Indentation of self-similar indenters: An FEM-assisted energy-based analysis
Introduction
The instrumented indentation technique is widely used for probing the local mechanical properties of materials. The most important result of an instrumented indentation experiment is the load–displacement curve (P–h curve). The P–h curve of a self-similar indenter leads to the following parameters: the loading curvature C, the ratio of elastic to total indentation work (or vice versa ), the ratio of residual to maximum indentation depth and the initial slope of the unloading curve S (Fig. 1). However, some researchers state that the P–h curve of a self-similar indenter leads, at most, to two independent indentation parameters (Cheng and Cheng, 1999, Tho et al., 2004, Loubet et al., 1984, Alcorta et al., 2005, Le, 2008, Le, 2009). Traditional analysis of P–h curves according to Oliver and Pharr allows for the calculation of important parameters such as hardness and Young׳s modulus (Oliver and Pharr, 1992, Oliver and Pharr, 2004).
Of the parameters, that are directly accessible from a P–h curve, there is a keen interest in relating the indentation results obtained from P–h curves to the parameters of constitutive laws (chiefly the Ludwik-power law, see Section 2) (Venkatesh et al., 2000, Dao et al., 2001). These material parameters characterize the stress–strain behavior of a material (mostly metals) which is normally derived by uniaxial tensile or compression tests. Due to the high complexity of the indentation of an elasto-plastic material – including non-linear material behavior, large strains, and contact – there is no one-to-one correlation between the indentation results and the material parameters of Ludwik power-law materials (Alcorta et al., 2005). In order to find relationships between indentation results and material parameters, different inverse methods have been developed based on dimensional analysis or optimization algorithms (Venkatesh et al., 2000, Dao et al., 2001, Giannakopoulos and Suresh, 1999, Cheng and Cheng, 2004, Ogasawara et al., 2006, Antunes et al., 2007, Yu et al., 2003, Cao and Lu, 2004, Cao and Lu, 2005, Tho et al., 2005, Zhao et al., 2006, Nakamura and Gu, 2007, Kopernik et al., 2008). These inverse methods can be used to calculate the material parameters from given P–h curves. Most of them are based on the concept of representative strain. However, the physical basis of the representative strain is contentious, although these methods work well for most combinations of material parameters (Tabor, 1951a, Chaudhri, 1998, Ogasawara et al., 2007, Branch et al., 2010). Different combinations of material parameters were found to result in equal or indistinguishable P–h curves. As a consequence, inverse methods can fail and result in non-unique inverse solutions (Chen et al., 2007, Tho et al., 2005, Cheng and Cheng, 1999, Capehart and Cheng, 2003, Tho et al., 2004, Liu et al., 2009). In this study, an energy-based approach is used to analyze the relationships between the indentation results (C and ) and the material parameters of the Ludwik-power law. These are discussed in terms of solving the forward and inverse indentation problem.
Section snippets
FEM modeling
FE-calculations were performed using the FE-software (version 6.11). The axisymmetric 2D FE-model is based on prior work (Pöhl et al., 2013). The geometry, mesh, and boundary conditions of the model are shown in Fig. 2. The indenter is modeled as an axisymmetric conical rigid tip with an included half-apex angle of 70.3°. A conical indenter with a half-apex angle of 70.3° leads to the same area-to-depth function as that of a Berkovich indenter. Numerous studies have shown that this
Energy-based analysis
The indentation of an isotropic, elasto-plastic material by a rigid, self-similar indenter is schematically illustrated in Fig. 3. The elasto-plastic behavior of the indented material is assumed to follow Ludwik-power law (see Section 2, Eq. (1)). A plastic zone and an elastic zone are induced into the material. The plastically deformed zone has volume and mean plastic strain . In this study, the mean plastic strain is defined according to Branch et al. as the average volumetric
Summary and conclusions
In this study, an energy-based approach was combined with FEM simulations to analyze indentation of elasto-plastic materials with self-similar indenters. Relationships between the indentation results and the parameters of Ludwik-power law materials (K, E, and n) were derived and analyzed. The following conclusions can be drawn:
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Kick׳s law is the natural outcome of the energy-based approach. The derived relationships are in agreement with Kick׳s law and confirm it analytically.
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The loading
Acknowledgment
The authors gratefully acknowledge financial support by the Kompetenzzentrum Hydraulische Strömungsmaschinen at the Ruhr-Universität Bochum (Germany).
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