Bayesian inference and uncertainty quantification for modeling of body-centered-cubic single crystals

TL;DR

This work tackles the challenge of predicting the deformation of bcc single-crystal Mo under quasi-static to shock loading by integrating two physics-based crystal plasticity models with a Bayesian calibration framework and a global sensitivity analysis. By coupling Bayesian parameter inference with variance-based Sobol indices, the authors quantify parameter uncertainties, reveal mechanistic differences between models (notably the role of mobile dislocations via the Orowan relation in Model 1 versus thermally activated glide in Model 2), and assess predictive capabilities across loading regimes including plate-impact tests. The combined UQ approach uncovers how loading conditions shape parameter importance and identifies missing physics—such as dislocation nucleation terms—that can improve thickness- and rate-dependent predictions. The results provide a principled pathway to refine continuum crystal plasticity models for a broad range of deformation mechanisms and extreme loading scenarios, with implications for materials design and high-f-rate applications.

Abstract

Uncertainties in the high-dimensional space of material parameters pose challenges for the predictive modeling of bcc single crystals, especially under extreme loading conditions. In this work, we identify the key physical assumptions and associated uncertainties in constitutive models that describe the deformation behavior of bcc single crystal molybdenum subjected to quasi-static to shock loading conditions. We employ two representative physics-based bcc single crystal plasticity models taken from our previous work (Nguyen et al. 2021a; Lee et al. 2023b), each prioritizing different key deformation mechanisms. The Bayesian model calibration (BMC) is used for probabilistic estimates of material parameters in both bcc crystal plasticity models. In conjunction with the BMC procedure, the global sensitivity analysis is conducted to quantify the impact of uncertainties in the material parameters on the key simulation results of quasi-static to shock responses. The sensitivity indices at various loading conditions clearly illustrate the physical basis underlying the predictive capabilities of the two distinct bcc crystal plasticity models at low to high strain rates. Both of the calibrated bcc models are then further validated beyond the calibration regime, by which we further identify critical physical mechanisms that govern the transient elastic-plastic responses of single crystal molybdenum under shock loading. The statistical inference framework demonstrated here facilitates the further development of continuum crystal plasticity models that account for a broad range of deformation mechanisms.

Figures (13)

• Figure 1: Comparison of cross-validated emulator predictions with the simulated stresses from the crystal plasticity models: (a) Model 1 and (c) Model 2. Each point on the scatter plot is colored according to its Gaussian kernel density estimate. The cross-validation comparisons show that the trained emulators exhibit excellent performance. Pair plots that represent posterior parameter distributions for (b) Model 1 and (d) Model 2. The plots on the diagonal of pair plots show the marginal probability distributions for each parameter, while the plots on the off-diagonal show the bivariate kernel density estimations between pairs of parameters. Overall, the model parameters are well calibrated through the BMC processes.
• Figure 2: Stress-strain responses of single crystal molybdenum from experiments, 100 training simulations, and 90% prediction interval (PI) of posterior emulator predictions after Bayesian calibration: (a) Model 1 and (b) Model 2. The solid lines show the mean of posterior emulator predictions. Information on the data is given in Table \ref{['tab:loading_conditions']}. The calibrated results show good agreement with the experiment data.
• Figure 3: Stress-strain responses of single crystal molybdenum from experiments and Model 1 using 100 parameter samples taken from the posterior distribution (e.g., Figure \ref{['fig1:pair']} (b)): (a) $\langle100\rangle$, (b) $\langle110\rangle$, (c) $\langle111\rangle$ and $\langle\bar{1}49\rangle$. The solid lines and dots represent the experimental data. The shaded regions show the 90% credible interval of the simulated responses. The dashed lines show the mean of the simulated responses.
• Figure 4: Stress-strain responses of single crystal molybdenum from experiments and Model 2 using 100 parameter samples taken from the posterior distribution (e.g., Figure \ref{['fig1:pair']} (d)): (a) $\langle100\rangle$, (b) $\langle110\rangle$, (c) $\langle111\rangle$ and $\langle\bar{1}49\rangle$. The solid lines and dots represent the experimental data. The shaded regions show the 90% credible interval of the simulated responses. The dashed lines show the mean of the simulated responses.
• Figure 5: Functional global sensitivity analysis results for [100] single crystal molybdenum using Model 1. Each color band indicates the evolution of the sensitivity index for a given parameter as a function of strain.