Some anomalies of the curved bar problem in classical elasticity

The classical formulation of the homogeneous problem of a curved bar loaded only by and end force involves the assumption of an appropriate stress function with four arbitrary constants and the determination of these constants from the boundary conditions. Since there are five boundary conditions, four on the curved edge and one at the end, the solution is only possible because the coefficient matrix of the resulting algebraic equations is singular. This in turn means that certain inhomogeneous problems in which the curved edges are loaded by sinusoidally varying tractions cannot be solved using apparently appropriate stress functions.Additional stress functions which resolve this difficulty are introduced and an example problem is solved, which exhibits qualitatively different behavior from that in more general cases of loading. The problem is then reconsidered as a limiting case of sinusoidal loading of arbitrary wavelength. It is shown that the solution of the latter problem appears to become unbounded as the special case is approached, but that when the end conditions have been correctly satisfied by superposing an appropriate multiple of the end-loaded solution, the limit can be approached regularly and the correct special solution is recovered. The limiting process reveals a general procedure for determining the additional stress functions required for the special case.Similar relationships between homogeneous and inhomogeneous solutions for other common geometries are discussed.     

Low-velocity viscous incompressible fluid flow around a hollow porous sphere

The problem of a viscous incompressible fluid flow around a hollow porous sphere in the Stokes approximation, in which the filtration flow through the sphere shell obeys the Darcy law, is solved. The force acting on the sphere from the fluid is calculated. The limiting cases are considered. The stream function is constructed.     

Mixed boundary value problems for a penny-shaped cut

The axially symmetric problem for a penny-shaped cut is solved in the case the displacements are prescribed on its upper surface and stresses on its lower surface. The solution is achieved by using integral transforms along with certain representations that reduce the general problem to the solution of uncoupled Hilbert problems. For polynomial loadings the Hilbert problems can be solved exactly by methods given by Muskhelishvili. As in the related plane problem two singularities at the edge of the disc are noted in the solution.     

Edge crack opening in an elastic plane with a wedge-like cut in contact with a punch

The equilibrium of an elastic plane with a wedge-like cut and an internal or edge crack on the symmetry axis was studied in [1] in the case of punch indentation in the lateral faces of the cut at a distance from the cut tip. In [1], the systemof singular integral equations of the problemwas solved numerically by the mechanical quadrature method. In this paper, the generalized Wiener-Hopf method [2] is used to obtain the analytic solution of a similar problem in the case of an edge crack under punch pressure on parts of the cut lateral faces adjacent to the cut tip. Some special cases of this problem were considered earlier without a crack [3, 4] or a punch [5, 6].     

Simulation of viscous fluid flow with consideration of boundary outflow and pressure drop

The problem of modeling a viscous fluid flow over the surface of a plate is considered when the pressure changes along the longitudinal coordinate according to a linear law. The corresponding boundary conditions are formulated for this problem. The Navier-Stokes equations are solved exactly in the problem of flow past the plate for the case of fluid outflow and a longitudinal pressure drop. Several formulas to determine the velocity profile are derived. The limiting cases are analyzed to study the consistency of various models. The corresponding pressure conditions are proposed for the case when the Navier-Stokes system has a known exact solution.     

On a pair of axisymmetric forces inside a sphere

In this paper, the stresses in a sphere under a pair of concentrated forces acting axisymmetrically inside the sphere are solved by using a Love's stress function. The method of solution is also applied to solve the problem of a pair of ring forces acting axisymmetrically inside the sphere. In either problem, the solution covers the limiting case in which the pair of forces or ring forces is acting on the sphere.     

