Journal Club for February 2023: Understanding Engineering Alloy Behavior by Combining 3D X-ray Characterization and Finite Element Modeling

Darren C. Pagan a, Romain Quey b, Matthew P. Kasemer c

a Materials Science and Engineering, The Pennsylvania State University, University Park PA 16802, US

b Mines Saint-Etienne, Univ. Lyon, CNRS, UMR 5307 LGF, F–42023 Saint-Etienne, France

c Mechanical Engineering, The University of Alabama, Tuscaloosa AL 35487, US

1. Introduction

Modeling the deformation response of polycrystalline metallic alloys has developed rapidly in the past three decades from traditional mean-field and adjacent methods (e.g., Taylor, Sachs, and self-consistent modeling) to full-field crystal plasticity methods which consider explicit representations of microstructures (e.g., crystal plasticity finite element method [CPFEM] and crystal plasticity fast fourier transform [CPFFT]). While modeling methods have enjoyed rapid development, the advent of full-field models was not immediately met with commensurate gains in experimental microstructural and micromechanical characterization methods. Largely, comparison of micromechanical simulations to experimental me- chanical data has been limited to the macroscopic response or (more rarely) orientation averages within aggregates [1]. Consequently, model calibration (and thus model predictions) has suffered from a degree of ambiguity, owing to the fact that non-unique parameter sets may lead to similar macroscopic responses, yet very different micromechanical responses [2]. As such, the relative lack of robust experimental micromechanical data has held back the 15 verification, validation, and proper progress of crystal plasticity modeling.

Within the past decade, however, the field has been witness to an extremely rapid development of experimental techniques to track the micromechanical response of materials. While 2D techniques (e.g., electron backscatter diffraction and high-resolution digital image correlation [3, 4]) have demonstrated the ability to track the development of elasticity and plasticity on the surfaces of alloy samples, the rise of non-destructive 3D measurement modalities have been particularly attractive. Chief among these methods are synchrotron 3D X-ray diffraction microscopy (3DXRD) [5], high energy X-ray diffraction microscopy (HEDM) [6, 7], diffraction contrast tomography (DCT) [8], and more recently laboratory X-ray [9] and neutron diffraction [10], which utilize large panel area detectors and rapid diffraction simulations to reconstruct microstructures and micromechanical response in 3D. With these methods, the micromechanical response of all individual grains in appreciable volumes of material (order mm3) are tracked in situ during deformation, allowing for an unmatched ability to track the spatiotemporal evolution of elasticity and plasticity. In addition, these techniques provide the ability to non-destructively characterize the three- dimensional geometric morphology of microstructures including the spatial distribution of orientations at intrangranular length scales. In conjunction, these methods may be used to offer a cohesive material state characterization and mechanical behavior measurement and modeling workflow [11].

Owing to the development of fast, wide dynamic-range X-ray detectors, precision load frames, synchrotron control software, and data reduction software, the once-arduous task of collecting these 3D data has begun transitioning from “experiment” to “measurement”. As these methods have matured and become more standard and accessible, along with expected continued progress in this regard, the ability to (more rapidly and more easily) gather mechanical data at relevant length scales presents a path forward to develop better deformation modeling frameworks. Model instantiation and calibration can now take place with grain-scale mechanical data, leading to the optimization of more-deterministic modeling parameter sets [12, 13], and consequently better micromechanical predictions. Beyond traditional uses, these can be integrated even more closely with modeling (i.e., data assimilation) and surrogate model training to provide novel predictive capabilities and insights 45 into alloy deformation.

Here, we give examples of how these new data and finite-element micromechanical modeling of metallic alloys can be brought together with increasing amounts of integration: instantiation, calibration, assimilation, and hybrid surrogate model development (transfer machine learning).

