Evaluation of an inverse methodology for estimating constitutive parameters in face-centered cubic materials from single crystal indentations
Introduction
The prediction of the mechanical response and internal structural evolution of crystalline solids is challenging due to the inherent anisotropy of the elastic and plastic properties of single crystals. Integration of anisotropy into crystal plasticity (CP) models of deformation has been quite successful (see Roters et al., 2010b) for a recent review) and is most relevant for metals that exhibit strong crystallographic texture resulting from processing or when one or more dimensions of an engineering component approach the internal grain size of the material. While the sophistication of the underlying constitutive description of deformation and structure kinetics varies among different CP approaches, for all of them the accuracy of prediction largely depends on the values selected for the adjustable constitutive parameters.
Correct identification of constitutive parameter values is an ongoing area of significant research interest as it involves the inverse problem of matching simulation outcomes to experimental reference. One way would be to use experimental reference from polycrystalline (macroscopic) deformation under unidirectional tension or compression in connection with either an isotropic plastic description (Kajberg and Lindkvist, 2004, Mahnken and Stein, 1996) or a phenomenological crystal plasticity material model (Herrera-Solaz et al., 2014). In (Herrera-Solaz et al., 2014), single crystal material parameters were extracted using a gradient-based Levenberg–Marquardt optimization algorithm that matched stress–strain responses of finite element simulations of representative volume elements to corresponding experiments. A reproducibility study for this approach revealed high reproducibility for some parameters as compared to others (Herrera-Solaz et al., 2015), indicating either the presence of multiple local minima in the objective function surface or lack of sensitivity for certain parameters over others, and thus, calling for further investigation. An alternative and more direct way is to use single crystal (microscopic) deformation since it excludes the problematic convolution caused by multiple grains deforming and interacting in parallel and appears, hence, to be a more promising venue. Instrumented nano-indentation experiments have been an efficient way to generate such a microscopic deformation response from multiple orientations in the form of load–displacement curves and surface topography after indentation.
Indentation experiments as a tool to identify material parameters was first proposed by Huber and Tsakmakis (1999) and Huber et al. (2001) by using an artificial neural network (ANN) and only indentation load–displacement response. Finite element (FE) simulations were performed based on an isotropic plasticity constitutive model including isotropic and kinematic hardening to generate simulated load–displacement curves. Input to the ANN included the simulated (reference) load–displacement curves while the output corresponded to the parameter values. Multiple FE simulations were performed to train the network. The effectiveness of this FE-ANN methodology was tested by implementing it to identify the material parameters for different materials using experimental load–displacement curves by Klötzer et al. (2006). It was observed that the methodology could only capture the elastic response with certainty, while there existed a high scatter among the identified plastic parameters. A modification to the FE-ANN methodology was proposed by Hajali et al. (2008) by considering only the loading part of the load–displacement response and using dimensionless material parameters for their identification. However, significant deviation from experimental response to that obtained from FE-ANN parameters were obtained especially in the plastic regime. A different approach, but still using only the indentation load–displacement response, was adopted by Nakamura et al. (2000) wherein the sequential stochastic Kalman filter algorithm was incorporated, used mostly in signal processing, to estimate Young's modulus and Poisson's ratio for functionally graded materials. The strategy was also extended to identify plastic parameters for transversely isotropic materials (Nakamura and Gu, 2007, Yonezu et al., 2009). Subsequently, with the advent and ease of access in measuring surface topographies, such as by atomic force microscopy or white light interferometry, Bolzon et al. (2004) proposed to include the residual imprint after indentation for estimating the constitutive parameters. They assumed an isotropic elasto-plastic material model and observed that inclusion of topography deviation in the objective function generally improved the accuracy of parameter estimation in the gradient-based deterministic optimization algorithm that was used. Bocciarelli et al., 2005, Bocciarelli and Maier, 2007 extended the scope to elastic-perfectly plastic material models with orthotropic symmetry (using a yield surface approach (Hill, 1948)) for orthotropic materials against both artificial reference data and real experimental data. The general shortcomings associated with gradient-based optimization and strong simplifications in the constitutive description in terms of the final optimization solution depending on the initial parameters was confirmed. Nevertheless, with a priori knowledge of viable plastic parameter ranges such an approach gave satisfactory results for multi-layered systems (Moy et al., 2011b) and aluminum alloys (Moy et al., 2011a). To reduce the computation cost of evaluating the gradient numerically, several studies focussed on estimating high-quality approximations of the solution field from a limited number of numerical results using proper orthogonal decomposition (POD) with radial basis functions (Arizzi and Rizzi, 2014, Bocciarelli et al., 2014, Bolzon and Talassi, 2013, Buljak and Maier, 2011, Hamim and Singh, 2017). Other simplifications to reduce computation time of calculating the gradient for gradient-based optimization algorithms have also been investigated using approaches such as parameter coupling (Rauchs and Bardon, 2011), solving the adjoint problem (Constantinescu and Tardieu, 2001) and making dimensionless equations based on a priori FEM simulations (Wang et al., 2010, Yonezu et al., 2010).
