Anomalous elasticity in classical glass formers

Selection principle for the screening parameters in the mechanical response of amorphous solids

The mechanical response of amorphous solids to external strains is riddled with plastic events that create topological charges in the resulting displacement field. It was recently shown that the latter lead to screening phenomena that are accompanied by the breaking of both translational and Chiral symmetries. The screening effects are quantified by two screening parameters κ_{e} and κ_{o}, which are inverse characteristic lengths that do not exist in classical elasticity. The screening parameters (and the associated lengths) are emergent, and it is important to understand how they are selected. This Letter explores the mechanism of selection of these characteristic lengths in two examples of strain protocols that allow analytic scrutiny.

Eshelby problem in amorphous solids

The Eshelby problem refers to the response of a two-dimensional elastic sheet to cutting away a circle, deforming it into an ellipse, and pushing it back. The resulting response is dominated by the so-called Eshelby kernel, which was derived for purely elastic (infinite) material, but has been employed extensively to model the redistribution of stress after plastic events in amorphous solids with finite boundaries. Here, we discuss and solve the Eshelby problem directly for amorphous solids, taking into account possible screening effects and realistic boundary conditions. We find major modifications compared to the classical Eshelby solution. These modifications are needed for modeling correctly the spatial responses to plastic events in amorphous solids.

Elastic to plastic transition in amorphous solids

The response of amorphous solids to mechanical loads is accompanied by plasticity that is generically associated with "non-affine" quadrupolar events seen in the resulting displacement field. To develop a continuum theory, one needs to assess when these quadrupolar events have a finite density, allowing the development of a field theory. Is there a transition, as a function of the material parameters and the nature of the loads, from isolated plastic events whose density is zero to a regime governed by a finite density? And if so, what is the nature of this transition? The aim of the paper is to explore this issue. The motivation for the present study stems from recent research in which it was shown that gradients of the quadrupolar fields act as dipole charges that can screen elasticity. Analytically soluble examples of mechanical loading that lead to screening and emergent length scales (that are absent in classical elasticity) have been analyzed and tested. However, "gradients of quadrupolar fields" make sense only when the density of quadrupoles is finite, and hence, the issue is central to this article. The article introduces a notion of polarizability under the strain of Eshelby quadrupoles and concludes that the onset of a density of such quadrupoles with random orientations can only appear when the polarizability is finite.

Dynamic screening by plasticity in amorphous solids

In recent work it was shown that elasticity theory can break down in amorphous solids subjected to nonuniform static loads. The elastic fields are screened by geometric dipoles; these stem from gradients of the quadrupole field associated with plastic responses. Here we study the dynamical responses induced by oscillatory loads. The required modification to classical elasticity is described. Exact solutions for the displacement field in circular geometry are presented, demonstrating that dipole screening results in essential departures from the expected predictions of classical elasticity theory. Numerical simulations are conducted to validate the theoretical predictions and to delineate their range of validity.

Intermediate phase between jammed and unjammed amorphous solids

A significant amount of attention was dedicated in recent years to the phenomenon of jamming of athermal amorphous solids by increasing the volume fraction of the microscopic constituents. At a critical value of the volume fraction, pressure shoots up from zero to finite values with a host of critical exponents discovered and discussed. In this paper, we advance evidence for the existence of a second transition, within the jammed state of two-dimensional granular systems, that separates two regimes of different mechanical responses. Explicitly, highly packed systems are quasielastic with quadrupole screening, and more loosely jammed systems exhibit anomalous mechanics with dipole screening. Evidence is given for a clear transition between these two regimes, reminiscent of the intermediate hexatic phase of crystal melting in two-dimensional crystals. Theoretical estimates of the screening parameters and the pressure where transition takes place are provided.

Dipole screening in pure shear strain protocols of amorphous solids

When amorphous solids are subjected to simple or pure strain, they exhibit elastic increase in stress, punctuated by plastic events that become denser (in strain) upon increasing the system size. It is customary to assume in theoretical models that the stress released in each plastic event is redistributed according to the linear Eshelby kernel, causing avalanches of additional stress release. Here we demonstrate that, contrary to the uniform affine strain resulting from simple or pure strain, each plastic event is associated with a nonuniform strain that gives rise to a displacement field that contains quadrupolar and dipolar charges that typically screen the linear elastic phenomenology and introduce anomalous length scales and influence the form of the stress redistribution. An important question that opens up is how to take this into account in elastoplastic models of shear induced phenomena like shear banding.

Geometric theory of mechanical screening in two-dimensional solids

Holes in mechanical metamaterials, quasilocalized plastic events in amorphous solids, and bound dislocations in a hexatic matter are different mechanisms of generic stress relaxation in solids. Regardless of the specific mechanism, these and other local stress relaxation modes are quadrupolar in nature, forming the foundation for stress screening in solids, similar to polarization fields in electrostatic media. We propose a geometric theory for stress screening in generalized solids based on this observation. The theory includes a hierarchy of screening modes, each characterized by internal length scales, and is partially analogous to theories of electrostatic screening such as dielectrics and Debye-Hückel theory. Additionally, our formalism suggests that the hexatic phase, traditionally defined by structural properties, can also be defined by mechanical properties and may exist in amorphous materials.

Anomalous elasticity and emergent dipole screening in three-dimensional amorphous solids

In recent work, we developed a screening theory for describing the effect of plastic events in amorphous solids on its emergent mechanics. The suggested theory uncovered an anomalous mechanical response of amorphous solids where plastic events collectively induce distributed dipoles that are analogous to dislocations in crystalline solids. The theory was tested against various models of amorphous solids in two dimensions, including frictional and frictionless granular media and numerical models of amorphous glass. Here we extend our theory to screening in three-dimensional amorphous solids and predict the existence of anomalous mechanics similar to the one observed in two-dimensional systems. We conclude by interpreting the mechanical response as the formation of nontopological distributed dipoles that have no analog in the crystalline defects literature. Having in mind that the onset of dipole screening is reminiscent of Kosterlitz-Thouless and hexatic transitions, the finding of dipole screening in three dimensions is surprising.

Anomalous linear elasticity of disordered networks

Continuum elasticity is a powerful tool applicable in a broad range of physical systems and phenomena. Yet, understanding how and on what scales material disorder may lead to the breakdown of continuum elasticity is not fully understood. We show, based on recent theoretical developments and extensive numerical computations, that disordered elastic networks near a critical rigidity transition, such as strain-stiffened fibrous biopolymer networks that are abundant in living systems, reveal an anomalous long-range linear elastic response below a correlation length. This emergent anomalous elasticity, which is non-affine in nature, is shown to feature a qualitatively different multipole expansion structure compared to ordinary continuum elasticity, and a slower spatial decay of perturbations. The potential degree of universality of these results, their implications (e.g. for cell-cell communication through biological extracellular matrices) and open questions are briefly discussed.