Energetics of two circular inclusions in anti-plane elastostatics

Honein, E.; Honein, T.; Herrmann, G.

doi:10.1016/s0020-7683(98)00286-8

Cited by 10 publications

(7 citation statements)

References 5 publications

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“…Constructive solution to two-particles problems in mechanics allows to describe the local interactions between particles expressed in terms of the flux around the particles. Series representations of the local fields were obtained for harmonic (heat N. Rylko and electric conduction, ideal fluids, dielectric permittivity) and biharmonic (elasticity and slow viscous fluids) equations [8,16,26,28] for some shapes of particles. The most complete results for two spheres were obtained in [1] by use of the bispherical coordinates.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, the general Dirichlet problem for a doubly connected domain was solved in [8,16,28], but its solution was presented in the form of a series which was not asymptotically analysed as δ tends to 0. The simplified version of such a series can be written in the form [30] precisely represents that race to infinity when one attempts to approximate the considerable flux between the disks by a sum of finite order terms.…”

Section: Introductionmentioning

confidence: 99%

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A Pair of Perfectly Conducting Disks in an External Field

Rylko

2015

Mathematical Modelling and Analysis

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A pair of non-overlapping perfectly conducting equal disks embedded in a two-dimensional background was investigated by the classic method of images, by Poincaré series, by use of the bipolar coordinates and by the elliptic functions in the previous works. In particular, successive application of the inversions with respect to circles were applied to obtain the field in the form of a series. For closely placed disks, the previous methods yield slowly convergent series. In this paper, we study the local fields around closely placed disks by the elliptic functions. The problem of small gap is completely investigated since the obtained closed form solution admits a precise asymptotic investigation in terms of the trigonometric functions when the gap between the disks tends to zero. The exact and asymptotic formulae are extended to the case when a prescribed singularity is located in the gap. This extends applications of structural approximations to estimations of the local fields in densely packed fiber composites in various external fields.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Pair of Perfectly Conducting Disks in an External Field

Rylko

2015

Mathematical Modelling and Analysis

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“…For the special cases, the image method [19,23] was employed to satisfy the boundary condition. To avoid the boundary integral along B 0 in Eq.…”

Section: Applications To Half-plane Problemsmentioning

confidence: 99%

“…Temperature in case of steady state heat conduction [1][2][3][4][5][6], electrostatic potential [7][8][9][10][11], velocity potential in a steady flow of an ideal fluid [12][13][14], the displacement of an infinite medium under remote uniformly shear [15][16][17][18][19], and the pure torsion of an elastic bar by equilibrated end torques [20,21] are examples in which the Laplace equation is satisfied. Circular geometries often appear in engineering structures.…”

Section: Introductionmentioning

confidence: 99%

Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

Chen

Shen

2008

Numerical Methods Partial

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In this article, a semi-analytical method for solving the Laplace problems with circular boundaries using the null-field integral equation is proposed. The main gain of using the degenerate kernels is to avoid calculating the principal values. To fully utilize the geometry of circular boundary, degenerate kernels for the fundamental solution and Fourier series for boundary densities are incorporated into the null-field integral equation. An adaptive observer system is considered to fully employ the property of degenerate kernels in the polar coordinates. A linear algebraic system is obtained without boundary discretization. By matching the boundary condition, the unknown coefficients can be determined. The present method can be seen as one kind of semianalytical approaches since error only attributes to the truncated Fourier series. For the eccentric case, vector decomposition technique for the normal and tangential directions is carefully considered in implementing the hypersingular equation in mathematical essence although we transform it to summability to divergent series. The five advantages, well-posed linear algebraic system, principal value free, elimination of boundary-layer effect, exponential convergence, and mesh free, are achieved. Several examples involving infinite, half-plane, and bounded domains with circular boundaries are given to demonstrate the validity of the proposed method.

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“…Honein et al [9,10] have considered the conjugation problem for two nonintersecting elastic circular inclusions perfectly bonded to an elastic matrix subjected to arbitrary loads (singularities) that produce an anti-plane deformation. We emphasize that in [9] the solution was derived only for the case when there are no singularities of f (z) inside both circles.…”

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confidence: 99%

Three–phase eccentric annulus subjected to a potential field induced by arbitrary singularities

Obnosov

2011

Quart. Appl. Math.

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Abstract. An infinite planar, three-component heterogeneous medium with a pair of circles as interfaces between homogeneous zones forming an eccentric annulus is considered for refraction of a potential field on the two interfaces. The velocity field is generated by an arbitrary system of singularities of arbitrary order, in congruity with the MilneThomson case of a two-component medium and a single circular interface. An exact analytical solution of the corresponding R-linear conjugation problem of two Laplacian fields in the eccentrical annulus structure is derived in the class of piecewise meromorphic functions with fixed principal part. Three general cases of loci of the singularities with respect to the interfaces are investigated. Flow nets (isobars and streamlines) are presented.

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Energetics of two circular inclusions in anti-plane elastostatics

Cited by 10 publications

References 5 publications

A Pair of Perfectly Conducting Disks in an External Field

A Pair of Perfectly Conducting Disks in an External Field

Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

Three–phase eccentric annulus subjected to a potential field induced by arbitrary singularities

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