arXiv subjects

Y. A. Antipov

Publications and source records attributed to Y. A. Antipov.

15 recordsLinked to original sources

arXiv2025

An edge crack and a crack close to the vertex of a wedge

Two model problems of an elastic wedge with an internal and edge crack are analyzed. The problem of an internal crack reduces to an order-4 vector Riemann-Hilbert problem whose matrix kernel entries are meromorphic functions and have exponential factors. When the internal crack is located along one of the wedge sides, an efficient method of solution is proposed. It requires a factorization of the order-2 matrix coefficient associated with the corresponding problem of an edge crack and the solution of an infinite system of linear algebraic system with an exponential rate of convergence of an approximate solution to the exact one. The order-2 Khrapkov's factorization is modified by splitting the matrix kernel into a scalar dominant function and a ``regular" matrix whose factorization is more convenient for numerical purposes. Expressions for the stress intensity coefficients and the potential energy released when the crack advances are derived. Asymptotic relations for the stress intensity coefficients and the potential energy when one of the crack tips is close the wedge vertex are obtained.

math.AP↗

arXiv2024

Dynamic problem of a power-law graded half-plane and an associated Carleman problem for two functions

A steady state plane problem of an inhomogeneous half-plane subjected to a load running along the boundary at subsonic speed is analyzed. The Lame coefficients and the density of the half-plane are assumed to be power functions of depth. The model is different from the standard static model have been used in contact mechanics since the Sixties and originated from the 1964 Rostovtsev exact solution of the Flamant problem of a power-law graded half-plane. To solve the governing dynamic equations with variable coefficients written in terms of the displacements, we propose a method that, by means of the Fourier and Mellin transforms, maps the model problem to a Carleman boundary value problem for two meromorphic functions in a strip with two shifts or, equivalently, to a system of two difference equations of the second order with variable coefficients. By partial factorization the Carleman problem is recast as a system of four singular integral equations on a segment with a fixed singularity and highly oscillating coefficients. A numerical method for its solution is proposed and tested. Numerical results for the displacement and stress fields are presented and discussed.

math.CV↗

arXiv2023

Scattering by a perforated sandwich membrane: method of Riemann surfaces

The model problem of scattering of a sound wave by an infinite plane structure formed by a semi-infinite acoustically hard screen and a semi-infinite sandwich panel perforated from one side and covered by a membrane from the other is exactly solved. The model is governed by two Helmholtz equations for the velocity potentials in the upper and lower half-planes coupled by the Leppington effective boundary condition and the equation of vibration of a membrane in a fluid. Two methods of solution are proposed and discussed. Both methods reduce the problem to an order-2 vector Riemann-Hilbert problem. The matrix coefficients have different entries, have the Chebotarev-Khrapkov structure and share the same order-4 characteristic polynomial. Exact Wiener-Hopf matrix factorization requires solving a scalar Riemann-Hilbert on an elliptic surface and the associated genus-1 Jacobi inversion problem solved in terms of the associated Riemann $θ$-function. Numerical results for the absolute value of the total velocity potentials are reported and discussed.

math-ph↗

arXiv2022

Two-dimensional contact of two different power-law graded elastic bodies

Previous study of contact of power-law graded materials concerned the contact of a rigid body (punch) with an elastic inhomogeneous foundation whose inhomogeneity is characterized by the Young modulus varying with depth as a power function. This paper models Hertzian and adhesive contact of two elastic inhomogeneous power-law graded bodies with different exponents. The problem is governed by an integral equation with two different power kernels. A nonstandard method of Gegenbauer orthogonal polynomials for its solution is proposed. It leads to infinite system of linear algebraic equations of a special structure. The integral representations of the system coefficients are evaluated, and the properties of the system are studied. It is shown that if the exponents coincide, the infinite system admits a simple exact solution that corresponds to the case when the Young moduli are different but the exponents are the same. Formulas for the length of the contact zone, the pressure distribution, and the surface normal displacements of the contacting bodies are obtain in the form convenient for computations. Effects of the mismatch in the Young moduli exponents are studied. A comparative analysis of the Hertzian and adhesive contact models clarifies the effects of the surface energy density on the contact pressure, the contact zone size, and the profile of the contacting bodies outside the contact area.

math.AP↗

arXiv2021

Riemann-Hilbert problem on an elliptic surface and a uniformly stressed inclusion embedded into a half-plane subjected to antiplane strain

