Optimal Plastic Design of a Cross Section under Torsion with Small Bending∗

A doubly connected cross section of a bar under torsion with bending is optimized in two variants; either the internal contour is given and the external one is subject to optimization, or only the inside area is given and both contours are optimized. Minimal cross-sectional area is the design objective. The material is assumed to be perfectly plastic, incompressible, and subject to the Huber-Mises-Hencky yield condition. A specially adopted perturbation method is used and only the range of small bending is analyzed. The resulting optimal bars are regarded as thick-walled, hence no wall stability constraints are applied.     

Origination of In‐Phase Oscillations of Thin Plates with Aeroelastic Interaction

Synchronization of oscillations of thin elastic plates that are walls of a gasfilled channel is considered. The gas motion is described by a system of Navier–Stokes equations, which is solved using the secondorder MacCormack method with time splitting. The motion of the channel walls is described by a system of geometrically nonlinear dynamic equations of the theory of this plates, which is solved by the finitedifference method. Kinematic and dynamic contact conditions are imposed at the interface between the media. A numerical experiment is performed to determine typical dynamic regimes and study the transition of the aeroelastic system to inphase oscillations.     

Longitudinal Free Vibration of Inhomogeneous Elastic Straight Strut with a Variable Cross Section

The scope of the present article,motivated by the case of the composite wooden propeller of anairplane,is to deal tentatively with the longitudinal free vibrationproblem of an elastic straight bar witha more general mathematical treatment.In this analysis,we have assigned to the modulus of elasticity,the bar cross section as well as themass per unit length of the bar an exponential function variation,and then found a general solution,wherein three parameters were considered as the main factors to affect the longitudinal free vibrationof the inhomogeneous elastic straight bar with a variable cross section.     

The Bending of Elastic Circular Ring of Non-homogeneous and Variable Cross Section under the Actions of Arbitrary Loads

On the basis of the stepped reduction method suggested in[1],we investigatethe problem of the bending of elastic circular ring of non-homogeneous andvariable cross section under the actions of arbitrary loads.The general solutionof this problem is obtained so that it can be used for the calculations of strengthand rigidity of practical problems such as arch,tunnel etc.In order to examineresults of this paper and explain the application of this new method,an exampleis brought out at the end of this paper.Circular ring and arch are commonly used structures in engineering.Timo-shenko,S.,Barber,J.R.,Tsumura Rimitsu et al.have studied theseproblems of bending,but,so far as we know,it has been solely restricted to thegeneral solution of homogeneous uniform cross section ring.The only knownsolution for the problems with variable cross section ones has been solelyrestricted to the solution of special case of flexural rigidity in linear functionof coordinates.On account of fundamental equations of the non-homoge     

Hyperbolicity of Steady‐State Equations of Gas Shear Flows in a Thin Layer

The steadystate threedimensional motion of an ideal gas in a thin layer of variable height is considered. In the longwave approximation, the equations of gas dynamics reduce to a system of integrodifferential equations. The generalized characteristics and hyperbolicity conditions of the obtained system are found.     

Applying the Generalized Step-function to Solve the Bending Problems of Nonuniform Beams with Variable Cross Section

The bending problems of nonuniform beams with variable cross-sectioncan be approximated by that of a step beam under sectionally uniform load(including both concentrated forces and couples).In this paper,the conceptof Heaviside function{x-a}~0 will be generalized,and a new function{x-a}~n,n=0,1,2……,will be defined,which may be named as a generalized stepfunction.The rules of operation will also be given to{x-a}~n{x-b}~0.Thereciprocal of the flexural of rigidity1/EJ and the bending moment M(x)can all be expressed in terms of{x-a}~n,and substituted into the differentialequation of the elastic curve of the beam respectively.Thus we mayestablish a set of unified method to solve various types of bending problemsof straight beams.The general solution of the deflection equation will begiven.     

Dust‐ion‐acoustic precursor of a shock wave

The effect of charged dust particles on the structure of the plasma precursor of a strong shock wave is studied. The conditions of formation of a weak discontinuity front are obtained. It is shown that resonant modes can occur in which the concentration of dust particles in the neighborhood of the front increases. In the case of positively charged particles of dust, the formation of a localized compaction region in the form of a soliton bunch is possible and the dependence of the amplitude of the soliton on shockwave velocity is nonmonotonic. In the case of negatively charged particles of dust, a rarefaction wave is formed. The indicated phenomena can substantially affect the concentration of the neutral component in a slightly ionized plasma.     

Some Applications of Perturbation Method in Thin Plate Bending Problems

In this paper,problems of bending of thin plates under the combined action of lateral loadingand in-plane forces are studied by means of perturbation method.     

Elastic Critical Buckling of Welded Thin Steel Rectangular Plates∗

A theoretical approach for the study of the effect of residual stresses due to welding on the elastic critical buckling behavior of thin steel rectangular plates is described. A finite difference technique is utilized for the determination of in-plane residual stresses due to a weld, and the Rayleight-Ritz method is used for the critical buckling problem, with stresses due to external loads being superimposed on the residual stresses. A number of illustrative examples are included, showing the possible detrimental effect of residual stresses due to welding. An approximately linear relationship is shown to exist between the square of the natural frequency of lateral vibration and the drop in buckling strength, for certain loading conditions     

Fiber Remodeling During Torsion of a Fiber Reinforced Hyperelastic Cylinder��Unloading Behavior

Previous studies introduced a constitutive theory for fiber reinforced hyperelastic materials that allows the fibers to undergo microstructural changes. In this theory, increasing deformation of the matrix leads to increasing stretch of the fibers that causes their gradual dissolution. The dissolving fibers reassemble in the direction of maximum principal stretch of the matrix. The implications of the constitutive theory were first studied for two homogeneous deformations: uniaxial extension along the fibers and simple shear in the direction normal to the fibers. The constitutive theory was then used in treatment of the non-homogeneous deformation of combined axial stretch and twisting. The emphasis was on the determination of the influence of increasing axial stretch and twist on the spatial distribution of fiber dissolution and reassembly within the cylinder and also on the axial force and torque applied to the end faces of the cylinder. The present work is concerned with another aspect of combined axial stretch and twisting of the cylinder, namely unloading following dissolution and reassembly of some of the fibers. In this case, the cylinder is given an initial twist until there is an inner core of original fiber/matrix material and an outer sheath of remodeled fiber/matrix material. A condition is established that determines the combinations of axial stretch and twist that cause no additional dissolution and reassembly of fibers during unloading. It is also shown that there is a residual axial stretch and twist if the axial force and torque become zero. A numerical example illustrates this for a particular choice of matrix and fiber properties.     

Calculation of Transport‐Equation Eigenvalues

The eigenvalue problem for the transport equation with variable coefficients in an arbitrary domain with a smooth boundary is considered. A saturationfree numerical algorithm is constructed. Examples of numerical calculations are given, which prove the effectiveness of the proposed procedure.     

Interaction of Large‐Scale and Small‐Scale Oscillations During Separation of a Laminar Boundary Layer

The mutual influence of shortwave oscillations (instability waves of the separated boundary layer) and longwave disturbances at the frequency of shedding of periodic largescale vortices is experimentally studied in flow separation behind a step. The possibility of controlling the process of vortex formation by exciting amplifying disturbances in the shear layer is demonstrated.     

Protective properties of hot‐air screens

An analytical and numerical study of protective properties of hotair screens is performed for a plane wave with a triangular profile. The reduction of the shockwave pressure behind the screen is only observed at temperatures higher a certain critical value; at lower temperatures, the pressure is higher than that in a shock wave traveling in an isothermal gas. With allowance for real properties of air, an analytical relation between the critical temperature of the hot screen and the incidentwave intensity is obtained.     

Model of Self‐Organization of an Ensemble of Cracks Radiating Sound

A model of selforganization of cracks arising in a rock specimen (granite) compressed by a press is proposed. The model is based on the assumption of acoustic wave interaction between the cracks. To construct the model of selforganization of cracks, solutions of the Fokker–Planck equation are used. The experimentally observed spontaneous increase in the activity of acoustic emission, spatial and temporal clusterization, and formation of a fractal structure in rock specimens under constant and slowly varying loads are explained.     

Re‐Entering of Bodies with a Positive Lift‐to‐Drag Ratio into the Earth's Atmosphere

The problem of gliding descent of a smooth blunted body with a positive lifttodrag ratio in the Earth's atmosphere is solved within the framework of the parabolized viscous shock layer model.     

Effect of free‐stream turbulence on the boundary‐layer structure with diffusion combustion of ethanol

The effect of turbulization of a subsonic air flow on the boundarylayer structure was experimentally studied during evaporation and combustion of ethanol behind an obstacle 3–6 mm high. It is shown that turbulization increases the thermal boundarylayer thickness by a factor of 2, where as the dynamic boundarylayer thickness changes weakly. For 1–18% turbulence at the entrance, the change in the momentum thickness along the channel is close to the change in the momentum thickness for a laminar isothermal boundary layer without injection. Local regions of elevated turbulence with a high intensity of heat and mass transfer arise in the case of combustion behind the obstacle at a distance of 40–50 obstacle heights.     

On the Propagation of Long‐Wave Perturbations in a Two‐Layer Free‐Boundary Rotational Fluid

A mathematical model for the propagation of longwave perturbations in a freeboundary shear flow of an ideal stratified twolayer fluid is considered. The characteristic equation defining the velocity of perturbation propagation in the fluid is obtained and studied. The necessary hyperbolicity conditions for the equations of motion are formulated for flows with a monotonic velocity profile over depth, and the characteristic form of the system is calculated. It is shown that the problem of deriving the sufficient hyperbolicity conditions is equivalent to solving a system of singular integral equations. The limiting cases of weak and strong stratification are studied. For these models, the necessary and sufficient hyperbolicity conditions are formulated, and the equations of motion are reduced to the Riemann integral invariants conserved along the characteristics.     

High‐Velocity Penetration of Plane Shaped‐Charge Jets into Nonlinear Media

This paper reports results of metallographic analysis of metal samples cut from targets penetrated by plane shapedcharge jets. It is shown that the plastic deformation due to penetration has a turbulent nature and, in some cases, it occurs in metals with fractal structure formed after passage of the shock wave running ahead of the jet. A penetration model is proposed that takes into account the nonlinear behavior of the target material and the fractality of its structure.     

Dispersion of Dynamically Passive Impurities in Non‐One‐Dimensional Liquid Flow

Equations that describe dispersion of a substance in a nononedimensional incompressible liquid flow through a plane channel are derived. The model under consideration extends the traditional Taylor model to the case where sources of the substance are present in the flow and relaxation transfer processes are taken into account. Additional conditions for the dispersion equations are obtained. The relation between the proposed model and the Taylor model is analyzed. Based on the equations obtained, the mass transfer between circulation regions in the flow is calculated and a system of cellularmodel equations for stagnant cavities is constructed.     

Electrical Characteristics of Three‐Component Dielectric Composites with Close‐Packed Inclusions

The electric field and effective permittivity are calculated for a twodimensional threecomponent dielectric material reinforced by cylindrical fibers. A composite material with a square close packing of inclusions is considered. The field in the periodic system is investigated using the exact solution of the model problem of interaction of two dissimilar cylindrical inclusions in an external homogeneous electric field. A diagram of the relative effective permittivity is constructed.