We solve the problem of finding the two-dimensional stress state of an elastic isotropic body containing a system of arbitrarily located cracks interacting with waves. The solution is based on the reduction of the original problem to a system of singular integrodifferential equations for the jumps of displacements on the crack surfaces. For the solution of this system, we propose to use an iterative method. This method enables us to avoid the difficulties connected with the necessity of solving the systems of integrodifferential equations of large dimensions. The analyzed examples demonstrate the convergence and stability of the proposed method also in the case of systems of densely located cracks of complex configurations.

We solve the problem of determination of the stress state of an infinitely long cylinder of an arbitrary cross section under the conditions of longitudinal shear vibrations in the presence of interaction between through defects (a crack and a thin rigid inclusion ) . We use a method that enables one to satisfy the conditions imposed on the surfaces of defects and the boundary conditions of time-harmonic loading. We propose approximate formulas for the evaluation of stress intensity factors with the help of which we investigate the influence of the geometric parameters of cross section of the cylinder on the values of resonance frequencies.

We study the stress concentration near cracks originating from the edges of a thin rigid inclusion at an arbitrary angle under the action of harmonic longitudinal shear waves. The original problem is reduced to a system of two singular integrodifferential equations and one integral equation whose kernels contain immobile singularities. These equations are solved by a numerical method in which the asymptotics of unknown functions are taken into account, and special quadrature formulas for singular integrals are applied.

We consider two-dimensional dynamic problems of the theory of elasticity, which can be reduced to singular integral or integrodifferential equations with immobile singularities. These problems include the problems of determination of the stressed state of bodies with edge defects and defects whose cross sections have the shape of broken line and some contact problems. For the solution of the obtained equations, we propose to apply a numerical method that takes into account the actual asymptotics of solutions and is based on the use of special quadrature formulas for singular integrals.

This paper is devoted to the analysis of the dynamic stress state of a finite cylinder with an internal circular crack. One of the cylinder’s bases is fixed, and a time-dependent torque is applied to the other base. For solving this problem, a modified finite difference method was applied. This method consists in the difference approximation of only the time derivative. As a result, the original problem is reduced to a set of sequentially solved homogeneous boundary value problems. Their solutions contain an unknown displacement jump on the crack. The integral equation for the unknown jump is reduced to the Fredholm integral equation of the second kind, which was solved numerically. This numerical solution made it possible to obtain an approximate formula for calculating the stress intensity factor (SIF).

We solve the problem of determination of the stress state formed in the vicinity of a tunnel rigid inclusion with cross section in the form of a broken line. The inclusion is located in the elastic space. It is assumed that plane harmonic longitudinal shear waves propagate in this space. The problem is reduced to a system of singular integral equations with fixed singularities. This system is solved approximately with the help of a numerical method with the use of the true asymptotics of unknown functions and special quadrature formulas for singular integrals.

An experimental study is made of electron tunneling in a resonant-tunneling diode in magnetic fields directed parallel and perpendicular to the planes of the GaAs/AlXGa1−XAs heterostructure layers. In particular, phonon replicas on the current–voltage characteristics of the diode are investigated. In the second current derivatives a fine structure of replicas is found. The transformation of the structure of replicas in a perpendicular magnetic field can be qualitatively understood as a manifestation of the transition of polaron states to magnetopolaronic ones.

A problem on the determination of the stress state in an infinitely long cylinder of arbitrary cross-section under the longitudinal shear oscillations is solved with account for the interaction between the through defects (i.e., a crack and a thin rigid inclusion). The employed method allows for the satisfaction of the conditions on the surfaces of defects along with satisfying the conditions of time-harmonic loading on the surface of the cylinder. Approximate formulas are obtained for calculating stress intensity factors. These formulas are used to analyze the influence of geometric parameters of the crosssectional geometry of the cylinder on the resonance frequencies. Cite as: O. I. Kyrylova, V. G. Popov, “The interaction between a crack and an inclusion in a cylindrical body under the longitudinal shear oscillations,” Mat. Met. Fiz. Mekh. Polya , 64 , No. 3, 131–141 (2021) (in Ukrainian), https://doi.org/10.15407/mmpmf2021.64.3.131-141

