Impact of a long thin body on a cylindrical cavity in liquid: A plane problem

An approach is developed to the investigation of the shock interaction between a long thin cylindrical body and a cylindrical cavity in an infinite compressible perfect liquid. This process accompanies the supercavitation of the body. Three typical cases of cross-sectional dimensions of the body and the cavity are examined. For each case, a mixed nonstationary boundary-value problem with an unknown moving boundary is formulated. The unknown quantities are expanded into Fourier series. An auxiliary problem is solved using the Laplace transform to establish the relationship between the pressure and the velocity on the cavity surface. As a result, the problem is reduced to an infinite system of Volterra equations of the second kind solved simultaneously with the equation of transverse motion and the equation of the contact boundary. An asymptotic solution valid at the initial stage of interaction is obtained for all the three cases, and a numerical solution is found for the most typical case __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 32–53, June 2006.     

Nonstationary Plane Elastic Contact Problem for Matched Cylindrical Surfaces

An approach is proposed to study a collision of a long cylinder with the inside surface of a circular cylindrical cavity in an elastic medium. The problem is solved in plane formulation. A nonstationary mixed initial–boundary-value problem with unknown boundaries moving with a variable velocity is formulated and then reduced to an infinite system of Volterra integral equations of the second kind or, in a simplified formulation, to a sequence of Volterra integral equations. The penetration velocity is determined as a function of the cylinder mass and initial conditions. It is established that the reaction force peaks instantaneously and then dies out     

Impact of a spherical rigid body on the surface of a cavity in a compressible liquid: An axisymmetric problem

The shock interaction of a spherical rigid body with a spherical cavity is studied. This nonstationary mixed boundary-value problem with an unknown boundary is reduced to an infinite system of linear Volterra equations of the second kind and the differential equation of motion of the body. The hydrodynamic and kinematic characteristics of the process are obtained __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 11–19, January 2008.     

Shrink fit of an elastic layer having a cylindrical cavity

Summary This paper deals with the problem of determining the stress distribution in an elastic layer with a cylindrical cavity when the mixed boundary conditions are prescribed on the curved surface of the cylinder. The problem is simplified to that of finding the solution of dual integral equations arising from the mixed boundary conditions. These dual integral equations are subsequently reduced to a singular integral equation. The solution of this integral equation is obtained numerically, and the quantities of physical interest are calculated.     

Diffraction of a plane acoustic wave by a rigid sphere in a cylindrical cavity: An axisymmetric problem

The paper studies the interaction of a rigid spherical body and a cylindrical cavity filled with an ideal compressible fluid in which a plane acoustic wave of unit amplitude propagates. The solution is based on the possibility of transforming partial solutions of the Helmholtz equation between cylindrical and spherical coordinates. Satisfying the interface conditions between the cavity and the acoustic medium and the boundary conditions on the spherical surface yields an infinite system of algebraic equations with indefinite integrals of cylindrical functions as coefficients. This system of equations is solved by reduction. The behavior of the system is studied depending on the frequency of the plane wave     

Plane Collision Problem for Two Identical Elastic Parabolic Bodies—Direct Central Impact

A direct central collision of two identical infinite cylindrical bodies is studied. A nonstationary plane elastic problem is solved. The variable boundary of the contact area is determined. A mixed boundary problem is formulated. Its solution is represented by Fourier series. An infinite system of Volterra equations of the second kind for the unknown expansion coefficients is derived by satisfying boundary conditions. The basic characteristics of the collision process are determined numerically depending on the curvature of the frontal surface of the bodies     

Integral formulation for a circular cylindrical cavity in infinite solid and a finite length coaxial cylindrical crack compressed axially

A method for solving problems of fracture of an infinite solid with a circular cylindrical cavity and a coaxial cylindrical crack near the surface under an uniform axial compression is proposed using a non-classical criterial approach associated with a mechanism of a local stability loss near the defect. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a system of linear algebraic equations with respect to the contraction parameter.     

