Mathematical Models of Elastic and Piezoelectric Fields in Two-Dimensional Composites

Filshtinsky, Leonid A.; Mityushev, Vladimir

doi:10.1007/978-1-4939-1124-0_8

Cited by 8 publications

(8 citation statements)

References 51 publications

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“…The next important step in the mathematical treatment of the 2D composites was made by V.Ya. Natanzon [54] in 1935 and L.A. Filshtinsky (Fil'shtinskii) in 1964 (see papers [24,25,16], his thesis [26], the fundamental books [17,18,19,23] and references in [27]). V.Ya.…”

Section: Methods Of Natanzon-filshtinskymentioning

confidence: 99%

Exact and “Exact” Formulae in the Theory of Composites

Andrianov

Mityushev

2018

Trends in Mathematics

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The effective properties of composites and review literature on the methods of Rayleigh, Natanzon-Filshtinsky, functional equations and asymptotic approaches are outlined. In connection with the above methods and new recent publications devoted to composites, we discuss the terms analytical formula, approximate solution, closed form solution, asymptotic formula etc... frequently used in applied mathematics and engineering in various contexts. Though mathematicians give rigorous definitions of exact form solution the term "exact solution" continues to be used too loosely and its attributes are lost. In the present paper, we give examples of misleading usage of such a term.

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Section: Methods Of Natanzon-filshtinskymentioning

confidence: 99%

Exact and “Exact” Formulae in the Theory of Composites

Andrianov

Mityushev

2018

Trends in Mathematics

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“…We seek the solution of the previous system of equations as 11) where pq N k are Y-periodic functions with k, p, q D 1, 2, 3. If we replace (3.11) in (3.10), we obtain the so-called local problems pq L in the unitary cell:…”

Section: Asymptotic Homogenization Methods and The Effective Coefficientsmentioning

confidence: 99%

“…The second approach considers operators of Cauchy type. These type of operators with reproducing kernel doubly periodic was used by Linkov, Lu and Fil'shtinskii , to solve plane problems of elasticity whose solutions were doubly periodic functions. In our case, our method is based by using this type of integrals to represent the KM potentials knowing only the jump of these functions in the interphase.…”

Section: Introductionmentioning

confidence: 99%

A solution for the local plane strain problems using elliptic integrals of Cauchy type

Felipe-Sosa

Otero

Solís

2016

Math Methods in App Sciences

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We consider a parallel fiber‐reinforced periodic elastic composite that present an imperfect contact of spring type between the fiber and the matrix. We use the elliptic integral of Cauchy type for solving the plane strain local problems that arise from the asymptotic homogenization method. Several general conditions are assumed, which include that the fibers are disposed of arbitrary manner in the local cell, that all fibers present contact perfect with different constants of imperfection, and that their cross section is a smooth closed arbitrary curve. We find that there are infinity solutions for these problems, and we find relations between these solutions and effective coefficients of the composite. Finally, we obtain analytic formulae for the circular fiber case and show some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

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“…1 ω rs from (39a) and (40) is also not convergent, but since it doesn't appear in the derivative of f m (z), it doesn't affect the heat flux and gradient fields. The relation (40) can be considered as a generalization of Filshtinsky's representation [1,7] for a doubly-periodic array of sole fibers to the case of a doubly-periodic array of fiber pairs.…”

Section: Average Flux Over a Unit Cellmentioning

confidence: 99%

“…Godin [15] developed a method based on the use of elliptic function for the cases of general doubly-periodic arrays. Andrianov and Mityushev [1] pointed out that this method is related to the method of Natanzon-Filshtinsky for conductivity problems [6,7]. The authors also developed an eigenfunction expansion-variational method (EEVM) [51] and a series method [50] extended from Rayleigh's solution to solve the effective conductivity for general periodic arrays, in order to obtain approximate analytical formulae with high accuracy and in a concise form.…”

Section: Introductionmentioning

confidence: 99%

Effective transport properties of composites with a doubly‐periodic array of fiber pairs and with a triangular array of fibers

Yan

Zhang

Chen³

et al. 2017

Z Angew Math Mech

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This paper presents a complex variable solution for the effective transport properties of composites with a doubly-periodic array of fiber pairs. By using the centrosymmetry of the problem, the method of Rayleigh and Natanzon-Filshtinsky's approach can be simply extended to the problems with two fibers per unit cell. The infinite system constructed in this paper only slightly complicates Rayleigh's system for the problems with one fiber per unit cell. Approximate analytical formulae of the effective transport properties for different fiber-pair arrays are obtained. The influence of pairwise interaction in fiber pairs on the effective transport properties is discussed in the numerical examples. As a special case of a doubly-periodic array of fiber pairs, effective transport property of composites with a triangular array of fibers is obtained. The obtained approximate analytical formulae are written in a concise form with good accuracy, thus are convenient for engineering application in most cases, except for those approaching the limit case of percolation when the perfectly conducting fibers become touching. Besides the square array and hexagonal array, the triangular fiber array (similar to carbon atom arrangement in graphene) is another special symmetric fiber array which results into transversely isotropic effective property. Therefore, the present solution for the triangular array is an extension of those for the square array and hexagonal array. The comparison of the results for the three symmetric fiber arrays reveals that the triangular fiber array has the highest conductivity. In addition, accuracy of the present solution is analyzed in the numerical examples.

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Mathematical Models of Elastic and Piezoelectric Fields in Two-Dimensional Composites

Cited by 8 publications

References 51 publications

Exact and “Exact” Formulae in the Theory of Composites

Exact and “Exact” Formulae in the Theory of Composites

A solution for the local plane strain problems using elliptic integrals of Cauchy type

Effective transport properties of composites with a doubly‐periodic array of fiber pairs and with a triangular array of fibers

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