Dislocation patterning in a two-dimensional continuum theory of dislocations

Groma, István; Zaiser, Michael; Ispánovity, Péter Dusán

doi:10.1103/physrevb.93.214110

Cited by 57 publications

(80 citation statements)

References 48 publications

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“…It describes the evolution of the density of 'positive' and 'negative' straight and parallel edge dislocations, which are represented by 'positive' and 'negative' points in two dimensions. This model, and elaborations of it [GVI15,GZI16] have been used in the engineering community to predict dislocation density profiles with surprising accuracy [YGG04, YG05, GPHK09, DPG15]. The original system of equations from [GB99] is central to this paper; we will refer to it as the Groma-Balogh equations, and it appears in generalized form as equations (1.4) below.…”

Section: Introductionmentioning

confidence: 99%

“…The derivations in [Gro97,GB99,GCZ03,GGK06] are based on an upscaling argument starting from a system of discrete dislocations. None of these results are rigorous, however, since they build on uncontrolled approximations such as exchanging averaging with nonlinearity [GB99] or postulating a closure relation in the BBGKY hierarchy [Gro97,GZI16].…”

Section: Introductionmentioning

confidence: 99%

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Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Garroni

Meurs

Peletier³

et al. 2019

Arch Rational Mech Anal

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We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals.Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale δn > 0, which converges to zero as the number of particles n tends to infinity.We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma-Balogh equations. In the proof we rely on both the theory of λ-convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a δn-regularised version of the Groma-Balogh equations, and a priori estimates for the Groma-Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n → ∞ we require δn to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if δn converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma-Balogh equations.These results show how the validity of the Groma-Balogh equations as the limit of manyparticle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Garroni

Meurs

Peletier³

et al. 2019

Arch Rational Mech Anal

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“…While most recent models exploit advances in computational power [18] or in kinematic averaging methods [17] in order to address the important problem of dislocation patterning in 3D and under conditions of multiple slip, the present authors have pursued a more simplistic yet more fundamental goal, namely to elucidate the respective influences of dynamic and energetic mechanisms on the patterning process. To this end we focus on a minimal system (2D, single slip) where an exact representation of the kinematics is possible and well-defined forms have been established both for the energy functional [13,15] and also for the effective mobility law [12]. As a consequence, we can dispense to a large extent of phenomenological assumptions and obtain a complete understanding of the interplay of energy minimization, external driving, and friction in driving the emergence of dislocation patterns.…”

Section: Introductionmentioning

confidence: 99%

Instability of dislocation fluxes in a single slip: Deterministic and stochastic models of dislocation patterning

Tüzes

Ispánovity

et al. 2018

Phys. Rev. B

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We study a continuum model of dislocation transport in order to investigate the formation of heterogeneous dislocation patterns. We propose a physical mechanism which relates the formation of heterogeneous patterns to the dynamics of a driven system which tries to minimize an internal energy functional while subject to dynamic constraints and state dependent friction. This leads us to a novel interpretation which resolves the old 'energetic vs. dynamic' controversy regarding the physical origin of dislocation patterns. We demonstrate the robustness of the developed patterning scenario by considering the simplest possible case (plane strain, single slip) yet implementing the dynamics of the dislocation density evolution in two very different manners, namely (i) a hydrodynamic formulation which considers transport equations that are continuous in space and time while assuming a linear stress dependency of dislocation motion, and (ii) a stochastic cellular automaton implementation which assumes spatially and temporally discrete transport of discrete 'packets' of dislocation density which move according to an extremal dynamics. Despite the huge differences between both kinds of models, we find that the emergent patterns are mutually consistent and in agreement with the prediction of a linear stability analysis of the continuum model. We also show how different types of initial conditions lead to different intermediate evolution scenarios which, however, do not affect the properties of the fully developed patterns.

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“…The extent of deviation of σ xy from the analytical solution for pure shear on linear (left panel) and logarithmic (right panel) scales. For the definition of p 1 , p 2 and p ∞ see equations(22),(23) and(24).…”

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confidence: 99%

Efficient numerical method to handle boundary conditions in 2D elastic media

Berta¹,

Groma²,

Ispánovity³

2020

Modelling Simul. Mater. Sci. Eng.

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A numerical method is developed to efficiently calculate the stress (and displacement) field in finite 2D rectangular media. The solution is expanded on a function basis with elements that satisfy the Navier-Cauchy equation. The obtained solution approximates the boundary conditions with their finite Fourier series. The method is capable to handle Dirichlet, Neumann and mixed boundary value problems as well and it was found to converge exponentially fast to the analytical solution with respect to the size of the basis. Possible application in discrete dislocation dynamics simulations is discussed and compared to the widely used finite element methods: it was found that the new method is superior in terms of computational complexity.

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Dislocation patterning in a two-dimensional continuum theory of dislocations

Cited by 57 publications

References 48 publications

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Instability of dislocation fluxes in a single slip: Deterministic and stochastic models of dislocation patterning

Efficient numerical method to handle boundary conditions in 2D elastic media

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