FFT-based disclination mechanics for high angle grain boundaries and comparisons with atomistic simulations

The paper addresses the question of determining local stress fields using the numerically efficient Fast Fourier Transform (FFT) method as an application of the continuum defect theory in the presence of interfacial defects, and specifically High Angle Grain Boundaries (HAGBs). First, the Field Dislocation and Disclination Mechanics (FDDM) equations are reported highlighting the use of the Stokes-Helmholtz orthogonal decomposition for both elastic strain and curvature tensors which are involved for disclination-type defects. Then, following this decomposition, both incompatible and compatible fields can be solved. Second, the Green’s function method for heterogeneous media is used to derive the stress polarization field, which integrates both dislocation and disclination densities. The compatible elastic strain tensor is numerically solved with the spectral method using the FFT (fast Fourier transform) algorithm together with finite difference (FD) schemes for computing first- and second-order spatial derivatives (FDDM-FFT). As applications of the method, the stress fields of two specific HAGBs (symmetric tilt GBs with [001] tilt axis), precisely $\Sigma 29\left(520\right)\left[001\right]46.40°$ and $\Sigma 149\left(1070\right)\left[001\right]20.02°$ are obtained from the Disclination Structural Unit Model (DSUM). They are calculated assuming both isotropic and anisotropic elasticity with the present FDDM-FFT numerical method and assuming periodic disclination density tensors. Quantitative comparisons are first performed for both HAGBs with analytical solutions obtained from specific combinations of disclination dipole walls in linear isotropic elasticity. Then, the effect of anisotropic elasticity is analyzed for both HAGBS considering two different FCC metals, namely Al and Ag. Lastly, some comparisons between the FDDM-FFT-based results with molecular statics (MS) simulations, using the virial stress method and an interpolation method based on Gaussian kernel are reported for both HAGBs applied to Al and Ag. It is shown that, despite their relative simplicity in describing HAGB defect cores, the FDDM-FFT results reproduce the major trends of MS-based results for both hydrostatic and shear stress components.