Identifiability of single crystal plasticity parameters from residual topographies in Berkovich nanoindentation on FCC nickel

Highlights

• A method is built to quantify the information richness of Berkovich nanoindentation imprints at grain scale.

• An identifiability index is used to design a well-posed parameter identification problem from Berkovich residual topographies.

• Combining imprints, and especially pile-ups, improves the conditioning of the inverse problem.

• The interaction matrix of FCC structure seems identifiable using three topographies.

Abstract

The information richness of imprints topographies obtained after Berkovich nanoindentation tests at grain scale is assessed for identifying all or part of the parameters of a single crystal plasticity law. In a previous paper (Renner et al., 2016), the strong potential of imprints topographies has been shown through a large experimental campaign conducted on nickel samples. A 3D crystal plasticity finite element modelling (CPFEM) of the nanoindentation experiment using the Méric-Cailletaud has also showed a large sensitivity of residual topographies to the indenter/grain orientation and to the plastic parameters, including the interaction matrix coefficients specifying the interactions between dislocations on different slip systems. This makes imprints topographies very good candidates to provide information for the single crystal parameters identification. The present paper focuses on the Méric-Cailletaud law parameters identifiability using residual topographies. A method is built to define the best well-posed inverse problem to ensure the parameters identification using a crystal plasticity finite element modelling updating (CPFEMU) method. An identifiability index proposed by Richard et al. (Richard et al., 2013) for measuring the information richness of the indentation curve is extended to the analysis of residual topographies. This index quantifies the possibility to achieve a stable/unstable solution using an inverse method. For the studied behaviour, the results show that eight of the nine Méric-Cailletaud law parameters can be identified using three topographies.

Introduction

Material parameters identification for single crystal plasticity laws remains a topical issue for a better understanding of metal behaviour at grain or polycrystalline scale (Méric et al., 1994; Fivel et al., 1997; Gérard, 2008; Gérard et al., 2013, 2009; Guilhem, 2011; Schwartz, 2011; Zambaldi et al., 2012; Guery, 2014; Guery et al., 2014; Tasan et al., 2014b, 2014a; Zambaldi et al., 2015). At the dislocation scale, some authors approached this issue using dislocation dynamics simulations (Fivel, 1997; Fivel et al., 1998; Forest and Fivel, 2001; Madec, 2001; Devincre et al., 2006). At the grain scale, the laws are also very complete and thus challenging when it comes to the material parameters identification from experimental data using inverse method. In this context, the often-missing concept of ill-posed problem, or identifiability, is essential to assess the stability and the uniqueness of the solution. The Méric-Cailletaud law is easily and effectively implemented in FE codes like ZeBuLon (Burlet and Cailletaud, 1991) [http://www.zset-software.com]. Strong nonlinearities and couplings between the dissipative phenomena during indentation make the identification of the parameters of the Méric-Cailletaud single crystal plasticity law very difficult, especially for the six interaction matrix components. These six parameters define the dislocation interactions hardening in the 12 FCC slip systems. Most recently, some works used FEM updating for identifying these interaction components, but one can notice some limitations. Méric et al. performed tensile tests on single and bi-crystals, which may be expensive and limited to materials that can be obtained macroscopically (Méric et al., 1994). Gérard et al. performed tensile tests on preloaded polycrystalline samples. These complex loadings activate the slip systems interactions. Even if most of isotropic hardening law parameters are identified, it does not provide enough information and some interaction components remain inaccessible (Gérard, 2008; Gérard et al., 2009). Guery et al. performed in-situ tensile tests on polycrystalline samples in which they measure macroscopic loads and displacement fields by digital image correlation (Guery, 2014; Guery et al., 2014). The authors have the merit of assessing the interaction components identifiability before identifying them. Besides the fact that the techniques used are quite complex, a gap subsists between experimental and numerical results. Moreover, the two latest studies are performed at the polycrystalline scale and must take the scale transition rules into account.

Based on the principle of hardness test, the instrumented indentation test, or nanoindentation test, offers three major advantages for the identification of single crystal plasticity laws. Firstly, it is easy of use. It performs continuous measurement of the applied load P on the indenter and of the indentation depth h. Thus, the indentation curve $\left(P-h\right)$ can be used to locally extract an elastic modulus and hardness when interpreted in the framework proposed in Sneddon et al., Harding and Sneddon (1945), Sneddon (1965), Oliver and Pharr (1992) and Vlassak and Nix (1993, 1994). Secondly, the stress field generated by the Berkovich indenter (three-sided based pyramid) penetration is multi-axial. One can think that the nanoindentation test activates simultaneously a large number of slip systems. Finally, it provides a direct mechanical measurement at intra-granular scale without any neighbouring grains interaction if the average grain size is large enough compared to the imprint size.

Indentation curves have been used by some authors for the identification of elastic-plastic parameters at a macroscopic scale. They mostly concluded that it is usually impossible to correctly identify plastic parameters for any kind of indenter shape (conical, pyramidal, spherical), even using “simple” elastic-plastic laws involving only two plastic parameters as the yield stress σ_y and the work-hardening exponent n of a power hardening law. Solutions are often unstable. Cheng and Cheng (1999) first pointed out this issue. Some authors have then numerically proved that, assuming that the Young modulus E of an isotropic material is known, both plastic parameters (σ_y,n) are not identifiable using the entire conical indentation curve (Capehart and Cheng, 2003; Tho et al., 2004; Alkorta et al., 2005). They showed that different materials governed by this law can lead to indistinguishable indentation curves (Fig. 1). To yield a unique solution, the dual nanoindentation technique using different conical indenter tips with half angles ranging from 60° to 80° has been proposed (Chen et al., 2007). But once again, they showed the existence of “mystical” materials which give almost indistinguishable indentation curves. This “mystical” material identification issue has been solved by some authors. Zhao et al. performed indentations on thin films which were indistinguishable by bulk indentation (Zhao et al., 2007). Thanks to the substrate effect, the elastoplastic properties can then be derived from an inverse analysis. Another solution was proposed by Ma et al. which is close to the approach considered in the present paper (Ma et al., 2012). They succeeded to correctly identify “mystical” materials using the indentation curves but also the residual imprint topographies in the inverse analysis. Other authors have examined this difficulty and have shown that non-uniqueness is an extreme case of instability of the solution (Cao, 2004; Phadikar et al., 2013).

Thus, the parameter identification of the Méric-Cailletaud law using the indentation curve is not a possible scenario since the identification of much simpler elastic-plastic laws at macroscopic scale is an issue. Bolzon et al. (2004) and Bocciarelli et al. (2005) used both the indentation curve and the imprint mapping for FEMU process. Zambaldi and Raabe said that “For the indentation of crystals, the topography of the free surface around the indent, the pile-up profile, can be used as a fingerprint of the underlying crystal deformation processes” (Zambaldi and Raabe, 2010). Indeed, pile-ups around the indentation imprint contain precious information for parameters identification. Targeting an extension of these results to crystal plasticity laws, this contribution is intended to quantify the identifiability of all or part of the Méric-Cailletaud parameters at grain scale using topographies of imprints (Fig. 2).

The paper is organized as follows. Firstly, the single crystal plasticity behaviour and the CPFEM of the nanoindentation test are described. Secondly, the identifiability method developed for measuring the information richness contained in indentation curves and residual topographies are described. Finally, the sensibility of the CPFEM and the identifiability of the plastic parameters are analysed and interpreted using one indentation curve only, one topography only and combinations of topographies.