# Workflow for integrating mesoscale heterogeneities in materials structure with process simulation of titanium alloys

- Ayman A Salem
^{1}Email author, - Joshua B Shaffer
^{1}, - Daniel P Satko
^{1}, - S Lee Semiatin
^{2}and - Surya R Kalidindi
^{3}

**3**:24

**DOI: **10.1186/s40192-014-0024-6

© Salem et al.; licensee Springer. 2014

**Received: **9 May 2014

**Accepted: **2 September 2014

**Published: **16 September 2014

## Abstract

In this paper, a generalized workflow is outlined for the necessary integration of multimodal measurements and multiphysics models at multiple hierarchical length scales demanded by an Integrated Computational Materials Engineering (ICME) approach to accelerated materials development. Recognizing that multiple choices or techniques are typically available in each of the main steps, several exemplary analyses are detailed utilizing mainly the alpha/beta titanium alloys as an illustrative case. It is anticipated that the use and further refinement of these workflows will promote transparency and engender intimate collaborations between materials experts and manufacturing/design specialists by providing an understanding of the various mesoscale heterogeneities that develop naturally in the workpiece as a direct consequence of the inherent heterogeneity imposed by the manufacturing history (i.e., different thermomechanical histories at different locations in the sample). More specifically, this article focuses on three main areas: (i) data science protocols for efficient analysis of large microstructure datasets (e.g., cluster analysis), (ii) protocols for extracting reduced descriptions of salient microstructure features for insertion into simulations (e.g., regions of homogeneity), and (iii) protocols for direct and efficient linking of materials models/databases into process/performance simulation codes (e.g., crystal plasticity finite element method).

### Keywords

ICME Microstructure informatics Higher-order statistics Materials big data Macrozones Region of homogeneity Representative orientation distribution Alpha/beta titanium alloys## Review

### Introduction

In order to provide for a greater understanding of the various operations and the flow of information throughout the workflow, detailed descriptions of exemplary illustrations are provided in the subsequent sections. In particular, discussion is focused on examples illustrating how a series of choices can be made to advance the incorporation of titanium microstructures into numerical simulations of a part production. The workflow and selected techniques are directed towards the emerging `Big-Data´ materials innovation ecosystem that utilizes modern data science techniques such as machine learning and computer vision [3],[4]. While the Ti-6Al-4V microstructure data presented here were generated with standard methods (e.g., electron backscatter diffraction (EBSD) and backscattered electron (BSE) imaging), they were recorded from large scan areas resulting in 100,000,000s of EBSD data points and 10,000s of high-resolution BSE images. Consequently, new tools have had to be developed for texture analysis, image segmentation, and quantification of microstructure metrics. In addition, salient microstructure descriptors (e.g., regions of homogeneity (ROH), representative orientation distributions (ROD), and microtextured regions or macrozones (MTR)) have been generated using a new generation of data analytics techniques.

### Microstructure of α-β titanium alloys

_{p}) particles and (α

_{s}) colonies based on vanadium partitioning (Figure 3b) [6]. However, applying these methods to large areas for practical applications is expensive and time-consuming. The use of data science approaches (as described in detail in subsequent sections) has enabled automated segmentation of 10,000s of EBSD and BSE datasets (Figure 3c).

### Data analytics for large microstructure datasets

The bimodal microstructure contains many microstructure features (α_{p} particles, α_{p} grains, α_{s} colonies, α_{s} laths, layers of alpha on beta grain boundaries, prior β grains, microtextured regions, etc.) that can affect the response of the material under loading. However, the data captured by typical characterization techniques (EBSD, BSE, spectroscopy, etc.) do not directly identify these microstructure features automatically. Rather, at each probed location, the data include signals from the materials that reflect the internal local state of the materials. These internal variables change with time under externally applied variables (e.g., temperature, cooling rate, strain). As such, one can assign the local state at each voxel location in the studied area using a list of internal variables that are related to the crystal structure and alloying elements of the underlying materials. The microstructure features mentioned earlier are then defined by certain morphological characteristics displayed by the local states. Consequently, it should be possible to use various cluster analysis techniques [7] to identify the features of interest (FoI) from large datasets (e.g., EBSD scan of 10 × 10 mm at 1 μm step size giving 10^{8} measurements of the local state). An example is the familiar spatial clustering of measurement points sharing a common orientation that is understood as a `grain´. Similarly, the spatial clustering of a group of grains indicates a microtextured region (macrozone), while the spatial clustering of similar chemical elements is indicative of micro/macrosegregation or precipitates. As such, taking advantage of recent advances [8] in data science has enabled the development of multiple algorithms and tools for quantifying the microstructural features of interest in computationally efficient ways [9]. In the next few sections, the main terminologies and methodologies used in these workflows are introduced.

