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Register a new userJamming graphs: A local approach to global mechanical rigidity
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We revisit the concept of minimal rigidity as applied to soft repulsive, frictionless sphere packings in two-dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Lamans theorem in two-dimensions. It constrains the global, average coordination number of the graph, for example. However, minimal rigidity does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local moves via the Henneberg construction such that these graphs fall under the jurisdiction of correlated percolation. We then probe how jamming graphs destabilize, or become unjammed, by deleting a bond and computing the resulting rigid cluster distribution. We also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra/redundant contacts destabilizes. The latter endeavor allows us to probe a disc packing in the rigid phase and uncover a potentially new diverging lengthscale associated with the random deletion of contacts as compared to the study of cut-out (or frozen in) subsystems.
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We discuss a microscopic scheme to compute the rigidity of glasses or the plateau modulus of supercooled liquids by twisting replicated liquids. We first summarize the method in the case of harmonic glasses with analytic potentials. Then we discuss how it can be extended to the case of repulsive contact systems : the hard sphere glass and related systems with repulsive contact potentials which enable the jamming transition at zero temperature. For the repulsive contact systems we find entropic rigidity which behaves similarly as the pressure in the low temperature limit: it is proportional to the temperature and tends to diverge approaching the jamming density with increasing volume fraction, which may account for experimental observations of rigidities of repulsive colloids and emulsions.
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.
This note gives a detailed proof of the following statement. Let $din mathbb{N}$ and $m,n ge d + 1$, with $m + n ge binom{d+2}{2} + 1$. Then the complete bipartite graph $K_{m,n}$ is generically globally rigid in dimension $d$.
Many textbooks dealing with surface tension favor the thermodynamic approach (minimization of some thermodynamic potential such as free energy) over the mechanical approach (balance of forces) to describe capillary phenomena, stating that the latter is flawed and misleading. Yet, mechanical approach is more intuitive for students than free energy minimization, and does not require any knowledge of thermodynamics. In this paper we show that capillary phenomena can be unmistakably described using the mechanical approach, as long as the system on which the forces act is properly defined. After reminding the microscopic origin of a tangential tensile force at the interface, we derive the Young-Dupr{e} equation, emphasizing that this relation should be interpreted as an interface condition at the contact line, rather than a force balance equation. This correct interpretation avoids misidentification of capillary forces acting on a given system. Moreover, we show that a reliable method to correctly identify the acting forces is to define a control volume that does not embed any contact line on its surface. Finally, as an illustration of this method, we apply the mechanical approach in a variety of ways on a classic example: the derivation of the equilibrium height of capillary rise (Jurins law).
Large scale modelling of fluid flow coupled with solid failure in geothermal reservoirs or hydrocarbon extraction from reservoir rocks usually involves behaviours at two scales: lower scale of the inelastic localization zone, and larger scale of the bulk continuum where elastic behaviour can be reasonably assumed. The hydraulic conductivities corresponding to the mechanical properties at these two scales are different. In the bulk elastic host rock, the hydraulic conductivity does not vary much with the deformation, while it significantly changes in the lower scale of the localization zone due to inelastic deformation. Increase of permeability due to fracture and/or dilation, or reduction of permeability due to material compaction can take place inside this zone. The challenge is to predict the evolution of hydraulic conductivities coupled with the mechanical behaviour of the material in all stages of the deformation process. In the early stage of diffuse deformation, the permeability of the material can be reasonably assumed to be homogenous over the whole Representative Volume Element (RVE) However, localized failure results in distinctly different conductivities in different parts of the RVE. This paper establishes a general framework and corresponding field equations to describe the hydro-mechanical coupling in both diffuse and localized stages of deformation in rocks. In particular, embedding the lower scale hydro-mechanical behaviour of the localization zone inside an elastic bulk, together with their corresponding effective sizes, helps effectively deal with scaling issues in large-scale modelling. Preliminary results are presented which demonstrate the promising features of this new approach.
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