Elastic Properties of Random Percolating Systems

Kantor, Yacov; Webman, Itzhak

doi:10.1103/physrevlett.52.1891

Cited by 608 publications

(310 citation statements)

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“…The irreversibility and non-locality of breaking processes produces rather complicated spatial and temporal correlations, giving rise to a potentially much richer phenomenology than the extensively studied steady-state properties of disordered media, such as the electrical conductivity or superconductivity [8][9][10][11], or the linear elasticity of random networks [12][13][14]. There is a vast literature on the general problem of mechanical failure and crack propagation in solids [1,6,7].…”

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confidence: 99%

“…2014 Breaking phenomena of disordered structures such as the tearing of unwoven textiles, the fracture of brittle materials, or the propagation of cracks in solids are receiving increasing attention due to their imminent technological relevance and due to the fundamental theoretical questions involved [1][2][3][4][5][6][7]. The irreversibility and non-locality of breaking processes produces rather complicated spatial and temporal correlations, giving rise to a potentially much richer phenomenology than the extensively studied steady-state properties of disordered media, such as the electrical conductivity or superconductivity [8][9][10][11], or the linear elasticity of random networks [12][13][14]. There is a vast literature on the general problem of mechanical failure and crack propagation in solids [1, 6,7].…”

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A random fuse model for breaking processes

1985

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Résumé. 2014 Breaking phenomena of disordered structures such as the tearing of unwoven textiles, the fracture of brittle materials, or the propagation of cracks in solids are receiving increasing attention due to their imminent technological relevance and due to the fundamental theoretical questions involved [1][2][3][4][5][6][7]. The irreversibility and non-locality of breaking processes produces rather complicated spatial and temporal correlations, giving rise to a potentially much richer phenomenology than the extensively studied steady-state properties of disordered media, such as the electrical conductivity or superconductivity [8][9][10][11], or the linear elasticity of random networks [12][13][14]. There is a vast literature on the general problem of mechanical failure and crack propagation in solids [1, 6,7]. Unfortunately much of the existing work involves very complicated models and/or calculations. Our motivation is to introduce a relatively simple and tractable model which captures at least some of the basic features of breaking processes. To this end, we consider the electrical analogue of breaking through the introduction of a random fuse network.(*) Supported in part by grants from the ARO, NSF, and ONR.Article published online by EDP Sciences and available at http://dx

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A random fuse model for breaking processes

1985

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“…In addition, the random walk is often used as a diffusion model that takes into account the morphological details of the system. Spring network models also provide a useful framework for analyzing the changes in the mechanical properties of the ECM (14,15).…”

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Dynamics of enzymatic digestion of elastic fibers and networks under tension

Araújo

Majumdar²,

Parameswaran³

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

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We study the enzymatic degradation of an elastic fiber under tension using an anisotropic random-walk model coupled with binding-unbinding reactions that weaken the fiber. The fiber is represented by a chain of elastic springs in series along which enzyme molecules can diffuse. Numerical simulations show that the fiber stiffness decreases exponentially with two distinct regimes. The time constant of the first regime decreases with increasing tension. Using a mean field calculation, we partition the time constant into geometrical, chemical and externally controllable factors, which is corroborated by the simulations. We incorporate the fiber model into a multiscale network model of the extracellular matrix and find that network effects do not mask the exponential decay of stiffness at the fiber level. To test these predictions, we measure the force relaxation of elastin sheets stretched to 20% uniaxial strain in the presence of elastase. The decay of force is exponential and the time constant is proportional to the inverse of enzyme concentration in agreement with model predictions. Furthermore, the fragment mass released into the bath during digestion is linearly related to enzyme concentration that is also borne out in the model. We conclude that in the complex extracellular matrix, feedback between the local rate of fiber digestion and the force the fiber carries acts to attenuate any spatial heterogeneity of digestion such that molecular processes manifest directly at the macroscale. Our findings can help better understand remodeling processes during development or in disease in which enzyme concentrations and/or mechanical forces become abnormal. diffusion | on rate | off rate | cleaving T he extracellular matrix (ECM), the biological structure that supports cells, is composed of elastic fibers such as elastin and collagen. The complex organization of these fibers undergoes a continuous maintenance that requires the catalytic action of enzymes, called proteases (1). In diseases, such as pulmonary emphysema, tissue destruction is thought to be a consequence of the imbalance between protease and antiprotease activity leading to degradation of elastin fibers (2). Biological tissues in vivo are also under tension that may interfere with the enzymatic activity. Indeed, mechanical stretch accelerates the rate of degradation of native ECM during elastase-induced digestion of lung tissue (3, 4), whereas it stabilizes type I collagen against in vitro digestion by collagenases (5, 6).The elastic and failure properties of single fibers have important biological functions, and these material properties depend on the hierarchical organization of the molecular constituents (7). During digestion, both the molecules and the cross-links in the fiber can be cleaved by enzymes reducing fiber stiffness. Furthermore, following cleavage, an enzyme can unbind, diffuse, bind at a different location and cleave again. This leads to the question: How are the diffusion and binding processes of the enzyme and the subsequent degradation ...

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“…Another such example of multi-path long-range ordering is rigidity percolation [2,3,4,5,6] where each occupied site on a lattice has g degrees of freedom. The degrees of freedom of the site become fixed as more neighboring sites become occupied-one occupied neighbor constrains one degree of freedom.…”

Section: Introductionmentioning

confidence: 99%

1∕dexpansion fork-core percolation

Harris

Schwarz

2005

Phys. Rev. E

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The physics of k-core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon.Examples of such a phenomenon range from orientational ordering in solid ortho-para H 2 mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary (k = 1) and biconnected (k = 2) percolation, the mean field k ≥ 3-core percolation transition is both continuous and discontinuous, i.e. there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a 1/d expansion for k-core percolation on the d-dimensional hypercubic lattice. We show that to order 1/d 3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field k-core transition survives in high dimensions.

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Elastic Properties of Random Percolating Systems

Cited by 608 publications

References 9 publications

A random fuse model for breaking processes

A random fuse model for breaking processes

Dynamics of enzymatic digestion of elastic fibers and networks under tension

1∕dexpansion fork-core percolation

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