Evaluation of an inverse methodology for estimating constitutive parameters in face-centered cubic materials from single crystal indentations

Highlights

• Updated Nelder–Mead simplex algorithm to avert infeasible points in parameter space.

• Combining force and topography in fitness improves reproducibility and robustness.

• Parameter identification is stable across multiple indented crystal orientations.

Abstract

The feasibility to determine the adjustable parameters of single crystal plasticity constitutive laws by an inverse approach that minimizes the deviation between the measured and simulated indentation response of individual grains (of a polycrystalline sample) is investigated for the case of face-centered cubic (fcc) lattice structure. Optimization uses the Nelder–Mead (NM) simplex algorithm, which was modified to navigate parameter space regions where objective function evaluations fail. A phenomenological power-law is assumed as the constitutive description for crystal plasticity. Simulated cases of indentation with prescribed constitutive parameter values serve as the virtual reference. A sensitivity analysis revealed that the initial and saturation slip resistance ${\tau }_0$ and ${\tau }_{\text{sat}}$ are the most dominant parameters while the hardening slope $h_0$ has less influence. Reproducibility and robustness are analyzed for different objective functions involving the load–displacement response and residual surface topography for several indentation crystal orientations. Concurrent optimization of load–displacement and topography consistently provided the least scatter in the optimized parameter values from the target solution compared to either one individually and essentially independent of indentation crystal orientation. Deviations in slip activity were typically of the same order of magnitude as the combined deviations of load–displacement and surface topography response. Optimization of more than one crystallographic indentation response at once did not improve the parameter estimation quality but proportionally increases the evaluation effort. It is concluded that for fcc materials one single crystal indentation experiment suffices to closely quantify the two most influential parameters of a phenomenological constitutive plasticity law when the objective function of the modified NM simplex algorithm proposed herein combines the load–displacement response and residual surface topography.

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.