A numerical spectral approach for solving elasto-static field dislocation and g-disclination mechanics

Abstract

A spectral approach is developed to solve the elasto-static equations of field dislocation and g-disclination mechanics in periodic media. Given the spatial distribution of Nye’s dislocation density and/or g-disclination density tensors in heterogeneous or homogenous linear elastic media, the incompatible and compatible elastic distortions are respectively obtained from the solutions of Poisson and Navier-type equations in the Fourier space. Intrinsic discrete Fourier transforms solved by the Fast Fourier Transform (FFT) method, which are consistent with the pixel grid for the calculation of first and second order spatial derivatives, are preferred and compared to the classical discrete approximation of continuous Fourier transforms when deriving elastic fields of defects. Numerical examples are provided for homogeneous linear elastic isotropic solids. For various defects, a regularized defect density in the core is considered and smooth elastic fields without Gibbs oscillations are obtained, when using intrinsic discrete Fourier transforms. The results include the elastic fields of single screw and edge dislocations, standard wedge disclinations and associated dipoles, as well as “twinning g-disclinations”. In order to validate the present spectral approach, comparisons are made with analytical solutions using the Riemann–Graves integral operator and with numerical simulations using the finite element approximation.