L.A. Filshtinsky's contribution to Applied Mathematics and Mechanics of Solids

Mityushev, Vladimir; Andrianov, Igor V.; Gluzman, S.

doi:10.1016/b978-0-32-390543-5.00006-2

Cited by 6 publications

(5 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Taking into account Ansatz (5), after an expansion in powers of a small parameter 𝜀 with regard to Equations ( 1)-( 3), the solution of the original problem is divided into two stages. At the first stage, the solution of the local problem is determined, that is, the problem for the periodically repeating cell of the composite (Figure 2):…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…Various mathematical techniques are used to determine the effective characteristics. We single out the Rayleigh method [2][3][4][5], the Natanzon-Filshtinsky method [5][6][7][8][9], and the method of functional equations [5,8]. The first two methods lead to an infinite system of linear algebraic equations, which is then reduced to a finite number of equations and solved numerically.…”

Section: Introductionmentioning

confidence: 99%

“…The claims of some authors who present the reduction of the original problem to an infinite system of linear algebraic equations as an ‘exact’ solution or a ‘closed‐form solution’ are groundless (see a detailed analysis of this problem in Refs. [4, 5]).…”

Section: Introductionmentioning

confidence: 99%

“…The method of functional equations allows to get a formally exact solution in which the Rayleigh and Natanzon–Filshtinsky coefficients are written out by formulas in the form of series [5, 8].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

Andrianov

Awrejcewicz

Starushenko³

et al. 2022

Z Angew Math Mech

View full text Add to dashboard Cite

Paper is devoted to the effective stationary heat conductivity for the fibre composite materials. We are aimed on getting on analytical expression for effective thermal conductivity coefficient. Asymptotic homogenization approach, based on the multiple scale perturbation method, is used. This allows to reduce the original boundary value problem in multiply connected domain to the sequence of boundary value problems in simply connected domains. These problems include: the local problem for the periodically repeated cell and homogenized problem with effective coefficient. It is shown that for densely packed and high contrast fibre composites, the cell problem can be solved analytically. For this aim, lubrication approach (asymptotics of thin layer) has been employed. We also generalise obtained solution to the case of medium‐sized inclusions in the framework of the Padé approximants.

show abstract

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

Andrianov

Awrejcewicz

Starushenko³

et al. 2022

Z Angew Math Mech

View full text Add to dashboard Cite

show abstract

“…There is almost a century of literature on the mathematical analysis of perforated plates and porous materials (see, e.g., [4,15,19,20,30], see also Mityushev et al [2] for a review). Examples can be found in the applications to the study of porous coatings and of interfacial coatings.…”

Section: Introductionmentioning

confidence: 99%

A Degenerating Robin-Type Traction Problem in a Periodic Domain

Dalla Riva,

Mishuris,

Musolino

2023

Mathematical Modelling and Analysis

View full text Add to dashboard Cite

We consider a linearly elastic material with a periodic set of voids. On the boundaries of the voids we set a Robin-type traction condition. Then, we investigate the asymptotic behavior of the displacement solution as the Robin condition turns into a pure traction one. To wit, there will be a matrix function b[k](·) that depends analytically on a real parameter k and vanishes for k = 0 and we multiply the Dirichlet-like part of the Robin condition by b[k](·). We show that the displacement solution can be written in terms of power series of k that converge for k in a whole neighborhood of 0. For our analysis we use the Functional Analytic Approach.

show abstract

Lattice sums for double periodic polyanalytic functions

Drygaś,

Mityushev

2023

Anal.Math.Phys.

View full text Add to dashboard Cite

In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, employing the Eisenstein summation, that the lattice sum $$S_2$$ S 2 is equal to $$\pi $$ π for the square array. Further, it became clear that such equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions, including the conditionally convergent Eisenstein series. In 1935, Natanzon used a polyharmonic function to solve the plane elasticity problem. This paper is devoted to the extension of the classic lattice sums to the lattice sums for double periodic (pseudoperiodic) polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums.

show abstract

L.A. Filshtinsky's contribution to Applied Mathematics and Mechanics of Solids

Cited by 6 publications

References 46 publications

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

A Degenerating Robin-Type Traction Problem in a Periodic Domain

Lattice sums for double periodic polyanalytic functions

Contact Info

Product

Resources

About