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In a recent paper [Phys. Rev. E {bf 104}, 024904] it was shown that mechanical strains in amorphous solids are screened via the formation of plastic events that are typically quadrupolar in nature. At low densities the screening effect is reminiscent of the role of dipoles in dielectrics, while the effect at high density has no immediate electrostatic analog, and is expected to change qualitatively the mechanical response, as seen for example in the displacement field. In this Letter we show that high-density screening results in undulating displacement field that strictly deviate from elasticity theory. We show that theoretical analysis, experimental measurements and numeric simulations of frictional granular amorphous assemblies are in agreement with each other and provide a strong support for the theory.
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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.
Spatial heterogeneity in the elastic properties of soft random solids is examined via vulcanization theory. The spatial heterogeneity in the emph{structure} of soft random solids is a result of the fluctuations locked-in at their synthesis, which also brings heterogeneity in their emph{elastic properties}. Vulcanization theory studies semi-microscopic models of random-solid-forming systems, and applies replica field theory to deal with their quenched disorder and thermal fluctuations. The elastic deformations of soft random solids are argued to be described by the Goldstone sector of fluctuations contained in vulcanization theory, associated with a subtle form of spontaneous symmetry breaking that is associated with the liquid-to-random-solid transition. The resulting free energy of this Goldstone sector can be reinterpreted as arising from a phenomenological description of an elastic medium with quenched disorder. Through this comparison, we arrive at the statistics of the quenched disorder of the elasticity of soft random solids, in terms of residual stress and Lame-coefficient fields. In particular, there are large residual stresses in the equilibrium reference state, and the disorder correlators involving the residual stress are found to be long-ranged and governed by a universal parameter that also gives the mean shear modulus.
We summarize current developments in the investigation of glassy matter using nonlinear dielectric spectroscopy. This work also provides a brief introduction into the phenomenology of the linear dielectric response of glass-forming materials and discusses the main mechanisms that can give rise to nonlinear dielectric response in this material class. Here we mainly concentrate on measurements of the conventional dielectric permittivity at high fields and the higher-order susceptibilities characterizing the 3-omega and 5-omega components of the dielectric response as performed in our group. Typical results on canonical glass-forming liquids and orientationally disordered plastic crystals are discussed, also treating the special case of supercooled monohydroxy alcohols.
Anomalous transport in a circular comb is considered. The circular motion takes place for a fixed radius, while radii are continuously distributed along the circle. Two scenarios of the anomalous transport, related to the reflecting and periodic angular boundary conditions, are studied. The first scenario with the reflection boundary conditions for the circular diffusion corresponds to the conformal mapping of a 2D comb Fokker-Planck equation on the circular comb. This topologically constraint motion is named umbrella comb model. In this case, the reflecting boundary conditions are imposed on the circular (rotator) motion, while the radial motion corresponds to geometric Brownian motion with vanishing to zero boundary conditions on infinity. The radial diffusion is described by the log-normal distribution, which corresponds to exponentially fast motion with the mean squared displacement (MSD) of the order of $e^t$. The second scenario corresponds to the circular diffusion with periodic boundary conditions and the outward radial diffusion with vanishing to zero boundary conditions at infinity. In this case the radial motion corresponds to normal diffusion. The circular motion in both scenarios is a superposition of cosine functions that results in the stationary Bernoulli polynomials for the probability distributions.
Inspired by spring-block models, we elaborate a minimal physical model of earthquakes which reproduces two main empirical seismological laws, the Gutenberg-Richter law and the Omori aftershock law. Our new point is to demonstrate that the simultaneous incorporation of ageing of contacts in the sliding interface and of elasticity of the sliding plates constitute the minimal ingredients to account for both laws within the same frictional model.
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