arXiv subjects

Patrick van Meurs

Publications and source records attributed to Patrick van Meurs.

29 records · Page 2Linked to original sources

arXiv2020

Atomistic origins of continuum dislocation dynamics

This paper focuses on the connections between four stochastic and deterministic models for the motion of straight screw dislocations. Starting from a description of screw dislocation motion as interacting random walks on a lattice, we prove explicit estimates of the distance between solutions of this model, an SDE system for the dislocation positions, and two deterministic mean-field models describing the dislocation density. The proof of these estimates uses a collection of various techniques in analysis and probability theory, including a novel approach to establish propagation-of-chaos on a spatially discrete model. The estimates are non-asymptotic and explicit in terms of four parameters: the lattice spacing, the number of dislocations, the dislocation core size, and the temperature. This work is a first step in exploring this parameter space with the ultimate aim to connect and quantify the relationships between the many different dislocation models present in the literature.

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arXiv2020

The continuum limit of interacting dislocations on multiple slip systems

In this paper we derive the continuum limit of a multiple-species, interacting particle system by proving a $Γ$-convergence result on the interaction energy as the number of particles tends to infinity. As the leading application, we consider $n$ edge dislocations in multiple slip systems. Since the interaction potential of dislocations has a logarithmic singularity at zero with a sign that depends on the orientation of the slip systems, the interaction energy is unbounded from below. To make the minimization problem of this energy meaningful, we follow the common approach to regularise the interaction potential over a length-scale $δ> 0$. The novelty of our result is that we leave the \emph{type} of regularisation general, and that we consider the joint limit $n \to \infty$ and $δ\to 0$. Our result shows that the limit behaviour of the interaction energy is not affected by the type of the regularisation used, but that it may depend on how fast the \emph{size} (i.e., $δ$) decays as $n \to \infty$.

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arXiv2020

Discrete-to-continuum limits of planar disclinations

In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition. Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof method is relaxation of the energy. The main mathematical novelty of our proof is a density theorem for the special boundary condition. In addition to our limit theorem, we construct examples of planar disclinations as solutions to numerical minimization of the model and show that classical results for wedge disclinations are recovered by our analysis.

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arXiv2018

Discrete-to-continuum limits of particles with an annihilation rule

In the recent trend of extending discrete-to-continuum limit passages for gradient flows of single-species particle systems with singular and nonlocal interactions to particles of opposite sign, any annihilation effect of particles with opposite sign has been side-stepped. We present the first rigorous discrete-to-continuum limit passage which includes annihilation. This result paves the way to applications such as vortices, charged particles, and dislocations. In more detail, the discrete setting of our discrete-to-continuum limit passage is given by particles on the real line. Particles of the same type interact by a singular interaction kernel; those of opposite sign interact by a regular one. If two particles of opposite sign collide, they annihilate, i.e., they are taken out of the system. The challenge for proving a discrete-to-continuum limit is that annihilation is an intrinsically discrete effect where particles vanish instantaneously in time, while on the continuum scale the mass of the particle density decays continuously in time. The proof contains two novelties: (i) the observation that empirical measures of the discrete dynamics (with annihilation rule) satisfy the continuum evolution equation that only implicitly encodes annihilation, and (ii) the fact that, by imposing a relatively mild separation assumption on the initial data, we can identify the limiting particle density as a solution to the same continuum evolution equation.

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arXiv2018

Convergence and non-convergence of many-particle evolutions with multiple signs

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 $δ_n>0$, 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 the theory of $λ$-convex gradient flows, a priori estimates for the Groma-Balogh equations and Orlicz spaces. The proof require $δ_n$ to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if $δ_n$ 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 many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

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arXiv2017

Many-particle limits and non-convergence of dislocation wall pile-ups

The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repelling force on each other, resulting in a nontrivial equilibrium configuration. This class of particle systems covers the setting of pile-ups of dislocation walls, which is an idealised setup for studying the microscopic origin of several dislocation density models in the literature. Such density models are used to construct constitutive relations in plasticity models. Our aim is to pass to the many-particle limit. The main challenge is the combination of the nonlocal nature of the interactions, the singularity of the interaction potential between particles of the same type, the non-convexity of the the interaction potential between particles of the opposite type, and the interplay between the length-scale of the domain with the length-scale $\ell_n$ of the decay of the interaction potential. Our main results are the $Γ$-convergence of the energy of the particle positions, the evolutionary convergence of the related gradient flows for $\ell_n$ sufficiently large, and the non-convergence of the gradient flows for $\ell_n$ sufficiently small.

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arXiv2016

Asymptotic analysis of boundary layers in a repulsive particle system

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. Our setting covers pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via Gamma-convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk.

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arXiv2016

Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy $E_γ$ modelling the interactions$-$at a typical length-scale of $1/γ$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_γ$ in powers of $1/γ$ by $Γ$-convergence, in the limit $γ\to\infty$. While the zero-order term in the expansion, the $Γ$-limit of $E_γ$, captures the `bulk' profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a `boundary-layer' energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.

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arXiv2016

Dynamics of screw dislocations: a generalised minimising-movements scheme approach

The gradient flow structure of the model introduced in [CG99] for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the "maximal dissipation criterion" that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in [CG99] is connected to a time-discrete evolution scheme which explicitly confines dislocations to move each time step along a single glide direction. It is proved that the time-continuous model in [CG99] is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalisation of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.

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arXiv2014

Upscaling of the dynamics of dislocation walls

We perform the discrete-to-continuum limit passage for a microscopic model describing the time evolution of dislocations in a one dimensional setting. This answers the related open question raised by Geers et al. in [GPPS13]. The proof of the upscaling procedure (i.e. the discrete-to-continuum passage) relies on the gradient flow structure of both the discrete and continuous energies of dislocations set in a suitable evolutionary variational inequality framework. Moreover, the convexity and $Γ$-convergence of the respective energies are properties of paramount importance for our arguments.

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arXiv2013

Is adding charcoal to soil a good method for CO2 sequestration? - Modeling a spatially homogeneous soil

Carbon sequestration is the process of capture and long-term storage of atmospheric carbon dioxide (CO2) with the aim to avoid dangerous climate change. In this paper, we propose a simple mathematical model (a coupled system of nonlinear ODEs) to capture some of the dynamical effects produced by adding charcoal to fertile soils. The main goal is to understand to which extent charcoal is able to lock up carbon in soils. Our results are preliminary in the sense that we do not solve the CO2 sequestration problem. Instead, we do set up a flexible modeling framework in which the interaction between charcoal and soil can be tackled by means of mathematical tools. We show that our model is well-posed and has interesting large-time behaviour. Depending on the reference parameter range (e.g. type of soil) and chosen time scale, numerical simulations suggest that adding charcoal typically postpones the release of CO2.

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