Motion of a slender body near the free surface of a compressible fluid

An analytic solution to the problem of motion of a slender rigid body in a semi-infinite domain of a compressible fluid is obtained for the case when the body moves in parallel to the free surface at a constant velocity. This problem is similar to the problem of motion of a hydrofoil ship whose wing-like device allows it to lift its hull above the water surface and to decrease the friction and drag forces limiting the speed of usual ships. During its motion in water, a hydrofoil produces a lift force. The obtained analytic solution allows one to derive explicit expressions for the drag force and for the lift force in the limiting cases of relatively small and large depths. When depth is small, the drag force is greater than that in an infinite medium, since the wave drag is additionally evolved. When the velocity increases and approaches the sound velocity, the forces exerted on the body increase without limit, which is typical for a linear formulation of the problem.     

Forces on oscillating bodies in viscous fluid

This paper describes a method for determining the fluid forces on oscillating bodies in viscous fluid when the corresponding flow problem has been solved using the finite element method. These forces are characterized by the concept of added mass, added damping and added force. Numerical results are obtained for several example body shapes. Comparison is made with exact analytical results and other finite element results for the limiting cases of Stoke's flow and inviscid flow, and good agreement is obtained. The results for finite values of the body amplitude parameter β show the appearance of added force from the steady streaming component of the flow for asymmetric bodies. Results are also obtained for the associated flow where the fluid remote from a fixed body is oscillating.     

Random vibrations of an isochronous SDOF bilinear system

A single-degree-of-freedom (SDOF) system with bilinear restoring force and damping characteristics is excited by a white-noise force. For the case of small damping the asymptotic method of quasi-conservative stochastic averaging is applied, which reduces the problem to that of a single Ito stochastic differential equation (SDE) for total energy of the response. A special case of the isochronous system is considered, where the static equilibrium position coincides with the slope discontinuity point of the system's characteristics, such as that of a moored body without slack and pretension in the mooring cable. For this case an analytical solution is obtained for autocorrelation function of the response energy. For the limiting case of a vibroimpact system the results are shown to agree with the exact solution.     

Elastic plane with a physical cut loaded by an antisymmetric system of forces

The problem of determining the stress-strain state of an elastic plane with a physical cut loaded by an antisymmetric system of forces is posed and solved under the condition that the tangential stresses are homogeneous across the thickness of the layer continuing the cut.     

Waves on the Surface of an Ideal Incompressible Heavy Fluid under Wind Loads

The gravity-forced motion of an ideal incompressible fluid of infinite depth is studied when a periodic pressure is applied to the surface of the fluid. This problem is solved on the basis of the small amplitude wave theory. The analytical solutions for the velocity potential, the velocity field, and the shape of the free surface are found. An expression for the horizontal force is obtained in the case of a traveling wave.     

Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of buoyancy force and thermal radiation

In this paper an analysis has been made to study heat and mass transfer in two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in the presence of buoyancy force and thermal radiation. The similarity solution is used to transform the problem under consideration into a boundary value problem of nonlinear coupled ordinary differential equations containing Prandtl number, Schmidt number and Sherwood number which are solved numerically with appropriate boundary conditions for various values of the dimensionless parameters. Comparison of the present numerical results are found to be in excellent with the earlier published results under limiting cases. The effects of various physical parameters on the boundary layer velocity, temperature and concentration profiles are discussed in detail for both the cases of assisting and opposing flows. The computed values of the skin friction coefficient, local Nusselt number and Sherwood number are discussed for various values of physical parameters. The tabulated results show that the effect of radiation is to increase skin friction coefficient, local Nusselt number and Sherwood number.     

The biaxial tension of an elastoplastic space with a prismatic inclusion

The problem on the elastoplastic state of a thick isotropic plate (the case of plane deformation) is solved. A larger prismatic inclusion is made a close fit in a polygonal hole in the plate. The plate is stretched at infinity by constant mutually perpendicular forces. The problem is solved by the small-parameter method and by the theory of ideal plasticity. The axisymmetric state of the plane with a circular hole stiffened by a round ring with a constant force applied to its inner contour is considered as a zero approximation. Some specific shapes of the hole and reinforcing elastic rigid ring are considered. Voronezh University, Russia. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 114–120, June, 2000.     

