On the theory of elastic nonhomogeneous media with a regular structure

Фильштинский, Л. А.

doi:10.1016/0021-8928(73)90033-6

Cited by 7 publications

(6 citation statements)

References 3 publications

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“…which proves (15) and (16). In (22) we have also used the fact that F ξ is continuous in H 1 (Y ), which is verified as follows.…”

Section: Proof If U ∈ C ∞mentioning

confidence: 64%

On the Elastic Deformation of Symmetric Periodic Structures

Lukkassen¹

2003

The Quarterly Journal of Mechanics and Applied Mathematics

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SummaryWe discuss locally anisotropic periodic elastic materials in R n which are symmetric with respect to a hyperplane. The cell problem for finding the effective elastic parameters is considered. It is shown that the components of the displacement satisfy either Neumann or Dirichlet conditions on the sides of the cell of periodicity parallel with the symmetry plane. We also prove that the corresponding homogenized material is monocyclic with respect to this plane. The proof for the general case is based on variational and weak formulations of the elasticity problem. For the special case of plates with a square array of holes or perfectly-bonded inclusions we present an independent proof based on Kolosov-Muskhelishvili analytic potentials. In computational practice the derived boundary conditions permit us to easily incorporate the periodic boundaryvalue problem into standard numerical schemes.

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“…which proves (15) and (16). In (22) we have also used the fact that F ξ is continuous in H 1 (Y ), which is verified as follows.…”

Section: Proof If U ∈ C ∞mentioning

confidence: 64%

On the Elastic Deformation of Symmetric Periodic Structures

Lukkassen¹

2003

The Quarterly Journal of Mechanics and Applied Mathematics

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“…Two real conjugations conditions (22) and (24) (21) and (17)]. Then (25) becomes the R-linear problem [11,56] …”

Section: Anti-plane Shear Problemsmentioning

confidence: 99%

“…2 [2]. Following [29,30] (originally see [20,21,34]) we represent the unknown analytic function via Cauchy's integral in a class of doubly periodic functions (Cauchy's integral on torus)…”

Section: Integral Equations For Anti-plane Problemsmentioning

confidence: 99%

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

Filshtinsky

Mityushev

2014

Mathematics Without Boundaries

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This paper is devoted to boundary value problems for harmonic and biharmonic equations which arise in modeling of elastic and piezoelectric fields in two-dimensional composites. All the problems are investigated by the method of complex potentials. The considered boundary value problems for analytic functions are reduced to integral equations. We discuss methods based on the integral equations for multiply connected domains and in the double periodic statement. Relations to the alternating scheme of Schwarz and to the method of perturbations are considered. Applications to calculation of the effective properties tensor are discussed. This paper also contains results published from 1964 in Russian and not known in English literature.

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“…"iT "rr where the Legendre relation (9) was also taken into account. Now, equation (3b) may be written in terms of the functions ~'(z) and T(z) as:…”

Section: N I Ioakimidis and P S Theocarismentioning

confidence: 99%

“…Koiter [5] has studied the problem of a doubly-periodic set of equal holes, based on the theory of Cauchy-type integrals for doubly-periodic functions, developed previously by him [6]. Furthermore, Fil'shtinskii has studied in more detail the problems of a doublyperiodic set of equal circular holes [7], of a doubly-periodic set of congruent groups of arbitrary holes [8] and of a doubly-periodic set of congruent groups of inclusions [9]. The method of Fil'shtinskii differs from that of Koiter mainly in the fact that he introduced a new doubly-periodic function, simplifying the whole development, but rendering the resulting singular integral equation more difficult for numerical applications.…”

Section: Introductionmentioning

confidence: 99%

Doubly-periodic array of cracks in an infinite isotropic medium

Ioakimidis

Theocaris

1978

J Elasticity

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The problem of a doubly-periodic array of curvilinear cracks in an infinite isotropic medium under conditions of generalized plane stress or plane strain is reduced, by using the method of complex potentials of Muskhelishvili [1], to a complex Cauchy-type singular integral equation on one of the cracks, which can be further numerically solved by reduction to a system of two real Cauchy-type singular integral equations and application of the Gauss or Lobatto-Chebyshev method of numerical solution of such equations. ZUSAMMENFASSUNGDas Problem einer zwei-periodischer Anordung gekriimmten Rissen in einem unendlichem isotropem Medium im allgemeinen ebener Spannung -bzw Dehnung -Zustand, unter Anwendung der Methode yon komplexen Potentialen yon Muskhelishvili [1], ist in einer Cauchyscher komplexer singul~irer IntegralGleichung auf einem der Rissen reduziert. Diese Gleichung kann weiter numerisch gel6st werden bei Transformation in einem System yon zwei reelen Cauchyschen singul/iren Integral-Gleichungen und bei Anwendung der Methoden yon Gauss oder Lobatto-Chebyshev fiir die numerische Resolution solcher Gleichungen.

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On the theory of elastic nonhomogeneous media with a regular structure

Cited by 7 publications

References 3 publications

On the Elastic Deformation of Symmetric Periodic Structures

On the Elastic Deformation of Symmetric Periodic Structures

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

Doubly-periodic array of cracks in an infinite isotropic medium

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