The correlation of indentation size effect experiments with pyramidal and spherical indenters
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The correlation of indentation size effect experiments with pyramidal and spherical indenters J. G. Swadener 1, E. P. George 2 and G. M. Pharr 2,3 1 Los Alamos National Laboratory, MST-8 MS-G755, Los Alamos, NM 87545 USA 2 Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA 3 Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996 USA ABSTRACT Experiments were conducted in annealed iridium using pyramidal and spherical indenters over a wide range of load. For a Berkovich pyramidal indenter, the hardness increased with decreasing depth of penetration. However, for spherical indenters, hardness increased with decreasing sphere radius. Based on the number of geometrically necessary dislocations generated during indentation, a theory that takes into account the work hardening differences between pyramidal and spherical indenters is developed to correlate the indentation size effects measured with the two indenters. The experimental results verify the theoretical correlation. INTRODUCTION Numerous indentation experiments over the last sixty years have shown that the hardness of crystalline materials measured by a pyramidal indenter increases with decreasing depth for small indents, which is known as the indentation size effect. Much of the early work has been reviewed by Mott [1]. Recent nanoindentation [2-4] studies have shown even greater increases in hardness for depths less than 1 µm. Ashby [5] proposed that indentation with a flat punch would produce geometrically necessary dislocations [6], which would lead to an increase in hardness. Nix and Gao [7] adapted Ashby’s concept to describe the geometrically necessary dislocations produced by a conical or pyramidal indenter. They incorporated Taylor hardening to develop a model that predicted an increase in hardness with decreasing depth in agreement with earlier nanoindentation experiments. More recent studies have shown only limited agreement with the Nix and Gao model [8-9]. It can be shown that this disagreement is due in part to the way in which the model accounts for the work hardening that occurs during indentation. Spherical indentation results are presented that do not show a size effect due to penetration depth, but rather a size effect based on the radius of the sphere. In addition, spherical indentation can be used to decouple work hardening and indentation size effects. Based on the concept of geometrically necessary dislocations, a relation is developed to correlate the size effects measured with spherical and pyramidal indenters. THEORY A summary of the Nix and Gao model is given first followed by an extension to spherical indenters. The theory is presented in brief with details and extensions to other geometries to be published separately [10]. The models assume the generation of geometrically necessary dislocations below the indent as shown schematically in Fig. 1. The Nix and Gao [7] model for a conical indenter estimates the density of geometrically necessary dislocations (ρG) as:
