Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Material under Triaxial Tension

The interaction of an elastic ellipsoidal inclusion with an elliptic crack in an infinite elastic medium under triaxial loading is analyzed. The stress state in the elastic space is represented as a superposition of the principal state and perturbed states, which are due to the presence and interaction of the inclusion and the crack. The analytical solution of the problem is found using the method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for an elliptic crack. The effect of triaxial loading on the stress intensity factors is analyzed     

The Action of an Arbitrary Load on the Surface of an Orthotropic Half-Space

The exact solution to the first boundary-value problem for a half-space is constructed on the basis of the general solution of the equilibrium equations for an orthotropic medium (nine elastic constants). The stress–strain state of an orthotropic half-space whose surface is under an arbitrarily applied concentrated force is described as an example. The well-known solution for the isotropic case is obtained by the same scheme, which confirms the reliability of the result.     

Stability of an infinite solid with a circular cylindrical crack under compression using the Treloar potential

The fracture stability of a circular cylindrical crack in an infinite incompressible solid subjected to an axial compression is considered. A state of subcritical initial strain is assumed. The failure criterion is based on the local stability loss. The investigation is carried out in a single form for the hyper-elastic bodies with an arbitrary type of an elastic potential. Critical loads are determined for axisymmetric forms of a stability loss in the region local to the crack. The linearized problem reduced to the eigenvalue problem is solved numerically. Numerical results are obtained for solids with Treloar potential.     

A partially implicit method for simulating viscous aerofoil flows

A partially implicit method for the unsteady compressible Navier-Stokes equations is developed. The method is based on an explicit treatment of streamwise fluxes and an implicit treatment of normal fluxes. This leads to a linear system which is generated by an efficient finite difference procedure and which is block pentadiagonal. The method is tested on a shock-induced oscillatory flow over an aerofoil. Parallel implementations of an explicit, fully implicit and partially implicit method are investigated.     

Analytical solution in the problem of constructing axisymmetric noses with minimum wave drag

An analytical procedure for determining the axisymmetric nose shapes that ensure minimum wave drag is developed. The solution is constructed as an improving variation on the conical shape. The target function is built up in an approximate form on the basis of the assumption that the relationship between the geometric and gasdynamic parameters is local in nature. It is shown that the optimal bodies are truncated power-law bodies with an exponent equal to 2/3. The bodies thus obtained are compared with those constructed in accordance with an exact formulation of the problem.     

Global bifurcations and chaos in a harmonically excited and undamped circular plate

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The method of multiple scales is used to obtain an autonomous system from a non-autonomous system of ordinary differential equations governing non-linear oscillations of an imperfect circular plate. The Melnikov's method for heteroclinic orbits of the autonomous system is used to obtain the criteria for chaotic motion. It is shown that the existence of heteroclinic orbits in the unperturbed system implies chaos arising from breaking of heteroclinic orbits under perturbation. The validity of the result is checked numerically. It is also observed numerically that chaos can appear due to breaking of invariant tori under perturbation.     

Non-linear argumental oscillators: Stability criterion and approximate implicit analytic solution

The behaviour of a space-modulated, so-called “argumental” oscillator, is studied. The oscillator is submitted to an external harmonic force, which is amplitude-modulated by the oscillator's position in space. An analytic expression of a stability criterion is given. Using the averaging method, an integrating factor and a Van der Pol representation in the (amplitude, phase)-space, an exact implicit analytic solution is given when there is no damping, and an approximate implicit analytic solution is given when there is damping, allowing the plotting of the separatrix curve. An attractor is identified.     

Example of motion of a cylindrical solid in a viscous liquid

The problem of motion of a pulsating solid (an infinitely long circular cylinder) in an oscillating viscous liquid in the presence (or absence) of an external stationary force is considered. The perturbation method is applied. It is found that the solution of the time-average motion of a body exists if and only if body pulsations, liquid vibrations, and external forces satisfy a certain relation. The presence of a plane analog of the phenomenon of predominantly unidirectional motion of a compressible solid in an oscillating liquid is established.     

Experimental Investigation of the Interaction between an Electric Arc and a Gas Flow

The results of investigating the shape of an electric arc carrying an electric current of less than 100 A and exposed to an air, nitrogen, or argon stream at a stream velocity of from zero to 20 m/s, when the interelectrode gap is less than 20 mm, are reported. The shape of the arc is qualitatively investigated as a function of the gas type, blowing velocity, and interelectrode gap. It is shown that an arc burning in a channel with insulated walls is much more stable under these conditions than an arc with the same parameters burning in free space.     

Ultrasonic scattering by a side-drilled hole

The scattering of elastic waves by a side-drilled hole (sdh) i.e. a circular cylindrical cavity, is considered. The scattering of plane or cylindrical waves by an sdh is an old subject; here the T matrix solution is adopted. The elastic waves are excited by an ultrasonic probe and a model of such a probe is thus used. The waves from the probes are expressed as a Fourier transform, i.e. as a superposition of plane waves. These plane waves are then transformed to the cylindrical system centred at the sdh. To obtain the signal in a receiving ultrasonic probe an electromechanical reciprocity relation is used. The signal response is obtained as a double wavenumber integral and an azimuthal summation. In the far field the integrals can be calculated approximately by the stationary phase approximation. Some numerical examples are given, in particular concentrating on when this approximation is valid.     