On moving heat sources

The two-dimensional thermal problem due to relative motion of a medium and a suddenly activated circular heat source is solved for several boundary conditions. The solutions can be interpreted as for a moving heat source in a stationary medium or a medium moving past a stationary heat source. Uniform and non-uniform temperature, and uniform and non-uniform heat flux boundary conditions are considered. The effect of velocity and radial direction on the temperature distribution is examined. Average, steady-state Nusselt numbers are derived. The transient response of a continuous line source is obtained as a limiting case of the prescribed heat flux solution. Received on 24 September 1996     

Cavitation bubble collapse in nonlinear viscous and viscoplastic media

The problem of the collapse of a bubble in nonlinear viscous and viscoplastic media under the influence of pressure at infinity is solved numerically. The pressure at which the radius of the bubble begins to decrease, the limiting radius of the bubble in the viscoplastic case and the critical collapse pressure where there is no plastic component are found. The critical pressure is found to be an order smaller than the corresponding value for a Newtonian viscous fluid. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 181–184, March–April, 1994.     

Shapes of a rotating rheonomic rod

The case of a rotating fluid mass is one of the classical fields of mechanics [1]. In particular, the solution of creep problems for a rotating mass is actual in geophysics in connection with Earth gravity force simulation on rotating samples under laboratory conditions [2]. A special case of a rheonomic rod in a potential field was studied in [3], where it was shown that the main problem about the rod shapes is the problem of determining the relations between the Lagrangian and Euler coordinates in the creep process.In what follows, we show how this problem can be solved for a rotating rod.     

The Hanging Rope of Minimum Elongation for a Nonlinear Stress-Strain Relation

We consider the problem of determining the shape that minimizes the elongation of a rope that hangs vertically under its own weight and an applied force, subject to either a constraint of fixed total mass or fixed total volume. The constitutive function for the rope is given by a nonlinear stress-strain relation and the mass-density function of the rope can be variable. For the case of fixed total mass we show that the problem can be explicitly solved in terms of the mass density function, applied force, and constitutive function. In the special case where the mass-density function is constant, we show that the optimal cross-sectional area of the rope is as that for a linear stress-strain relation (Hooke's Law). For the total fixed volume problem, we use the implicit function theorem to show the existence of a branch of solutions depending on the parameter representing the acceleration of gravity. This local branch of solutions is extended globally using degree theoretic techniques.     

Two contact problems for a wedge with a symmetric cut on the edge

In this paper, the solutions of two contact problems for a wedge with a symmetric cut on the edge are presented. First, special approximation methods and orthogonal polynomials are used to solve the auxiliary problem on the action of a lumped force on the cut edge. The obtained solutions are compared with known solutions in some special cases.     

The Stress State of a Transversally Isotropic Piezoceramic Body with a Parabolic Crack in a Uniform Heat Flow

The explicit solution is constructed for a static thermoelastic problem for an infinite transversally isotropic piezoceramic body containing a heat-insulated parabolic crack in the isotropy plane. The crack surface is assumed free of forces. The body is under a uniform heat flow, which is perpendicular to the crack surface and is far from the crack itself. The problem is solved for two cases of electric conditions on the crack surface. In the first case, an electric potential is absent on the crack surface and, in the second case, the normal component of the electric-displacement vector is equal to zero. The intensity factors, which depend on the heat flow, crack geometry, and the thermoelectroelastic properties of the piezoceramic body, are determined for the force field and electric displacement near the crack tip     

三维电动力绳系子卫星轨道转移的最优控制

考虑复杂状态和控制约束的作用,研究了倾斜轨道上三维电动力绳系子卫星轨道转移的最优控制问题.借助Gauss伪谱算法,将绳系子卫星轨道转移的连续时间最优控制问题离散为大规模动态规划问题,并利用非线性规划方法进行求解.通过数值仿真计算了最优控制时间、子星最优转移轨道及最优控制张力和电流,同时讨论了轨道倾角对最优控制量的影响....