Development of a new consistent discrete green operator for FFT-based methods to solve heterogeneous problems with eigenstrains

In this paper, a new expression of the periodized discrete Green operator using the Discrete Fourier Transform method and consistent with the Fourier grid is derived from the classic Continuous Green Operator (CGO) in order to take explicitly into account the discreteness of the Discrete Fourier Transform methods. It is shown that the easy use of the conventional continuous Fourier transform of the modified Green operator (CGO approximation) for heterogeneous materials with eigenstrains leads to spurious oscillations when computing the local responses of composite materials close to materials discontinuities like interfaces, dislocations. In this paper, we also focus on the calculation of the displacement field and its associated discrete Green operator which may be useful for materials characterization methods like diffraction techniques. We show that the development of these new consistent discrete Green operators in the Fourier space named Discrete Green Operators (DGO) allows to eliminate oscillations while retaining similar convergence capability. For illustration, a DGO for strain-based modified Green tensor is implemented in an iterative algorithm for heterogeneous periodic composites with eigenstrain fields. Numerical examples are reported, such as the computation of the local stresses and displacements of composite materials with homogeneous or heterogeneous elasticity combined with dilatational eigenstrain or eigenstrain representing prismatic dislocation loops. The numerical stress and displacement solutions obtained with the DGO are calculated for cubic-shaped inclusions, spherical Eshelby and inhomogeneity problems. The results are discussed and compared with analytical solutions and the classic discretization method using the CGO.