Anomalous elasticity and plastic screening 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.

Scale Separation of Shear-Induced Criticality in Glasses

In a sheared steady state, glasses reach a nonequilibrium criticality called yielding criticality. We report that the qualitative nature of this nonequilibrium critical phenomenon depends on the details of the system and that responses and fluctuations are governed by different critical correlation lengths in specific situations. This scale separation of critical lengths arises when the screening of elastic propagation of mechanical signals is not negligible. We also discuss the determinant of the impact of screening effects from the viewpoint of the microscopic dissipation mechanism.

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.

Dynamical yield criterion for granular matter from first principles

We investigate, using a recently developed model of liquid state theory describing the rheology of dense granular flows, how a yield stress appears in granular matter at the yielding transition. Our model allows us to predict an analytical equation of the corresponding dynamical yield surface, which is compared to the usual models of solid fracture. In particular, this yield surface interpolates between the typical failure behaviors of soft and hard materials. This work also underlines the central role played by the effective friction coefficient at the yielding transition.

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.

Tensor electromagnetism and emergent elasticity in jammed solids

The theory of mechanical response and stress transmission in disordered, jammed solids poses several open questions of how nonperiodic networks-apparently indistinguishable from a snapshot of a fluid-sustain shear. We present a stress-only theory of emergent elasticity for a nonthermal amorphous assembly of grains in a jammed solid, where each grain is subjected to mechanical constraints of force and torque balance. These grain-level constraints lead to the Gauss's law of an emergent U(1) tensor electromagnetism, which then accounts for the mechanical response of such solids. This formulation of amorphous elasticity has several immediate consequences. The mechanical response maps exactly to the static, dielectric response of this tensorial electromagnetism with the polarizability of the medium mapping to emergent elastic moduli. External forces act as vector electric charges, whereas the tensorial magnetic fields are sourced by momentum density. The dynamics in the electric and magnetic sectors naturally translate into the dynamics of the rigid jammed network and ballistic particle motion, respectively. The theoretical predictions for both stress-stress correlations and responses are borne out by the results of numerical simulations of frictionless granular packings in the static limit of the theory in both 2D and 3D.

Finite-Disorder Critical Point in the Yielding Transition of Elastoplastic Models

Upon loading, amorphous solids can exhibit brittle yielding, with the abrupt formation of macroscopic shear bands leading to fracture, or ductile yielding, with a multitude of plastic events leading to homogeneous flow. It has been recently proposed, and subsequently questioned, that the two regimes are separated by a sharp critical point, as a function of some control parameter characterizing the intrinsic disorder strength and the degree of stability of the solid. In order to resolve this issue, we have performed extensive numerical simulations of athermally driven elastoplastic models with long-range and anisotropic realistic interaction kernels in two and three dimensions. Our results provide clear evidence for a finite-disorder critical point separating brittle and ductile yielding, and we provide an estimate of the critical exponents in 2D and 3D.

Anomalous elasticity in classical glass formers

Amorphous solids under mechanical strains are prone to plastic responses. Recent work showed that in amorphous granular systems these plastic events, that are typically quadrupolar in nature, can screen the elastic response. When the density of the quadrupoles is high, the gradients of the quadrupole field act as emergent dipole sources, leading to qualitative changes in the mechanical response, as seen for example in the displacement field. In this paper we examine the effect of screening in classical glass formers. These are made of point particles that interact via binary forces. Both inverse power law forces and Lennard-Jones interactions are examined, and it is shown that in both cases the elastic response can be strongly screened, in agreement with the novel theory. The degree of deviation from classical elasticity theory is parametrized by a proposed measure that is shown to have a functional dependence on the amount of energy lost to plastic responses.

Direct measurement of dipoles in anomalous elasticity of amorphous solids

Recent progress in studying the physics of amorphous solids has revealed that mechanical strains can be strongly screened by the formation of plastic events that are typically quadrupolar in nature. The theory stipulates that gradients in the density of the quadrupoles act as emergent dipole sources, leading to strong screening and to qualitative changes in the mechanical response, as seen, for example, in the displacement field. In this Letter we first offer direct measurements of the dipole field, independently of any theoretical assumptions, and second we demonstrate detailed agreement with the recently proposed theory. These two goals are achieved by using data from both simulations and experiments. Finally, we show how measurements of the dipole fields pinpoint the theory parameters that determine the profile of the displacement field.