The probability distribution of internal stresses in externally loaded 2D dislocation systems

Ispánovity, Pé ter Dusán; Groma, István

doi:10.1088/1742-5468/2008/12/p12009

Cited by 5 publications

(11 citation statements)

References 38 publications

(111 reference statements)

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“…In a simplified model used in this paper only parallel edge dislocations are considered with a single glide plane. Although in crystals dislocations usually form complex three-dimensional networks, this model proved capable of reproducing many experimentally observed phenomena related to plasticity, like, e.g., strain avalanche statistics [1], Andrade-creep exponents [2,3], and properties of X-ray profiles [4,5].…”

Section: Introductionmentioning

confidence: 96%

Critical behavior in dislocation systems: power-law relaxation below the yield stress

Ispánovity¹

2011

Preprint

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Plasticity of two-dimensional discrete dislocation systems is studied. It is shown, that at some threshold stress level the response becomes stress-rate dependent. Below this stress level the stress-plastic strain relation exhibits power-law type behavior. In this regime the plastic strain rate induced by a constant external stress decays to zero as a power-law, which stems from the scaling of the dislocation velocity distribution. The scaling is cut-off at a time only dependent on the system size and the scaling exponent depends on the external stress and on the initial correlations present in the system. These results show, that the dislocation system is in a critical state everywhere we studied below the threshold stress.

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Section: Introductionmentioning

confidence: 96%

Critical behavior in dislocation systems: power-law relaxation below the yield stress

Ispánovity¹

2011

Preprint

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“…It was also shown that the central part of the PDF is Gaussian and Cauchy for homogeneous and uncorrelated systems of straight infinite edge dislocations and dipolar configurations, respectively [40]. Both the central and asymptotic parts of stress distribution are shown to be affected by the external shear stress [41]; the central part becomes asymmetric and shifted by an amount proportional to the applied stress, and the asymptotic part becomes dependent on an extra term that decays with inverse forth power of stress. In a related work, an analytical expression of pair correlation of internal stress was proposed in [42] in which the stress fluctuation is directly connected to the dislocation density fluctuation.…”

Section: Introductionmentioning

confidence: 98%

“…The works reported in [39][40][41] are of direct relevance to the current work for they include an analysis of the internal shear stresses of straight parallel dislocation arrangements in terms of probability density functions (PDF). Under zero external stress, it is was shown that the asymptotic behavior of the PDF of stress is only dependent on the local dislocation density, and it decays with the inverse third power of the stress [39].…”

Section: Introductionmentioning

confidence: 99%

Statistics of internal stress fluctuations in dislocated crystals and relevance to density-based dislocation dynamics models

Vivekanandan¹,

Anderson²,

Pachaury³

et al. 2022

Modelling Simul. Mater. Sci. Eng.

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A statistical analysis of internal stress fluctuations, defined as the difference between the local mean stress and stress on dislocations, is presented for deforming crystals with 3D discrete dislocation systems. Dislocation realizations are generated using dislocation dynamics simulations and the associated stress field is computed as a superposition of a regularized stress field of dislocation lines within the domain of the solution and a complementary stress field computed via a finite-element boundary value problem. The internal stress fluctuations of interest are defined by an ensemble of the difference between the stress on dislocation lines and the local mean field stress in the crystal. The latter is established in a piecewise fashion over small voxels in the crystal thus allowing the difference between the local average stress and stress on segments to be easily estimated. The results show that the Schmid stress (resolved shear stress) and Escaig stress fluctuations on various slip systems sampled over a random set of points follow a Cauchy (Lorentz) distribution at all strain levels, with the amplitude and width of the distribution being dependent on the strain. The implications of the Schmid and Escaig internal stress fluctuations are discussed from the points of view of dislocation cross-slip and the dislocation motion in continuum dislocation dynamics.

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“…This idea led to the development of a statistical description of dislocation dynamics based on pair correlation functions, as first done by Groma [36] in the context of X-ray line broadening due to disordered distributions of dislocations and, subsequently, by Groma and Bakó [37] with stochastic ensembles of parallel edge dislocations. Further work by El Azab [18], Zaiser et al [38], Kratochvíl and Sedlácek [19] amongst many others, have focused on computing the pair correlation functions describing the dislocation density distribution and the general probability distribution functions, aiming at describing plasticity evolution laws (see [39][40][41]), and the energetics of dislocation ensembles [42][43][44]. Recently, Berdichevsky [45] has offered an expression for the regularised energy of a random set of dislocation lines, Zaiser [44] used an elastic energy functional approximation to study two and three dimensional ensembles of dislocations.…”

Section: Introductionmentioning

confidence: 99%

“…In the statistical approach developed by Groma and coworkers [37], the stress state is obtained from the probability distribution of the stress field, which in turn is dependent on the conditional probability to find dislocations at a given location, and is therefore not available in a general case [46], but may be obtained from X-ray diffraction data, dislocation dynamics simulations [39,40], or for specific kinds of dislocation ensembles [41,47]. For instance, Csikor and Groma [47] computed it for disperse arrangements of uncorrelated dislocation dipoles.…”

Section: Introductionmentioning

confidence: 99%

A stochastic study of the collective effect of random distributions of dislocations

Gurrutxaga-Lerma

2019

Journal of the Mechanics and Physics of Solids

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The effect that random populations of dislocations have on a material is examined through stochastic integration of a random cloud of dislocations lying at some distance away from a material point. The problem is studied in one, two, and three dimensions. In 1D, the cloud consists of individual edge dislocations placed along the real line; in 2D, of edge dislocations and edge dipoles on the plane; in 3D, of dislocation loops. In all cases, the dislocation cloud is randomly distributed in space, associated to which several relevant physical parameters, including the material's slip geometry, the dislocation's sign, and its relative orientation, are also stochastically treated. A fully disordered population, i.e., one where the dislocation's signatures and orientations are entirely random, is first studied. It is shown that such disordered systems entail a strong indeterminacy in the collective stress fields, which here is solved by enforcing mass conservation locally. In 2D, this is achieved by modelling a cloud of edge dipoles instead of individual dislocations; in 3D, this is naturally guaranteed by the modelling of closed dislocation loops. The long-range fields of the dipoles in 2D and of the loops in 3D is modelled via their multipolar force expansions, which greatly simplifies the analytical treatment of the problem. The cloud's effect is then studied by performing the stochastic integration of the multipolar fields via Campbell's theorem. The local order, but not the magnitude of the dislocation density, is shown to be critical in contributing to the plastic relaxation of the material: fully disordered systems are shown to self-attenuate, leading to plastic neutrality; ordered and partially ordered systems, achieved when dislocation signatures are aligned, display a direct relationship between the dislocation density and the average stress shielding the material. We establish and generalise the conditions that a system of dislocations must fulfil to display Taylor's equation and the Hall-Petch relation, and offer adequate scaling laws related to this. * bg374@cam.ac.uk

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The probability distribution of internal stresses in externally loaded 2D dislocation systems

Cited by 5 publications

References 38 publications

Critical behavior in dislocation systems: power-law relaxation below the yield stress

Critical behavior in dislocation systems: power-law relaxation below the yield stress

Statistics of internal stress fluctuations in dislocated crystals and relevance to density-based dislocation dynamics models

A stochastic study of the collective effect of random distributions of dislocations

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