Field models of anisotropic dislocation systems
Mehran Monavari
Institute for Materials Simulation, WW8
FAU Erlangen-Nürnberg
Wednesday, 27. November 2013, 17.00
WW8, Room 2.018-2, Dr.-Mack-Str. 77, Fürth
The collective dynamics of dislocations provides the physical foundation for plastic deformation processes in metals. Discrete Dislocation Dynamics (DDD) simulations are able to predict the evolution of the dislocation microstructure in great detail but are limited to small length scales, strains and/or dislocation densities, because their computational cost increases rapidly as systems enlarge.
To overcome this limitation, the Continuum Dislocation Dynamics (CDD) theory was developed which represents the kinematics of curved dislocation lines in terms of statistically averaged quantities (e.g. densities). In this approach the dislocation density is considered a function of the local dislocation line orientation – in case of dislocation motion by glide only, in each material point it becomes a function on the unit circle. CDD provides kinematically closed evolution equations for the densities and associated orientation distribution functions. To reduce the number of variables and arrive at a theory that is amenable to efficient numerical implementation it is, however, desirable to approximate the orientation distribution in terms of a finite number of moments, e.g. in terms of Fourier expansion coefficients or so-called alignment tensors. These define novel field variables for which CDD allows to derive a hierarchy of evolution equations. To relate the approach to more traditional classical ways of envisaging dislocation arrangements, we note that the lowest- order coefficient is the total dislocation density, the first-order coefficients contain the information about density and orientation of geometrically necessary excess dislocations, while the second- order coefficients embody information about preferred dislocation orientations (e.g. screw or edge or 60o) irrespective of sign.
CDD allows to derive, in a kinematically exact manner, an infinite hierarchy of equations for such field variables. In this paper, we discuss different approaches such as the maximum entropy method for condensing this hierarchy into a closed set of kinematic equations which are capable of characterizing the evolution of strongly anisotropic dislocation arrangements as encountered when dislocations pile up against extended obstacles (grain boundaries), in BCC metals deforming below the transition temperature, or in fatigue of FCC metals. We investigate numerically how CDD performs in these simulations and evaluate the consequences of different closure approximations by benchmarking the results of our numerical studies against data from DDD simulations.