Converging spherical and cylindrical shock waves

The self-similar solutions for converging spherical and cylindrical strong shock waves in a non-ideal gas satisfying the equation of state of the Mie-Gruneisen type are investigated. The equations governing the flow, which are highly non-linear hyperbolic partial differential equations, are first reduced to a Poincaré-type ordinary differential equation with suitable approximation. Such an approximation helps in obtaining the self-similar solutions and the similarity exponent numerically by phase-plane analysis.     

Formation of cylindrical shock waves in radiation-magneto gas dynamics

The propagation of acceleration waves in radiation-magneto gas dynamic flow which is induced by the motion of a piston advancing with finite acceleration into a constant state of rest has been studied along with the characteristic path by using the characteristics of the governing quasilinear system as the reference coordinate system. It is shown that a linear solution in the characteristic plane can exhibit nonlinear behaviour in the physical plane. A differential equation governing the growth and decay of an acceleration wave is derived. The critical time is obtained when all the characteristics will pile up at the wave front to form a shock wave. It is found that the effect of magnetic field on compressive waves which owe their origin to radiation is to cause an early shock formation, while on those of expansion waves which owe their origin to radiation the effect of magnetic field is to decrease the decay rate. However, the effect of coupling of radiation and magnetic field on waves (compressive) which are emanating from piston movement is to slow down the motion of a breakdown point and thus increase the cylindrical shock formation time, while on those of expansion waves, the effect is to enhance the decay rate.     

The transient responses of two-layered cylindrical shells attacked by underwater explosive shock waves

An approximation solution is introduced for the dynamic response of a two-layered cylindrical shell of circular cross-section subjected to an underwater explosive shock wave. The solution is obtained within the framework of the Flugge thin shell theory and the reflected-afterflow-virtual-source (RAVS) method is used to account for the fluid–structure interaction. Detailed numerical computations are carried out, in dimensionless form, for the cases of infinitely long two-layered cylindrical shells. Time histories of nondimensional radial velocity, mid-surface strain, 0th mode radial displacement and 1st mode radial velocity are presented in graphical form and the effects of elastic modulus, shell radius and thickness on the transient response characteristics of the shells are investigated.     

Self-similar flows with increasing energy-3. Radiation-driven shock wave

Self-similar flows behind a radiation-driven shock wave have been investigated. Approximate analytical solutions are presented when the flow is adiabatic. These solutions are agreeing well with the exact numerical solutions. A simple method to obtain the shock propagation law has been presented. Also explicit approximate analytical solutions are obtained when the flow behind the shock is taken to be isothermal.     

Dislocation dynamics and plastic shock waves

In a previous paper (Weertman and Follansbee (1983)) steady-state finite amplitude elastic plastic waves of moderate amplitude were treated as summed steady-state infinitesimal elastic plastic waves, each traveling at slightly different velocities with respect to the crystal lattice but at the same velocity with respect to the laboratory coordinate system. In this paper finite amplitude elastic-plastic waves are again treated as summed infinitesimal elastic-plastic waves. The infinitesimal waves, however, are not steady-state waves. For the constant dislocation density case it is shown that the governing wave equation for the infinitesimal wave is (∂/∂t){(∂²σ/∂x²)-(1/a₀²)(∂²σ/∂t²)} = −(a₀/Λ){(∂²σ/∂x²)-(1/a₁²) (∂²σ/∂t²)}, where σ is stress, a₀ is the elastic wave velocity, a₁ is the bulk wave velocity, and Λ is a characteristic length. The steady-state finite amplitude wave profile (for a constant dislocation density) is given by the equation a′(∂σ_x/∂x) = [a₁ + u](∂σ_x/∂x)-D(∂²σ_x/∂x²), where a′ is the finite amplitude wave velocity and μ is the particle velocity (μ is proportional to σ) and D=(a₀²-a₁²)a₁²Λ/2a₀³. Similar equations hold for the case in which the density increases. The tendency of the infinitesimal wave to spread is counteracted by the squeezing effect of the particle velocity to give a steady-state finite amplitude wave. The width of the steady-state finite amplitude wave is inversely proportional to the maximum stress if the dislocation density is constant and inversely proportional to the cube of the stress if the density increases with stress. The finite amplitude wave velocity is slightly greater than a₁ for the constant dislocation density case and lies between a₁ and a₀ when the density is not constant.The plastic deformation that occurs behind a strong shock wave is also treated in this paper.     