### Feature vector

Each voxel in a materials dataset is assigned an n-dimensional feature vector [10] of variables needed to obtain a concise mathematical representation of all the distinct local states in the dataset. Such representation facilitates image processing, statistical analysis, and utilization of numerous algorithms from the pattern recognition and machine learning communities [11] to extract salient information about the material. The list of variables used in the feature vectors can range from scalar variables (e.g., chemical composition) to tensorial variables (e.g., crystal lattice orientation). These feature vectors play an important role in identifying proper materials models for process simulations (e.g., thermomechanical processing of Ti alloys).

### Identification of features of interest

As noted earlier, the local states do not necessarily identify the microstructure features of interest directly, necessitating additional analysis. As such, prior knowledge in the field will help identify the specific microstructural FoI (presumably these features control the performance characteristics of the final finished product) that need to be tracked in the process simulations. Automated protocols for the extraction of selected features of interest are needed to obtain consistency and reproducibility independent of operator bias and with minimum computational cost. Due to the importance of the FoI as the main building block for further analysis and modeling efforts, computer vision algorithms built on supervised and unsupervised machine learning techniques [11] should be employed for automated feature identification and extraction.

_{p}particle (Figure 5a) may engulf multiple α

_{p}grains (Figure 5b), though in the case of a microstructure that has been fully spherodized during the ingot breakdown process, a particle may include only one grain. As such, while identification of particles can be accomplished directly from BSE images (Figure 2c) by finding the remnant beta layer separating particles, further differentiation of α

_{p}grains within a particle requires orientation information in addition to BSE information (Figure 3c, Figure 5b).

Independent of the length scale, a feature of interest may be defined as *a region in physical space* (*3D*) *that depicts similar characteristic feature vectors as some other regions in the microstructure*. This definition enables setting up automated extraction tools that are mainly dependent on the components of the feature vector. In the case mentioned above, the feature vector for alpha `particle´ identification needs only one element (BSE gray level), while a larger feature vector is necessary to define primary alpha `grains´ (BSE gray level + orientation). To achieve automation and to increase the speed of extraction, the identification is conducted in a two-step process in two separate domains. The first step is conducted in the feature vector domain (i.e., feature space) in which volumes with similar feature vectors are classified using cluster analysis [7]. The second step is conducted in physical space by mapping the classified clusters from the first step to corresponding spatial locations in real space. To demonstrate the FoI concept practically on Ti-6Al-4V, EBSD, and BSE data within a feature vector are used to extract information about MTRs which can be defined as 3D regions in physical space that contain similarly oriented primary alpha grains [14].

^{7}EBSD data points covering hundreds of square millimeters and >10

^{5}BSE images). However, the two-step process [14] described above has enabled the rapid identification of MTR clusters, each distributed about a common texture component with a defined misorientation range (<10° in this case) within each cluster. The cluster analysis was conducted with a feature vector of 551 dimensions in domain of the generalized spherical harmonics (GSH), a mathematical construct commonly used to analytically describe the distribution of crystallographic orientations [16]-[21]. The normal direction (ND) inverse pole figure (IPF) maps in Figure 6a,b show the variability of the length scale from primary alpha grains in (b) to MTRs in (a). One of the identified MTR cluster families is shown in real space in Figure 6c, along with the corresponding GSH-space projection (Figure 6d), demonstrating the orientation clustering of various MTR families.

