Couette geometry refers to two concentric rings in two dimensions (or cylinders) in three dimensions with a medium in between. Typically, the inner and outer rings (or cylinders) rotate at different rates and the response of the medium is studied. Here we study a medium which is a two-dimensional amorphous solid, and we rotate the inner ring quasistatically. As stress accumulates, plastic avalanches can result in shear localization, characterized by adjacent parts of the system rotating in opposite directions, with the maximum shear localized between them. We derive an analytic theory that describes and explains the shear localization, providing a priori predictions for the angle-averaged displacement field associated with the plastic drops and the shear localization.

A question of some fundamental importance is whether a given assembly of frictional granules (say sand or powder) will exhibit stress autocorrelations with long-range anisotropic decay as determined by the elastic Green’s function. In 2-dimensional Hamiltonian systems with central forces, mechanical balance and material isotropy demand the stress auto-correlation matrix to be fully determined by the pressure auto-correlation only. If the local pressure fluctuations are normal, it follows that stress autocorrelations decay at long distance like the elastic Green’s function. With friction, Hamiltonian symmetry is lost, and one may expect more constraints. Indeed, it was shown recently that for frictional amorphous solids mechanical balance and material isotropy demand the stress auto-correlation matrix to be fully determined by twospatially isotropic functions: the pressure and torque auto-correlations. Elastic-like decay of the stress autocorrelations follows from normal fluctuations of the pressure and from the torque fluctuations being hyperuniform. The theoretical discovery of these conditions required experimental confirmation, to test whether these conditions are generically obeyed in actual frictional amorphous solids. Recently the confirmation was announced for 2-dimensional amorphous assemblies of frictional disks under isotropic load, in which torque is caused by tangential forces only. In this paper we review that case and report confirmation of the theoretical predictions in 2-dimensional systems of disks under shear and in isotropically loaded frictional ellipses, where contributions to torque come also from normal forces. The paper ends with physical explanations of the hyperuniformity of the torque fluctuations and predictions for how the results are expected to extend to d-dimensions.

Dissipation, or the loss of energy conservation, is necessarily related to symmetry breaking. Here we study the symmetry breaking that is responsible for dissipation in the mechanical response of amorphous solids to quasi-static strains. To this aim we consider carefully just one cycle of strain, to reveal the source of dissipation. In general the response can be conservative or dissipative, with a sharp transition between these options as a function of preparation parameters, accompanied by translation and rotational symmetry breaking and the onset of screening. We choose an example for which the mesoscopic theory can be solved exactly, and the microscopic physics can be revealed by numerical simulations. The mechanism of energy dissipation (when it exists) is explored in detail, showing that in this case it is due to the breaking of rotational symmetry.

A recent purely theoretical prediction stated that in frictional granular packings, mechanical balance and material isotropy constrain the stress auto-correlation matrix to be fully determined by two spatially isotropic functions: the pressure and torque auto-correlations. Moreover, and unexpectedly, the torque fluctuations were predicted to be hyperuniform, a condition for the stress auto-correlation to decay as the elastic Green's function. In this letter we present first experimental evidence for the hyperuniformity of the stress fluctuations. We propose that quite generally the variance of torque fluctuations in a d-dimensional ball of radius R will increase exactly like Rd-1, satisfying the definition of hyperuniformity. Copyright c 2026 EPLA

Shear banding is a well-known and widespread instability in strained solids: under external strain, the deformation localizes along a line in two dimensions or a plane in three dimensions. Developing a proper theoretical description of this phenomenon is key to understanding mechanical failure in solid materials. Very recently, a nonlinear theory extending classical elasticity to include plastic deformations as topological charges was proposed, offering detailed predictions on the nature and consequences of the shear-banding instability. The theory derives a Hessian operator whose lowest eigenvalue vanishes at the onset of instability, and the corresponding critical eigenmode describes the displacement field across the shear band. The resulting soft mode possesses the selected localization scale and subsequently saturates into a finite-width shear band. The aim of this Letter is to examine this theory numerically, establishing the role of topological screening and nonlinear instability as the mechanisms governing shear banding during athermal quasistatic deformation. We show that the displacement profile around the shear band is directly determined by the screening parameter and the nonlinear coefficient, thereby quantitatively verifying the theoretical predictions. Our results demonstrate that shear banding differs fundamentally from fracture: it arises from a nonlinear instability of an elastic field screened by plastic deformations. This establishes topological screening as the essential mechanism governing shear banding in amorphous solids.

The aim of this paper is to offer an analytic theory of the shear banding instability in amorphous solids that are subjected to athermal quasi-static shear. To this aim we derive nonlinear equations for the displacement field, including the consequences of plastic deformation on the mechanical response of amorphous solids. The plastic events collectively induce distributed dipoles that are responsible for screening effects and the creation of typical length-scales that are absent in classical elasticity theory. The nonlinear theory exposes an instability that results in the creation of shear bands. By solving the weakly nonlinear amplitude equation we present analytic expressions for the displacement fields that is associated with shear bands, explaining the role of the elastic moduli that determine the width of a shear band from ductile to brittle characteristics. We derive an energy functional whose Hessian possesses an eigenvalue that goes to zero at the shear-banding instability, providing a prediction for the critical value of the accumulated stress that results in an instability.

