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In the current work, a number of algorithms are developed and compared for the numerical solution of periodic (quasi-static) linear elastic mechanical boundary-value problems (BVPs) based on two different discretizations of Fourier series. The first is standard and based on the trapezoidal approximation of the Fourier mode integral, resulting in trapezoidal discretization (TD) of the truncated Fourier series. Less standard is the second discretization based on piecewise-constant approximation of the Fourier mode integrand, yielding a piecewise-constant discretization (PCD) of this series. Employing these, fixed-point algorithms are formulated via Green-function preconditioning (GFP) and finite-difference discretization (of differential operators; FDD). In particular, in the context of PCD, this includes an algorithm based on the so-called "discrete Green operator" (DGO) recently introduced by Eloh et al. (2019), which employs GFP, but not FDD. For computational comparisons, the (classic) benchmark case of a cubic inclusion embedded in a matrix (e.g., Suquet, 1997; Willot, 2015) is employed. Both discontinuous and smooth transitions in elastic stiffness at the matrix-inclusion (MI) interface are considered. In the context of both TD and PCD, a number of GFP- and FDD-based algorithms are developed. Among these, one based on so-called averaged-forward-backward-differencing (AFB) is shown to result in the greatest improvement in convergence rate. As it turns out, AFB is equivalent to the "rotated scheme" (R) of Willot (2015) in the context of TD. In the context of PCD, comparison of the performance and convergence behavior of AFB/R- and DGO-based algorithms shows that the former is more efficient than the latter for larger phase contrasts., Comment: 35 pages, 10 figures

To study the nanoscopic interaction between edge dislocations and a phase boundary within a two-phase microstructure the effect of the phase contrast on the internal stress field due to the dislocations needs to be taken into account. For this purpose a 2D semi-discrete model is proposed in this paper. It consists of two distinct phases, each with its specific material properties, separated by a fully coherent and non-damaging phase boundary. Each phase is modelled as a continuum enriched with a Peierls--Nabarro (PN) dislocation region, confining dislocation motion to a discrete plane, the glide plane. In this paper, a single glide plane perpendicular to and continuous across the phase boundary is considered. Along the glide plane bulk induced shear tractions are balanced by glide plane shear tractions based on the classical PN model. The model's ability to capture dislocation obstruction at phase boundaries, dislocation pile-ups and dislocation transmission is studied. Results show that the phase contrast in material properties (e.g. elastic stiffness, glide plane properties) alone creates a barrier to the motion of dislocations from a soft to a hard phase. The proposed model accounts for the interplay between dislocations, external boundaries and phase boundary and thus represents a suitable tool for studying edge dislocation--phase boundary interaction in two-phase microstructures.

We investigate non-linear elastic deformations in the phase field crystal model and derived amplitude equations formulations. Two sources of non-linearity are found, one of them based on geometric non-linearity expressed through a finite strain tensor. It reflects the Eulerian structure of the continuum models and correctly describes the strain dependence of the stiffness. In general, the relevant strain tensor is related to the left Cauchy-Green deformation tensor. In isotropic one- and two-dimensional situations the elastic energy can be expressed equivalently through the right deformation tensor. The predicted isotropic low temperature non-linear elastic effects are directly related to the Birch-Murnaghan equation of state with bulk modulus derivative $K'=4$ for bcc. A two-dimensional generalization suggests $K'_{2D}=5$. These predictions are in agreement with ab initio results for large strain bulk deformations of various bcc elements and graphene. Physical non-linearity arises if the strain dependence of the density wave amplitudes is taken into account and leads to elastic weakening. For anisotropic deformations the magnitudes of the amplitudes depend on their relative orientation to the applied strain., Comment: 16 pages