Predicting Microstructural Pattern Formation Using Stabilized Spectral Homogenization

Abstract

Instability-induced patterns are ubiquitous in nature, from phase transformations and ferroelectric switching to spinodal decomposition and cellular organization. While the mathematical basis for pattern formation has been well-established, autonomous numerical prediction of complex pattern formation has remained an open challenge. This work aims to simulate realistic pattern evolution in material systems exhibiting non-(quasi)convex energy landscapes. These simulations are performed using fast Fourier spectral techniques, developed for high-resolution numerical homogenization. In a departure from previous efforts, compositions of standard FFT-based spectral techniques with finite-difference schemes are used to overcome ringing artifacts while adding grid-dependent implicit regularization. The resulting spectral homogenization strategies are first validated using benchmark energy minimization examples involving non-convex energy landscapes. The first investigation involves the St. Venant-Kirchhoff model, and is followed by a novel phase transformation model and finally a finite-strain single-slip crystal plasticity model. In all these examples, numerical approximations of energy envelopes, computed through homogenization, are compared to laminate constructions and, where available, analytical quasiconvex hulls. Subsequently, as an extension of single-slip plasticity, a finite-strain viscoplastic formulation for hexagonal-closed-packed magnesium is presented. Microscale intragranular inelastic behavior is captured through high-fidelity simulations, providing insight into the micromechanical deformation and failure mechanisms in magnesium. Studies of numerical homogenization in polycrystals, with varying numbers of grains and textures, are also performed to quantify convergence statistics for the macroscopic viscoplastic response. In order to simulate the kinetics of pattern evolution, stabilized spectral techniques are utilized to solve phase-field equations. As an example of conservative gradient-flow kinetics, phase separation by anisotropic spinodal decomposition is shown to result in cellular structures with tunable elastic properties and promise for metamaterial design. Finally, as an example of nonconservative kinetics, the study of domain wall motion in polycrystalline ferroelectric ceramics predicts electromechanical hysteresis behavior under large bias fields. A first-principles approach using DFT-informed model constants is outlined for lead zirconate titanate, producing results showing convincing qualitative agreement with in-house experiments. Overall, these examples demonstrate the promise of the stabilized spectral scheme in predicting pattern evolution as well as effective homogenized response in systems with non-quasiconvex energy landscapes.