest in atomic force microscopy measurements and also for indentation tests at high load if the interfacial energy is large (;0.1 J/m2) and the material is soft (Young’s modulus #100 MPa).
I. INTRODUCTION
A goal of indentation characterization is to extract the elastic modulus and indentation hardness of materials from load-displacement curves. Given that the basic parameter measured in indentation testing to extract hardness and elasticity is the contact stiffness, the relationship between the contact stiffness and the elastic modulus is of key importance. On the basis of the Sneddon contact solutions without adhesion, Oliver and Pharr1,2 derived simple contact stiffness relations for a general axisymmetric indenter and hence developed an improved method to measure the hardness and elastic modulus of a material from indentation load-displacement data. The relations have been widely used in indentation characterization. In the Oliver–Pharr analysis,1,2 the effect of adhesion between the indenter and the material surface is not taken into consideration. The influence of adhesion between contacting surfaces becomes important for smooth surfaces and for biological or soft materials.3–5 The incorporation of adhesion in elastic contact was addressed in the Johnson–Kendall–Roberts (JKR) model,6 which is now generally used for contact problems with compliant materials. A review of JKR and other adhesive contact models can be found in Ref. 7. Subsequently, on the basis of physical relationships between the adhesive contact and the fracture mechanics, Maugis and Barquins8–10 extended the JKR model using the Sneddon solutions11 to adhesive contacts of general axisymmetric indenters on an elastic half-space. The Maugis–Barquins (MB) solutions can be applied to obtain general relations including adhesive effects for indentation analysis, and results for some specific indenter geometries12,13,17 have been obtained. a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2011.132 1406
J. Mater. Res., Vol. 26, No. 11, Jun 14, 2011
http://journals.cambridge.org
Downloaded: 16 Mar 2015
The adhesive contact surface in the MB solutions8–10 is defined by the projected area of contact between the indenter and the surface (see Fig. 1), not the real area of contact between the surfaces. This difference is not a concern for a flat-ended indenter as in this case, the MB solutions are exact. The MB solutions are also a good approximation for a spherical indenter of large radius.6 However, errors may
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