Shear and necking instabilities in nonlinear elasticity

This paper examines the behavior of a class of nonlinearly elastic rods under tension. These rods can suffer longitudinal extension, transverse contraction, shear, and flexure. The deformation is governed by an eighth order, quasilinear system of ordinary differential equations. A convergent perturbation process shows that there is a very rich collection of solutions that bifurcate from the trivial affine configuration. Global results are obtained for hyperelastic rods. These solutions describe instabilities of the shear and necking type and can exhibit hysteresis.

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Résumé On considère le comportement d'une classe de poutres nonlinéairement élastiques en état de tension. Ces poutres peuvent souffrir une extension longitudinale, une contraction transversale, un cisaillement et une fiexion. La déformation est gouvernée par un système quasi-linéaire d'équations différentielles ordinaires du huitième ordre. Un procès convergeant de perturbations montre qu'il y a une collection très riche de solutions qui bifurquent de la configuration banale affine. Dans le cas hypérélastique on obtient des résultats globaux. On montre que ces solutions décrivent des instabilités du type de cisaillement et d'étranglement et peuvent exhiber une hystérésis.

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The research of the first author was supported by National Science Foundation Grant MPS 73-08587 A02. Some of the results reported here were obtained by the second author in his master's thesis.     

Prediction of clay soil compaction

Field measurements were made of soil density and moisture patterns under different vehicle tire paths with varying external pressures and number of passes. In addition, laboratory index tests were performed to determine the compaction behaviour of the same soil. Using these results, a prediction equation of dry density in terms of applied pressure and moisture content was obtained for the clay soil. A previously developed equation for sandy soil was modified for the complete range of moisture contents encountered. Estimation of shear strength for the clay soil was made using plastic and liquid limits.     

A study of the dimensions of the wake and of the radial distribution of the gas velocity in the wake behind a blunt body moving at hypersonic velocity

We present a method for experimentally studying in a ballistic range the dimensions of the turbulent wake behind a blunt body moving in air at hypersonic speed; included also in our results is the radial distribution of the gas velocity in the wake.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 165–168, May–June, 1976.The author thanks N. N. Baulin, A. S. Mkrtumov, and Yu. N. Urbanovich for their help in carrying out the experimental work.     

Optimization of elastic bars in torsion

The paper considers the problem of optimization of mechanical systems described by partial differential equations. The shape of the region of integration of these equations is not specified beforehand but is to determined from the condition that a certain integral functional attains an extremal value. The mathematical optimization problem is reduced to a variational one having no differential constraints and the necessary optimality conditions are derived. The latter are used for seeking the cross-sectional shape of elastic bars of maximum torsional rigidity. Exact and approximate analytical solutions are given and the effectiveness of the optimal solutions is estimated.     

Asymptotic behavior of solutions of the Orr-Sommerfeld equations in regions adjacent to two branches of the neutral curve

The asymptotic behavior of the upper and lower branches of the neutral stability curve of a boundary layer found by Lin [1] was determined more accurately by various authors [2–4], who, on the basis of the linearized Navien-Stokes equations, analyzed the higher approximations in the Reynolds number R. In the limit R , neutral perturbations have wavelengths that exceed in order of magnitude the boundary layer thickness. The long-wavelength asymptotic behavior of the Orr-Sommerfeld equation is, in particular, of interest because the characteristic solutions of the linearized equations of free interaction (triple-deck theory) [5–7] are a limiting form of Tollmierr-Schlichting waves in an incompressible fluid with critical layers next to the wall [8–9]. At the same time, the dispersion relation, which is identical to the secular equation of the Orr-Sommerfeld problem, contains an entire spectrum of solutions not considered in the earlier studies [2–4]. The first oscillation mode in the spectrum may be either stable or unstable. In the present paper, solutions are constructed for each of the subregions (including the critical layer) into which the perturbed velocity field in the linear stability problem is divided at large Reynolds numbers. Dispersion relations describing the neighborhood of the upper and lower branches of the neutral curve for the boundary layer are derived. These relations, which contain neutral solutions as a special case, go over asymptotically into each other in the unstable region between the two branches.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–11, July–August, 1984.     

Subsonic axisymmetric wake in a viscous gas

The problem of the subsonic axisymmetric flow of a compressible viscous perfect gas in the wake behind a cylindrical body with a flat base section is considered under the condition that the stream parameters are given at infinity and at some distance x_w upstream of the base section. (Let us note that the possibility of the existence of an axisymmetric wake at moderate Reynolds numbers has been shown experimentally [1].) The problem is solved by the numerical build-up method in a cylindrical x, y coordinate system on the basis of the Navier-Stokes equations, by a method elucidated in [2, 3]. Equations obtained from the fundamental system by a passage to the limit while taking into account the symmetry conditions on the axis y=0 are used on the axis of symmetry.     

Kinetic droplet evaporation in a turbulent air jet

Droplet evaporation in a turbulent air jet is considered for conditions such that the evaporation rate is determined by the evaporation kinetics for the individual droplets rather than by the diffusion rate of the air in the jet. Numerical solution of the equations by computer has shown that the mean air speed in the jet has little effect on the droplet evaporation in the range covered. A simplified solution is presented for the dispersal of an evaporating impurity in a jet. Experiments confirm that this solution is correct, and they show that the turbulent pulsations play a large part, with the result that an initially monodisperse system becomes more and more polydisperse away from the jet.     

Dynamics of the cavitation zone during an underwater explosion near a free surface

The problem of the development of the cavitation zone and the rarefaction wave profile in the region of regular reflection of the spherical shock wave of an underwater explosion from a free surface is analyzed for the axisymmetric formulation within the framework of a model of the two-phase medium consisting of a liquid with cavitation nuclei of the free gas uniformly distributed in it. An example of the calculation of the rarefaction wave profile and the zone of visible cavitation at different times is given for the case of the explosion of 1-g charge at depths of 3 and 5.3 cm for an initial volumetric gas concentration of 10^?11 and an initial cavitation nucleus radius of 5 · 10^?5 cm. The results of the calculation are compared with experiment.     

Model of a gas of solid particles with allowance for inelastic collisions

A study is made of the structure of the interaction region of a flow of solid particles with a surface (or another flow) in the case when the width of this region is much greater than the mean free path of the particles between collisions. The disperse phase is described by means of a model of an ideal non-heat-conducting gas of imperfectly elastic spherical particles. For simplicity, the influence of the carrier medium on the motion of the particles is ignored.