### Representative descriptions of microstructure

#### Regions of homogeneity

Identifying specific subsets that represent the whole material has been the target of many research efforts including finding a representative volume element (RVE) [22],[23] and/or the statistical volume element (SVE) [24]. One of the defining features of an RVE or SVE description is the linking of materials properties of the ensemble to the properties of the defined RVE or SVE. For example, Drugan and Willis [23] defined an RVE as `a smallest material volume element of the composite for which the usual spatially constant `overall modulus´ macroscopic constitutive representation is a sufficiently accurate model to represent mean constitutive response´. However, estimating the materials response depends on the defined RVE/SVE, so defining the RVE or SVE that is based on the response of the material to use it in models that predict the response of the material becomes a challenging task. Furthermore, many of the traditional methods used to define RVEs/SVEs are based on highly simplified metrics (e.g., average grain size and its distribution) and ignore the spatial correlations of individual FoI (e.g., complex morphologies that may not fit a standard geometrical shape such as an ellipse). While the traditional methods may work for materials with homogeneous spatial distributions of the FoI, it may not be efficient for heterogeneous spatial distributions of FoI such as MTRs (Figure 6c) or gradient microstructures. In both cases, there is a need for microstructure representation that captures the essence of spatial and morphological heterogeneities. Presented below is a new microstructure descriptor (regions of homogeneity) that has been developed based on the spatial homogeneity of the two-point statistics that were calculated for sampled microstructure regions.

The hierarchical materials systems described here exhibit salient features at different length scales. For example, in the Ti alloys described here, one scale of heterogeneity occurs at the length scale of individual α_{p} grains (on the order of 10 to 30 μm) and another scale of heterogeneity occurs at the scale of each MTR (up to several mm in length). In multiscale modeling, one typically identifies different length scales where one might be able to homogenize the materials response by aggregating in some way all of the inherent heterogeneity below that length scale. In materials with complex structures, one has to identify suitable hierarchical length scales for homogenization, which are called regions of homogeneity. ROH can be established objectively at different hierarchical length scales by carefully quantifying spatial correlations (e.g., using two-point spatial correlations [17],[25]-[30]) and finding suitable window sizes in the microstructure that capture the inherent heterogeneity (FoI and their spatial distributions) to a desired accuracy. It is also desired to keep these ROH small enough to enable cost-effective modeling (the computational cost rises steeply with increases in the number of voxels needed to capture the ROH).

### Representative orientation distribution

^{−17}in the orientation distribution function (ODF)) the same texture as the 12,467,961 (Figure 13) orientations measured in the Ti-6Al-4V sample given in Figure 6. To give an estimate of the time savings, a Taylor-type crystal plasticity modeling of simple compression to strain of −1.0 was executed in 1.6 mins on a standard desktop computer (quad-core 3.0 GHz) using the ROD. Under an approximation of linear time complexity with the number of orientations evolved, the identical simulation is estimated to take more than 25 days to run with the original dataset on the same computer.

### Materials models: crystal plasticity

Titanium exhibits highly anisotropic properties at the single-crystal level, which can be attributed to the operation of different deformation mechanisms under different external stimuli (temperature, strain rate, etc.). In addition to the numerous competing deformation mechanisms [36], there also exist additional challenges due to the fact that an allotropic transformation occurs from alpha-HCP to beta-BCC at high temperatures (beta transus is dependent on alloying elements). The addition of alloying elements causes the alpha and beta phases to coexist with varying ratios and morphologies based on the temperature and the amount of alpha or beta stabilizers. This consequently alters the activity of various deformation mechanisms depending on the resultant microstructure. For example, the Ti-6Al-4V with bimodal microstructure shown previously (Figure 2) accommodates plastic deformation through slip in both alpha and beta phases, with the latter contribution increasing as the deformation temperature increases. As such, accurate modeling of the mechanical behavior and texture evolution of this material requires an understanding on the crystal level of the various deformation mechanisms of each phase, the interaction between both phases, and their evolution as a function of temperature.

