Stress concentration near a thin elastic inclusion under interaction with harmonic waves in the case of smooth contact

We solve the problem on the interaction of plane harmonic waves with a thin elastic plate-shaped inclusion. The ambient medium is assumed to be in plane strain. The smooth contact conditions are satisfied on both sides of the inclusion. The bending displacements of the inclusion are determined from the corresponding differential equation. In the statement of boundary conditions for this equation, one should take into account the transverse forces and bending moments applied to the lateral edges of the inclusion, while the boundary conditions are posed on the midplane of the inclusion. Using the discontinuous solution method, we reduce the problem to a system of two singular integral equations, which are solved numerically by the mechanical quadrature method. We obtain approximate formulas for the stress intensity coefficients near the ends of the inclusion and for the transverse forces and moments applied to the inclusion.     

Interaction of plane harmonic waves with a thin elastic inclusion of zero flexural rigidity

We solve the problem on the interaction of plane elastic harmonic waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) that is under plane strain conditions. The normal forces applied by the medium to the inclusion side edges are taken into account. Because of the small thickness of the inclusion, we assume that its flexural rigidity is zero and that the shear displacements at any of its points coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself can be found from the corresponding equation of the theory of plates. The solution method consists in representing the displacements as discontinuous solutions of the Lamé equations and then determining the unknown jump from a singular integral equation. This equation is solved numerically by the collocation method, and formulas for the approximate calculation of the stress intensity factors near the inclusion ends are obtained.     

含孔von Karman板中非线性波散射与边值问题

基于ｖｏｎ Ｋａｒｍａｎ板大挠度弯曲理论,利用小参数摄动法,分析研究了含孔ｖｏｎＫａｒｍａｎ板的非线性波散射与动应力集中问题,其中一类可看成是薄板弯曲波动问题的控制方程。当有单频波入射时,由于弯曲应力与膜应力状态的非线性耦合,孔洞会产生高次谐波散射现象。建立了求解本问题的边界积分方程法,利用积分方程法交替求求这两类问题,最终可获得问题的近似分析解。     

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lamé equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived     

Interaction of a plane harmonic wave with a thin rigid inclusion of the shape of a cylindrical shell

The paper presents the solution of the problem of determining the stress state in an elastic matrix containing a rigid inclusion of the shape of a thin cylindrical shell. It is assumed that harmonic vibrations occur in the matrix under the conditions of axial symmetry (the symmetry axis is the inclusion axis) and the conditions of full adhesion between the inclusion and the matrix are satisfied. The vibrations are caused by the propagation of a plane wave whose front is perpendicular to the inclusion axis. The solution method is based on representing the displacements in the matrix as discontinuous solutions of the equations of axisymmetric oscillations of an elastic medium with unknown stress jumps on the inclusion surface. The realization of the boundary conditions for these jumps leads to a system of integral equations. Its solution is constructed numerically by the mechanical quadrature method with the use of special quadrature formulas for specific integrals. It is numerically investigated how the ratio of the inclusion geometric dimensions and the propagating wave frequency affect the stress concentration near the inclusion.     

Scattering of harmonic waves by a disk-shaped rigid inclusion in a three-dimensional elastic matrix

We consider a three-dimensional problem on the interaction of harmonic waves with a thin rigid movable inclusion in an infinite elastic body. The problem is reduced to solving a system of two-dimensional boundary integral equations of Helmholtz potential type for the stress jump functions on the opposite surfaces of the inclusion. We propose a boundary element method for solving the integral equations on the basis of the regularization of their weakly singular kernels. Using the asymptotic relations between the amplitude-frequency characteristics of the wave farzone field and the obtained boundary stress jump functions, we determine the amplitudes of the shear plane wave scattering by a circular disk-shaped inclusion for various directions of the wave incident on the inclusion and for a broad range of wave numbers.     

Diffraction of a sound wave by a slit

The diffraction of a sound wave by a slit in an unbounded plane is analyzed as an initial-boundary-value problem with a moving boundary for the two-dimensional wave equation. The initial-boundary-value problem is solved by the formation and inversion of Volterra integral equations. A solution is obtained in closed form in quadratures for an arbitrary angle of inclination of the incident wave front relative to the plane. The solution is presented in the form of recursion formulas, which take into account the influence of diffraction waves occurring in succession at the boundaries of the slit.     

