Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions

Czapla, Roman; Nawalaniec, Wojciech; Mityushev, Vladimir

doi:10.1016/j.commatsci.2012.05.058

Cited by 50 publications

(41 citation statements)

References 21 publications

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“…The computationally effective formulae and algorithms for the absolutely convergent sums and ( > 2) were developed in [13,14]. However, the numerical implementation of the conditionally convergent series (74)-(75) for = 2 and (76) for = 3 requires further investigations.…”

Section: Numerical Simulationsmentioning

confidence: 99%

Contrast Expansion Method for Elastic Incompressible Fibrous Composites

Drygaś

Mityushev

2017

Advances in Mathematical Physics

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Contrast parameter expansion of the elastic fields for 2D composites is developed by Schwarz's method and by the method of functional equations for the case of circular inclusions. A computationally efficient algorithm is described and implemented in symbolic form to compute the local fields in 2D elastic composites and the effective shear modulus for macroscopically isotropic composites. The obtained new analytical formula contains high-order terms in the contrast parameter and explicitly demonstrates dependence on the location of inclusions. As a numerical example, the hexagonal array is considered.

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Section: Numerical Simulationsmentioning

confidence: 99%

Contrast Expansion Method for Elastic Incompressible Fibrous Composites

Drygaś

Mityushev

2017

Advances in Mathematical Physics

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“…The method of Rayleigh was extended to elastic problems by Filshtinskii [20,29] and further by McPhedran et al [30][31][32][33]. Mityushev et al [34][35][36][37][38][39] extended the latter results for conductivity to arbitrary locations of disks per periodicity cell by a method of functional equations. High-order approximate analytical formulae for the effective conductivity were derived and exact formulae for regular arrays were deduced (see the survey [37]).…”

Section: Introductionmentioning

confidence: 99%

Effective anti-plane properties of piezoelectric fibrous composites

Rylko

2013

Acta Mech

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Anti-plane shear of piezoelectric fibrous composites is theoretically investigated. The geometry of composites is described by the 2-dimensional geometry in a section perpendicular to the unidirectional fibers. The previous constructive results obtained for scalar conductivity problems are extended to piezoelectric antiplane problems. First, the piezoelectric problem is written in the form of the vector-matrix R-linear problem in a class of double periodic functions. In particular, application of the zeroth-order solution to the R-linear problem yields a vector-matrix extension of the famous Clausius-Mossotti approximation. The vector-matrix problem is decomposed into two scalar R-linear problems. This reduction allows us to directly apply all the known exact and approximate analytical results for scalar problems to establish high-order formulae for the effective piezoelectric constants. Special attention is paid to non-overlapping disks embedded in a two-dimensional background.

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“…The hexagonal lattice serves as the domain Q, where random composite is generated as a probabilistic distribution of disks of radius r (particles), by means of some Monte-Carlo algorithm (protocol) [11].…”

Section: Random Composite From Hexagonal Representative Cellmentioning

confidence: 99%

“…Expression (1) was derived using only terms up to the 12th order in concentration. The expansion of (1) is characterized by a rather regular behavior of the coefficients, σ reg (x) = 1 + 2x + 2x 2 + 2x 3 + 2x 4 + 2x 5 + 2x 6 + 2.15084x 7 + 2.30169x 8 + 2.45253x 9 + 2.60338x 10 + 2.75422x 11 + 2.90506x 12 + O(x 13 ).…”

Section: Introductionmentioning

confidence: 99%

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Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders

Gluzman

Mityushev

Nawalaniec

et al. 2016

Contributions in Mathematics and Engineering

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Effective conductivity of a 2D composite corresponding to the regular hexagonal arrangement of superconducting disks is expressed in the form of a long series in the volume fraction of ideally conducting disks. According to our calculations based on various re-summation techniques, both the threshold and critical index are obtained in good agreement with expected values. The critical amplitude is in the interval (5.14, 5.24) that is close to the theoretical estimation 5.18. The next order (constant) term in the high concentration regime is calculated for the first time, and the best estimate is equal to (−6.229). Final formula is derived for the effective conductivity for arbitrary volume fraction.

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Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions

Cited by 50 publications

References 21 publications

Contrast Expansion Method for Elastic Incompressible Fibrous Composites

Contrast Expansion Method for Elastic Incompressible Fibrous Composites

Effective anti-plane properties of piezoelectric fibrous composites

Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders

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