Strength and elasticity of iron and copper at high shock-wave compression pressures

The elastic and strength parameters of iron and copper were determined experimentally at high shock-wave compression pressures of 1–2 Mbar. The attenuation of shock waves created by the impact of thin plates in blocks of the investigated materials was recorded in the experiments. The Poisson ratios, bulk moduli, shear moduli, and yield strength Y for iron at 1.11 and 1.85 Mbar and for copper at 1.22 Mbar were determined from the experimentally observed amplitudes and velocites of the unloading shock waves. The shape of the curve of the change of the yield strength of copper with an increase of pressures to states of shock-wave compression causing melting was determined on the basis of the results obtained and data of other investigators. The curve has a maximum at P 800 kbar corresponding to Y =280 kg/mm². The yield strengths for iron are located on the ascending branch of the curve Y(P) and are numerically equal to 110 kg/mm² at 1.11 Mbar and 270 kg/mm² at 1.85 Mbar. The measured values of Y exceed the yield strengths of uncompressed metals by a factor of 5–7. The authors also recorded a substantial increase of Poisson's ratios with increase of pressures in the investigated metals.Deceased.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 159–166, November–December, 1971.     

Shock Compression of a Plate on a Wedged–Shaped Target

Problems of compression of a plate on a wedge–shaped target by a strong shock wave and plate acceleration are studied using the equations of dissipationless hydrodynamics of compressible media. The state of an aluminum plate accelerated or compressed by an aluminum impactor with a velocity of 5—15 km/sec is studied numerically. For a compression regime in which a shaped–charge jet forms, critical values of the wedge angle are obtained beginning with which the shaped–charge jet is in the liquid or solid state and does not contain the boiling liquid. For the jetless regime of shock–wave compression, an approximate solution with an attached shock wave is constructed that takes into account the phase composition of the plate material in the rarefaction wave. The constructed solution is compared with the solution of the original problem. The temperature behind the front of the attached shock wave was found to be considerably (severalfold) higher than the temperature behind the front of the compression wave. The fundamental possibility of initiating a thermonuclear reaction is shown for jetless compression of a plate of deuterium ice by a strong shock wave.     

Structure of the heating layer ahead of the front of a strong intensely emitting shock wave

The problem of the structure and brightness of strong shock waves arises in the investigation of such phenomena as the motion of large meteoroids in the atmosphere, optical and electrical discharges, the development of strong explosions, and other similar processes and in the creation of powerful radiation sources based on them. This problem also has a general physics interest. As the propagation velocity of a strong shock wave increases the gas temperature behind its front and the role of emission grow. Part of the radiation emitted by the gas heated and compressed in a shock wave is absorbed ahead of the front, forming the so-called heating layer. The quasisteady structure of a strong intensely emitting shock wave was studied in [1, 2]. In this case a diffusional approximation and the assumption of a gray gas were used to describe the radiation transfer. They introduced the concept of a wave of critical amplitude, when the maximum temperature T_- in the heating layer reaches the temperature T_a determined on the basis of the conservation laws, i.e., from the usual shock adiabat; it is shown that behind a compression shock moving through an already heated gas there is a temperature peak in which the maximum temperature T₊ exceeds T_a. The problem of the quasisteady structure of an emitting shock wave in air of normal density was solved numerically in [3]. The angular distribution of the radiation was approximately taken into account — it was assigned by a simple cosinusoidal law. The spectral effects were taken into account in a multigroup approximation. They introduced 38 spectral intervals, which is insufficient to describe a radiation spectrum with allowance for the numerous lines and absorption bands.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 86–92, September–October, 1978.     

Numerical study of spherical blast-wave propagation and reflection

The objective of this study is to understand the flow structures of weak and strong spherical blast waves either propagating in a free field or interacting with a flat plate. A 5th-order weighted essentially non-oscillatory scheme with a 4th-order Runge-Kutta method is employed to solve the compressible Euler/Navier-Stokes equations in a finite volume approach. The real-gas effects are taken into account when high temperature occurs. A shock-tube problem with the real-gas effect is first tested in order to verify the solver accuracy. Moreover, unsteady shock waves moving over a stationary wedge with various wedge angles, resulting in different types of shock wave reflections, are also tested. It is found that the computed results agreed well with the existing data. Second, the propagation of a weak spherical blast wave, created by rupture of a high-pressure isothermal sphere, in a free field is studied. It is found that there are three minor shock waves moving behind the main shock. Third, the problem of a strong blast wave interacting with a flat plate is investigated. The flow structures associated with single and double Mach reflections are reported in detail. It is found that there are at least three local high-pressure regions near the flat plate. Received 27 July 2000 / Accepted 25 January 2002 – Published online 17 June 2002     

