A Fast Fourier Transform-Based Method for Static Field Dislocation Mechanics Within a First Strain Gradient Elasticity Theory of Mindlin Type
Stéphane Berbenni 1*, Thomas Böhlke 2
1 Université de Lorraine, CNRS, Arts et Métiers Institute of Technology, LEM3, Metz, F-57000, France; 2 Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT), Karlsruhe, 76131, Germany
Keywords: field dislocation mechanics, strain gradient elasticity, Mindlin, fast Fourier transform method
Here, a fast Fourier transform (FFT)-based micromechanical method for static Field Dislocation Mechanics (FDM) within a first Strain Gradient Elasticity (SGE) theory of Mindlin type is developed. The kinematic equations of dislocation defects are first presented in the framework of static FDM for periodic media. Using Stokes-Helmholtz decomposition, the incompatible elastic distortion is solved from a prescribed dislocation density field with a Poisson equation. The thermodynamic-based constitutive relationships are reported for stress and double stress tensors as functions of elastic strain and elastic strain gradient tensors, respectively. Homogeneous centrosymmetric materials are considered and are specified for cubic and isotropic materials. Then, the compatible elastic distortion field is obtained from solving equilibrium equations through a generalized Navier-type equation. The Fourier-based method is applied, which defines a generalized Green's function in Fourier space for first SGE of Mindlin type. Thus, the solutions for incompatible and compatible fields can be derived in Fourier space. The efficient Fast Fourier Transform (FFT) algorithm is used based on intrinsic Discrete Fourier Transforms (DFT). The numerical scheme is well adapted to the discrete grid to compute higher order partial derivatives with finite differences, which are transformed to the Fourier space. Therefore, using the inverse FFT, regularized dislocation density fields obtained from the present method are numerically computed for single straight dislocations like pure screw and edge dislocations. Stress and double stress fields are also calculated and compared with analytical results from the literature. Finally, some applications of the method focus on more complex dislocation configurations with distributions of dislocations, as dislocation dipoles and configurations with dislocation walls like low-angle grain boundaries (LAGBs).