Variational approach in dislocation theory

Groma, István; Györgyi, G.; Ispánovity, Péter Dusán

doi:10.1080/14786430903401073

Cited by 23 publications

(21 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, for any given slip plane spacing s, it is always the configurations consisting of pure dislocation dipoles (ρ g = 0) that deliver the strongest flow stress, and this phenomenon has also been mentioned in other works (e.g. Groma et al, 2010). Fig.…”

Section: Effective Flow Stresssupporting

confidence: 67%

Continuum dynamics of the formation, migration and dissociation of self-locked dislocation structures on parallel slip planes

Zhu

Niu

Xiang

2016

Journal of the Mechanics and Physics of Solids

View full text Add to dashboard Cite

In continuum models of dislocations, proper formulations of short-range elastic interactions of dislocations are crucial for capturing various types of dislocation patterns formed in crystalline materials. In this article, the continuum dynamics of straight dislocations distributed on two parallel slip planes is modelled through upscaling the underlying discrete dislocation dynamics. Two continuum velocity field quantities are introduced to facilitate the discrete-to-continuum transition. The first one is the local migration velocity of dislocation ensembles which is found fully independent of the short-range dislocation correlations. The second one is the decoupling velocity of dislocation pairs controlled by a threshold stress value, which is proposed to be the effective flow stress for single slip systems. Compared to the almost ubiquitously adopted Taylor relationship, the derived flow stress formula exhibits two features that are more consistent with the underlying discrete dislocation dynamics: i) the flow stress increases with the in-plane component of the dislocation density only up to a certain value, hence the derived formula admits a minimum inter-dislocation distance within slip planes; ii) the flow stress smoothly transits to zero when all dislocations become geometrically necessary dislocations. A regime under which inhomogeneities in dislocation density grow is identified, and is further validated through comparison with discrete dislocation dynamical simulation results. Based on the findings in this article and in our previous works, a general strategy for incorporating short-range dislocation correlations in continuum models of dislocations is proposed.

show abstract

Section: Effective Flow Stresssupporting

confidence: 67%

Continuum dynamics of the formation, migration and dissociation of self-locked dislocation structures on parallel slip planes

Zhu

Niu

Xiang

2016

Journal of the Mechanics and Physics of Solids

View full text Add to dashboard Cite

show abstract

“…We have shown in earlier work that the evolution equations for the two densities of positive and negative dislocations as studied above can be cast into the framework of phase field theories [22,23,44,45]. The terms proportional to τ a , however, were not included into the earlier considerations.…”

Section: Variational Approachmentioning

confidence: 99%

Dislocation patterning in a two-dimensional continuum theory of dislocations

2016

View full text Add to dashboard Cite

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.

show abstract

“…So, terms accounting for correlations have to be added to P sc to arrive at a physically relevant model. As it is explained in detail in [40][41][42] because of the stress screening observed by DDD simulations for close to neutral systems (κ = ρ + − ρ − is much smaller than ρ = ρ + + ρ − ) correlations can be well accounted for by adding a quadratic term in κ to P sc . With this term, the potential reads as…”

mentioning

confidence: 99%

“…As it was shown in detail in [40][41][42] the equation of motion of the dislocations Eq. (1) can be obtained from the variational "plastic" potential…”

mentioning

confidence: 99%

See 1 more Smart Citation

Scale-Free Phase Field Theory of Dislocations

2015

View full text Add to dashboard Cite

According to recent experimental and numerical investigations if the characteristic size of a specimen is in the submicron size regime several new interesting phenomena emerge during the deformation of the samples. Since in such a systems the boundaries play a crucial role, to model the plastic response of submicron sized crystals it is crucial to determine the dislocation distribution near the boundaries. In this paper a phase field type of continuum theory of the time evolution of an ensemble of parallel edge dislocations with identical Burgers vectors, corresponding to the dislocation geometry near boundaries, is presented. Since the dislocation-dislocation interaction is scale free (1/r), apart from the average dislocation spacing the theory cannot contain any length scale parameter. As shown, the continuum theory suggested is able to recover the dislocation distribution near boundaries obtained by discrete dislocation dynamics simulations. One may expect, however, that for a large number of problems not all the details accounted by DDD simulations are important, the response of the dislocation network can be well described on a continuum level. Although several such continuum theories of dislocations have been developed, [27-37] most of them correspond either to mean field approximation or are based on completely phenomenological grounds. However, the role of dislocation-dislocation correlation, crucial because of the long range nature of dislocation-dislocation interaction, is far from understood. Correlation effects are taken into account in a systematic manner only in the limit when the signed dislocation density κ (geometrically necessary dislocation (GND) density) is much smaller than the stored density ρ. [38][39][40][41][42].With the advance of nanotechnology the characteristic size of the microstructure of crystalline materials reduced to the submicron level. As a consequence, the role of boundaries (sample surface, grain boundary, etc.) has become even more important than earlier. So, to model the plastic response of samples with features on the submicron scale it is crucial to determine the dislocation distribution near the boundaries. Close to a boundary the GND density is often comparable to the stored one, so the assumption |κ| ρ is not valid. The dislocation distribution near a boundary is traditionally described by the 1D pile-up of the dislocations [43]. For many real dislocation configurations, however, the interaction between dislocations in different slip planes is important requiring to go up to modeling in minimum of 2D. In this paper a phase field type theory is suggested for the simplest possible 2D dislocation arrangement consisting of straight parallel dislocations with single slip. The evolution equations of the dislocation densities are obtained from a functional of the dislocation densities and the stress potential. In contrast to other approaches suggested recently, where a set of walls of dislocations with equidistant slip distances is considered to model the dislocation config...

show abstract

Variational approach in dislocation theory

Cited by 23 publications

References 23 publications

Continuum dynamics of the formation, migration and dissociation of self-locked dislocation structures on parallel slip planes

Continuum dynamics of the formation, migration and dissociation of self-locked dislocation structures on parallel slip planes

Dislocation patterning in a two-dimensional continuum theory of dislocations

Scale-Free Phase Field Theory of Dislocations

Contact Info

Product

Resources

About