5,812 research outputs found
Beyond Sperner's lemma
The present paper is devoted to a recent beautiful and ingenious proof of
Brouwer's fixed point theorem due to mathematical economists H. Petri and M.
Voorneveld. The heart of this proof is an analogue of Sperner's lemma motivated
by Shapley-Scarf model of markets of agents with preferences over indivisible
goods.
The goal of the paper is to present a relatively abstract version of
Petri-Voorneveld proof which makes transparent both its similarities and its
differences with the classical proof based on Sperner's lemma and a well known
Knaster-Kuratowski-Mazurkiewich argument.Comment: 9 page
Plastic deformation of a model glass induced by a local shear transformation
The effect of a local shear transformation on plastic deformation of a
three-dimensional amorphous solid is studied using molecular dynamics
simulations. We consider a spherical inclusion, which is gradually transformed
into an ellipsoid of the same volume and then converted back into the sphere.
It is shown that at sufficiently large strain amplitudes, the deformation of
the material involves localized plastic events that were identified based on
the relative displacement of atoms before and after the shear transformation.
We found that the density profiles of cage jumps decay away from the inclusion,
which correlates well with the radial dependence of the local deformation of
the material. At the same strain amplitude, the plastic deformation becomes
more pronounced in the cases of weakly damped dynamics or large time scales of
the shear transformation.Comment: 19 pages, 7 figure
The effect of a reversible shear transformation on plastic deformation of an amorphous solid
Molecular dynamics simulations are performed to investigate the plastic
response of a model glass to a local shear transformation in a quiescent
system. The deformation of the material is induced by a spherical inclusion
that is gradually strained into an ellipsoid of the same volume and then
reverted back into the sphere. We show that the number of cage-breaking events
increases with increasing strain amplitude of the shear transformation. The
results of numerical simulations indicate that the density of cage jumps is
larger in the cases of weak damping or slow shear transformation. Remarkably,
we also found that, for a given strain amplitude, the peak value of the density
profiles is a function of the ratio of the damping coefficient and the time
scale of the shear transformation.Comment: 19 pages, 7 figure
Dynamical heterogeneity in periodically deformed polymer glasses
The dynamics of structural relaxation in a model polymer glass subject to
spatially-homogeneous, time-periodic shear deformation is investigated using
molecular dynamics simulations. We study a coarse-grained bead-spring model of
short polymer chains below the glass transition temperature. It is found that
at small strain amplitudes, the segmental dynamics is nearly reversible over
about cycles, while at strain amplitudes above a few percent, polymer
chains become fully relaxed after a hundred cycles. At the critical strain
amplitude, the transition from slow to fast relaxation dynamics is associated
with the largest number of dynamically correlated monomers as indicated by the
peak value of the dynamical susceptibility. The analysis of individual monomer
trajectories showed that mobile monomers tend to assist their neighbors to
become mobile and aggregate into relatively compact transient clusters.Comment: 22 pages, 9 figure
The potential energy states and mechanical properties of thermally cycled binary glasses
The influence of repeated thermal cycling on mechanical properties,
structural relaxation, and evolution of the potential energy in binary glasses
is investigated using molecular dynamics simulations. We consider a binary
mixture with strongly non-additive cross interactions, which is annealed across
the glass transition with different cooling rates and then exposed to one
thousand thermal cycles at constant pressure. We found that during the first
few hundred transient cycles, the potential energy minima after eachcycle
gradually decrease and the structural relaxation proceeds via collective,
irreversible displacements of atoms. With increasing cycle number, the
amplitudes of the volume and potential energy oscillations are significantly
reduced, and the potential energy minima saturate to a constant value that
depends on the thermal amplitude and cooling rate. In the steady state, the
glasses thermally expand and contract but most of the atoms return to
theircages after each cycle, similar to limit cycles found in periodically
driven amorphous materials. The results of tensile tests demonstrate that the
elastic modulus and the yielding peak, evaluated after the thermal treatment,
acquire maximum values at a particular thermal amplitude, which coincides with
the minimum of the potential energy.Comment: 23 pages, 10 figure
- …