Quasilocalized states of self stress in packing-derived networks

States of self stress (SSS) are assignments of forces on the edges of a network that satisfy mechanical equilibrium in the absence of external forces. In this work we show that a particular class of quasilocalized SSS in packing-derived networks, first introduced in [D. M. Sussman, C. P. Goodrich, and A. J. Liu, Soft Matter 12, 3982 (2016)], are characterized by a lengthscale $\ell_c$ that scales as $1/\sqrt{z_c-z}$ where $z$ is the mean connectivity of the network, and $z_c\!\equiv\!4$ is the Maxwell threshold in two dimensions, at odds with previous claims. Our results verify the previously proposed analogy between quasilocalized SSS and the mechanical response to a local dipolar force in random networks of relaxed Hookean springs. We show that the normalization factor that distinguishes between quasilocalized SSS and the response to a local dipole constitutes a measure of the mechanical coupling of the forced spring to the elastic network in which it is embedded. We further demonstrate that the lengthscale that characterizes quasilocalized SSS does not depend on its associated degree of mechanical coupling, but instead only on the network connectivity.Comment: 8 pages, 4 figure

Rigidity and auxeticity transitions in networks with strong bond-bending interactions

A widely-studied model for gels or biopolymeric fibrous materials are networks with central force interactions, such as Hookean springs. Less commonly studied are materials whose mechanics are dominated by non-central force interactions such as bond-bending potentials. Inspired by recent experimental advancements in designing colloidal gels with tunable interactions, we study the micro- and macroscopic elasticity of two-dimensional planar graphs with strong bond bending potentials, in addition to weak central forces. We introduce a theoretical framework that allows us to directly investigate the limit in which the ratio of characteristic central-force to bending stiffnesses vanishes. In this limit we show that a generic isostatic point exists at $z_c=4$, coinciding with the isostatic point of frames with central force interactions in two dimensions. We further demonstrate the emergence of a stiffening transition when the coordination is increased towards the isostatic point, which shares similarities with the strain-induced stiffening transition observed in biopolymeric fibrous materials, and coincides with an auxeticity transition above which the material's Poisson's ratio approaches -1 when bond-bending interactions dominate.Comment: 11 pages, 8 figure

Nonlinear plastic modes in disordered solids

We propose a framework within which a robust mechanical definition of precursors to plastic instabilities, often termed `soft-spots', naturally emerges. They are shown to be collective displacements (modes) $\hat{z}_0$ that correspond to local minima of the `barrier function' $b(\hat{z})$. The latter is derived from the cubic approximation of the variation $\delta U_{\hat{z}}(s)$ of the potential energy upon displacing particles a distance $s$ along $\hat{z}$. We show that modes $\hat{z}_0$ corresponding to low-lying minima of $b(\hat{z})$ lead to transitions over energy barriers in the glass, and are therefore associated with highly asymmetric variations $\delta U_{\hat{z}}(s)$ with $s$. We further demonstrate how a heuristic search for local minima of $b(\hat{z})$ can a-priori detect the locus and geometry of imminent plastic instabilities with remarkable accuracy, at strains as large as $\gamma_c-\gamma \sim 10^{-2}$ away from the instability strain $\gamma_c$, where the non-affine displacements under shear are still largely delocalized. Our findings suggest that the a-priori detection of plastic instabilities can be effectively carried out by the investigation of the landscape of $b(\hat{z})$.Comment: 8 pages, 8 figure

Note: Simple argument for emergent anisotropic stress correlations in disordered solids

It is now well-established that mechanical equilibrium in athermal disordered solids gives rise to anisotropic spatial correlations of the coarse-grained stress field that decay in space as $1/r^d$, where $r$ is the distance from the origin, and $d$ denotes the spatial dimension. In this note we present a simple, geometry based argument for the scaling form of the emergent spatial correlations of the stress field in disordered solids.Comment: 2 pages, no figures. v2: expanded discussio

Statistical Physics of the Yielding Transition in Amorphous Solids

The art of making structural, polymeric and metallic glasses is rapidly developing with many applications. A limitation to their use is their mechanical stability: under increasing external strain all amorphous solids respond elastically to small strains but have a finite yield stress which cannot be exceeded without effecting a plastic response which typically leads to mechanical failure. Understanding this is crucial for assessing the risk of failure of glassy materials under mechanical loads. Here we show that the statistics of the energy barriers \Delta E that need to be surmounted changes from a probability distribution function (pdf) that goes smoothly to zero to a pdf which is finite at \Delta E=0. This fundamental change implies a dramatic transition in the mechanical stability properties with respect to external strain. We derive exact results for the scaling exponents that characterize the magnitudes of average energy and stress drops in plastic events as a function of system size.Comment: 4 pages, 5 figure

Finite-size effects in the nonphononic density of states in computer glasses

The universal form of the density of nonphononic, quasilocalized vibrational modes of frequency $\omega$ in structural glasses, ${\cal D}(\omega)$, was predicted theoretically decades ago, but only recently revealed in numerical simulations. In particular, it has been recently established that, in generic computer glasses, ${\cal D}(\omega)$ increases from zero frequency as $\omega^4$, independent of spatial dimension and of microscopic details. However, in [E. Lerner, and E. Bouchbinder, Phys. Rev. E 96, 020104(R) (2017)] it was shown that the preparation protocol employed to create glassy samples may affect the form of their resulting ${\cal D}(\omega)$: glassy samples rapidly quenched from high temperature liquid states were shown to feature ${\cal D}(\omega)\!\sim\!\omega^\beta$ with $\beta\!<\!4$, presumably limiting the degree of universality of the $\omega^4$ law. Here we show that exponents $\beta\!<\!4$ are only seen in small glassy samples quenched from high-temperatue liquid states --- whose sizes are comparable to or smaller than the size of the disordered core of soft quasilocalized vibrations --- while larger glassy samples made with the same protocol feature the universal $\omega^4$ law. Our results demonstrate that observations of $\beta\!<\!4$ in the nonphononic density of states stem from finite-size effects, and we thus conclude that the $\omega^4$ law should be featured by any sufficiently large glass quenched from a melt.Comment: 5 pages, 3 figures. v2: data for larger systems included in fig.

Theory for Swap Acceleration near the Glass and Jamming Transitions

Swap algorithms can shift the glass transition to lower temperatures, a recent unexplained observation constraining the nature of this phenomenon. Here we show that swap dynamic is governed by an effective potential describing both particle interactions as well as their ability to change size. Requiring its stability is more demanding than for the potential energy alone. This result implies that stable configurations appear at lower energies with swap dynamics, and thus at lower temperatures when the liquid is cooled. \maa{ The magnitude of this effect is proportional to the width of the radii distribution, and decreases with compression for finite-range purely repulsive interaction potentials.} We test these predictions numerically and discuss the implications of these findings for the glass transition.We extend these results to the case of hard spheres where swap is argued to destroy meta-stable states of the free energy coarse-grained on vibrational time scales. Our analysis unravels the soft elastic modes responsible for the speed up swap induces, and allows us to predict the structure and the vibrational properties of glass configurations reachable with swap. In particular for continuously poly-disperse systems we predict the jamming transition to be dramatically altered, as we confirm numerically. A surprising practical outcome of our analysis is new algorithm that generates ultra-stable glasses by simple descent in an appropriate effective potential.Comment: 8 pages, 7 figures in the main text, 3 pages 4 figures in the supplemental material. We improved the theoretical discussion in the v3. In particular, we added a section with an extended discussion of the implications of our findings for the glass transitio