A singular integral equation with a fixed singularity to which the problem of contact interaction of two quarters of spaces in the conditions of harmonic oscillations of longitudinal shear is reduced is considered. A quarters of the space is situated so that the half-space composed of them has a stepped boundary. In the contact area, the conditions for ideal coupled are satisfied. The unknown function in this equation is the contact stresses. For the numerical solution of this equation, a method that takes into account the asymptotic behavior of contact stresses at the edge point is proposed. The basis of this method is the use of special quadrature formulas for singular integrals obtained in the article. When obtaining these formulas, the unknown function was approximated by an interpolation polynomial, in which the roots of the Laguerre polynomials are the points of interpolation. The values of the unknown function at the interpolation points are found by the collocation method, herewith the collocation points of collocationare the roots of the special function. An approximate formula for calculating contact stresses can have practical application. The effectiveness of the proposed method is demonstrated by the numerical example.

An elastic cylinder of finite length, one of the ends of which is perfectly coupled to the surface of the elastic half-space is considered. A round rigid plate of the same radius is coupled to the other end of the cylinder, and is loaded the torsion moment that is harmonic depend of time. The surface of the half-space outside the contact area with the cylinder and the side surface of the cylinder are been unload. The formulated boundary problem is reduced to a singular integral equation for a function related to stresses in the contact area of the cylinder and half-space. Since the kernel of this integral equation contains fixed singularities, a numerical method for solving this equation the is main result. After solving the integral equation, approximate formulas for calculating the contact stresses .

Abstract The problem the determining of stressed state in the vicinity of the tunnel rigid inclusion when it’s cross-section a broken is considered. The inclusion is located in an infinite elastic medium and harmonic shear force impacts on it. It is supposed the inclusion is fully coupled with the medium. The problem is reduced to the solution of at system singular integral equations with fixed singularities. A numerical method of the solving of this system with regard to true asymptotic of the unknown functions is developed.

Abstract The axisymmetric dynamic problems of determining the stress state in the vicinity of the delamination in a finite cylinder partly coupled with the rigid base were solved. In contrast to the traditional solution methods based on the use of the integral Laplace transform, the proposed method consists in the difference approximation only of the time derivative. As a result, the original problem is replaced by a sequence of homogeneous boundary value problems for the Helmholtz equation that reduce to the Fredholm integral equation of the second kind. The numerical solution found made it possible to obtain an approximate formula for calculating the stress intensity factor.

Электрохимические системы очень перспективны для разработки новой элементной базы для микроэлектроники и для использования в широком спектре инженерных задач. Мы разработали новую микроэлектронную технологию для изготовления электрохимических преобразователей (ЭХП) и новые приборы на основе новых электрохимических микроэлектронных чипов. Планарные электрохимические преобразователи могут использоваться в акселерометрах, сейсмических датчиках, датчиках вращения, гидрофонах и датчиках давления. Electrochemical systems are very promising for the development of a new element base for microelectronics, and for use in a wide range of engineering applications. We have developed a new microelectronic technology for manufacturing electrochemical transducers (ECP) and new devices based on new electrochemical microelectronic chips. Planar electrochemical transducers are used in accelerometers, seismic sensors, rotation sensors, hydrophones and pressure sensors.

The problem of determining the stress state near the through-cracks in an infinite hollow cylinder of arbitrary cross-section under oscillations of longitudinal shear is solved. The method allows satisfying the conditions separately on the surface of cracks and on the borders of the cylinder. The solution scheme is based on the use of discontinuous solutions of equations of motion of elastic medium with jumps of displacements on the surface of defects. For this displacement are represented by the sums of discontinuous solutions, built for each defect, and an unknown characteristic function. Designed presentation enables fulfilling separately the boundary conditions on the surface of defects that leads to the set of systems of integral equations, which don't depend from the shape of the boundaries of the body. Then the unknown coefficients of represented characteristic function are determined from the conditions on the boundaries of the body by the collocation method.

