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

A characteristic energy scale in glasses

Glasses feature a broad distribution of relaxation times and activation energies without an obvious characteristic scale. At the same time, macroscopic quantities such as Newtonian viscosity and nonlinear plastic deformation, are interpreted in terms of a characteristic energy scale, e.g. an effective temperature-dependent activation energy in Arrhenius relations. Nevertheless, despite its fundamental importance, such a characteristic energy scale has not been robustly identified. Inspired by the accumulated evidence regarding the crucial role played by soft quasilocalized excitations in glassy dynamics, we propose that the bulk average of the glass response to a localized force dipole defines such a characteristic energy scale. We show that this characteristic glassy energy scale features remarkable properties: $(i)$ It increases dramatically with decreasing temperature of equilibrium supercooled states, significantly surpassing the corresponding increase in the shear modulus, dismissing the common view that structural variations in supercooled liquids upon vitrification are minute $(ii)$ Its variation with annealing and system size is very similar in magnitude and form to that of the energy of the softest non-phononic vibrational mode, thus establishing a very unusual relation between a rare glassy fluctuation and a bulk average $(iii)$ It exhibits striking dependence on spatial dimensionality and system size, due to the long-ranged fields associated with quasilocalization, which are speculated to be related to peculiarities of the glass transition in two dimensions. In addition, we identify a truly-static growing lengthscale associated with the characteristic glassy energy scale, and discuss possible connections between the increase of this energy scale and the slowing down of dynamics near the glass transition temperature. Open questions and future directions are discussed.Comment: 16 pages, 13 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

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

Plasticity-Induced Anisotropy in Amorphous Solids: the Bauschinger Effect

Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner `remember' that direction also when relaxed back to a state with zero mean stress. We address the question `what is the order parameter that is responsible for this memory?' and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stress-strain curve, and propose that it and its probability distribution function must form a necessary component of a theory of elasto-plasticity.Comment: 6 pages, 6 figure

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

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