### Single-crystal deformation behavior

Deformation mechanisms in the alpha phase (HCP) of unalloyed titanium and the two-phase materials are limited to a finite number of slip systems and/or deformation twinning which results in pronounced anisotropic behavior and strong deformation texture [36]. In particular, deformation can be accommodated by prism < a > and basal < a > slip which result in only four independent slip systems [36]. Hence, extension or contraction of the HCP c-axis requires activation of pyramidal < c + a > slip and/or deformation twinning, both of which exhibit high resistance to activation. The activity of any slip or twin system occurs once the resolved shear stress exceeds the critical resolved shear stress (CRSS) on that system. The values of the CRSS can be calibrated using experimental stress-strain data [37],[38]. Once these values are estimated for the materials under a selected deformation path, the validated values can be subsequently used to predict the mechanical behavior and texture evolution under any applied deformation using crystal plasticity modeling [39].

On the other hand, the beta phase (BCC) is known to accommodate plastic deformation exclusively by slip on various reported slip systems. Some simulations have used pencil glide on any slip plane containing <111 > slip systems [40],[41]. Others have assumed slip on {110}, {112}, and {123} planes [42],[43]. The abovementioned options should be available in any practical crystal plasticity model.

_{1}> and prism < a

_{3}> slip systems (Figure 14) [45].

**F**. To separate elastic and plastic deformations, Kröner [46] suggested the multiplicative decomposition of the deformation gradient into elastic (

**F**

^{ * }) and plastic (

**F**

^{ p }) components (Figure 15). The plastic component

**F**

^{ p }causes the permanent deformation of the materials, and it is applied in an intermediate configuration which maintains a perfect lattice. As such, estimating how a single crystal accommodates plastic deformation by crystallographic slip or twinning (pseudo slip [47]) can be conducted in the intermediate configuration using the starting orientation of the single crystal [37]. Further details of the crystal plasticity theory are presented in Figure 15 and have been explained elsewhere [39].

### Polycrystalline deformation behavior

Most metallic parts are made of polycrystalline and/or multiphase alloys which require homogenization methods to predict their response starting with the single-crystal constitutive equations. Furthermore, calibrating single-crystal CRSSs and its evolutions (strain hardening) require comparisons with experimental measurements which are mostly conducted on polycrystalline materials.

In addition to homogenization methods, strain partitioning between coexisting phases is also used to capture the overall mechanical response and texture evolution of two phase materials (e.g., Ti-6Al-4V) [48].

### Homogenization method: Taylor approach

### Spectral crystal plasticity

### Full-field crystal plasticity finite element simulations

*dΔσ*/

*dΔε*). As such, once the appropriate user subroutine is verified to work for a crystal structure, further validation and calibration can be done by users for various materials within the same crystal structure. Adopting this method results in full-field simulations, also known as CPFEM simulations. While they provide higher accuracy in predicting the materials behavior, they are known to be slow for large simulations. The use of spectral crystal plasticity [60] has demonstrated an improvement of the calculation speed more than 30× compared to the conventional approaches (Figure 18). Further increase in the speed of calculation can be achieved by using the ROH and the ROD concepts described above.

In certain situations, it becomes necessary to simulate multiscale coupled phenomena at two well-separated length scales. As an example, consider the simulation of a complex processing operation where different macroscale spatial locations in the workpiece experience different thermomechanical histories (often an unavoidable consequence of the boundary conditions imposed at the macroscale). Consequently, strong variations in the microstructure should be expected at different locations in the workpiece. In other words, it is not enough to track the evolution of a single representative microstructure for the entire workpiece. The development of such microstructure heterogeneities can be expected to influence the macroscale simulation by altering the local effective properties at different locations in the workpiece. In such a situation, it is necessary to track independently microstructures at multiple macroscale locations in the workpiece and pass high value information in both directions (between the microscale and the macroscale). This is extremely difficult, if not impossible, using any of the currently employed computational strategies.

