Frictionless and adhesive nanoindentation: Asymptotic modeling of size effects I. Argatov Laboratory of Friction and Wear, Institute for Problems in Mechanical Engineering, V.O., Bolshoy pr., 61, 199178 St. Petersburg, Russia article info Article history: Received 26 December 2009 Keywords: Nanoindentation Elastic layer Spherical indenter Receding contact Asymptotic modeling abstract Widely used the Bulychev–Alekhin–Shorshorov relation for analyzing nanoindentation load–displacement data to determine elastic modulus of a thin specimen does not account for the size of specimen since the BASh relation is based on analytical solutions of the con- tact problems for an elastic half-space. In order to model the substrate effect, the unilateral contact problem for a spherical indenter pressed against an elastic layer on an elastic half- space is analyzed for different types of boundary conditions imposed at the interface between the specimen and the substrate. Approximate (asymptotically exact) solutions are obtained in explicit form. The influence of the substrate effect on the incremental con- tact stiffness is described in terms of the asymptotic constants possessing information about the thickness of the specimen and depending on the relative stiffness of the substrate. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Depth-sensing nanoindentation tests are widely used for obtaining mechanical properties of small specimens and thin films (Fischer-Cripps, 2004). Recent experiments on measuring the stiffness of articular cartilage at the nanometre scale using the indentation type atomic force microscopy showed that the nanoindentation test may potentially be developed into a minimally invasive diagno- sis technique for early detection of osteoarthritis in situ (Stolz et al., 2009). The following well-known formula constitutes the basis of depth-sensing indentation technique (Bulychev et al., 1976): dP dw ¼ 2 ffiffiffi p p ffiffiffi A p E eff : ð1Þ Here, P is the external load, w is the indentation depth of the indenter tip, A is the contact area, E eff is the effective (or reduced) elastic modulus which combines elastic prop- erties of the specimen and indenter. A comprehensive analysis of the literature related to the so-called Buly- chev–Alekhin–Shorshorov (BASh) relation was performed by Borodich and Keer (2004) and Poon et al. (2008). Formula (1) is valid for axisymmetric frictionless inden- tation, and it assumes that A = pa 2 , where a is the radius of the contact area. It was rigorously proved by Pharr et al. (1992) that the relationship (1) is not depend on the geom- etry of the axisymmetric indenter. In the case of non-axisymmetric indenter, King (1987) suggested to use the shape factor to account for the geom- etry of the contact area. Thus, introducing the shape factor U c normalized by the condition U c = 1 for a circular con- tact area, we will have dP dw ¼ U c 2 ffiffiffi p p ffiffiffi A p E eff : ð2Þ Pharr et al. (1992) observed that King’s shape factor b ¼ 2U c = ffiffiffi p p does not obey the normalization condition and, therefore, it does not represent the measure of devia- tion of the geometry of the contact area from a circle. In the present paper, we clarify the geometrical meaning of the shape factor U c appeared in formula (2) for an arbitrary contact area. 0167-6636/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2010.04.002 E-mail addresses: argatov@home.ru, ivan.argatov@gmail.com Mechanics of Materials 42 (2010) 807–815 Contents lists available at ScienceDirect Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat