A Computational Model for Discrete-to-Continuum Dislocation-Based Crystal Plasticity

In metals, the evolution of the underlying dislocation microstructure is responsible for most of their mechanical properties of engineering interest. However, in spite of the microscopic origin of plasticity, models of plastic flow are still mostly based on phenomenological equations. These models not only are limited to the range spanned by the supporting database, but also fail to predict characteristic phenomena of plastic flow such as size effects and formation of dislocation patterns.;The objective of this research is to develop and implement a self-consistent computational framework for dislocation-based plasticity at the mesoscale. In the proposed framework, the collective motion of dislocations, governed by mutual dislocation interactions and applied stress field, determines the plastic component of strain during the deformation process of crystals. In turn, plastic strain affects the stress field entering as a source of eigenstrains in the displacement boundary value problem. In order to formulate rigorous kinetic equations governing the time evolution of dislocation densities and plastic flow, the model builds on the well-established incompatible kinematics of the theory of continuously distributed dislocations. Based on thermodynamic considerations, we develop a constitutive relationship that allows to determine the average dislocation flux as a function of the self-consistent stress field and dislocation densities.;As a first special case, we express the two dimensional version of the computational framework in both its strong and weak forms. Coupled finite element simulations of plane strain micro-indentation experiments are performed, where we solve simultaneously for the displacement field, dislocation densities and plastic components of deformation. Experimental comparison is provided.;As a second special case, we formulate the framework for discrete dislocation densities, recovering the equation of motion used in discrete dislocation dynamics simulations. In order to study the rate of dislocation reactions we implement a graph theory- based finite element method for discrete dislocation dynamics. According to this formulation the dislocation configuration is discretized in its entangled structure using vertex degrees of freedom and edge shape functions, chosen to maximize accuracy and minimize the computational cost. Dislocation reactions (such as cross-slip, junction formation and annihilation) are expressed in terms of flow-conserving network operations. As an application of the method we analyze the rate of cross-slip if fcc metals, study the dependence of the activation energy on the resolved shear stress and compare with experiments.