On a modification of the wave equation for a layered medium

A new form of the wave equation in inhomogeneous media is presented which does not contain derivatives of the medium parameters in its coefficients. Hence this equation can be used not only for the case of smooth but also for the case of abrupt changes of the parameters with the coordinates. The equation can be used for waves of different nature.

To illustrate the advantages of the new form of the wave equation four problems have been solved. They are: scattering of a plane sound wave by weak inhomogeneities; excitation of a lateral wave; the symmetry of the plane-wave transmission coefficient with respect to inversion of the path of the wave; and plane were reflection from a thin inhomogeneous layer.     

Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion

We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lamé equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion. Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions.     

Interaction of plane elastic nonstationary waves with an elastic inclusion under complete adhesion

We solve the problem on the interaction of plane elastic nonstationary waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) which in under conditions of plane strain. It is assumed that the condition of perfect adhesion between the inclusion and the matrix is satisfied. Because of the small thickness of the inclusion we assume that the bending and shear displacements at any inclusion point coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself are found from the corresponding equations of the theory of plates. The statement of the boundary conditions for these equations takes into account the forces and moments acting on the inclusion edges from the matrix. The solution method is based on representing the displacements in the space of Laplace transforms as a discontinuous solution of the Lame’ equations for the plane strain with subsequent determining the transforms of the unknown jumps from integral equations. The passage to the original functions is performed numerically by methods based on replacement of the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors for the inclusion. These formulas are used to study the time dependence of the stress intensity factors and the influence of the inclusion rigidity on their values. We also study the possibility of treating inclusions of high rigidity as absolutely rigid inclusions.     

Electromagnetic diffraction by a thin conducting circular disk placed at the plane interface of two different media

In this paper, the problem of diffraction of time harmonic, electromagnetic waves by a thin ideally conducting disk lying at the plane interface of two different media is considered. In this analysis, the incident wave is a plane wave travelling in a direction perpendicular to the plane interface of the two media. A Hertz vector formulation is applied to reduce our electromagnetic diffraction problem to a system of two scalar problems which are solved by the help of two pairs of Fredholm integral equations of the second kind. Low frequency approximations to the tangential components of the magnetic intensity associated with the diffracted field at the surfaces of the disk, the induced surface current density on the disk and the scattering cross section are obtained.     

Scattering of flexural waves by a cracked Mindlin plate of soft ferromagnetic material in a uniform magnetic field

Consider the impingement of time harmonic flexural waves on a through crack in a soft ferromagnetic plate the surface of which is subjected to a uniform magnetic field at normal incidence. Mindlin's plate theory is used to account for the magneto-elastic interaction. For an incident wave that gives rise to moments symmetric about the crack plane, Fourier transforms are applied reducing the mixed boundary value problem to a Fredholm integral equation that can be solved numerically. The dynamic moment intensity factor versus frequency is computed to exhibit the influence of the magnetic field.     

Solving the elastic bending problem for a plate with mixed boundary conditions

The bending problem for an arbitrarily outlined thin plane with mixed boundary conditions is solved. A technique based on the methods of potentials and balancing loads is proposed for constructing Green’s function for the Germain-Lagrange equation. This technique ensures high accuracy of approximate solutions, which is checked against Levi’s solution for rectangular plates __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 104–112, May 2006.     

