### Abstract

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.

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.

Original language | English |
---|---|

Pages (from-to) | 803-822 |

Number of pages | 20 |

Journal | Mathematics and Mechanics of Solids |

Volume | 18 |

Issue number | 8 |

Early online date | 13 Sep 2012 |

DOIs | |

Publication status | Published - Nov 2013 |

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## Cite this

Das, A., Acharya, A., Zimmer, J., & Matthies, K. (2013). Can equations of equilibrium predict all physical equilibria? A case study from Field Dislocation Mechanics.

*Mathematics and Mechanics of Solids*,*18*(8), 803-822. https://doi.org/10.1177/1081286512451940