A quantitative analysis for the stress field around an elastoplastic indentation/contact
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Shaoxing Qu Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China; and International Center for New-Structured Materials, Zhejiang University, Hangzhou 310027, China
Yonggang Huang Department of Civil and Environmental Engineering and Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 61208
William D. Nix Department of Materials Science and Engineering, Stanford University, Stanford, California 94305 (Received 12 July 2008; accepted 17 September 2008)
In our previous paper [G. Feng et al., Acta Mater. 55, 2929 (2007)], an analytical model is proposed to estimate the stress field around an elastoplastic indentation/contact, matching nicely with the finite element analysis. The model is related to an embedded center of dilatation (ECD) in a half-space. In this paper, we focus on determining the ECD strength B* and the ECD depth x. By matching an expanding cavity model and the ECD model, we find that B* Yc3/6 and x 0.4c, where Y is the yield strength and c is the plastic zone radius. We provide a method to predict Y, c, and thereby B* as well as x through nanoindentation data, and we also demonstrate that pileup is the physical reason for the existence of the upper limit for the ratio of hardness to Y. Thus, our ECD model is completed by combining our previous paper (the analytical expression) and this paper (the essential parameters). I. INTRODUCTION
Compared with many mechanical testing techniques, the indentation method has several advantages, such as the simplicity of sample preparation, the characterization capability under wet environments,1 and the quantitative characterization at the nanometer scales.2,3 Knowing the indentation stress field is important for theoretical interest and practical applications, for example, the understanding of contact-induced fracture and delamination. Because of highly confined deformation, the indentation process generally involves plasticity and induces a plastic zone, and the associated stress solution is a longstanding classic contact mechanics problem.4 To solve this problem, there are mainly two sets of approaches: computational methods5,6 and analytical methods.7–10 Computational methods, for example, finite element analysis (FEA), can quantitatively determine the stress field; however, it is not practical to perform computational analyses for every individual indentation, and more importantly, computational methods generally lack analytical tractability and direct physical insights. On the a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2009.0097
704
http://journals.cambridge.org
J. Mater. Res., Vol. 24, No. 3, Mar 2009 Downloaded: 18 Mar 2015
other hand, because of the complexity of the problem, most of the analytical approaches can only be solved numerically.8–10 There are two brilliant closed-form analytical models: Johnson’s expanding cavity model4 and Yoffe’s surface blister model.11 By generalizing Yoffe’s model,11 Feng et al.12 proposed an embedded c
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