Can equations of equilibrium predict all physical equilibria? A case study from Field Dislocation Mechanics

Numerical solutions of a one dimensional model of screw dislocation walls (twist boundaries) are explored. The model is an exact reduction of the 3D system of partial differential equations of Field Dislocation Mechanics. It shares features of both Ginzburg-Landau (GL) type gradient flow equations as well as hyperbolic conservation laws, but is qualitatively different from both. We demonstrate such similarities and differences in an effort to understand the equation through simulation. A primary result is the existence of spatially non-periodic, extremely slowly evolving (quasi-equilibrium) cell-wall dislocation microstructures practically indistinguishable from equilibria, which however cannot be solutions to the equilibrium equations of the model, a feature shared with certain types of GL equations. However, we show that the class of quasi-equilibria comprising spatially non-periodic microstructure consisting of fronts is larger than that of the GL equations associated with the energy of the model. In addition, under applied strain-controlled loading, a single dislocation wall is shown to be capable of moving as a localized entity as expected in a physical model of dislocation dynamics, in contrast to the associated GL equations. The collective evolution of the quasi-equilibrium cell-wall microstructure exhibits a yielding-type behavior as bulk plasticity ensues, and the effective stress-strain response under loading is found to be rate-dependent. The numerical scheme employed is non-conventional since wave-type behavior has to be accounted for, and interesting features of two different schemes are discussed. Interestingly, a stable scheme conjectured by us to produce a non-physical result in the present context nevertheless suggests a modified continuum model that appears to incorporate apparent intermittency.