Numerical analysis of the indentation size effect using a strain gradient crystal plasticity model D. González, J. Alkorta, J.M. Martínez-Esnaola ⇑ , J. Gil Sevillano CEIT and TECNUN (University of Navarra), P. Manuel Lardizabal 15, 20018 San Sebastián, Spain article info Article history: Received 18 July 2013 Received in revised form 24 September 2013 Accepted 1 October 2013 Available online 30 October 2013 Keywords: Indentation size effect (ISE) Crystal plasticity Strain gradient Geometrically necessary dislocations Hardness measurement Finite element method abstract This work presents a finite element analysis of the indentation size effect (ISE) experimentally observed in tests performed at submicron scale. A 3D model of a conical rigid surface indenting on a Nb single crystal at different depths has been developed. The bcc Nb material has been characterized within a finite-strain framework through a crystal plasticity model incorporating strain-gradient hardening. The hardness evolution for different material orientations and for different initial dislocation densities has been studied. The numerical results are compared with predictions of existing analytical models and with experimental results. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction From an experimental point of view, the indentation size effect (ISE) is a well-known phenomenon that becomes apparent at submicron scale with a decrease in the hardness measure as the indentation depth increases. Numerous works have reported the ISE experimentally, with different types of materials and arrange- ments [1–6]. The analytical model of Nix and Gao [5] suggests a linear rela- tionship between the square of the hardness and the inverse of the penetration depth. Many studies have confirmed this linearity above a certain depth; however, more detailed inspections at ex- tremely low indentation depths show a deviation from this linear- ity [7–10]. In this work, we try to cast some light on this problem. The objective is to study the ISE from a numerical point of view with the aid of a finite element (FE) model of an indentation exper- iment. For this purpose, a 3D model consisting of a conical rigid surface with a round tip indenting on a single crystal of Nb has been constructed. The bcc material has been simulated by means of a crystal plasticity model, implemented within a finite-strain framework. Of capital importance, the size effect phenomenon present in the experiments has been captured in the simulation work incorporating a strain-gradient formulation in the constitutive equations of the material model. The general-purpose FE program ABAQUS has been used in this analysis, complemented via FORTRAN user subroutines. Among the pioneering analysis on strain-gradient plasticity, of special relevance is the numerical work of Fleck and Hutchinson [11,12]. Their plastic models capture the strain-gradient effects using a set of the so-called ‘material lengths’ within a coupled- stress general framework. These phenomenological length mea- sures are necessary in their formulation to calculate an equivalent local strain, acting as scaling factors of a close-related measure of the strain-gradient value, obtained with the second derivatives of the displacement field. A dependence of the material hardening on other magnitudes apart from the strain is therefore established, in contrast with the conventional plasticity theories. As previously mentioned, the material model defined here also considers a strain gradient measure, though a different approach is used to estimate its influence in the material hardening equations. The constitutive equations descend to the crystallographic slip nat- ure of plastic deformation, and, under this crystallographic frame- work, a local density of geometrically necessary dislocations (GND) is defined. The GND concept appears in this model closely linked to the strain gradient measure, as the works of Nye [13] and Fleck et al. [14] show. In order to prove and explain the ISE effect with this material model, a series of different indentation depths, ranging from 0.01 lm to 66 lm, have been performed on the Nb material model described. The numerical results obtained have been compared with those predicted by the analytical models of Nix and Gao [5] – calculated for a perfectly sharp tip – and Alkorta et al. [6] – that improves the previous one by adding a rounded tip to the conical indenter and 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.10.004 ⇑ Corresponding author. Tel.: +34 943 212800. E-mail addresses: dgonzalez@ceit.es (D. González), jalkorta@ceit.es (J. Alkorta), jmesnaola@ceit.es (J.M. Martínez-Esnaola), jgil@ceit.es (J. Gil Sevillano). Computational Materials Science 82 (2014) 314–319 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci