Numerical analysis of axisymmetric buckling of conical shells

Nonlinear boundaryvalue problems of axisymmetric buckling of conical shells under a uniformly distributed normal pressure are solved by the shooting method. The problems are formulated for a system of six firstorder ordinary differential equations with independent rotation and displacement fields. Simply supported and clamped cases are considered. Branching solutions of the boundaryvalue problems are studied for different pressures and geometrical parameters of the shells. The nonmonotonic and discontinuous curves of equilibrium states obtained show that collapse, i.e., snapthrough instability is possible. For a simply supported shell, multivalued solutions are obtained for both external and internal pressure. For a clamped thinwalled shell, theoretical results are compared with experimental data.     

Numerical Analysis of Branched Shapes of Arches in Bending

Nonlinear boundaryvalue problems of plane bending of elastic arches under a uniformly distributed load are solved by the shooting method. The problems are formulated for a system of six firstorder ordinary differential equations with a finiterotation field independent of displacements. Simply supported and clamped cases are considered. Branching solutions of the boundaryvalue problems are obtained. For a simply supported arch, a set of solutions describes symmetric and nonsymmetric shapes of bending, which correspond to positive, negative, and zero loads. For a clamped arch, the set of solutions consists of symmetric shapes that occur only for positive loads.     

Exact solutions and mathematical properties of boundary‐value problems for dynamic‐diffusion boundary layers

The paper studies boundaryvalue problems for dynamicdiffusion boundary layers occurring near a vertical wall at high Schmidt numbers and for dynamic boundary layers whose inner edge is adjacent to the dynamicdiffusion layers. Exact solutions for boundary layers at small and large times are derived. The wellposedness of the boundaryvalue problem for a steady dynamicdiffusion layer is studied.     

Effect of Dynamic Prehistory on Aerodynamics of a Laminar Separated Flow in a Channel Behind a Rectangular Backward‐Facing Step

The effect of dynamic prehistory of the flow and the channelexpansion ratio on aerodynamics of a steady separated laminar flow behind a rectangular backwardfacing step located in a planeparallel channel is numerically studied. It is shown that the boundary layer upstream of the flow separation exerts a strong effect on flow characteristics behind the step. A decrease in the boundarylayer thickness in the cross section of the step leads to a decrease in the separationregion length, and an increase in the channelexpansion ratio with a fixed initial boundarylayer thickness and Reynolds number leads to an increase in the separationregion length.     

Averaging of a Layered Elastic Medium with Low Dynamic Dissipation at the Interlayer Boundary

Averaged relations for a layered medium with dynamic dissipation at the interlayer boundary are constructed for dynamic problems of longitudinal shear and twodimensional theory of elasticity. For low dissipation, the averaged boundaryvalue problem is shown to be singularly perturbed. The degeneration of the boundaryvalue problem is studied qualitatively.     

Estimation of the Effective Absorptivity of Metals from the Observation of Their Martensitic Restructuring under Laser Radiation

A new experimental method is proposed for determining the effective absorptivity of a metal under pulsed laser radiation. The method is based on solving an axisymmetric boundaryvalue heatconduction problem for a halfspace with the use of metallographically measured sizes of the polymorphictransformation zone in the irradiated material. The method is tested on singlecrystal cobalt and St.45 steel samples.     

Unsteady Radiative–Convective Heat Transfer in a Flow Emitting‐Absorbing and Scattering Medium around an Ablating Plate

A conjugate problem of radiative–convective heat transfer in a turbulent hightemperature gasdisperse flow around a thermally thin ablating plate is considered. The plate experiences intense radiative heating by an external source, which is a blackbody. The temperature fields and the distributions of heat fluxes along the plate under unsteady conditions are calculated. The data gained make it possible to examine the effect of the Stark number and phasetransition heat in the plate material on the time evolution of the thermal state of the boundarylayer medium and the plate itself being heated by a hightemperature radiation source.     

Effect of Free‐Stream Turbulence on the Flow Structure near a Wedge and the Windward Side of an Airfoil

The flow structure behind wire grids is studied for flows with a low subsonic velocity, and the effect of grids on the boundarylayer flow structure is considered. It is shown that the meanvelocity inhomogeneity induced by the grid does not disappear until a distance of 925 calibers downstream of the grid is reached. Liquidcrystal thermography combined with hotwire measurements made it possible to find the source of steady largescale streamwise vortex structures in the boundary layer on a wedge and on an airfoil and to determine the parameters of these structures.     

Model of a plastically compressible material and its application to the analysis of the compaction of a porous body

This paper considers a model of a plastically compressible porous medium with a cylindricaltype yield condition and its associated constitutive relations, which ensure independent mechanisms of shear and compaction of the porous material. This allows one to use the wellknown theorems of plastic theory to analyze plastically compressible media and obtain analytical solutions for a number of boundaryvalue problems, including those taking into account conditions on strongdiscontinuity surfaces. Results from fullscale studies of the structural periodicity of noncompact materials using wavelet analysis were employed to choose a physical model for a porous body and determine the properties and dimensions of a representative volume. The problem of extrusion of a porous material through a conical matrix was solved.     