2. Instantiating Crystal-Plasticity Simulations with 3D Experimental Data

The availability of 3D data of the microstructure of polycrystalline materials, and quantitative comparison to simulations, have motivated the development and use of novel methods for the generation of accurate representations of polycrystalline microstructures. For simulations, 3D measurement techniques provide quantitative information on the microstructures that generally cannot be reproduced by the models historically used in numerical simulations of polycrystal deformation, which were often based on purely mathematical models such as regular-shaped grains or Voronoi tessellations. The microstructural information provided by the new experimental techniques can be of different types. Far-field techniques provide accurate information of the average orientation, centroid location, and size of the grains within a polycrystal, while near-field techniques provide accurate information of the polycrystal morphology. Quey and Renversade [14] presented a general and fully-automated method to generate a polycrystal morphology from the various types of information that these experimental characterization techniques may provide, which is implemented in the Neper code [15]. The method is based on a (constraint-free) optimization of Laguerre tessellations (or “weighted Voronoi tessellations”), which can virtually represent any polycrystal morphology, in the framework of convex cells. Using the minimization function of the optimization approach, different types of microstructural information can be provided as input (statistical distributions of grain sizes, positions and volumes of the grains, images of the grains), and a geometrical tessellation conforming to said input is generated. This applies for the standard (single-phase or multi-phase) polycrystal, while an extension of the method allows for the representation of the multiscale microstructures found in many industrial alloys [4]. The 3D tessellations can then be meshed for CPFEM simulations, at the desired resolution (in terms of number of elements per grain), or “rasterized” for CPFFT simulations. A schematic of the various steps of moving from raw 3D microstructural data to a computable mesh is given in Fig. 1.

Figure 1: Schematic of the various levels of processing from raw 3D microstructural data to cleaned microstructural data to a tesselation to a mesh upon which finite element simulations can be performed (left to right) [14].

While instantiation and meshing precede the actual simulation and post-processing, this typical workflow also benefits from close dialogue with the associated numerical tools. With this in mind, the Neper code [15] for polycrystal generation and meshing and the FEPX code [16] for parallel crystal-plasticity finite-element simulation have recently come together to form an extensive and homogeneous ensemble for polycrystal plasticity studies [17]. Beyond the direct benefits resulting from the convergence of the two codes, the project has also led to the standardization of the typical workflow of polycrystal plasticity studies. The post-processing capabilities include operations such as the computation of grain (or mesh) average values or other statistical treatments. The same post-processing operations can be applied to the experimental data, which makes direct comparisons between experiments and simulations straightforward.

3. Calibrating Micromechanical Models

While maybe obvious, it is worth stating that the accuracy of a micromechanical simulation is influenced by the accuracy of the material properties used in the simulation. Even with a perfectly accurate constitutive model, micro and macroscopic mechanical predictions will be inaccurate if the wrong material parameters are used. An on-going challenge for the crystal plasticity (and broader micromechanics) community is the determination of accurate microscale material parameters. Typically, micromechanical material parameters (e.g., elastic moduli, slip system strengths, rate sensitivities) are determined by fitting macroscopic data. However, it is well established that non-unique sets of microscale parameters can be used to fit a macroscopic response [2], so while the macroscopic response may be correct, confidence in conclusions drawn regarding micromechanical response is diminished. A benefit of these 3D micromechanical data is that model parameters in simulations can be fit at the length scales at which they influence the mechanical response. In a recent study by Boyce et al., a novel methodology was presented that combined HEDM data with elasticity FEM modeling to optimize single crystal moduli for Inconel 625 [18]. While often overlooked, accurate single crystal elastic moduli, particularly for standard engineering alloys, are often difficult to find in the literature. Historically, to measure single crystal elastic moduli, large single crystals of various orientations needed to be grown (often a major undertaking, and for some alloys an impossible task), and then single crystal elastic moduli were determined from either mechanical testing or acoustic measurements. In the work of Boyce et al., the hundreds of grain elastic strain responses measured during in situ uniaxial tension were used to avoid arduous single crystal testing. Through a minimization process, single crystal moduli that best predicted the elastic strain response of all grains (spanning a wide range of orientations) with respect to the measured elastic strain tensors were found. The applicability of this approach is wide, as many (most) engineering alloys in literature are missing rigorous and appropriate experimental measurement to provide accurate moduli for micromechanical modeling.