Stability of periodic waves of finite amplitude on the surface of a deep fluid

We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface (r, t) and the hydrodynamic potential (r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case.In section 2, using a method similar to van der Pohl's method, we obtain simplified equations describing nonlinear waves in the small amplitude approximation. These equations are particularly simple if we assume that the wave packet is narrow. The equations have an exact solution which approximates a periodic wave of finite amplitude.In section 3 we investigate the instability of periodic waves of finite amplitude. Instabilities of two types are found. The first type of instability is destructive instability, similar to the destructive instability of waves in a plasma [5, 6], In this type of instability, a pair of waves is simultaneously excited, the sum of the frequencies of which is a multiple of the frequency of the original wave. The most rapid destructive instability occurs for capillary waves and the slowest for gravitational waves. The second type of instability is the negative-pressure type, which arises because of the dependence of the nonlinear wave velocity on the amplitude; this results in an unbounded increase in the percentage modulation of the wave. This type of instability occurs for nonlinear waves through any media in which the sign of the second derivative in the dispersion law with respect to the wave number (d²/dk²) is different from the sign of the frequency shift due to the nonlinearity.As announced by A. N. Litvak and V. I. Talanov [7], this type of instability was independently observed for nonlinear electromagnetic waves.The author wishes to thank L. V. Ovsyannikov and R. Z. Sagdeev for fruitful discussions.     

Self-switching of displacement waves in elastic nonlinearly deformed materialsSelf-commutation d'ondes de déplacement dans les milieux élastiques non-linéaires

The problem of self-switching plane waves in elastic nonlinearly deformed materials is formulated. Reduced and evolution equations, which describe the interaction of two waves the power pumping wave and the faint signal wave are obtained. For the case of wave numbers matching the pumping and signal waves, a procedure of finding the exact solution of evolution equations is described. The solution is expressed by elliptic Jacobi functions. The existence of the power wave self-switching is shown and commented. To cite this article: J. Rushchitsky, C. R. Mecanique 330 (2002) 175–180.     

Lifting bodies using the flow behind axisymmetric conical shock waves

One of the methods of designing aircraft with supersonic flight speeds involves solving an inverse problem by means of the well-known flow schemes and the substitution of rigid surfaces for the flow surfaces. Lifting bodies using the flows behind axisymmetric shock waves belong to these configurations. All lifting bodies using the flow behind a conical shock wave can be divided into two types [1]. Bodies whose leading edge passes through the apex of the conical shock wave pertain to the first type and those whose leading edge lies below the apex of the conical shock wave, to the second. For small apex angles of the basic cone at hypersonic flow velocities an approximate solution of the variation problem was obtained, which showed that the lift-drag ratio of lifting bodies of the second type is higher than that of the first [2]. The present paper gives a numerical solution of the problem for flow past lifting bodies of the second type using the flow behind axisymmetric conical shock waves with half-angles of the basic cone _S=9.5 and 18° The upper surfaces of the bodies are formed by intersecting planes parallel to the velocity vector of the oncoming flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–138, March–April, 1986.     

Shock wave interaction in a dusty gas and the appearance of fully dispersed waves

A problem of regular (symmetric and asymmetric) interaction of plane shock waves in a steady-state dusty-gas flow is considered. The possibility of the formation of wave structures is revealed, in which either all or some of the incident or reflected waves degenerate into fully dispersed waves, i.e. zones in which the parameters of both phases vary continuously. Using the Rankine-Hugoniot relations for a one-velocity “effective-gas” model, the ranges of nondimensional governing parameters (the Mach number, the angles between the incident waves and the free stream, the phase specific-heat ratio, and the particle mass concentration) are found, which correspond to different wave configurations. In the framework of a two-fluid dusty-gas model, the flow structure in the region of symmetric interaction of the shocks is calculated numerically for typical configurations containing fully dispersed waves. The flow in the region of a normal fully dispersed wave is also calculated. Good agreement between the calculated wave structure and the data known in the literature is obtained. A range of governing parameters in which the carrier-phase temperature has a local maximum inside the wave structure is found.     

The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

Two-dimensional stress wave analysis in incompressible elastic solids

Two-dimensional stress waves in a general incompressible elastic solid are investigated. First, basic equations for simple waves and shock waves are presented for a general strain energy function. Then the characteristic wave speeds and the associated characteristic vectors are deduced. It is shown that there usually exist two simple waves and two shock waves. Finally, two examples are given for the case of plane strain deformation and antiplane strain deformation, respectively. It is proved that, in the case of plane strain deformation the oblique reflection problem of a plane shock is not solvable in general.     

