Size effects in nanoindentation

Abstract

Microindentation hardness experiments have repeatedly shown that the indentation hardness increases with the decrease of indentation depth, i.e., the smaller the harder [3, 5, 9, 30, 31, 37, 43, 51]. On the basis of the Taylor dislocation model [48, 49] and a model of geometrically necessary dislocations (GND) underneath an indenter tip shown in the inset of Fig. 2.1, Nix and Gao established the relation between the microindentation hardness H and the indentation depth h (H/H0)2 =1+ h*/h, (2.1) where h* is a characteristic length on the order of micrometers that depends on the properties of indented material and the indenter angle, and H0 is the indentation hardness for a large indentation depth (e.g., h≫h*) [33]. The above relation predicts a linear relation between H2 and 1/h, which agrees well with the microindentation hardness data for single crystal and polycrystalline copper [31] as shown in Fig. 2.1, as well as for single crystal silver [30]. The nanoindentation hardness data, however, do not follow (2.1) [10, 12, 13, 15, 23, 28, 29, 45, 46]. Here nanoindentation and microindentation typically refer to the indentation depth below and above 100 nm, respectively. As shown in Fig. 2.2, Lim and Chaudhri's nanoindentation hardness data for annealed copper start to deviate from (2.1) (the dotted straight line of the Nix-Gao model for annealed copper (Figure presented) in Fig. 2.2) when the indentation depth h is on the order of 100 nm [28]. Even though the indentation hardness continues to increase as h decreases, the hardness data are significantly lower than the straight line predicted by [34]. Swadener et al. also showed that the nanoindentation hardness data for annealed iridium are smaller than that given by (2.1) when the indentation depth h becomes submicrometer (see the dotted straight line of the Nix-Gao model for iridium in Fig. 2.2) [45]. Recently, Feng and Nix and Elmustafa and Stone found that, once the indentation depth is less than 0.2 μm, (2.1) does not hold in MgO (see the dotted straight line of the Nix-Gao model for MgO in Fig. 2.2) and in annealed a-brass and aluminum, respectively [11, 15]. There are two main factors for the discrepancy between (2.1) and the nanoindentation hardness data. (i) Indenter tip radius. Equation (2.1) holds only for "sharp," pyramid indenters since the effect of indenter tip radius (typically around 50 nm) has not been accounted for [23, 39]. Qu et al. studied pyramid indenters with spherical tips, and found that the finite tip radius indeed gives smaller indentation hardness than (2.1) [39]. Qu et al. also studied spherical indenters, and established an analytic relation between the indentation hardness and indentation depth that is very different from (2.1) [41]. On the basis of the maximum allowable GND density Huang et al. established a model for the effect of finite tip radius in nanoindentation [20, 21]. (ii) Storage volume for GNDs. Equation (2.1) assumes that all GNDs are stored in a hemisphere of radius a, where a is the contact radius of indentation. Such an assumption may not hold in nanoindentation. Swadener et al., Feng and Nix, and Durst et al. proposed to modify this storage volume for GNDs [10, 15, 45]. Huang et al. established a model of nanoindentation based on the maximum allowable GND density [20]. Other factors may also contribute to the discrepancy between (2.1) and nanoindentation hardness data, such as the intrinsic lattice resistance or friction stress [38], surface roughness [29, 52], and long-range stress associated with GNDs [13, 14]. The objective of this chapter is to review the analytic models based on the maximum allowable GND density to study the nanoindentation size effect, and validate them by comparing with the experiments. The analytical models give simple relations between the indentation hardness H and the indentation depth h or contact radius in nanoindentation. This chapter is outlined as follows. The Taylor dislocation model [48, 49], which has been widely used to explain the indentation size effect, is summarized in Sect. 2.2. The conventional theory of mechanism-based strain gradient plasticity established from the Taylor dislocation model is reviewed in Sect. 2.3. Sections 2.4 and 2.5 review the nanoindentation models for the sharp, conical indenters and for spherical indenters, respectively. © Springer Science+Business Media, LLC, 2008.