The plane asymmetric problem of immersion of a rigid circular cylinder in a fluid

The plane asymmetric problem of vertical impact of a rigid body against the surface of a compressible fluid is considered. It is assumed that the penetrating circular cylinder has a shifted center of mass. Solution of the boundary-value problem is reduced to solution of an infinite system of linear integral Volterra equations of the second kind. We present an analysis of the results as a function of initial values of the angle of asymmetry. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Ukranian Transportation Institute, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 61–70, August, 1999.     

Stress singularities for three-dimensional corners using the boundary integral equation method

The boundary integral equation method is developed to study three-dimensional asymptotic singular stress fields at vertices of a pyramidal notch or inclusion in an isotropic elastic space. Two-dimensional boundary integral equations are used for the infinite body with pyramidal notches and inclusions when either stresses or displacements are specified on its surface. Applying the Mellin integral transformation reduces the problem to one-dimensional singular integral equations over a closed, piece-wise smooth line. Using quadrature formulas for regular and singular integrals with Hilbert and logarithmic kernels, these integral equations are reduced to a homogeneous system of linear algebraic equations. Setting its determinant to zero provides a characteristic equation for the determination of the stress singularity power. Numerical results are obtained and compared against known eigenvalues from the literature for an infinite region with a conical notch or inclusion, for a Fichera vertex, and for a half-space with a wedge-shaped notch or inclusion.     

Impact of a solid on a liquid surface: Asymmetric plane case

The vertical impact of a rigid elliptical cylinder on a compressible liquid surface is considered in a plane asymmetric formulation. Solution of the boundary problem reduces to solving an infinite system of linear Volterra integral equations of the second kind. The results are analyzed in terms of the initial asymmetry angle. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 75–84, September, 1999.     

Unsteady flow over a rough bottom

The unsteady motion of an ideal incompressible fluid with a free surface, developing from a state of rest, is considered. The flow is assumed to be irrotational, continuous and two-dimensional; it may be the result either of an initial disturbance of the free boundary or of a given boundary pressure distribution. The rigid boundaries of the flow region are fixed, and the free surface does not cross them at any time during the motion. The fluid is located in a uniform gravity force field and there is no surface tension. A method which in the case of localized roughness of the bottom makes it possible to find the shape of the free surface at any moment of time with predetermined accuracy is proposed. The method involves reducing the initial linear problem to a Volterra integral equation of the second kind, the kernel of this equation being a nonlocal operator. This operator has a smoothing effect, which makes it possible to reduce the solution of the initial problem to the solution of an infinite, perfect lyregular system of Volterra integral equations for a denumerable set of auxiliary functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–119, November–December, 1989.The author is grateful to I. V. Sturova and B. E. Protopopov for useful discussions and criticism.     

多洞室对矢量波散射引起半空间表面位移的边界元解

采用边界元方法研究了半空间中近表面多洞室对矢量波的散射问题,给出了以全空间格林函数为基本解且半空间表面离散的边界积分方程,在这一边界各分方程中,较好的消除了主值积分,在半空间表面进行离散时,采用无限单元与有限单元相结合的方法,大大减少了计算量,提出了精度。     

Stress intensity factors for a moving cylindrical crack in a nonhomogeneous cylindrical layer in composite materials

Stresses are determined for a finite cylindrical crack that is propagating with a constant velocity in a nonhomogeneous cylindrical elastic layer, sandwiched between an infinite elastic medium and a circular elastic cylinder made from another material. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. An internal gas pressure is then applied to the crack surfaces. The solution is derived by dividing the nonhomogeneous interfacial layer into several homogeneous cylindrical layers with different material properties. The boundary conditions are reduced to two pairs of dual integral equations. These equations are solved by expanding the differences in the crack surface displacements into a series of functions that are equal to zero outside the crack. The Schmidt method is then used to solve for the unknown coefficients in the series. Numerical calculations for the stress intensity factors were performed for speeds and composite material combinations.     