An inverse problem of elasticity of $n$ elastic inclusions embedded into an elastic half-plane is analyzed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the conditions of ideal contact hold in the interfaces between the inclusions and the half-plane. The shapes of the inclusions are not prescribed and have to be determined by enforcing uniform stresses inside the inclusions. The method of conformal mappings from a slit domain onto the $(n+1)$-connected physical domain is worked out. It is shown that to recover the map and therefore the inclusions shapes, one needs to solve a vector Riemann-Hilbert problem on a genus-$n$ hyperelliptic surface. In a particular case of loading of a single inclusion in a half-plane, the problem is equivalent to two scalar Riemann-Hilbert problems on two slits on an elliptic surface. In addition to three parameters of the model the conformal map possesses a free geometric parameter. Results of numerical tests which show the impact of these parameters on the inclusion shape are presented.

math.CV↗

arXiv2021

Sadovskii vortex in a wedge and the associated Riemann-Hilbert problem on a torus

Reconstruction of conformal mappings from canonical slit domains onto multiply-connected physical domains with a free boundary is of interest in many different models arising in fluid mechanics. In the present paper, an exact formula for the conformal map from the exterior of two slits onto the doubly connected flow domain is obtained when a fluid flows in a wedge about a Sadovskii vortex. The map is employed to determine the potential flow outside the vortex and the vortex domain boundary provided the circulation $Γ$ around the vortex and constant speed $U$ on the vortex boundary are prescribed, and there are no stagnation points on the walls. The map is expressed in terms of a rational function on an elliptic surface topologically equivalent to a torus and the solution to a symmetric Riemann-Hilbert problem on a finite and a semi-infinite segments on the same genus-1 Riemann surface. Owing to its special features, the Riemann-Hilbert problem requires a novel analogue of the Cauchy kernel on an elliptic surface. Such a kernel is proposed and employed to derive a closed-form solution to the Riemann-Hilbert problem and the associated Jacobi inversion problem. The final formula for the conformal map possesses a free geometric parameter and two model parameters, the wedge angle $α$ and $Γ/U$. It is shown that when $α<π$ the solution exists and the vortex has two cusps, while the solution does not exist when the wedge angle exceeds $π$.

math.CV↗

arXiv2020

Annular and circular rigid inclusions planted into a penny-shaped crack and factorization of triangular matrices

Analytical solutions to two axisymmetric problems of a penny-shaped crack when an annulus-shaped (model 1) or a disc-shaped (model 2) rigid inclusion of arbitrary profile are embedded into the crack are derived. The problems are governed by integral equations with the Weber--Sonin kernel on two segments. By the Mellin convolution theorem the integral equations associated with the models 1 and 2 reduce to vector Riemann-Hilbert problems with and 3x3 and 2x2 triangular matrix coefficients whose entries consist of meromorphic and of infinite indices exponential functions. Canonical matrices of factorization are derived and the partial indices are computed. Exact representation formulas for the normal stress, the stress intensity factor, and the normal displacement are obtained and the results of numerical tests are reported.

math.AP↗

arXiv2018

Method of automorphic functions for an inverse problem of antiplane elasticity

A nonlinear inverse problem of antiplane elasticity for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings of circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky groups a series-form representation of a ($3n-4$) parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.

math.CV↗

arXiv2018

Integral relations associated with the semi-infinite Hilbert transform and applications to singular integral equations

Integral relations with the Cauchy kernel on a semi-axis for the Laguerre polynomials, the confluent hypergeometric function, and the cylindrical functions are derived. A part of these formulas is obtained by exploiting some properties of the Hermite polynomials, including their Hilbert and Fourier transforms and connections to the Laguerre polynomials. The relations discovered give rise to complete systems of new orthogonal functions. Free of singular integrals, exact and approximate solutions to the characteristic and complete singular integral equations in a semi-infinite interval are proposed. Another set of the Hilbert transforms in a semi-axis are deduced from integral relations with the Cauchy kernel in a finite segment for the Jacobi polynomials and the Jacobi functions of the second kind by letting some parameters involved go to infinity. These formulas lead to integral relations for the Bessel functions. Their application to a model problem of contact mechanics is given. A new quadrature formula for the Cauchy integral in a semi-axis based on an integral relation for the Laguerre polynomials and the confluent hypergeometric function is derived and tested numerically. Bounds for the remainder are found.

