RESUMEN
We present a universal characterization of stress correlations in athermal systems, across crystalline to amorphous packings. Via numerical analysis of static configurations of particles interacting through harmonic as well as Lennard-Jones potentials, for a variety of preparation protocols and ranges of microscopic disorder, we show that the properties of the stress correlations at large lengthscales are surprisingly universal across all situations, independent of structural correlations, or the correlations in orientational order. In the near-crystalline limit, we present exact results for the stress correlations for both models, which work surprisingly well at large lengthscales, even in the amorphous phase. Finally, we study the differences in stress fluctuations across the amorphization transition, where stress correlations reveal the loss of periodicity in the structure at short lengthscales with increasing disorder.
RESUMEN
We analytically study friction and dissipation of a driven bead in a 1D harmonic chain, and analyze the role of internal damping mechanism as well as chain length. Specifically, we investigate Dissipative Particle Dynamics and Langevin Dynamics, as paradigmatic examples that do and do not display translational symmetry, with distinct results: For identical parameters, the friction forces can differ by many orders of magnitude. For slow driving, a Goldstone mode traverses the entire system, resulting in friction of the driven bead that grows arbitrarily large (Langevin) or gets arbitrarily small (Dissipative Particle Dynamics) with system size. For a long chain, the friction for DPD is shown to be bound, while it shows a singularity (i.e. can be arbitrarily large) for Langevin damping. For long underdamped chains, a radiation mode is recovered in either case, with friction independent of damping mechanism. For medium length chains, the chain shows the expected resonant behavior. At the resonance, friction is non-analytic in damping parameterγ, depending on it asγ-1. Generally, no zero frequency bulk friction coefficient can be determined, as the limits of small frequency and infinite chain length do not commute, and we discuss the regimes where 'simple' macroscopic friction occurs.
RESUMEN
We derive exact results for the fluctuations in energy produced by microscopic disorder in near-crystalline athermal systems. Our formalism captures the heterogeneity in the elastic energy of polydisperse soft disks in energy-minimized configurations. We use this to predict the distribution of interaction energy between two defects in a disordered background. We show that this interaction energy displays a disorder-averaged power-law behavior ãδEãâ¼Δ^{-4} at large distances Δ between the defects. These interactions upon disorder average also display the sixfold symmetry of the underlying reference crystal. Additionally, we show that the fluctuations in the interaction energy encode the athermal correlations introduced by the disordered background. We verify our predictions with energy-minimized configurations of polydisperse soft disks in two dimensions.
RESUMEN
We introduce a perturbation expansion for athermal systems that allows an exact determination of displacement fields away from the crystalline state as a response to disorder. We show that the displacement fields in energy-minimized configurations of particles interacting through central potentials with microscopic disorder can be obtained as a series expansion in the strength of the disorder. We introduce a hierarchy of force-balance equations that allows an order-by-order determination of the displacement fields, with the solutions at lower orders providing sources for the higher-order solutions. This allows the simultaneous force-balance equations to be solved, within a hierarchical perturbation expansion to arbitrary accuracy. We present exact results for an isotropic defect introduced into the crystalline ground state at linear order and second order in our expansion. We show that the displacement fields produced by the defect display interesting self-similar properties at every order. We derive a |δr|â¼1/r and |δf|â¼1/r^{2} decay for the displacement fields and excess interparticle forces at large distances r away from the defect. Finally, we derive nonlinear corrections introduced by the interactions between defects at second order in our expansion. We verify our exact results with displacement fields obtained from energy-minimized configurations of soft disks.
RESUMEN
We derive exact results for displacement fields that develop as a response to external pinning forces in two-dimensional athermal networks. For a triangular lattice arrangement of particles interacting through soft potentials, we develop a Green's function formalism which we use to derive exact results for displacement fields produced by localized external forces. We show that in the continuum limit the displacement fields decay as 1/r at large distances r away from a force dipole. Finally, we extend our formulation to study correlations in the displacement fields produced by the external pinning forces. We show that uncorrelated pinned forces at each vertex give rise to long-range correlations in displacements in athermal systems, with a nontrivial system size dependence. We verify our predictions with numerical simulations of athermal networks in two dimensions.
RESUMEN
We show that a flat two dimensional network of connected vertices, when stretched, may deform plastically by producing "pleats", system spanning linear structures with width comparable to the lattice spacing, where the network overlaps on itself. To understand the pleating process, we introduce an external field that couples to local non-affine displacements, i.e., those displacements of neighbouring vertices that cannot be represented as a local affine strain. We obtain both zero and finite temperature phase diagrams in the strain-field plane. Pleats occur here as a result of an equilibrium first-order transition from the homogeneous network to a heterogeneous phase where stress is localised within pleats and eliminated elsewhere. We show that in the thermodynamic limit, the un-pleated state is always metastable at vanishing field for infinitesimal strain. Plastic deformation of the initially homogeneous network is akin to the decay of a metastable phase via a dynamical transition. We make predictions concerning local stress distributions and thermal effects associated with pleats which may be observable in suitable experimental systems.