A similar approach was adopted by Pagan et al. to determine the strengths of the various families of slip systems in Ti-7Al [19]. Like single crystal moduli, the traditional approach for determining slip system strengths relies on measuring the critical resolved shear stresses for slip system activation by loading single crystals along different crystallographic orientations. Measuring slip system strengths, however, is even more difficult than elastic moduli because, ideally, only a single slip system should be activated at yield, which isn’t always possible. For example, hexagonal-symmetry crystals, which exhibit a propensity to slip on different families of slip systems (prismatic, basal, and pyramidal), often have large differences in CRSS, making activation of high-strength families alone not possible. Instead, Pagan et al. used the evolving grain scale stresses of over five hundred grains to estimate the bounds of a single crystal yield surface and in turn determine evolving slip system strengths of prismatic, basal, and pyramidal slip systems (see Fig. 2A). This work highlights the need for micromechanical data, because while the macroscopic response showed relatively little work hardening, significant amounts of softening and hardening (depending on the family of slip systems) were observed at the microscale. If only fitting simulation parameters to the macroscopic response, this microscale slip behavior would not have necessarily been captured by the model’s predictions. The ultimate consequence ultimate need for these data is showcased by the fact that the inclusion of this microscale softening behavior (only observable with new 3D techniques) in a CPFEM framework was found to increase plastic deformation localization, which is linked to fatigue and fracture failure (illustrated in Fig. 2B).

Figure 2: A) Measured evolving slip system strengths (τ*) in various families of slip systems in Ti-7Al measured from far-field HEDM data. B) Effect of increased plastic deformation localization (shading) with inclusion of slip system softening into crystal plasticity model. [19]

4. Data Assimilation Approaches

In a traditional micromechanical modeling approach, the evolution of a stress field is determined using either an implicit or explicit formulation based solely on initial and boundary conditions. Accuracy is then a function of the material parameters, constitutive model, and solution scheme. However, with 3D micromechanical data, interesting opportunities to gain new insights into mechanical response are possible by directly combining micromechanical modeling with measured data. This approach, dubbed “data assimilation”, is common in the geosciences [20] and meteorology [21] fields, where data is plentiful, but predictive models are known to lack perfect accuracy. By including data continuously into the model predictions, accuracy can be greatly enhanced without new constitutive models, which can be of great value to engineering applications of micromechanics.

As a micromechanics example, formulations have long existed for determining the stress fields that arise due to incompatible plastic deformation [22]. However, the utility of such a formulation is limited because, historically, there were no means of determining what this plastic deformation field is. This has changed with the 3D characterization techniques now available. For example, Pagan and Beaudoin combined high-resolution 3D far-field diffraction data with continuum dislocation mechanics to solve for the stress field around a pair of shear bands [23]. High-resolution diffraction data was utilized to reconstruct the plastic deformation fields around the shear bands, and importantly, the incompatible portion of this plastic deformation field was used as input for a continuum dislocation formulation, which allowed for the stress fields around these slip bands to be solved for. Figure 3A shows both the reconstructed 3D deformation and stress fields. In this case, measurement and modeling were used together in a fully integrated fashion to address a long standing challenge associated with the use of continuum dislocation mechanics.

Figure 3: A) Combining plastic deformation fields from 3D X-ray data (γ) and continuum dislocation mechanics enables reconstruction of stress fields (σVM) around shear bands in a Cu single crystal [23]. B) Combining continuum dislocation mechanics with measured 3D grain-average elastic strain data (ϵ) enables full intragranular stress fields (σ) to be reconstructed in deforming polycrystals [24]. C) Combining measured 3D orientation fields with crystal plasticity enables reconstruction of crystallographic slip fields (Γ).

This combination of combining 3D X-ray data with continuum dislocation mechanics was extended by Naragani et al. to reconstruct intergranular stress fields in deforming polycrystals without evolving plasticity descriptions [24]. The full 3D orientation field of an Inconel 625 polycrystal was measured using near-field HEDM [25] along with the grain average elastic strain tensors in situ during monotonic plastic deformation. Typically, far-field HEDM is only capable of probing the average elastic strain and stress states of individual grains; however, combining these data with a micromechanics formulation derived to maintain compatibility and equilibrium, more information can be extracted from the data than can be measured directly (as seen in Fig. 3B, which shows the raw 3D and reconstructed intragranular stress fields). Of particular value is the ability to now probe the stress concentrations that develop around grain boundaries which may ultimately influence macroscopic 170 fatigue and fracture properties.

