Depth-sensing instrumented indentation with dual sharp indenters: stability analysis and corresponding regularization schemes Yan Ping Cao, Jian Lu * LASMIS, Universit e de Technologie de Troyes 12, rue Marie Curie, BP 2060-10010 Troyes Cedex, France Received 29 September 2003; received in revised form 4 November 2003; accepted 5 November 2003 Abstract In this paper, the stability of the reverse dual indenter algorithms proposed by Chollacoop et al. [Acta Mater. 51 (2003) 3713] to determine the plastic properties of materials has been explored further by using the mathematical theory of inverse problems. To identify the representative stress r er , an explicit form of the condition number has been derived which can be used directly to ex- amine ill-conditioned or ill-posed cases in the inverse problem. Corresponding regularization schemes have been proposed to produce stable results. To determine the plastic properties of materials according to the identified representative stress, the sensitivity matrix properties have been varied to achieve more accurate results. Two methods have been suggested, i.e., introduction of prior knowledge of the model space using the well-known Tikhonov regularization scheme, and systematic investigation of the influence of the tip apex angles on the accuracy of the identified plastic properties. Guidelines for selecting the tip apex angles have also been presented. Ó 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Indentation; Dual indenter algorithms; Condition number; Ill-posed/ill-conditioned problems; Regularization schemes 1. Introduction This paper lies within the framework of research on methods used to determine the material properties based on an indentation test initially used to gauge hardness. The idea of relating the mechanical properties of mate- rials to their hardness dates back to the work of Tabor [2] in 1951. Later, Johnson [3,4] proposed a spherical cavity model for the conical indentation of elastic– perfectly plastic solids which can be used to predict the relationship between hardness and yield strength. Dur- ing the past two decades, technological advances and the need to measure the mechanical properties of materials on a small scale have led to an increasing interest in the development of systematic methods to deduce material properties from depth-sensing instrumented indentation experiments. Doerner and Nix [5] and Oliver and Pharr [6] have proposed successful methods to obtain the hardness and YoungÕs modulus from the loading and unloading P –h curves. Hill et al. [7] developed a self- similar solution for the spherical indentation of a power law plastic material and the extension of this approach to sharp indentation has been reported by Giannako- poulos et al. [8] and Larsson et al. [9]. By extending the computational models described in [8,9], Giannakopo- ulos and Suresh [10] have proposed a systematic framework for obtaining the elastic–plastic properties from the P –h data, but their results are mainly based on small deformation finite element analysis. By using di- mensional analysis and large deformation FEM, Cheng and Cheng [11,12] have derived several useful scaling relationships for conical indentation in elastic–perfectly plastic solids and elastic–plastic solids with work-hard- ening. These relationships provide new insight into the shape of indentation curves and are also useful as a guide to the FE computation of conical indentation. * Corresponding author. Tel.: +33-3-25715650; fax: +33-3-25715675. E-mail address: lu@utt.fr (J. Lu). Acta Materialia 52 (2004) 1143–1153 www.actamat-journals.com 1359-6454/$30.00 Ó 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2003.11.001