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 $10^4$ 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