Piezoelectrics and ferroelectrics have been widely used in modern industries because of their peculiar electromechanical coupling properties, quick response, and compact size. In this work, we give a comprehensive review of our works and others' works in the past decade on the multiscale computational mechanics methods for electromechanical coupling behavior of piezoelectrics and ferroelectrics. The methods are classified into three types based on their applicable scale (i.e., macroscopic methods, mesoscopic methods, and atomic-level methods). In macroscopic methods, we first introduce the basic linear finite element method and employ it to analyze the crack problems in piezoelectrics. Then, the nonlinear finite element methods are presented for electromechanically coupled deformation and the domain switching processes were simulated. Based on our developed nonlinear electromechanically coupled finite element method, the domain switching instability problem was specially discussed and a constrained domain-switching model was proposed to overcome it. To specially address the crack problem in piezoelectrics, we further proposed a meshless electromechanical coupling method for piezoelectrics. In mesoscopic methods, the phase field methods (PFM) were firstly presented and the simulation results on the defects effect and size effect of deformation in ferroelectrics were given. Then, to solve the computational complexity problem of PFM in polycrystals, we proposed an optimization-based computational method taking the interactions between grains in an Eshelby inclusion manner. The domain texture evolution process can be calculated, and the Taylor's rule of plasticity has been reproduced well by this optimization-based model. Alternatively, the domain switching in polycrystalline ferroelectrics can be simulated by a proposed Monte Carlo method, which treated domain switching as a stochastic process. In atomic-level methods, we firstly introduce the first-principles method to calculate polarization and studied the topological polarization and strain gradient effect in ferroelectrics. Then, we present a modified electromechanically coupled molecular dynamic (MD) method for ferroelectrics based on the shell model and investigated the size effect of electromechanical deformation in ferroelectric thin films and nanowires. Finally, we introduced our recently proposed novel atomic finite element method (AFEM), which has higher computational efficiency than the MD. The deformation as well as domain evolution processes in ferroelectrics calculated by AFEM were also presented. The development of electromechanically coupled computational mechanics methods at multiscale is greatly beneficial, not only to the deformation and fracture of piezoelectrics/ferroelectrics, but also to structural design and reliability analysis of smart devices in engineering.
Crystal plasticity finite element method modelling of indentation size effect