Contact Interaction of the Faces of a Rectangular Crack under Normally Incident Tension–Compression Waves

A three-dimensional problem on the contact interaction between the faces of a rectangular crack under a normally incident harmonic tension–compression wave is considered. The problem is solved by using the method of boundary integral equations and an iterative algorithm. The contact forces and the discontinuity in the displacement of the crack faces are studied. The results obtained are compared with those for a finite plane crack.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

Vertical vibrations of a rigid edge inclusion under a harmonic load

The paper considers the problem of vibrations of a rigid edge inclusion, which lies in an elastic half-plane and emerges on the surface perpendicular to that half-plane. The vibrations are initiated by a harmonic force acting on the end of the inclusion, which emerges on the surface. The field of translations in the half-plane is shown to be represented by the superposition of two discontinuous solutions with discontinuities at the boundary between the half-plane and the line of the inclusion. The unknown discontinuities are determined from the boundary conditions and the conditions of the inclusion-medium interaction. The problem is thus reduced to one of solving a singular integral equation with an immobile singularity for the jump in shear stresses on the line of the inclusion. The equation obtained is solved numerically by the method of mechanical quadratures. The amplitudes of the inclusion vibrations and the stressed state of the medium near it are studied.Odessa State Marine Academy, Odessa, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 46–55, July, 1995.     

二维直角平面内固定圆形夹杂对稳态入射反平面剪切波的散射

利用复变函数法、多极坐标及傅立叶级数展开技术求解了二维直角平面内固定圆形夹杂对稳态入射反平面剪切（shearing horizontal, SH）波的散射问题。首先构造出介质内不存在夹杂时的入射波场和反射波场,然后建立介质内存在夹杂时由夹杂边界产生的能够自动满足直角边应力自由条件的散射波解，从而利用叠加原理写出介质内的总波场。利用夹杂边界处位移条件和傅立叶级数展开方法列出求解散射波中未知系数的无穷代数方程组，在满足计算精度的前提下通过有限项截断，得到相应有限代数方程组的解，最后通过算例具体讨论了二维直角平面水平边界点的位移幅度比和相位随量纲一波数、入射波入射角及夹杂位置的不同而变化的情况，结果表明了算法的有效实用性。     

Interaction of plane nonstationary waves with a thin elastic inclusion under smooth contact conditions

We solve the problem of determining the stress state near a thin elastic inclusion in the form of a strip of finite width in an unbounded elastic body (matrix) with plane nonstationary waves propagating through it and with the forces exerted by the ambient medium taken into account. We assume that the matrix is in the plane strain state, and the smooth contact conditions are realized on both sides of the inclusion. The method for solving this problem consists in using the integral Laplace transform with respect to time and in representing the stress and displacement images in terms of the discontinuous solution of Lamé equations in the case of plane strain. As a result, the initial problem is reduced to a system of singular integral equations for the transforms of the unknown stress and displacement jumps. To invert the Laplace transform, we use a numerical method based on replacing the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors (SIF) for the inclusion, which are used to study the SIF time-dependence and its influence on the values of the inclusion rigidity. We also studied the possibility of considering the inclusions of higher rigidity as absolutely rigid inclusions.     

Interaction of general plane P wave and cylindrical inclusion partially debonded from its viscoelastic matrix

The interaction of a general plane P wave and an elastic cylindrical inclusion of infinite length partially debonded from its surrounding viscoelastic matrix of infinite extension is investigated. The debonded region is modeled as an arc-shaped interface crack between inclusion and matrix with non-contacting faces. With wave functions expansion and singular integral equation technique, the interaction problem is reduced to a set of simultaneous singular integral equations of crack dislocation density function. By analysis of the fundamental solution of the singular integral equation, it is found that dynamic stress field at the crack tip is oscillatory singular, which is related to the frequency of incident wave. The singular integral equations are solved numerically, and the crack open displacement and dynamic stress intensity factor are evaluated for various incident angles and frequencies. The project supported by the National Natural Science Foundation of China (19872002) and Climbing Foundation of Northern Jiaotong University     

Sound propagation in a plane wave guide with an elastic wall section

Problems of acoustic wave propagation in a plane wave guide whose walls are assumed to be undeformed with the exception of a section of finite length whose bending is described by the thin plate theory equations in the framework of the Kirchhoff-Love hypotheses are considered. The sound-proofing characteristics of the wave guide described and the stability of the forced oscillations of the system considered are investigated. Formulations of the problem of active vibroacoustic protection and the problem for the peristaltic pump are given. Soundproofing in wave guides has been considered in a number of papers, a fairly complete review of which is given in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 1, pp. 132–139, January–February, 1986.