Transformation of radially traveling cylindrical waves between two skew cylindrical coordinate systems

The transformation of radially traveling cylindrical waves between two cylindrical coordinate systems with skew (nonparallel) axes is derived for the first time to our knowledge. The analytical procedure is based on the complex integral representation of the Hankel function and appropriate contour deformation and change of variables to obtain a final Fourier transform expression of the cylindrical wave in the new system. Scalar and vector waves are considered. This new result is a powerful tool for the rigorous analysis of scattering and coupling in nonparallel optical fiber configurations.     

Spall in non-one-dimensional shock waves

Consideration is given to non-stationary shock wave on spallation. Spall strength can be written as (i) pressure — particle velocity diagram in a one-dimensional case, and (ii) pressure — angle of turn flow diagram in a two-dimensional case. Experimental procedure involves (i) loading of the explosive by sliding detonation and (ii) orthogonal flash X-raying. In this way, some metals in particular low-melting ones and plastics as well as liquids have been studied based on the present results and those of other researchers. The dependences of spall strength — deformation rate have been obtained for substances studied in the form of power or linear functions. Some aspects of spallation and of accompanying effects on the base relations obtained are discussed.     

Deformation of cylindrical shells by thermal shock

In a plane two-dimensional statement the article investigates numerically the processes of deformation and rupture of cylindrical shells under asymmetric thermal shock. It studies the effect of the thickness of the shell on the nature of deformation and rupture. Computing difficulties in the solution of similar problems are pointed out.Translated from Problemy Prochnosti, No. 6, pp. 64–69, June, 1990.     

Plastic deformation behind strong shock waves

The dislocation produced plastic deformation that must occur behind the front of strong shock waves is analysed in this paper. (A strong shock has a driving stress that is large compared with the bulk elastic modulus). Because the front of strong shock waves must be exceedingly narrow, there is a problem of how the effective shear stress, which is probably of the order of the theoretical shear strength at the immediate front itself, is relaxed in the region behind the shock front. Our analysis (Weertman and Follansbee, 1983)) of moderate strength shock waves is used for the relaxing region behind a strong shock wave front. It is concluded that a strong shock wave front has the following dislocation structure: The shock front, of atomic dimensions, consists of a Smith dislocation interface of dislocations that keep up with the front by moving at transonic or supersonic velocities. Immediately behind the Smith interface is a region of moving dislocations that do not keep up with the shock front. In this region normal dislocation motion and multiplication takes place. Within this region of normal plastic deformation the shock pressure rises by an amount that is quite small compared with the shock pressure itself.     

Interference of nonstationary oblique shock waves

Special features of calculation of the flow parameters behind a nonstationary oblique shock wave moving in a stream of absolutely nonviscous gas are considered. The wave intensity at which the stream behind the shock wave may exhibit singularities is determined. The problem of calculating a nonstationary shock wave configuration formed during the interaction of a supersonic jet with an obstacle is solved.     

Converging shock waves in porous media

We have numerically solved several problems related to converging shock waves, including (i) one-dimensional spherical and cylindrical waves with cumulation limited to a ball or cylinder of small radius and (ii) shock-wave flow in a cone-shaped solid target. The passage from a continuous loaded substance to a porous medium in these problems leads to a significant increase in both temperature and pressure in the sample. This character of pressure variation depending on the porosity qualitatively differs from the case of plane waves of constant intensity, for which an increase in the sample porosity under otherwise equal conditions of loading always leads to a decrease in the pressure.     

On reflection of shock waves and shock-wave configurations

A new approach to the problem of unsteady reflection of shock waves is proposed. It is shown that the type and parameters of the shock wave reflection configuration depend on the type of an object (single shock versus shock-wave configuration) incident on the reflecting surface. The concept of the reflection of a shockwave configuration is introduced, and data are presented to illustrate the fact that theoretically established boundaries of the existence of various types of configurations do not represent the transitions lines if shockwave configurations rather than single shocks interact with the surface.     

Materials synthesis and phase transitions under shock waves

Shock wave compression of materials is accompanied by high pressure, high-strain-rate loading, elevated temperatures, large shear stresses and excessive plastic deformation. All these features create an unusual state in materials that is not possible in any processing method. Thus, in multi-component system of powders, these can lead to enhanced reactivity because of defect promoted diffusional flow of the reactants in a solid-solid chemical reaction for materials synthesis. On the other hand, in a one component system, these shock compression features may significantly influence the onset pressure, the kinetics, the mechanism and the reversibility of a phase transformation. We shall illustrate these using examples on the synthesis ofβ-C₃N₄ and our work on the crystal to amorphous phase transitions inq-GeO₂ andα-FePO₄.     

