153 research outputs found
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 -convergence of the respective
energies are properties of paramount importance for our arguments
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 -convergence result on the
interaction energy as the number of particles tends to infinity. As the leading
application, we consider 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 . The novelty of our
result is that we leave the \emph{type} of regularisation general, and that we
consider the joint limit and . 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., ) decays as .Comment: 34 page
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.Comment: 17 pages, 2 figures http://cvgmt.sns.it/paper/2781
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 modelling the
interactionsat a typical length-scale of of the walls subjected
to a constant shear stress, we derive a first-order approximation of the energy
in powers of by -convergence, in the limit
. While the zero-order term in the expansion, the
-limit of , 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.Comment: 25 page
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
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
Convergence rates for energies of interacting particles whose distribution spreads out as their number increases
We consider a class of particle systems which appear in various applications
such as approximation theory, plasticity, potential theory and space-filling
designs. The positions of the particles on the real line are described as a
global minimum of an interaction energy, which consists of a nonlocal,
repulsive interaction part and a confining part. Motivated by the applications,
we cover non-standard scenarios in which the confining potential weakens as the
number of particles increases. This results in a large area over which the
particles spread out. Our aim is to approximate the particle interaction energy
by a corresponding continuum interacting energy. Our main results are bounds on
the corresponding energy difference and on the difference between the related
potential values. We demonstrate that these bounds are useful to problems in
approximation theory and plasticity. The proof of these bounds relies on
convexity assumptions on the interaction and confining potentials. It combines
recent advances in the literature with a new upper bound on the minimizer of
the continuum interaction energy.Comment: 28 page
Discrete-to-continuum convergence of charged particles in 1D with annihilation
We consider a system of charged particles moving on the real line driven by
electrostatic interactions. Since we consider charges of both signs, collisions
might occur in finite time. Upon collision, some of the colliding particles are
effectively removed from the system (annihilation). The two applications we
have in mind are vortices and dislocations in metals.
In this paper we reach two goals. First, we develop a rigorous solution
concept for the interacting particle system with annihilation. The main
innovation here is to provide a careful management of the annihilation of
groups of more than two particles, and we show that the definition is
consistent by proving existence, uniqueness, and continuous dependence on
initial data. The proof relies on a detailed analysis of ODE trajectories close
to collision, and a reparametrization of vectors in terms of the moments of
their elements.
Secondly, we pass to the many-particle limit (discrete-to-continuum), and
recover the expected limiting equation for the particle density. Due to the
singular interactions and the annihilation rule, standard proof techniques of
discrete-to-continuum limits do not apply. In particular, the framework of
measures seems unfit. Instead, we use the one-dimensional feature that both the
particle system and the limiting PDE can be characterized in terms of
Hamilton--Jacobi equations. While our proof follows a standard limit procedure
for such equations, the novelty with respect to existing results lies in
allowing for stronger singularities in the particle system by exploiting the
freedom of choice in the definition of viscosity solutions.Comment: 51 page
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