A Fast Fourier Transform-Based Approach for Generalized Disclination Mechanics Within a Couple Stress Theory

Abstract

Recently, a small-distortion theory of coupled plasticity and phase transformation accounting for the kinematics and thermodynamics of generalized defects called generalized disclinations (abbreviated g-disclinations) has been proposed by Acharya and Fressengeas (2012, 2015). Then, a first numerical spectral approach has been developed to solve the elasto-static equations of field dislocation and g-disclination mechanics set out in this theory for periodic media and for linear elastic media using the classic Hooke’s law within a Cauchy stress theory (Berbenni et al. 2014). Here, given a spatial distribution of generalized disclination density tensors in a homogenous linear higher order elastic media, a couple stress theory with elastic incompatibilities of first and second orders is developed. The incompatible and compatible elastic second and first distortions are obtained from the solution of Poisson and Navier-type equations in the Fourier space. The efficient Fast Fourier Transform (FFT) algorithm is used based on intrinsic Discrete Fourier Transforms (DFT) that are well adapted to the discrete grid to compute higher order partial derivatives in the Fourier space. Therefore, stress and couple stress fields can be calculated using the inverse FFT. The numerical examples are given for straight wedge disclinations and associated wedge disclination dipoles which are of importance to geometrically describe tilt grain boundaries at fine scales in polycrystalline solids.