Dynamics and Patterning of Screw Dislocations in Two Dimensions (original) (raw)

Homogeneous nucleation of dislocations as a pattern formation phenomenon

European Journal of Mechanics - A/Solids

Dislocation nucleation in homogeneous crystals initially unfolds as a linear symmetry-breaking elastic instability. In the absence of explicit nucleation centers, such instability develops simultaneously all over the crystal and due to the dominance of long range elastic interactions it advances into the nonlinear stage as a collective phenomenon through pattern formation. In this paper we use a novel mesoscopic tensorial model (MTM) of crystal plasticity to study the delicate role of crystallographic symmetry in the development of the dislocation nucleation patterns in defect free crystals loaded in a hard device. The model is formulated in 2D and we systematically compare lattices with square and triangular symmetry. To avoid the prevalence of the conventional plastic mechanisms, we consider the loading paths represented by pure shears applied on the boundary of the otherwise unloaded body. These loading protocols can be qualified as exploiting the 'softest' and the 'hardest' directions and we show that the associated dislocation patterns are strikingly different.

Dislocation patterning in a two-dimensional continuum theory of dislocations

Physical Review B

Understanding the spontaneous emergence of dislocation patterns during plastic deformation is a long standing challenge in dislocation theory. During the past decades several phenomenological continuum models of dislocation patterning were proposed, but few of them (if any) are derived from microscopic considerations through systematic and controlled averaging procedures. In this paper we present a two-dimensional continuum theory that is obtained by systematic averaging of the equations of motion of discrete dislocations. It is shown that in the evolution equations of the dislocation densities diffusionlike terms neglected in earlier considerations play a crucial role in the length scale selection of the dislocation density fluctuations. It is also shown that the formulated continuum theory can be derived from an averaged energy functional using the framework of phase field theories. However, in order to account for the flow stress one has in that case to introduce a nontrivial dislocation mobility function, which proves to be crucial for the instability leading to patterning.

Thermodynamically consistent phase field approach to dislocation evolution at small and large strains

Journal of The Mechanics and Physics of Solids, 2015

A thermodynamically consistent, large strain phase field approach to dislocation nucleation and evolution at the nanoscale is developed. Each dislocation is defined by an order parameter, which determines the magnitude of the Burgers vector for the given slip planes and directions. The kinematics is based on the multiplicative decomposition of the deformation gradient into elastic and plastic contributions. The relationship between the rates of the plastic deformation gradient and the order parameters is consistent with phenomenological crystal plasticity. Thermodynamic and stability conditions for homogeneous states are formulated and satisfied by the proper choice of the Helmholtz free energy and the order parameter dependence on the Burgers vector. They allow us to reproduce desired lattice instability conditions and a stress-order parameter curve, as well as to obtain a stress-independent equilibrium Burgers vector and to avoid artificial dissipation during elastic deformation. The Ginzburg-Landau equations are obtained as the linear kinetic relations between the rate of change of the order parameters and the conjugate thermodynamic driving forces. A crystalline energy coefficient for dislocations is defined as a periodic step-wise function of the coordinate along the normal to the slip plane, which provides an energy barrier normal to the slip plane and determines the desired, mesh-independent height of the dislocation bands for any slip system orientation. Gradient energy contains an additional term, which excludes the localization of a dislocation within a height smaller than the prescribed height, but it does not produce artificial interface energy. An additional energy term is introduced that penalizes the interaction of different dislocations at the same point. Non-periodic boundary conditions for dislocations are introduced which include the change of the surface energy due to the exit of dislocations from the crystal.

Phase field approach to dislocation evolution at large strains: Computational aspects

International Journal of Solids and Structures, 2016

Computational aspects of the phase field simulations of dislocation nucleation and evolution are addressed. The complete system of equations for the coupled phase field approach to dislocation nucleation and evolution and nonlinear mechanics for large strains is formulated. Analytical solutions for a stationary and propagating single dislocation, dislocation velocity, core energy, and core width are found. Dislocation parameters for nickel are identified based on existing molecular dynamics simulations. In contrast to all previous efforts that are based on the spectral approach, finite element method (FEM) is utilized, which allowed us to treat large strain problems and non-periodic boundary conditions. The single dislocation order parameter profile and the stationary distance between two neighboring dislocations at a semicoherent sharp austenite-martensite interface are in perfect agreement with analytical expressions. The main focus is on proving that the new points of the developed theory can be confirmed in simulations, including possibility of obtaining the desired dislocation height for aligned and inclined dislocations, eliminating spurious stresses, resolving dislocation cores and interaction between cores of different dislocations. Mesh independence of the solutions is demonstrated and the effect of approximating finite element polynomials is analyzed, exhibiting possibility of significant numerical errors when special care is not taken of. Problems of nucleation and evolution of multiple dislocations along the single and multiple slip systems near martensitic lath, and along the sharp austenite-martensite interface, the activity of dislocations with two different orientations in a nanograined material under shear and pressure, and the interaction between two intersecting dislocation systems are studied. Surface-modified partial dislocation was revealed. These problems represent the first step in the future study of interaction of phase transformation and dislocations.

On the formation and stability of dislocation patterns—I: One-dimensional considerations

International Journal of Engineering Science, 1985

distinguishing among mobile and immobile dislocations and operating within the framework of continuum mechanics it is possible to derive a set of partial differential equations of the diffusion-reaction type for the evolution of dislocation species. On examining the competition between gradient dependent terms modelling the motion of dislocations and nonlinear terms modelling their interactions, it is shown that stable solutions are possible. The wavelength turns out to be a material property in agreement with observations. The discussion is limited to one dimension, that is to glide of straight dislocations in the slip direction, and the model corresponds physically to the ladder-like structure of persistent slip bands.