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

Amit Das, Amit Acharya, Johannes Zimmer, Karsten Matthies

Research output: Contribution to journalArticle

4 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)803-822
Number of pages20
JournalMathematics and Mechanics of Solids
Volume18
Issue number8
Early online date13 Sep 2012
DOIs
Publication statusPublished - Nov 2013

Fingerprint

Dislocation
Mechanics
Ginzburg-Landau Equation
Predict
Microstructure
Cell Wall
Dislocation Dynamics
Screw Dislocation
Hyperbolic Conservation Laws
Gradient Flow
Ginzburg-Landau
Intermittency
Continuum Model
Systems of Partial Differential Equations
Screw dislocations
One-dimensional Model
Physical Model
Twist
Plasticity
Numerical Scheme

Cite this

Can equations of equilibrium predict all physical equilibria? A case study from Field Dislocation Mechanics. / Das, Amit; Acharya, Amit; Zimmer, Johannes; Matthies, Karsten.

In: Mathematics and Mechanics of Solids, Vol. 18, No. 8, 11.2013, p. 803-822.

Research output: Contribution to journalArticle

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