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Random composite: stirred or shaken?

Gluzman S., Mityushev V., Nawalaniec W., Sokal G.

Archives of Mechanics

2016

Vol. 68, nr 3

229--241

A James Bond’s (JB) catchphrase “shaken, not stirred" is explored for the problem of effective conductivity of composites. The superconductivity critical index s for the conductivity of random non-overlapping disks turns out to be distinctly different for shaking and stirring protocols. In the case of stirring modeled by random walks the formula s(τ) = 0.5 + 0.8 3√τ is deduced for evolution of the critical index with the normalized time 0 ≤ τ ≤ 1, which is proportional to the number of random walks and serving as the disorder measure. Strikingly, the coefficient 0.8 is very close to the critical index for shaking protocol and 0.5 is the critical index for regular lattices. The obtained formula for s is based on the analytical solution to the 2D conductivity problem of randomly distributed disks up to O(x19), where x denotes the concentration of inclusions and its extension to special 3D composites.

Homogenization of fiber-reinforced composites with random properties using the weighted least squares response function approach

Kamiński M.

Archives of Mechanics

2011

Vol. 63, nr 5-6

479-505

The main aim of the paper is a determination of the basic probabilistic characteristics for the effective elasticity tensor of the periodic fiber-reinforced composites, using the generalized stochastic perturbation technique. An evaluation of the generalized stochastic perturbation method of the analytical formulas and the Monte-Carlo simulation technique is provided for the 1D periodic structure with random material parameters. The higher-order terms are determined using numerical determination of the response functions between the effective tensor components and the given random input variables. It is carried out with the use of the Least Squares Method (LSM), applied for the series of computational experiments consisting of the Finite Element Method (FEM) solutions to the cell problems for the randomized input parameters. The key problem is the weighting LSM procedure worked out to speed up the probabilistic convergence of the homogenization results.