Unified approach to numerical transfer matrix methods for disordered systems : applications to mixed crystals and to elasticity percolation

Lemieux, M. A.; Breton, P.; Tremblay, A.–M. S.

doi:10.1051/jphyslet:019850046010100

Cited by 90 publications

(20 citation statements)

References 11 publications

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“…First, we comment on the case X/p -1 already considered by other authors. We note that our result for the percolation threshold p Ce is in very good agreement with the most accurate estimate reported up to now, 8 The results for two additional values of X/p are also given in Table I. First, we note that p Ce increases with X/p.…”

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confidence: 90%

Percolation in Isotropic Elastic Media

1988

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An investigation is presented of elasticity percolation in isotropic media with arbitrary elastic constants. A model which combines the central-force potential energy and a triangular lattice with a large unit cell allows us to vary the ratio between the two Lame coefficients. Results of numerical simulations indicate that the critical properties of the percolation transition do strongly depend on the elastic constants.

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confidence: 90%

Percolation in Isotropic Elastic Media

1988

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“…Nevertheless their numerically determined exponent ratio: T ce =t ce ¼ 3:0 AE 0:4, is still very close to that for bond-bending percolation. However, as noted by FENG (1985), two other estimates of T ce by FENG andSEN (1984) andLEMIEUX et al (1985) do not agree. Additionally CHAKRABATI and BENGUIGUI (1997) quote a value for T ce =t ce ¼ 1:12 AE 0:05 in two dimensions.…”

Section: Consistency With Previous Investigations Of Scaling Of Modulusmentioning

confidence: 87%

Spatial Scaling of Effective Modulus and Correlation of Deformation Near the Critical Point of Fracturing

Heffer

King

2006

Pure appl. geophys.

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Many observations point to the lithosphere being metastable and close to a critical mechanical point. Exercises in modelling deformation, past or present, across subsurface reservoirs need to take account of this criticality in an efficient way. Using a renormalization technique, the spatial scaling of effective elastic modulus is derived for 2-D and 3-D bodies close to the critical point of through-going fracturing. The resulting exponent, d l , of spatial scaling of effective modulus with size, L dl , takes the values $ )2.5 and )4.2 in two-and three-dimensional space, respectively. The exponents are compatible with those for scaling of effective modulus with fracture density near the percolation threshold determined by other workers from numerical experiments; the high absolute values are also approximately consistent with empirical data from a) fluctuations in depth of a seismic surface; b) '1/k' scaling of heterogeneities observed in one-dimensional well-log samples; c) spatial correlation of slip displacements induced by water injection. The effective modulus scaling modifies the spatial correlation of components of displacement or strain for a domain close to the critical point of fracturing. This correlation function has been used to geostatistically interpolate components of the strain tensor across subsurface reservoirs with the prime purpose of predicting fracture densities between drilled wells. Simulations of strain distributions appear realistic and can be conditioned to surface depths and observations at wells of fracture-related information such as densities and orientations, welltest permeabilities, changes in well-test permeabilities, etc.

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“…8,12 It has become clear that they have different critical elastic behavior but the full nature of this difference and its underlying causes have not yet been generally appreciated. In this Letter we address this issue: We present the first analysis of the geometry of the percolating rigid clusters for the two-dimensional central-force model.…”

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confidence: 99%

Rigid Backbone: A New Geometry for Percolation

Day

Tremblay

1986

Phys. Rev. Lett.

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It is shown that the diluted two-dimensional central-force problem belongs to a new class of percolation problems. Geometric properties such as the fractal dimension of the backbone, the correlation-length exponent, and the connectivity are completely different from those of previously studied percolation problems. Explicit calculations of the backbone and the construction of an algorithm which identifies the infinite rigid cluster clearly demonstrate the absence of singly connected bonds, the overwhelming importance of loops, and the long-range nature of the rigidity.PACS numbers: 64.60. Cn, 05.70.Jk, 63.70. + h The percolation model 1 has, for a number of years, been a source of insight into many diverse physical phenomena. 2 Recently, it has been employed to understand elastic properties of tenuous media such as gels, 3 ' 4 sinters, 5 or even certain glasses. 6,7 It was originally suggested by de Gennes 3 in the polymer context that the critical properties of the elastic moduli at the percolation threshold would be the same as those of the conductivity. It has become evident, however, that this is not generally true, 8 " 10 one of the reasons lying in the different tensorial character of the problem.Recent work has concentrated on two models of elasticity percolation, the bond-bending model 9,11 and the central-force model. 8,12 It has become clear that they have different critical elastic behavior but the full nature of this difference and its underlying causes have not yet been generally appreciated. In this Letter we address this issue: We present the first analysis of the geometry of the percolating rigid clusters for the two-dimensional central-force model. While the backbone geometry of the bond-bending universality class is the same as that of ordinary percolation, 9 we show here that for the central-force model, the geometry is both quantitatively and qualitatively different. We compute the backbone fractal dimension 13,14 to exhibit a quantitative difference and present graphic and algorithmic evidence which show qualitative differences such as the absence of singly connected bonds, the importance of loops, and the essentially long-range nature of elastic connectivity.The best-understood model for elasticity percolation is the so-called bond-bending model. 9,11 That model is the simplest realization of a general class of models 15 where geometric connection implies elastic connection. For these models the threshold, p c , and the percolation critical exponents £ p , v p , y p , and a p are identical to those of the equivalent geometric problem. "Dynamic" exponents, however, depend on the problem: Namely, the exponent /which governs the elastic moduli is different from the exponent t which governs the conductivity. Nevertheless, the basic physics which controls these exponents can be readily understood in both cases from the "nodes-linksblobs" picture 14 of percolation clusters. In fact, it is fair to say that in the vast majority of percolation problems encountered to date, even continuum percolation,...

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Unified approach to numerical transfer matrix methods for disordered systems : applications to mixed crystals and to elasticity percolation

Cited by 90 publications

References 11 publications

Percolation in Isotropic Elastic Media

Percolation in Isotropic Elastic Media

Spatial Scaling of Effective Modulus and Correlation of Deformation Near the Critical Point of Fracturing

Rigid Backbone: A New Geometry for Percolation

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