Physical Review B

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Journal of Physics Condensed Matter: Preface

Computational Materials Science X

Dynamic fracture mechanics

We describe a model based on continuum mechanics that reduces the study of a significant class of problems of discrete dislocation dynamics to questions of the modern theory of continuum plasticity As applications, we explore the questions of the existence of a Peierls stress in a continuum theory, dislocation annihilation, dislocation dissociation, finite-speed-of-propagation effects of elastic waves vis-a-vis dynamic dislocation fields, supersonic dislocation motion, and short-slip duration in rupture dynamics

http://faculty.ce.cmu.edu/acharya/files/2013/07/layer_problem.pdf

A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations

A continuum model for dislocation pile-up problems

Finite element approximation of finite deformation dislocation mechanics

A mathematical theory of time-dependent dislocation mechanics of unrestricted geometric and material nonlinearity is reviewed. Within a ‘small deformation’ setting, a suite of simplied, but interesting, models, namely a nonlocal Ginzburg Landau, a nonlocal level set, and a nonlocal generalized Burgers equation are derived. In the nite deformation setting, it is shown that an additive decomposition of the total velocity gradient into elastic and plastic parts emerges naturally from a micromechanical starting point that involves no notion of plastic deformation but only the elastic distortion, material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin tensor emerges naturally as well.

/pdf/from-dislocation-motion-to-an-additive-velocity-gradient-j33nyhbroc.pdf

From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics

Nonsingular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. It is well known that the universally accepted Oseen-Frank energy is infinite for configurations that contain disclination line defects. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where, under certain conditions, infinite director gradients have zero associated energy cost, as would be necessary for modeling half-integer strength disclinations within the framework of the director theory. Equilibria and dynamics (in the absence of flow) of line defects are studied within the proposed model. Using appropriate initial/boundary data, the gradient-flow dynamics of this energy leads to non-singular, line defect equilibrium solutions, including those of half-integer strength. However, we demonstrate that the gradient flow dynamics for this energy is not able to adequately describe defect evolution. Motivated by similarity with dislocation dynamics in solids, a novel 2D-model of disclination dynamics in nematics is proposed. The model is based on the extended Oseen-Frank energy and takes into account thermodynamics and the kinematics of conservation of defect topological charge. We validate this model through computations of disclination equilibria, annihilation, repulsion, and splitting. We show that the energy function we devise, suitably interpreted, can serve as well for the modeling of equilibria and dynamics of dislocation line defects in solids making the conclusions of this paper relevant to mechanics of both solids and liquid crystals.

/pdf/a-non-traditional-view-on-the-modeling-of-nematic-3fts9cirma.pdf

Xiaohan Zhang

Papers

A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations

A continuum model for dislocation pile-up problems

Finite element approximation of finite deformation dislocation mechanics

From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics

A non-traditional view on the modeling of nematic disclination dynamics