流体可压缩性对Rayleigh-Taylor不稳定性影响机理分析

在等熵方程为压力是密度的任意单值函数形式情况下 ,分析了R T(Rayleigh Taylor)不稳定性中流体可压缩性的作用。在没有边界效应的条件下所作的分析表明 :在重力场作用下流体可压缩性形成的密度分布是R T不稳定性中的致稳因素 ,而扰动流体的膨胀 (收缩 )效应助长R T不稳定性的发展 ;上层重流体的可压缩性是稳定因素 ,而下层轻流体的可压缩性是失稳因素。从扰动发展驱动力和扰动带动的等效质量两个方面对该结论的物理机制进行了分析。     

The electromagnetic torque on axially symmetric rotating metal cylinders and spheres

The torque is calculated on an electrically resistive rotating cylinder and sphere through which a current is fed by means of sliding contacts. The torque on a rotating cylinder is proven to vanish identically for arbitrary angular velocity. The torque on the sphere is shown to vanish up to second order in an expansion with the angular velocity as the expansion parameter. The nonzero torque in first order found by Gruenberg is shown to be due to an algebraic error.The author wrote this article in honour of Bert Broer, who he considers an exponent of classical physics viewed from a modern and general point of view. Bert would undoubtedly have written on the present subject in a broader sense.     

Contribution to the determination of the global minimum time for the brachistochronic motion of a holonomic mechanical system

The problem of the brachistochronic motion of a holonomic scleronomic mechanical system is analyzed. The system moves in an arbitrary field of known potential forces. The problem is formulated as an optimal control task, where generalized speeds are taken as control variables. The problem considered is reduced to solving the corresponding two-point boundary-value problem (TPBVP). In order to determine the global minimal solution of the TPBVP, an appropriate numerical procedure based on the shooting method is presented. The global minimal solution represents the solution with the minimum time of motion. The procedure is illustrated by an example of determining the brachistochronic motion of a disk that performs plane motion in a vertical plane in a homogeneous field of gravity.     

Oscillations of a cylindrical body submerged in a fluid with ice cover

The linear plane problem of oscillations of an elliptic cylinder in an ideal incompressible fluid of finite depth in the presence of an ice cover of finite length is solved. The ice cover is modeled by an elastic plate of constant thickness. The hydrodynamic loads acting on the body are determined as functions of the oscillation frequency and the positions of the cylinder and plate.     

An internal crack problem for an infinite elastic layer

The elastostatic plane problem of an infinite elastic layer with an internal crack is considered. The elastic layer is subjected to two different loadings, (a) the elastic layer is loaded by a symmetric transverse pair of compressive concentrated forces P/2, (b) it is loaded by a symmetric transverse pair of tensile concentrated forces P/2. The crack is opened by an uniform internal pressure p ₀ along its surface and located halfway between and parallel to the surfaces of the elastic layer. It is assumed that the effect of the gravity force is neglected. Using an appropriate integral transform technique, the mixed boundary value problem is reduced to a singular integral equation. The singular integral equation is solved numerically by making use of an appropriate Gauss–Chebyshev integration formula and the stress-intensity factors and the crack opening displacements are determined according to two different loading cases for various dimensionless quantities.     

The edge dislocation in a three-quarter plane. Part I: Influence functions

The problem of an edge dislocation, having arbitrary Burgers vector, and lying on one of the projection lines of a three-quarter plane is studied. The complete internal state of stress is found, with particular attention focused on the correct asymptotic behaviour. The solution is found in a form which makes it particularly suitable as the kernel of an integral equation formulation for an interface crack.     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

Theory of rolling: Solution of the Coulomb problem

A theory of rolling of round bodies in the normal mode with adhesion conditions satisfied on the entire contact area is proposed. This theory refines the classical Coulomb’s theory of rolling in which the rolling moment is directly proportional to the pressing force (e.g., the weight of the rolling body). The rolling moment of cylinders is found to be directly proportional to the pressing force raised to a power of 3/2, and the rolling moment of balls and tori is proportional to the pressing force raised to a power of 4/3. It is shown that the normal mode of uniform rolling can only be provided for a certain ratio of the elastic constants of the materials of the round body and the base forming an ideal pair. The Coulomb problem is solved for the cases of rolling of an elastic cylinder over an elastic half-space, of an elastic ball over an elastic half-space, of an elastic torus over an elastic half-space, and of a cylinder and ball over a tightly stretched membrane. The rolling law is derived for such cases. The rolling friction coefficients, the rolling moment, and the rolling friction force are calculated.     

Estimation of thermal stresses in the region with an eccentric hole by means of an analogy

The analogy used to solve some thermoelastic problems is extended to include a region with an eccentric hole. The relation is shown to be a little more complicated in this case than in the one for the concentric hole. Three bending tests of an analogous thin plate are required. As an experimental proof, an eccentric circular region is considered and mental proof, an eccentric circular region is considered and the analogous thin plate is prepared. The heat-conduction equation is first solved to give the required temperature terms. The measured strains on the plate and the coefficients specified from the transverse loads and moments applied to the plate are combined effectively to get the estimated thermal stress. The experimentally estimated thermal stress of the region and the numerical result are shown to be in agreement.Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

Closed-form solutions for the nonlinear problem of an air-inflated,heavy membrane tube rolling down an incline

This study considers an air-inflated, heavy membrane tube section rolling down an incline without slipping. This problem is modeled as four simultaneous nonlinear ordinary equations with an undetermined boundary condition. The boundary condition at the upper separation point is carefully examined. Through an analytic approach, the general solutions for shape and tension are expressed in closed form. The unknown acceleration of the tube section is determined by imposing the boundary condition. The effect of the velocity is reflected in the so-called apparent tension. In addition, the acceleration results in an exponential decay of the apparent tension along the arc length. Some specific cases for shape and apparent tension are presented and discussed.