The AFRL Additive Manufacturing Modeling Challenge: Predicting Micromechanical Fields in AM IN625 Using an FFT-Based Method with Direct Input from a 3D Microstructural Image

Abstract

The efficacy of an elasto-viscoplastic fast Fourier transform (EVPFFT) code was assessed based on blind predictions of micromechanical fields in a sample of Inconel 625 produced with additive manufacturing (AM) and experimentally characterized with high-energy X-ray diffraction microscopy during an in situ tensile test. The blind predictions were made in the context of Challenge 4 in the AFRL AM Modeling Challenge Series, which required predictions of grain-averaged elastic strain tensors for 28 unique target (Challenge) grains at six target stress states given a 3D microstructural image, initial elastic strains of Challenge grains, and macroscopic stress–strain response. Among all submissions, the EVPFFT-based submission presented in this work achieved the lowest total error in comparison with experimental results and received the award for Top Performer. A post-Challenge investigation by the authors revealed that predictions could be further improved, by over 25% compared to the Challenge-submission model, through several model modifications that required no additional information beyond what was initially provided for the Challenge. These modifications included a material parameter optimization scheme to improve model bias and the incorporation of the initial strain field through both superposition and eigenstrain methods. For the first time with respect to EVPFFT modeling, an ellipsoidal-grain-shape Eshelby approximation was tested and shown to improve predictive capability compared to previously used spherical-grain-shape assumptions. Lessons learned for predicting full-field micromechanical response using the EVPFFT modeling method are discussed.

Appendices

Appendix

Appendix A. Global and Local Model Convergence Study

During the post-Challenge analysis, a convergence study was completed to determine the optimal image resolution (voxel size) with respect to global and local error. The global error was calculated using the mean squared error (MSE) method given in Eq. (16), and the local error was calculated using the Challenge grading metric described in Eq. (1). The model inputs from Sect. 3.3.3 were used for the convergence study, aside from microstructure image resolution changes. For the purposes of the convergence study, a beta version of MASSIF with concatenated loading processes implemented was used. Utilizing MASSIF thus allowed for more data points (i.e., geometries that do not satisfy the $2^n$ requirement) to be included in the convergence study than if the serial EVPFFT code were used. The local and global errors were calculated for seven different resolutions of the cuboidal voxel microstructure discussed in Sect. 3.3.2. The local and global error was additionally calculated for the original microstructure resolution provided for the Challenge (roughly 33 million voxels), i.e., a microstructure with no geometry modifications. Figure 12 shows the global and local error data for these eight microstructure resolutions.

Fig. 12

Convergence plot showing the total number of voxels versus global error (MSE) and local error (L2). The global error was calculated with Eq. (16), and the local error was calculated with Eq. (1). The top horizontal axis indicates the mean number of voxels comprising each Challenge grain. The two lowest resolution microstructures entirely removed one or more Challenge grains, so an average error from the remaining grains was used to fill those values

From Fig. 12, the global error reached convergence before the local error. The global error reached convergence at about 1 million voxels or an average of about three voxels per grain (not shown in the plot), while the local error reached convergence at about 2.1 million voxels or an average of about 200 voxels per Challenge grain. Based on these results, the microstructure resolution used in all models presented herein did not have any significant effect on the model predictions.

Appendix B. Eigenstrain Calculations for Ellipsoidal Grains

The following calculations show the process of calculating the constant eigenstrain tensor using the dimensions of the best-fit ellipsoid of an individual grain. These calculations are implemented in DREAM.3D as the “Compute Eigenstrains by Feature (Grain/Inclusion)” filter as of version 6.5.151. The DREAM.3D filter does not have an option to subtract the mean initial elastic strain tensor as in Eq. (18), so that must be performed manually if desired. The first calculation is to change the basis of the elastic strain tensor into the best-fit ellipsoid (local) reference frame, i.e.,

$$\begin{aligned} \varepsilon ^{\mathrm {e}'}_{kl} = Q_{ki} Q_{lj} \varepsilon ^{\mathrm {e}}_{ij}, \end{aligned}$$

(22)

where $\varepsilon ^{\mathrm {e}'}_{kl}$ is the elastic strain tensor in the local reference frame and $Q_{ki}$ is the orientation matrix corresponding to the orientation of the best-fit ellipsoid reference frame. The prime ($^\prime$) superscript indicates a tensor in the local reference frame and the absence of such indicates a tensor in the global reference frame. Next, the fourth-rank Eshelby tensor is calculated via Eshelby’s solution [37] for isotropic ellipsoidal inclusions using Eqns. 11.16–11.19 given by Mura [34], where Eshelby’s tensor, $S_{ijkl}$, is a function of the ellipsoid semi-axis lengths, $a\ge b\ge c$, and Poisson’s ratio, $\nu$. Edge cases where the grain shape is a sphere or spheroid are handled using Eqns. 11.21, 11.28, and 11.29 given by Mura [34]. Within Eshelby’s solution, there are two elliptic integrals that are numerically integrated using 32-point Gaussian quadrature (arbitrarily chosen as execution time is negligible). Once Eshelby’s tensor is calculated, the uniform eigenstrain tensor is calculated in the local reference frame as follows:

$$\begin{aligned} \varepsilon ^{*'}_{ij} = (S_{ijkl}-I_{ijkl})^{-1} \varepsilon ^{\mathrm {e}'}_{kl}, \end{aligned}$$

(23)

where $\varepsilon ^{*'}_{ij}$ is the eigenstrain tensor in the local reference frame. This fourth-rank tensor inversion is performed on a flattened 9 × 9 matrix and then rebuilt into a fourth-rank tensor. The final calculation is to transform the eigenstrain tensor back into the global reference frame to complete the eigenstrain calculations as follows:

$$\begin{aligned} \varepsilon ^{*}_{kl} = Q_{ik} Q_{jl} \varepsilon ^{*'}_{ij}, \end{aligned}$$

(24)

where $\varepsilon ^{*}_{kl}$ is the final eigenstrain tensor. The correctional matrix can then be applied through Eq. 20, if desired. These calculations are then repeated for every grain in the microstructure, which is 29,663 grains in the case of the Challenge microstructure.