An efficient numerical method for "all-dislocation" density dynamics

Abstract

Dislocations are linear defects of crystals with a crucial role in the plastic deformation of metals. Although dislocations play a significant role in the deformation of metals in various scales, it has always been a challenge to resolve dislocation effects when simulating the deformation of samples larger than a few micrometers. This thesis introduces an efficient and accurate numerical for mesoscopic plasticity. It preserves dislocation contents in the form of “all-dislocation” density, which results in interesting features for the proposed numerical method. The utilization of “all-dislocation” density provides us with the opportunity to preserve the value of geometrically necessary dislocations (GNDs) and statically stored dislocations (SSDs) in order to apply them for devising a dynamics closure. In the first chapter, a partial differential equation (PDE) for the kinematics of "all dislocation" density or scalar density is introduced and subsequently solved. The derived numerical method contains using a Gaussian interpolation function for mining the data of discrete dislocation lines and a finite difference method for solving the PDE. The proposed method can detect the density evolution in different characteristic lengths from ones finer than the core of dislocation lines to those larger than the average distance between dislocation lines. It is noteworthy that this numerical solver is minimalist since it only contains two sufficient variables, namely, the density and character of the dislocations. In the second chapter, a dynamics closure scheme is developed to calculate the approaching and tilting velocity of dislocation segments. It employs Mura’s formula for the calculation of mutual interaction among dislocation lines after singularity removal. Also, the forces within the core of dislocations are calculated through a phenomenological scheme that can specify the core shape and width of the dislocations. To verify the accuracy and applicability of the numerical method for various scales, multi-slip systems, intergranular slip transfer, Orowan looping, Frank-reed source, and dislocation loop expansion are simulated. The results show that the proposed numerical method has the potential to be applied as a simulator for the simulation of plasticity in mesoscale for different materials in different conditions.