The plane problem on vertical impact of a rotating circular cylindrical shell against the surface of a liquid

The plane nonsymmetric problem on impact against and immersion into a compressible fluid of a thin electic cylindrical shell is considered. The shell rotates about its axis with a given angular velocity. The boundary-value problem is reduced to an infinite system of integral Volterra equations of the second kind. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 103–113, May, 2000.     

Wave processes in a perfect liquid layer impacted on by a blunted solid

The problem of impact of a smooth blunt solid upon a compressible-liquid layer of finite depth is addressed. A mixed initial-boundary-value problem with an unknown moving boundary is formulated. In the general case, the problem is reduced to an infinite system of integral Volterra equations of the second kind. It is solved numerically, using quadrature formulas and truncation method. An exact analytic solution to the problem is obtained in the special case where the body moves with a constant velocity at the initial stage of submergence. This solution makes it possible to examine the effect of successive wave reflections on the pressure at the frontal point and inside the layer __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 37–47, March 2007.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Relation Between Non-Steady Supply Pressure and Flux for a Double Packer Conductivity Meter: An Approximate Analytical Solution

Measurement of the hydraulic coefficients of soil or rock is carried out with the aid of double packer conductivity meters in wells or boreholes. The evaluation of tests is based on the theory of creeping flow through a porous medium. The theory leads to a non-steady cylindrically and axially symmetric heat conduction problem in an infinity domain outside a circular hole with mixed boundary conditions. With the aid of a simple but efficient approximation, which describes the singularity of the boundary value, two of the three integrations in the solution can be evaluated in closed form. The analysis shows that the problem can be reduced to a simple Volterra integral equation, which is easily solved with numerical means. The solution is characterized by one single parameter, namely the aspect ratio of the cylindrical inner cavity.     

Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials

Axisymmetric stresses around a cylindrical crack in an interfacial cylindrical layer between an infinite elastic medium with a cylindrical cavity and a circular elastic cylinder made of another material have been determined. The material constants of the layer vary continuously from those of the infinite medium to those of the cylinder. Tension surrounding the cylinder and perpendicular to the axis of the cylinder is applied to the composite materials. To solve this problem, the interfacial layer is divided into several layers with different material properties. The boundary conditions are reduced to dual integral equations. The differences in the crack faces are expanded in a series so as to satisfy the conditions outside the crack. The unknown coefficients in the series are solved using the conditions inside the crack. Numerical calculations are performed for several thicknesses of the interfacial layer. Using these numerical results, the stress intensity factors are evaluated for infinitesimal thickness of the layer.     

Collision of dissimilar blunt elastic bodies: a plane problem

The paper deals with a direct central impact of two infinite cylindrical bodies having differently shaped cross sections and made of different materials. A nonstationary plane problem of elasticity is solved. The contact boundary is moving and determined during the solution. A mixed boundary-value problem is formulated. Its solution has the form of Fourier series. Satisfying mixed boundary conditions gives an infinite system of Volterra equations of the second kind for the unknown coefficients of the series. The basic characteristics of the impact process and their dependence on the physical and mechanical properties of the bodies are determined numerically Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 36–45, February 2009.     

Development of a computer program for the prediction of flow in a rotating cavity

A computer program has been developed to predict laminar source-sink flow in a rotating cylindrical cavity. Although the program is based on a standard finite difference technique for recirculating flow, it incorporates two novel features. Step changes in grid size are employed to obtain sufficient resolution in the boundary layers and special treatment is given to the solution of the pressure correction equations, in the ‘SIMPLE’ algorithm, in order to improve the convergence properties of the method. Results are presented both for the flow in an infinite rotating cylindrical annulus and a finite rotating cylindrical cavity, with the inner cylindrical surface acting as a uniform source and the outer cylinder as a sink. These show good agreement with existing analytical solutions and illustrate some of the problems associated with the computation of rapidly rotating flows.