An uncertainty quantification framework for multiscale parametrically homogenized constitutive models (PHCMs) of polycrystalline Ti alloys

Abstract

Abstract This paper develops an uncertainty quantified, parametrically homogenized constitutive model (UQ-PHCM) for microstructure-sensitive modeling and simulation at the structural scale. The PHCMs are thermodynamically consistent, macroscopic constitutive models, whose parameters are explicit functions of Representative Aggregated Microstructural Parameters or (RAMPs) that represent statistical distributions of morphological and crystallographic descriptors of the microstructure. The forms of the PHCM equations are chosen to reflect the fundamental deformation characteristics of aggregated response of crystal plasticity finite element model (CPFEM) simulations of microstructural statistically equivalent RVEs. Machine learning tools operate on datasets generated by CPFEM to obtain these functional forms. Significantly reduced number of solution variables in the PHCM simulations, compared to direct numerical simulations of micromechanical models, make them several orders of magnitude more efficient with good accuracy. The UQ-PHCM framework is built from computational homogenization of CPFE simulations performed on a large set of microstructures and load paths, followed by Bayesian inference from these results to derive probabilistic microstructure-dependent constitutive laws of the macroscopic material response. The framework addresses three sources of uncertainty that accrue at the model development and response prediction stages, viz: (i) model reduction error (MRE), (ii) data sparsity (DS), and (iii) microstructural variability (MSV) of the material. A series expansion-based uncertainty propagation (UP) method, which is much more efficient than Monte Carlo sampling, is developed to propagate uncertainties to the material response variables. The UQ-PHCM framework is validated by comparing the stochastic predictions with a collection of CPFEM-based results and limited experimental data on the\u2009 α -phase Ti alloy, Ti-7Al. Finally the UQ-PHCM is used to simulate an engine blade under operating conditions to test its viability in real applications. Results are discussed from the perspective of extreme values and microstructure-informed predictions of fatigue nucleation.