The challenge described above can be addressed with modest computational resources using a data science approach called materials knowledge systems (MKS) [61]-[66]. In the MKS framework, the focus is on localization (i.e., opposite of homogenization) relationships that capture the spatial distribution of the response field of interest (e.g., stress or strain rate fields) at the microscale (on an RVE) for an imposed loading condition at the macroscale. In this approach, the localization relationships are expressed as calibrated metamodels that take the form of a simple algebraic series whose terms capture the individual contributions from a hierarchy of local microstructure descriptors. Each term in this series expansion is expressed as a convolution of the appropriate local microstructure descriptor and its corresponding influence at the microscale. The series expansion in the MKS framework is in complete accord with the series expansion obtained in the physics-based statistical continuum theories [30],[67]-[70]. The MKS approach dramatically improves the accuracy of these expressions by calibrating the convolution kernels using results from previously validated physics-based models. The most impressive benefit of the MKS approach lies in the dramatic reduction of the computational cost, often by several orders of magnitude compared to numerical approaches typically employed in microstructure design problems. In various preliminary demonstrations, the MKS methodology has been successfully applied to capturing thermoelastic stress (or strain) distributions in composite RVEs [64],[65],[71] and multiscale structures[61], rigid-viscoplastic deformation fields in composite RVEs [72], and the evolution of the composition fields in spinodal decomposition of binary alloys [62]. Efforts are currently underway to extend the MKS to address multiscale plastic deformation of complex alloys such as Ti alloys.

On the other hand, the use of uncoupled CPFEM simulations may also result in some computational cost savings by eliminating the feedback loop between the crystal plasticity calculations and the FEM and using only the crystal plasticity after the FEM simulation is finished [73]. In this approach, the FEM exports post-simulation strain tensor and rigid-body rotations at each integration point which is then input to the crystal plasticity code to estimate texture evolution. The main drawback of this method is the lack of coupling.

In the cases where commercial FEM codes use only empirical constitutive laws (e.g., Hill yield surface), a virtual testing laboratory method [74] using a RVE approach can be used to overcome some limitations of empirical laws by fitting the yield surface to series of simulated tests. Such a numerical test protocol predicts the shape of a yield locus and then uses it to calibrate empirical constitutive models. The accuracy of this method is dependent on the ability to find best fit for the limited number of preset yield surface variables. In addition, a new yield locus needs to be generated for each new texture (starting or evolving).

## Conclusions

A workflow to incorporate microstructure morphology and crystallography into modeling of TMP of titanium alloys has been summarized based on the application of data analytics for generating representative descriptors of large microstructure datasets coupled with novel techniques that may increase throughput of multiscale materials models. Representing texture using weighted orientations (based on GSH representations of ODF) enables fast data mining of orientation information and feature of interest extraction. Using the concept of a region of homogeneity defined based on the spatial distribution of the two-point correlations enables efficient insertion of a detailed quantitative microstructure description into materials models without the need to know a priori the properties of the ROH. Reducing the size of crystallographic orientation datasets using the representative orientation distribution allows for significant increases in the speed of crystal plasticity calculations while maintaining the accuracy of predictions.

A number of options exist for solution of the crystal plasticity constitutive equations during deformation of a sample volume. Uncoupled Taylor-type simulations provide a simple methodology for estimation of final texture. For fully coupled integration of crystal plasticity into FEM simulations, the FE codes need to allow user developed subroutines that allow specification of a broad range of materials constitutive descriptions. Once again, smartly constructed databases might prove to be very practical in integrating these sophisticated materials descriptions with typically used process simulation codes.

## Authors' contributions

AAS conceived the workflow concept and created the initial draft. JBS and DPS helped with manuscript writing and analysis of exemplary datasets. SLS provided materials science guidance and expertise. SRK contributed to the overall development of the main concepts presented in this paper. All authors read and approved the final manuscript.

## Declarations

### Acknowledgements

Support from the Air Force Research Laboratory and Air Force Office of Scientific Research is gratefully acknowledged. In particular, AAS, JBS, and DPS were partially supported by contract no. FA865009D5600 (Dr. J. Calcaterra, program manager). SRK was supported by the Air Force Office of Scientific Research, MURI contract no. FA9550-12-1-0458.

## Authors’ Affiliations

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