Uncertainty Quantification in Discrete Dislocation to Strain Gradient Plasticity Multiscale Modeling of Microscale Materials

This contribution develops a hierarchical discrete-to-continuum multiscale model to predict micro-pillars' mechanical behavior with quantified uncertainty in microstructure and macroscopic plasticity responses. The multiscale model consists of discrete dislocation dynamics (DDD) simulations and a continuum strain-gradient plasticity (SGP) model. The DDD model directly accounts for material size effect on the plastic responses as it captures the microscopic material behaviors naturally within the dynamics of dislocation evolution.  Due to DDD's high computation cost, this model is only accessible in a limited range of length scales and time scales. The SGP continuum model includes two (energetic and dissipative) length scales, accounting for size-dependent yield stress and kinematic hardening.  The SGP model addresses the main features in microscale plasticity with low computational cost relative to discrete simulations without constraint in domain geometry.

In this study, the DDD simulations comprise metallic micro-pillars under uniaxial compression and over a range of sizes, initial dislocation contents, and spatial distributions of dislocation. The SGP model governed by a highly nonlinear system of partial differential equations. For the SGP model's numerical solution, a dual-mixed finite element numerical algorithm is employed to enable the parallel solution of such governing equations on a multiprocessor platform for computationally intense uncertainty analyses. A comprehensive uncertainty analysis is then performed to assess the multiscale model's predictive capability, starting with the variance-based global sensitivity analysis (Sobol indices), which determines the relative effects of parameter uncertainties on predictive behaviors of the SGP model. The sensitivity analyses show that the SGP parameters significantly vary with the micro-pillars' sizes. Subsequently, a novel Bayesian calibration framework is implemented that enables training the SGP model using the high-fidelity synthetic data furnished by DDD simulations while quantifying uncertainties due to microstructural randomness in DDD simulations (density and spatial distributions of dislocations), inherent stochasticity of the DDD, and inadequacy of the SGP in capturing the dislocation heterogeneity. Different sequences of the training and testing data sets are taken into account to access the validity of the SGP model in predicting the observations outside the range of calibration data.

The outcomes indicate that the multiscale model can accurately simulate the plastic deformation of micro-pillars, despite the significant uncertainty in the discrete simulations. The SGP model parameters are adequately learned from DDD simulations of micropillars. Additionally, depending on the macroscopic features of the discrete dislocation data, the strain gradient plasticity can reliably predict the size effect of the micropillars plasticity responses below 10 percent of error.