Unsteady pressure waves and shock waves in elastic tubes containing bubbly air-water mixtures

Summary The flow of a two-phase fluid through elastic tubes is more complex than that of a single phase fluid. The mathematical model is based on an one-dimensional approach to the flow of a liquid-gas mixture. The one-dimensional equations for transient two-phase flow through elastic tubes are a system of nonlinear hyperbolic partial differential equations if the bubbles and the liquid particles move with the same velocity. Included in the model are the effects of wall elasticity, compressibility of the gas and the liquid, the surface tension and the variable area change. The propagation of finite pressure waves and shock waves in a liquid containing gas bubbles has been investigated. The results show a differently strong influence of the parameters on the wave propagation speed and on the shock wave relations.

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Zur Ausbreitung von Druck- und Stoßwellen in instationären Blasen-Flüssigkeitsströmungen durch elastische Leitungen

Zusammenfassung Es ist bekannt, daß bei der mathematischen Beschreibung einer Zweiphasenströmung insofern Schwierigkeiten auftreten können, als unter bestimmten Voraussetzungen sowohl reelle als auch komplexe charakteristische Richtungen auftreten können. Für den Fall gleicher Geschwindigkeiten von Blasen und Flüssigkeit erhält man aus den instationären Gleichungen ein nichtlineares hyperbolisches Differentialgleichungssystem. Berücksichtigt werden die Elastizität der Wandungen, die Kompressibilität des Gases und der Flüssigkeit sowie die Oberflächenspannung. Wellenausbreitungsgeschwindigkeiten und Stoßrelationen werden angegeben. Die Resultate zeigen einen unterschiedlich starken Einfluß der verschiedenen Parameter auf die Wellenaus-breitungsgeschwindigkeit und die Stoßrelationen.

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With 13 Figures     

Some new possibilities for using waves from cylindrical emitters

An analysis is made of a new principle for flow rate measurements based on using cylindrical emitters, which has significant advantages over known methods. The advantages of using these emitters in waveguide technology are discussed. Pis’ma Zh. Tekh. Fiz. 25, 53–56 (March 12, 1999)     

Thermal waves in an elastic highly stretched cylindrical bar

Summary The propagation of thermoelastic waves in a cylindrical bar of incompressible material and circular cross-section is investigated assuming that the bar, kept at a uniform constant temperature, undergoes a finite initial stretch, has an infinite length and the superposed disturbance is axially symmetric and of small amplitude. Three governing equations for the infinitesimal radial and longitudinal displacements as well as for the secondary pressure function and the secondary transient temperature field are derived. The remaining field equation is supplied by the incompressibility condition. The system of coupled second order partial differential equations involving three independent variables is solved by the method ofFrobenius assuming a propagation of simple harmonic waves. Three approximations to the phase velocity are found that in the isothermal case coincide with the known results both of the classical and the finite deformation theory.

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Thermische Wellen in einem elastischen, stark vorgedehnten zylindrischen Stab

Zusammenfassung Die Ausbreitung thermoelastischer Wellen in einem kreiszylindrischen Stab aus inkompressiblem Material wird untersucht. Es wird angenommen, daß der unendlich lange Stab auf gleichförmiger konstanter Temperatur gehalten wird, einer endlichen Anfangsdehnung unterworfen ist, und daß die überlagerte Störung axialsymmetrisch und von kleiner Amplitude ist. Drei Gleichungen für die infinitesimalen radialen und longitudinalen Verschiebungen sowie für die sekundäre Druckfunktion und das sekundäre instationäre Temperaturfeld werden hergeleitet. Die verbleibende Feldgleichung wird durch die Inkompressibilitätsbedingung geliefert. Das System gekoppelter partieller Differentialgleichungen zweiter Ordnung in drei unabhängigen Variablen wird mittels derFrobeniusschen Methode unter Annahme einfacher harmonischer Wellen gelöst. Es werden drei Näherungen für die Phasengeschwindigkeit gefunden, die im isothermen Fall in die wohlbekannten Ergebnisse sowohl der klassischen als auch der Theorie endlicher Verformungen übergehen.

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With 1 Figure

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This research has been supported by a grant from the National Science Foundation.