The problem of the diffraction field determination is arising as a result of the longitudinal shear wave interaction with the thin rigid inclusions system arbitrarily situated in an infinity body was solved. Inclusions are considered to be fully coupled to the elastic medium and are moving. Unknown amplitudes of inclusions are determined from the equations of motion. The solution method is based on the submission diffraction field displacement as sum of discontinuous solutions to the Helmholtz equation, the constructed for each inclusion. As result the original problem is reduced to the system of the singular integral equations for unknown jumps of stresses on the inclusions surface, The iterative method of this system solving, where the zero approximation are the solutions of the integral equations for the single inclusions, is proposed. This integral equation for single inclusions are numerical solved the mechanical quadrature method. The final result is the approximate formulas for calculating stress intensity factors and the amplitudes of the oscillations.

This paper solves the problem of determining the stress state near cracks in an infinite hollow cylinder of arbitrary cross section during longitudinal shear oscillations. We propose an approach that allows us to separately satisfy conditions both on the cracks and boundaries of a cylinder. The problem reduces to the equations of motion in a flat domain with the defects bounded by arbitrary smooth closed curves under anti-plane deformation conditions. The solution scheme is based on the use of discontinuous solutions to the equations of motion of an elastic medium with displacement jumps on the surfaces of defects. Displacements in a cylinder with defects are represented both as a sum of discontinuous solutions constructed for each defect and an unknown specific function ensuring that the conditions of a harmonic load on the body boundaries are met. This function is sought as a linear combination of linearly independent solutions to the equations of the theory of elasticity in the frequency domain with unknown coefficients. The constructed representation makes it possible to separately satisfy the boundary conditions on the surfaces of defects, which results in a set of systems of integral equations that differ only in their right-hand sides and do not depend on the body boundary shape. The resulting systems of integral equations can be solved by the method of mechanical quadratures. After that, the conditions on the boundaries of the cylindrical body are satisfied, from which the unknown coefficients of the introduced specific function are determined by a collocation method. Using the approach proposed, the stress intensity factors in the vicinity of defects were calculated. With the help of those calculations, we investigated the effect of the frequency and location of the defects on the stress intensity coefficient values.

The axisymmetric dynamic problem of determining the stress state in the vicinity of a ring-shaped crack in a finite cylinder is solved. The source of the loading is the rigid circular plate, which is joined with one of the cylinder ends and loaded by the time-dependent torque. The proposed method consists in the difference approximation of only the time derivative. To do this, specially selected non-equidistant nodes and special representation of the solution in these nodes are used. Such an approach allows the original problem to be reduced to a sequence of boundary value problems for the homogeneous Helmholtz equation. Each such problem is solved by using integral Fourier and Hankel transforms, with their subsequent reversal. As a result, integral representations were obtained for the angular displacement through unknown tangential stresses in the plane of the crack. From boundary condition on a crack, an integral equation is obtained, which, as a result of using the Weber-Sonin integral operator and a series of transformations, is reduced to the Fredholm integral equation of the second kind. The numerical solution found made it possible to obtain an approximate formula for calculating the stress intensity factor (SIF).

There is a thin absolutely rigid inclusion that in a cross-section represents three segments broken line in an infinite elastic medium (matrix) that is in the conditions of antiplane strain. The inclusion is under the action of harmonic shear force Pe^{iwt} along the axis Oz. Under the conditions of the antiplane strain the only one different from 0 z-component of displacement vector W (x; y) satisfies the Helmholtz equation. The inclusion is fully couple with the matrix. The tangential stresses are discontinuous on the inclusion with unknown jumps. The method of the solution is based on the representation of displacement W (x; y) by discontinuous solutions of the Helmholtz equation. After the satisfaction of the conditions on the inclusion the system of integral equations relatively unknown jumps is obtained. One of the main results is a numerical method for solving the obtained system, which takes into account the singularity of the solution and is based on the use of the special quadrature formulas for singular integrals.