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order , the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's Weakly Nonlinear Geometrical Optics method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced. Communicated by C. C. Wang     

Multiple scattering of flexural waves in a semi-infinite thin plate with a cutout

The multiple scattering of flexural waves and dynamic stress concentration in a semi-infinite thin plate with a cutout are investigated, and the expressions of this problem are obtained. The analytical solutions of wave fields are expressed by employing the wave function expansion method and the expanded mode coefficients are solved by satisfying the boundary condition of the cutout. The image method is used to satisfy the traction free boundary condition of the plate. As an example, the numerical results of dynamic stress concentration factors are graphically presented and discussed. Numerical results show that the analytical results of the scattered waves and dynamic stress in semi-infinite plates are significantly different from those in infinite plates when the ratio of distance b/a is relatively little. In the region of low frequency and long wavelength, the maximum dynamic stress concentration factors occur on the illuminated side of the scattering body with θ = π, but not at the edge of the cutout with θ = π/2. As the incidence frequency increases (the wavelength becomes short), the dynamic stress on the illuminated side of the cutout decreases, however, the dynamic stress on the shadow side increases.     

Navier–Stokes Computations in Micro Shock Tubes

Micro shock tube flows were simulated using unsteady 2D Navier–Stokes equations combined with boundary slip velocities and temperature jumps conditions. These simulations were performed using the parallel version of a multi-block finite-volume home code. Different initial low pressures and shock tube diameters allow to have the scaling ratio ReD/4L vary. The numerical results show a strong attenuation of the shock wave strength with a decrease of the hot flow values along the tube. When the scaling ratio decreases the shock waves can transform into compression waves. Comparison to the existing 1D models also shows the limit of these models.     

Smooth particle hydrodynamics method for modeling cavitation-induced fracture of a fluid under shock-wave loading

It is demonstrated that the method of smoothed particle hydrodynamics can be used to study the flow structure in a cavitating medium with a high concentration of the gas phase and to describe the process of inversion of the two-phase state of this medium: transition from a cavitating fluid to a system consisting of a gas and particles. A numerical analysis of the dynamics of the state of a hemispherical droplet under shock-wave loading shows that focusing of the shock wave reflected from the free surface of the droplet leads to the formation of a dense, but rapidly expanding cavitation cluster at the droplet center. By the time t = 500 µs, the bubbles at the cluster center not only coalesce and form a foam-type structure, but also transform to a gas-particle system, thus, forming an almost free rapidly expanding zone. The mechanism of this process defined previously as an internal “cavitation explosion” of the droplet is validated by means of mathematical modeling of the problem by the smoothed particle hydrodynamics method. The deformation of the cavitating droplet is finalized by its decomposition into individual fragments and particles.     

Resonance in rarefied gases

Dispersion and damping of ultrasound waves are a standard test for mathematical models of rarefied gas flows. Normally, one considers waves in semi-infinite systems in relatively large distance of the source. For a more complete picture, ultrasound propagation in finite closed systems of length L is studied by means of several models for rarefied gas flows: the Navier-Stokes-Fourier equations, Grad’s 13 moment equations, the regularized 13 moment equations, and the Burnett equations. All systems of equations are considered in simple 1-D geometry with their appropriate jump and slip boundary conditions. Damping and resonance are studied in dependence of frequency and length. For small L, all wave modes contribute to the solution.     

Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p _e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p _e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.     

Propagation of Quasiacoustic Pulses in an Elastoplastic Medium

Expressions for the velocity of a plastic shock wave and phase velocity of longitudinal waves in an elastoplastic medium with hardening are obtained in a quasiacoustic approximation. An analytical solution of the problem of shockpulse attenuation is constructed. A special feature of the amplitude of the attenuating plastic shock wave is that it reaches the amplitude of the elastic precursor in a finite time, whereas in hydrodynamics, the amplitude of a quasiacoustic shock pulse tends to zero asymptotically.     

Numerical confirmation of the hysteresis phenomenon in the regular to the Mach reflection transition in steady flows

Numerical calculations based on the Navier-Stokes equations are carried out to investigate the reflection of shock waves over straight reflecting surfaces in steady flows. The results for a flow Mach number of M₀=4.96 confirm the recent experimental findings of Chpoun et al. (1995) concerning the transition from regular to Mach reflection. Numerical calculations as well as experimental results show a hysteresis phenomenon during this transition and the regular reflection is found to be stable in the dual-solution domain in which theoretically both regular and Mach reflection wave configurations are possible.