In all above studies, deformation behavior has been approximated by simplified material models that hardly consider plastic anisotropy of the material. Sánchez-Martín et al. (2014) postulated values for crystal plasticity constitutive parameters (of a particular Mg alloy, MN11) by comparing hardness values from grain indentation simulations using sets of parameters reported for similar materials in the literature, and choosing that one giving the closest fit to the indentation experiments. Comparing responses between experiment and simulation of axisymmetric instrumented nano-indentation into individual grains by incorporation of a crystal plasticity model was first performed by Zambaldi and Raabe (2010) on face-centered tetragonal γ-TiAl. Their study lead to the inverse methodology of identifying crystal plasticity parameters using gradient-free Nelder–Mead (NM) simplex optimization based on matching the simulated load–displacement and surface topography (Zambaldi et al., 2012) to measured data for hexagonal commercially pure titanium (cp-Ti). However, the constitutive parameters obtained for cp-Ti differed significantly from other reports (see (Li et al., 2013) for summary). This discrepancy motivates the current study to evaluate the effectiveness of identifying constitutive parameters from indentation responses. The present study implements the inverse methodology suggested by Zambaldi et al. (2012) on face-centered cubic (fcc) crystal systems using “pseudo-experimental” reference data to determine the robustness of such an approach with material orientation.
After outlining the simulation set up including the material model and optimization method, the efficacy of the used fitness function is examined. Following, a sensitivity analysis is performed for seven crystal orientations distributed across the orientation space to determine the most influential adjustable parameters that can then be used as design variables for optimization. Finally, the single crystal indentation inverse analysis using fitness obtained from pairs of crystal orientation was performed to check whether any additional improvement can be achieved.
Section snippets
Finite element discretization
Similar to previous work (Zambaldi et al., 2012), a three-dimensional finite element (FE) model of conospherical indentation (see Fig. 1) was generated using the open source toolkit “STABiX” (Mercier and Zambaldi, 2014). The single crystalline cylindrical substrate of height 4 μm and diameter 8 μm was discretized by 4596 linear hexahedral elements based on a mesh sensitivity study (Section 4.1). Nodal displacements were fixed on the bottom and outer surfaces. The indenter of radius 1 μm and
Universal optimization module
Fig. 2 illustrates the optimization strategy that iteratively adjusts the constitutive parameters, performs crystal plasticity finite element simulation(s) of single crystal indentation, and compares the resulting load–displacement data and/or surface topography to their reference until the deviation meets a given tolerance. The optimizer used in this study was implemented as a general Python class that can be equipped with different stochastic and deterministic optimization algorithms such as
Results
The pseudo-experimental reference indentations were obtained with a parameter set chosen as , , , , , and , based on those of aluminum (Kalidindi and Schoenfeld, 2000). The six dislocation interaction parameters were based on (Kubin et al., 2008) with values of self = 1.4, coplanar = 1.4, collinear = 3.0, Hirth lock = 1.0, glissile junction = 1.4, Lomer lock = 1.4. In response to the performed sensitivity analysis, the three most influential
Discussion
The analysis of reproducibility and robustness for different objective functions reveals a few noteworthy points that are addressed in the following.
The combination of both original objective functions, i.e. , appears to increase the overall steepness of the resulting fitness hypersurface, particularly in the vicinity of the correct parameter set (target). In other words, increases faster with “distance” from the target compared to either or . This can be
Conclusion
Based on the comprehensive analysis performed in this study, it can be concluded that inverse indentation analysis using single crystal nanoindentation is a reliable avenue towards successful identification of material parameters in fcc materials. The sensitivities of the four adjustable parameters of the phenomenological crystal plasticity constitutive law used, , , , and a, were indifferent to both crystal orientation and the stress exponent n that characterizes the material flow
Acknowledgments
Financial support from the National Science Foundation through grant DMR-1411102 is gratefully acknowledged.
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