Fast Fourier transform approach to Strain Gradient Crystal Plasticity: Regularization of strain localization and size effect

Abstract

The Strain Gradient Crystal Plasticity (SGCP) model, based on cumulative shear strain, is developed to regularize and simulate the size effect behavior of polycrystalline aggregates, specifically addressing the formation of localization bands, such as slip and kink bands, influenced by strain softening during the initial stages of plastic deformation. In this respect, the thermodynamically consistent derivation of the SGCP equations is presented, establishing their connection to the kinematics of classical crystal plasticity (CCP) framework. The governing balance equations are solved using the fixed-point algorithm of the fast Fourier transform (FFT)-homogenization method, involving explicit coupling between the classical and SGCP balance equations. To address this problem, a strong 21-voxel finite difference scheme is established. This scheme is considered to solve the higher order balance equation inherent to SGCP. Additionally, three types of interface conditions are implemented to explore the impact of grain boundaries on the transmission of localization bands. These conditions yield consistent intragranular/transgranular localization patterns in the MicroFree and MicroContinuity cases, while in the MicroHard condition all localization bands are intragranular with stress concentrations appearing at the grain boundaries.

Analytical solutions corresponding to different material behaviors are developed and compared with numerical results to validate the numerical implementation of the FFT fixed-point algorithm. It is observed that both the macroscopic behavior and microscopic variables in CCP framework are highly influenced by grid resolutions (non-objective), leading to numerical instabilities arising from the material softening and subsequent formation of localization bands, both in single crystals and polycrystalline aggregates. Remarkably, the developed SGCP model provides results that are independent of grid resolutions (objective) and effectively regularizes the material behavior on local scale. Moreover, the non-local parameter of the model is capable of controlling the localization band widths. Finally, the proposed SGCP model, together with employed MicroHard condition on grain boundaries, is demonstrated to qualitatively reproduce main microstructural features of irradiated polycrystalline materials.