In a similar fashion, Pagan et al. combined near-field HEDM and far-field HEDM along with constitutive modeling to extract more from the data than would be available directly [26]. In this case, crystal plasticity was used to reconstruct the slip field in a deforming Ti-7Al polycrystal. Direct characterization of crystallographic slip through diffraction methods has traditionally not been possible, as dislocations moving through a crystal lattice leave the lattice unchanged [27]. As the lattice is what is interrogated with diffraction, complete crystallographic slip is generally “invisible”. However, by looking at heterogeneities of orientation that remain after heterogeneous plastic flow through the lens of crystal plasticity kinematics, determining the slip that must have occurred for the deformation to happen can be possible. Figure 3C shows the total crystallographic slip and slip resolved on to different slip system families in the Ti-7Al polycrystal after 2% strain. With these extended slip fields, it was found that slip through the polycrystal was dictated not just by local grain orientation and immediate grain neighborhoods (following prevailing understanding), but by extended networks of grains favorably oriented for slip.

5. Reduced-Order Surrogate Modeling

While CPFEM and CPFFT simulations present the state-of-the-art with respect to prediction of the complex heterogeneous deformation fields in polycrystalline materials, they come at a relatively high computational cost. Indeed, high-fidelity simulations often require (at the least) high-powered computational workstations and (more often) multi-nodal computational clusters. While the level of fidelity afforded by these simulation methods allow for inspection at intragranular scales, this is not always necessary for specific applications and likewise may be too computationally expensive to embed in component-scale simulations. Consequently, the development of reduced-order—or surrogate—models offers a path forward in including the user-desired essence of full-field models at reduced computational cost.

Machine learning methods have been increasingly attractive to help distill these complex trends into relatively efficient computational models. One such approach is to utilize a graph neural network (GNN) framework. GNNs model the response (generally) of networks of “nodes” and “edges” connecting those nodes, which both may be described by “features”, and together define a graph. This adheres naturally to the microstructural topology of a polycrystal, where grains or crystals correspond to nodes, and the granular connectivity or boundaries to edges). In [28], a GNN framework was employed to model the grainaveraged elastic response of grains in polycrystalline materials of varying degrees of elastic anisotropy. A transfer learning approach was demonstrated, where the GNN is trained with CPFEM data and used to predict the micromechanical response of an experimental specimen’s micromechanical response measured via far-field HEDM (an overview of the process is presented in Fig. 4). Results of this study yielded predictions which correspond well to both CPFEM and experimental data, and surpass the predictions from mean-field approaches, though at significantly reduced computational cost compared to CPFEM simulations. In the future, both CPFEM and 3D data can be used in tandem for surrogate model training with enhanced accuracy for engineering applications.

Figure 4: Overview of a transfer learning approach to move GNN surrogate models between simulated and measured 3D deformation data in polycrystalline metallic alloys [28].

6. Summary and Outlook

In summary, the existing and nascent 3D X-ray characterization and micromechanical measurement capabilities offer an exciting path forward to more directly interface with predictive modeling frameworks. The progress of 3D predictive models has been historically limited by the lack of robust experimental data to properly characterize, verify, and validate against, instead comparing to bulk material response. This has limited both the predictive capabilities of models, which inherently had high error due to ambiguity in optimized parameter sets, as well as a physically and experimentally guided progress of crystal plasticity models. As the above-described 3D X-ray methods become increasingly commonplace and accessible, we expect the synergy between these data and crystal plasticity simulation methods to increase further. In the coming decade, we expect that reduced-order models— especially those derived from machine learning techniques—which capture the essence of computationally complex full-field models will become more commonplace. As models are only as good as the physics they contain, the training of said reduced-order models will lean even more heavily on data derived from 3D experimental methods to better capture complex polycrystalline behaviors that may not be explicitly considered in full-field models.

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