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Register a new userEnergetic rigidity II: Applications in examples of biological and underconstrained materials
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Added by
Ojan Khatib Damavandi
Publication date
2021
fields
Physics
and research's language is
English
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This is the second paper devoted to energetic rigidity, in which we apply our formalism to examples in two dimensions: underconstrained random regular spring networks, vertex models, and jammed packings of soft particles. Spring networks and vertex models are both highly underconstrained, and first-order constraint counting does not predict their rigidity, but second-order rigidity does. In contrast, spherical jammed packings are overconstrained and thus first-order rigid, meaning that constraint counting is equivalent to energetic rigidity as long as prestresses in the system are sufficiently small. Aspherical jammed packings on the other hand have been shown to be jammed at hypostaticity, which we use to argue for a modified constraint counting for systems that are energetically rigid at quartic order.
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Rigidity regulates the integrity and function of many physical and biological systems. This is the first of two papers on the origin of rigidity, wherein we propose that energetic rigidity, in which all non-trivial deformations raise the energy of a structure, is a more useful notion of rigidity in practice than two more commonly used rigidity tests: Maxwell-Calladine constraint counting (first-order rigidity) and second-order rigidity. We find that constraint counting robustly predicts energetic rigidity only when the system has no states of self stress. When the system has states of self stress, we show that second-order rigidity can imply energetic rigidity in systems that are not considered rigid based on constraint counting, and is even more reliable than shear modulus. We also show that there may be systems for which neither first nor second-order rigidity imply energetic rigidity. The formalism of energetic rigidity unifies our understanding of mechanical stability and also suggests new avenues for material design.
Motivated by the formal argument that a non-zero shear modulus is the result of averaging over a constrained configurations space, we demonstrate that the shear modulus calculated over a range of temperatures and averaging times can be expressed (relative to its infinite frequency value) as a single function of the mean squared displacement. This result is shown to hold for both a glass-liquid and a crystal-liquid system.
We experimentally investigate the fluidization of a granular material subject to mechanical vibrations by monitoring the angular velocity of a vane suspended in the medium and driven by an external motor. On increasing the frequency we observe a re-entrant transition, as a jammed system first enters a fluidized state, where the vane rotates with high constant velocity, and then returns to a frictional state, where the vane velocity is much lower. While the fluidization frequency is material independent, the viscosity recovery frequency shows a clear dependence on the material, that we rationalize by relating this frequency to the balance between dissipative and inertial forces in the system. Molecular dynamics simulations well reproduce the experimental data, confirming the suggested theoretical picture.
Yield stress fluids display complex dynamics, in particular when driven into the transient regime between the solid and the flowing state. Inspired by creep experiments on dense amorphous materials, we implement mesocale elasto-plastic descriptions to analyze such transient dynamics in athermal systems. Both our mean-field and space-dependent approaches consistently reproduce the typical experimental strain rate responses to different applied steps in stress. Moreover, they allow us to understand basic processes involved in the strain rate slowing down (creep) and the strain rate acceleration (fluidization) phases. The fluidization time increases in a power-law fashion as the applied external stress approaches a static yield stress. This stress value is related to the stress over-shoot in shear start-up experiments, and it is known to depend on sample preparation and age. By calculating correlations of the accumulated plasticity in the spatially resolved model, we reveal different modes of cooperative motion during the creep dynamics.
We explore the compaction dynamics of a granular pile after a hard quench from a liquid into the glassy regime. First, we establish that the otherwise athermal granular pile during tapping exhibits annealing behavior comparable to glassy polymer or colloidal systems. Like those other systems, the pile undergoes a glass transition and freezes into different non-equilibrium glassy states at low agitation for different annealing speeds, starting from the same initial equilibrium state at high agitation. Then, we quench the system instantaneously from the highly-agitated state to below the glass transition regime to study the ensuing aging dynamics. In this classical aging protocol, the density increases (i.e., the potential energy of the pile decreases) logarithmically over several decades in time. Instead of system-wide, thermodynamic measures, here we identify the intermittent, irreversible events (quakes) that actually drive the glassy relaxation process. We find that the event rate decelerates hyperbolically, which explains the observed increase in density when the integrated contribution to the downward displacements is evaluated. We argue that such a hyperbolically decelerating event rate is consistent with a log-Poisson process, also found as a universal feature of aging in many thermal glasses.
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