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Plastic slip avalanches.
Avalanche phenomena have been observed in a wide
variety of disordered systems that exhibit crackling noise
near a depinning transition, including plastic deformations in single crystals due to the collective
motion of dislocations. Recent experimental studies of slip avalanches in mesocopic
crystal plasticity have reported that the distribution of the maximum amplitude
of the acoustic emission (AE) signal from each avalanche follows a power law with an
exponent close to -2. Owing to the proportionality between the AE
amplitude and the collective velocity of dislocations,
their corresponding distributions should be characterized
by the same scaling exponents and scaling functions.
We provide a theoretical calculation of the maximum velocity distribution
in a mean field model of interface depinning and show how it relates to
the known classes of extreme value statistics of correlated variables. The distribution of maximal
velocities is determined by the distribution conditioned on fixed avalanche durations, which we show has a
universal scaling form.
References:
Distribution of maximum velocities in avalanches near the depinning transition
M. LeBlanc, L. Angheluta, K. Dahmen and N. Goldenfeld,
Physical Review Letters 109, 105702 (2012)
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(Top)Time series of the velocity fluctuations as function of different driving rates obtained
in a mean field model of interface depinning. (Bottom) Universal scaling function of maximal velocities
in avalanches of fixed durations.
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Fracture propagation.
Albeit many studies on the brittle and ductile regimes, a fundamental understanding
of brittle-to-ductile (BTD) transition is still lacking. Experiments and molecular dynamics simulations
support the scenario of a sharp transition at a critical temperature from a brittle (at low temperatures)
to a ductile failure (above the critical point). The transition is often assigned
to thermally nucleated dislocations and their mobility. A generalization
of the Kosterlitz-Thouless transition in the presense of stress has been proposed
as a mechanism for the temperature dependent density of dislocations ahead of
fractures where the stresses are higher.
When this happens, the fracture dissipation mechanism is
mainly through dislocations dynamics. We study the consequences of this theory on the fracture growth
as a function of crystal softness, dependent on how far away the crystal is from the critical point.
We also developed a numerical model using the phase field crystal approach to study fracture propagation
with a tunable temperature.
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Phase field crystal simulation of a brittle fracture in a single crystal.
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Rayleigh Bernard convection.
When a fluid is heated from below in the presence of a
gravitational field, the static state with thermal conduction
can become unstable towards a succession of instabilities,
ultimately leading to turbulence if the buoyancy-induced driving force is sufficiently
greater than the viscous drag and diffusion of heat. In turbulent Rayleigh-Benard convection,
a large scale circulation (LSC) develops and is maintained by rising and falling plumes detaching
from the unstable
thermal boundary layers. Rare but large fluctuations in the LSC amplitude can lead to
extinction of the LSC known as a cessation event, followed by the
re-emergence of another LSC with a different azimuthal orientation.
Using a low-dimensional stochastic model we derive statistical distributions of the
fluctuations in the temperature amplitude and azimuthal orientation of the LSC and study
the effect of a weak rotation on the LSC statistics and the frequency of cessations.
References:
Effect of weak rotation on large-scale circulation cessations in turbulent
convection
M. Assaf, L. Angheluta and N. Goldenfeld,
Physical Review Letters 109, 074502 (2012).
Rare fluctuations and large circulation
cessations in turbulent convection
M. Assaf, L. Angheluta and N. Goldenfeld,
Physical Review Letters 107, 044502 (2011)
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Temperature amplitude fluctuations of the LSC. A cessation event occurs when the LSC
amplitude drops below a low threshold represented by the dashed line.
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Anomalous diffusion.
A characteristics of turbulent flows is the anomalous, enhanced diffusion of advecting
particles. Pair-particle diffusion is typically described by a nonlinear diffusion equation
with a scale depedent diffusivity, which yields Richardson's scaling of the mean square
separation as a function of time. Our study showed that simple toy models, such as shell models
in real space, can capture the Richardson's diffusion. Moreover, this law also falls out from
a simple stochastic formulation of relative dispersion, where the Langevin's type particles
move in a random force field, mimicking the turbulent kinks with the only constrain of
a constant energy dissipation rate.
References:
Kolmogorov scaling from random fields
M. H. Jensen, K. Sneppen and L. Angheluta,
Europhysics Letters 97 (1), 10011 (2008).
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Two particles being advected in a random force field generated by a shell model of turbulence.
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Interfaces in stressed solids. Stylolites are one of
fascinating geological interfaces that develop mainly in sedimentary rocks, e.g.
sandstones, limestones, during their compactification.
They are rough seams filled with residual materials, typically clays, and are organized in almost parallel surfaces to the
main compaction direction.
A pressure solution process is often used to phenomenologically
describe their formation, as due to a local dissolution of the rock in the more stressed regions,
a transport through the rock pores and a deposition in the less stressed regions.
Supposedly, this
leads to a positive feedback that generates localized dissolution surfaces marked by a higher
concentration of the insoluble material. The thermodynamics of coupling non-hydrostatic stresses
with dissolution/precipitation kinetics and transport
is however still not uniquelly formulated.
We developed a toy model to study the onset of roughening of solid-solid interfaces.
Inspired initially by the stylolite patterns, our
model is relevant, owed to its simplicity, to any interface
between stressed solids, e.g. grain boundaries, faults, etc.
The model consists of two elastically
stressed solids separated by a sharp interface which moves with a normal velocity
that is proportional to the jump in material properties and the elastic energy density.
We find a kind of Mullings-Sekerka instability, but in elastostatic solids, where
the interface develops typical dendritic patterns,
with the fingers alligned along the principal direction of compaction.
References:
Thermodynamics and roughening of solid-solid interfaces
L. Angheluta, E. Jettestuen and J. Mathiesen,
Physical Review E 79, 031601 (2009).
Stress driven stress transformation
and the roughening of solid-solid interfaces
L. Angheluta, E. Jettestuen, J. Mathiesen, F. Renard and B. Jamtveit,
Physical Review Letters 100, 096105 (2008).
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(Top) The elastic energy density (in logarithmic scale), lighter color corresponds
to higher energy. The bottom figure shows the initial interface with a small roughness, which develops at later times
into a dendritic-like pattern as illustrated in the figure above.
(Bottom) The interface profile plotted at various times
starting from a periodic initial perturbation.
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