Amorphous materials driven away from equilibrium display a diverse repertoire of complex, history-dependent behaviors. One striking feature is a failure to return to equilibrium after an abrupt change in otherwise static external conditions. Instead, amorphous materials often exhibit physical aging: an ever-slowing, nonexponential relaxation that can span a huge range of timescales. Here, we examine the aging behavior of three different amorphous materials subjected to slow periodic driving. The results reveal a generic aging phenomenon characterized by a logarithmic decay of dissipation per cycle. This observation is evaluated against several mesoscopic models of amorphous matter that successfully capture aging under static conditions: i) a collection of noninteracting relaxation processes ii) a noisy hysteron model with random pairwise interactions, and iii) a structural model consisting of a random network of bistable elastic bonds. We find that only the latter model reproduces all experimental findings and relates its success to its persistent, slow exploration of a complex energy landscape with clear signatures of replica symmetry breaking. Thus, cyclic driving emerges as a simple yet powerful protocol to characterize amorphous materials, probe their complex energy landscapes, and distinguish between different models.

Disordered granular matter exhibits mechanical responses that occupy the boundary between fluids and solids, lacking a complete description within a continuum theoretical framework. Recent studies have shown that, in the quasi-static limit, the mechanical response of disordered solids to external perturbations is anomalous and can be accurately predicted by the theory of "odd dipole screening." In this work, we investigate responsive granular matter, where grains change size in response to stimuli such as humidity, temperature, or other factors. We develop a geometric theory of odd dipole-screening, incorporating the growth field into the equilibrium equation. Our theory predicts an anomalous displacement field in response to non-uniform growth fields, confirmed by molecular dynamics simulations of granular matter. Although the screening parameters in our theory are phenomenological and not derived from microscopic physics, we identify a surprising relationship between the odd parameter and Poisson's ratio. This theory has implications for various experimental protocols, including non-uniform heating or wetting, which lead to spatially varying expansion fields.

Understanding how a fluid turns into an amorphous solid is a fundamental challenge in statistical physics, during which no apparent structural ordering appears. In the athermal limit, the two states are connected by a well-defined jamming transition, near which the solid is marginally stable. A recent mechanical response screening theory proposes an additional transition above jamming, called a plastic-to-elastic transition here, separating anomalous and quasielastic mechanical behavior. Through numerical inflation simulations in two dimensions, we show that the onsets of long-range radial and angular correlations of particle displacements decouple, occurring, respectively, at the jamming and plastic-to-elastic transitions. The latter is characterized by a power-law diverging correlation angle and a power-law spectrum of the displacements along a circle. This work establishes two-step transitions on the mechanical properties during "decompression melting" of an athermal overjammed amorphous solid, reminiscent of the two-step structural melting of a crystal in two dimensions. In contradistinction with the latter, the plastic-to-elastic transition exists also in three dimensions.

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.

The Kosterlitz-Thouless and the Hexatic phase transitions are celebrated examples of dipole (vortex, dislocation) induced transitions in condensed matter physics. For very clear reasons, these important “topological" transitions are restricted to 2-dimensions. Here we present a genuine dipole-induced transition in the 3-dimensional response of (athermal) amorphous solids to applied strain. Similarly to the existence of a hexatic phase between normal solid and fluid, we identify an intermediate phase between a phase of normal elastic response at high pressure, and fluid matter at zero pressure. The mechanical response in the intermediate phase is accompanied by plasticity that is generically associated with “non-affine" quadrupolar events seen in the resulting displacement field. Gradients of the quadrupolar fields act as dipole charges that screen elasticity, breaking both translational and Chiral symmetries. We highlight angular correlations that exhibit diverging correlation lengths at this transition and determine the critical scaling exponents.

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.

The aim of this short review is to summarize the developing theory aimed at describing the effect of plastic events in amorphous solids on its emergent mechanics. Experiments and simulations present anomalous mechanical response of amorphous solids where quadrupolar plastic events collectively induce distributed dipoles that are analogous to dislocations in crystalline solids. The novel theory is described, and a number of pertinent examples are provided, including the comparison of theoretical prediction to simulations or experiments.

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.

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.

Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

The inflation of an inner radial (or spherical) cavity in an amorphous solid confined in a disk (or a sphere) served as a fruitful case model for studying the effects of plastic deformations on the mechanical response. It was shown that when the field associated with Eshelby quadrupolar charges is nonuniform, the displacement field is riddled with dipole charges that screen elasticity, reminiscent of Debye monopoles screening in electrostatics. In this paper we look deeper into the screening phenomenon, taking into account the consequences of irreversibility that are associated with the breaking of Chiral symmetry. We consider the equations for the displacement field with the presence of "odd dipole screening," solve them analytically and compare with numerical simulations. Suggestions how to test the theory in experiments are provided.

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.

Applying very small purely radial strains on amorphous solids in radial geometry one observes elastic responses that break the radial symmetry. Without any plasticity involved, the responses indicate mode coupling contributions even for minute strains. We show that these symmetry-breaking responses are due to disorder, typical to amorphous configurations. The symmetry-breaking responses are quantitatively explained using the classical Michell solutions which are excited by mode coupling.

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 non-uniform 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 elasto-plastic models of shear induced phenomena like shear-banding.