math.CV↗

arXiv2016

A crack induced by a thin rigid inclusion partly debonded from the matrix

The interaction of a thin rigid inclusion with a finite crack is studied. Two plane problems of elasticity are considered. The first one concerns the case when the upper side of the inclusion is completely debonded from the matrix, and the crack penetrates into the medium. In the second model, the upper side of the inclusion is partly separated from the matrix, that is the crack length $2a$ is less than $2b$, the inclusion length. It is shown that both problems are governed by a singular integral equation of the same structure. Derivation of the closed-form solution of this integral equation is the main result of the paper. The solution is found by solving the associated vector Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient. A feature of the method proposed is that the vector Riemann-Hilbert problem is set on a finite segment, while the original Khrapkov method of matrix factorization is developed for a closed contour. In the case, when the crack and inclusion lengths are the same, the solution is derived by passing to the limit $b/a\to 1$. It is demonstrated that the limiting case $a=b$ is unstable, and when $a<b$, and the crack tips approach the inclusion ends, the crack tends to accelerate in order to penetrate into the matrix.

math.AP↗

arXiv2016

Helmholtz equation in a semi-infinite strip with impedance boundary conditions of the third and fifth orders

Two boundary value problems for the Helmholtz equation in a semi-infinite strip are considered. The main feature of these problems is that, in addition to the function and its normal derivative on the boundary, the functionals of the boundary conditions possess tangential derivatives of the second and fourth orders. Also, the setting of the problems is complimented by certain edge conditions at the two vertices of the semi-strip. The problems model wave propagation in a semi-infinite waveguide with membrane and plate walls. A technique for the exact solution of these fluid-structure interaction problems is proposed. It requires application of two Laplace transforms with respect to both variables with the parameter of the second transform being a certain function of the first Laplace transform parameter. Ultimately, this method yields two scalar Riemann-Hilbert problems with the same coefficient and different right-hand sides. The dependence of the existence and uniqueness results of the physical model problems on the index of the Riemann-Hilbert problem is discussed.

math.AP↗

arXiv2015

Singular integral equations with two fixed singularities and applications to fractured composites

A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector Riemann-Hilbert problem on a real axis with a piecewise constant matrix coefficient that has two points of discontinuity. A condition of solvability and a closed-form solution to the integral equation are derived. For the Chebyshev polynomials of the first kind in the right hand-side, the solution of the integral equation is expressed in terms of two nonorthogonal polynomials with associated weights. Based on this new generalized spectral relation for the singular operator with two fixed singularities an approximate solution to the complete singular integral equation is derived by recasting it as an infinite system of linear algebraic equations of the second kind. The method is illustrated by solving two problems of fracture mechanics, the antiplane and plane strain problems for a finite crack in a composite plane. The plane is formed by a strip and two half-planes; the elastic constants of the strip are different from those of the half-planes. The crack is orthogonal to the interfaces, and it is located in the strip with the ends lying in the interfaces. Numerical results are reported and discussed.

math.CV↗

arXiv2015

Fundamental solution and the weight functions of the transient problem on a semi-infinite crack propagating in a half-plane

The two-dimensional transient problem that is studied concerns a semi-infinite crack in an isotropic solid comprising an infinite strip and a half-plane joined together and having the same elastic constants. The crack propagates along the interface at constant speed subject to time-independent loading. By means of the Laplace and Fourier transforms the problem is formulated as a vector Riemann-Hilbert problem. When the distance from the crack to the boundary grows to infinity the problem admits a closed-form solution. In the general case, a method of partial matrix factorization is proposed. In addition to factorizing some scalar functions it requires solving a certain system of integral equations whose numerical solution is computed by the collocation method. The stress intensity factors and the associated weight functions are derived. Numerical results for the weight functions are reported and the boundary effects are discussed. The weight functions are employed to describe propagation of a semi-infinite crack beneath the half-plane boundary at piecewise constant speed.

math.AP↗

arXiv2014

Diffraction of an obliquely incident electromagnetic wave by an impedance right-angled concave wedge

Scattering of a plane electromagnetic wave by an anisotropic impedance right-angled concave wedge at skew incidence is analyzed. A closed-form solution is derived by reducing the problem to a symmetric order-2 vector Riemann-Hilbert problem (RHP) on the real axis. The problem of matrix factorization leads to a scalar RHP on a genus-3 Riemann surface. Its solution is derived by the Weierstrass integrals. Due to a special symmetry of the problem the associated Jacobi inversion problem is solved in terms of elliptic integrals, not a genus-3 Riemann è-function. The electric and magnetic field components are expressed through the Sommerfeld integrals, and the incident and reflected waves are recovered.

math-ph↗

arXiv2009

Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an $(N+1)$-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetry Schottky group. We prove the existence theorem for the second, quasimultiplicative, kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with $N+1$ circular holes with electrodes and dielectrics on the walls when the conductor is placed at right angle to the applied magnetic field.

math.CV↗