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Register a new userElastic avalanches reveal marginal behaviour in amorphous solids
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Mechanical deformation of amorphous solids can be described as consisting of an elastic part in which the stress increases linearly with strain, up to a yield point at which the solid either fractures or starts deforming plastically. It is well established, however, that the apparent linearity of stress with strain is actually a proxy for a much more complex behavior, with a microscopic plasticity that is reflected in diverging nonlinear elastic coefficients. Very generally, the complex structure of the energy landscape is expected to induce a singular response to small perturbations. In the athermal quasistatic regime, this response manifests itself in the form of a scale free plastic activity. The distribution of the corresponding avalanches should reflect, according to theoretical mean field calculations (Franz and Spigler, Phys. Rev. E., 2017, 95, 022139), the geometry of phase space in the vicinity of a typical local minimum. In this work, we characterize this distribution for simple models of glass forming systems, and we find that its scaling is compatible with the mean field predictions for systems above the jamming transition. These systems exhibit marginal stability, and scaling relations that hold in the stationary state are examined and confirmed in the elastic regime. By studying the respective influence of system size and age, we suggest that marginal stability is systematic in the thermodynamic limit.
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We show that the low-frequency regime of the density of states of structural glass formers is crucially sensitive to the stress-ensemble from which the configurations are sampled. Specifically, in two dimensions, an exactly isotropic ensemble with zero shear stress fluctuations, displays a $D(omega_{min}) sim omega^{5}_{min}$ regime, as opposed to the $omega^{4}_{min}$ regime observed under unstrained conditions. We also study an ensemble of strained amorphous solids near a plastic event. We show that the minimum eigenvalue distribution at a strain $gamma$ near the plastic event occurring at $gamma_{P}$, displays a collapse when scaled by $sqrt{gamma_P - gamma}$, and with the number of particles as $N^{-0.22}$. Notably, at low-frequencies, this scaled distribution displays a robust $D(omega_{min}) sim omega^{6}_{min}$ power-law regime, which survives in the large $N$ limit. Finally, we probe the universal properties of this ensemble through a characterization of the second and third eigenvalues of the Hessian matrix near a plastic event.
Sound attenuation in low temperature amorphous solids originates from their disordered structure. However, its detailed mechanism is still being debated. Here we analyze sound attenuation starting directly from the microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping coefficient and verify that it agrees with results of earlier sound attenuation simulations. The small wavevector analysis of this expression shows that sound attenuation is primarily determined by the non-affine displacements contribution to the wave propagation coefficient coming from the frequency shell of the sound wave.
The mechanical response of naturally abundant amorphous solids such as gels, jammed grains, and biological tissues are not described by the conventional paradigm of broken symmetry that defines crystalline elasticity. In contrast, the response of such athermal solids are governed by local conditions of mechanical equilibrium, i.e., force and torque balance of its constituents. Here we show that these constraints have the mathematical structure of a generalized electromagnetism, where the electrostatic limit successfully captures the anisotropic elasticity of amorphous solids. The emergence of elasticity from local mechanical constraints offers a new paradigm for systems with no broken symmetry, analogous to emergent gauge theories of quantum spin liquids. Specifically, our $U(1)$ rank-2 symmetric tensor gauge theory of elasticity translates to the electromagnetism of fractonic phases of matter with the stress mapped to electric displacement and forces to vector charges. We corroborate our theoretical results with numerical simulations of soft frictionless disks in both two and three dimensions, and experiments on frictional disks in two dimensions. We also present experimental evidence indicating that force chains in granular media are sub-dimensional excitations of amorphous elasticity similar to fractons.
We show that the distribution of elements $H$ in the Hessian matrices associated with amorphous materials exhibit singularities $P(H) sim {lvert H rvert}^{gamma}$ with an exponent $gamma < 0$, as $lvert H rvert to 0$. We exploit the rotational invariance of the underlying disorder in amorphous structures to derive these exponents exactly for systems interacting via radially symmetric potentials. We show that $gamma$ depends only on the degree of smoothness $n$ of the potential of interaction between the constituent particles at the cut-off distance, independent of the details of interaction in both two and three dimensions. We verify our predictions with numerical simulations of models of structural glass formers. Finally, we show that such singularities affect the stability of amorphous solids, through the distributions of the minimum eigenvalue of the Hessian matrix.
The holographic principle has proven successful in linking seemingly unrelated problems in physics; a famous example is the gauge-gravity duality. Recently, intriguing correspondences between the physics of soft matter and gravity are emerging, including strong similarities between the rheology of amorphous solids, effective field theories for elasticity and the physics of black holes. However, direct comparisons between theoretical predictions and experimental/simulation observations remain limited. Here, we study the effects of non-linear elasticity on the mechanical and thermodynamic properties of amorphous materials responding to shear, using effective field and gravitational theories. The predicted correlations among the non-linear elastic exponent, the yielding strain/stress and the entropy change due to shear are supported qualitatively by simulations of granular matter models. Our approach opens a path towards understanding complex mechanical responses of amorphous solids, such as mixed effects of shear softening and shear hardening, and offers the possibility to study the rheology of solid states and black holes in a unified framework.
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