Asymptotic Modeling of Nonlinear Wave Processes in Shock‐Loaded Elastoplastic Materials

Nonlinear wave processes in shockloaded elastoplastic materials are modeled. A comparison of the results obtained with experimental data and numerical solutions of exact systems of dynamic equations shows that the model equations proposed qualitatively describe the stressdistribution evolution in both the elasticflow and plasticflow regions and can be used to solve one and twodimensional problems of pulsed deformation and fracture of elastoplastic media.     

Adjoint problems of mechanics of continuous media in gas‐laser cutting of metals

A mathematical model of gaslaser cutting of metal plates in an inert gas is proposed. The formation and flow of the liquid metal melt film at the cutting front is considered within the framework of incompressible boundarylayer equations. Based on the resultant analytical solution, a local law of energy conservation on the cutting surface is derived, which takes into account the meltfilm thickness and the temperature dependence of thermophysical parameters of the metal. The problem of the cutting shape and depth is solved in the twodimensional formulation. A comparison with experimental data is made in terms of the cutting depth and maximum cutting velocity for carbon and alloy steel.     

Variational principle of thermoelectroelasticity and its application to the problem of vibrations of a thin‐wall member

The dynamic behavior of thinwall members manufactured from materials with the pyroelectric effect was studied. A variational formulation of the problem is used, and a variational principle is formulated that differs from the wellknown one. Correct boundaryvalue problems describing the tension, compression, and bending of a thinwall pyroelectric member are constructed using the variational principle and a number of hypotheses on the distribution of the components of physical fields along the width of the member.     

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.     

Numerical Simulation of the Fall of a Body with a Corrugated Bottom on Water

A numerical model is proposed for the potential flow of an ideal incompressible fluid produced by impact of a body with concave bottom on water. Compression of the entrapped air is taken into account. The algorithm is based on joint solution of the equations of motion for the body and the fluid by the finite difference method with approximation in time. At each time, the boundaryvalue problem for the Laplace equation is solved by the boundaryelement method. Calculation results are given. The effects of the air layer, dimensions and shape of the corrugations, initial velocity, and other parameters on the impact process are shown.     

Stationary Simple Waves on a Barotropic Liquid Shear Flow

Stationary threedimensional flows of a barotropic liquid in a gravity field are considered. In the shallowwater approximation, the Euler equations are transformed into a system of integrodifferential equations by the EulerLagrange change of coordinates. A system of simplewave equations is obtained, for which the theorem of existence of a solution attached to a given shear flow is proved. As an example, a particular solution analogous to the solution of the problem of a gas flow around a convex angle is given.     

Heat Transfer on the Surface on a Spherically Blunted Cone Exposed to a Supersonic Spatial Flow with Injection of a Coolant Gas

The heattransfer processes in a supersonic spatial flow around a spherically blunted cone with allowance for heat overflow along the longitudinal and circumferential coordinates and injection of a coolant gas are studied numerically. The prospects of using highly heatconducting materials and injection of a coolant gas for reduction of the maximum temperatures at the body surface are demonstrated. The solutions of the direct and inverse problems in one, two, and threedimensional formulations for different shell materials are compared. The error of the thinwall method in determining the heat flux on the heatloaded boundary of the body is estimated.     

On the Quasi‐Steadiness Hypothesis as Applied to Gas Exhaustion from a Receiver

A semiempirical method of determining the stabilization time for a quasisteady mode of gas exhaustion from a receiver after sudden opening of the nozzle and the time evolution of the real flow rate at the stage of the transitional process are considered. The numerical solution of the equations of exhaustion gas dynamics in a twodimensional formulation and the results of model experiments demonstrated that the method can be used to estimate the conditions of applicability of the quasisteadiness hypothesis and to determine the discharge coefficient of the nozzle with controlled accuracy.     

Stresses in an elastic body under nonlinear antiplane deformation

The stress field in a cylindrical elastic body under antiplane deformation and certain constraints imposed on volume and surface forces is studied in a nonlinear formulation in actualstate variables. A boundaryvalue problem for independent stress components is formulated in Cartesian and complex variables, sufficient ellipticity conditions for this problem are indicated, and constraints on surface loading are imposed. Analytical solutions are given for linear and weak nonlinear elastic potentials. Similarity to a plane subsonic idealgas flow is established. An approximate method for the solution of the problem is developed.     

Numerical Analysis of Axisymmetric Buckling of Plates under Radial Compression

Nonlinear boundaryvalue problems of axisymmetric buckling of simply supported and clamped plates under radial compression are formulated for a system of six firstorder ordinary differential equations with independent fields of finite displacements and rotations. Multivalued solutions are obtained by the shooting method with specified accuracy. Bifurcation of the solutions of the problem is studied, and a parametric bifurcation diagram is constructed for various values of the loading parameter. Curves of buckling modes are given for three branches of the solution. The numerical results agree with available theoretical data.