Computational Approximation of Mesoscale Field Dislocation Mechanics at Finite Deformation

Abstract

This work involves the modeling and understanding of mechanical behavior of crystalline materials using nite deformation Mesoscale Field Dislocation Mechanics (MFDM). MFDM is a Partial Differential Equation (pde) based model to understand meso-macroscale plasticityin solids as it arises from dislocation motion, interaction, and nucleation within the material. Speci fically, the work is divided into the following two parts:  The fi rst part of the work deals with the development of a novel, parallel computationalframework for nite deformation MFDM. Beyond the development, implementation, and verfii cation of novel algorithms for the stated purpose, the computational frame-work involves the use of various state-of-the-art open source libraries such as Deal.ii, PetSc, P4est, MUMPS, ParMetis, and MPI. This accomplishment stands as a fi rstcomputational implementation of a partial differential equation based model of the mechanics of dislocations at nite deformations in the whole literature, including time-dependent behavior. The second part involves the veri fication and qualitative validation of the developed MFDM framework by applying it to study problems of signi ficant scienti fic and technological interest. We compare the stress fields obtained from nite deformation FDM with the small deformation analytical result for fields of screw and edge dislocations embedded in an in nite medium. The code is also veri fied against the sharply contrastingpredictions of geometrically linear and nonlinear theories for the stress field of a spatially homogeneous dislocation distribution in the body. This ability to calculate stress fields for an arbitrary distribution of dislocation density in the geometrically nonlinear setting is then used to quantify the change in volume of the body upon introductionof dislocations, and to study the stress- field path followed in a body corresponding to the presence of a sequence of dislocation con figurations comprising a single dislocationto a stress-free dislocation wall (or a grain boundary). Stringent veri fication tests of the time-dependent numerics are also undertaken such as i) studying the evolution ofa prescribed dislocation density in the absence of a flux of dislocations ii) accurately reproducing classical hyperelastic response, including the prediction of no hysteresis in a loading-unloading cycle, all up to nite ( 100%) strains. The model is also used to predict size effects, dislocation patterning, and the occurrence of stressed dislocation microstructures both under applied loads and in unloaded bodies using crystal andJ2 plasticity MFDM. Finally, we demonstrate the potential of the computational tool by modeling the longitudinal propagation of localized bands of plastic deformation inmetals, in particular, shear bands. This research presents results of a fi rst computational framework of (mesoscale) plasticity of unrestricted geometric and material nonlinearities with the following features:1. Computation of nite deformation stress fields of arbitrary (evolving) dislocation distributions in nite bodies of arbitrary shape and elastic anisotropy, under general statically admissible traction boundary conditions;2. Prediction of size effect and stressed dislocation pattern formation including dipolar dislocation walls under load;3. Prediction of stress-free dislocation microstructures and unloaded stressed metastable dislocation microstructures;4. Modeling of longitudinal propagation of plastic wavefront as a fundamental kinematical feature of plastic flow; 5. No involvement of a multiplicative decomposition of the deformation gradient, a plastic distortion tensor, or a choice of a reference con guration to describe the micromechanicsof plasticity arising from dislocation motion and predicts plastic spin with isotropic J2 plasticity assumptions.