Shock Amplitude Research Articles

The evolutionary behaviour of plane transverse weak nonlinear shock waves in unstrained incompressible isotropic elastic non-conductors

Asymptotic evolution laws for the amplitudes of a plane transverse shock wave of the type alluded to in the title and its accompanying second order discontinuity are derived in this paper using two different approximation methods. The first method is a combination of singular surface theory and the method of multiple scales. Singular surface theory yields a set of coupled evolution equations for the shock amplitude and the amplitudes of higher order discontinuities which accompany the shock. These equations are then solved using the method of multiple scales to obtain uniformly valid solutions. The other method is the shock-fitting method based on simple wave theory. These two methods are compared by considering a specific loading programme for a half-space. We deduce from the evolution laws that the effects of nonlinearity on the evolution of the transverse shock waves are cumulative, becoming most pronounced for distances of travel of order (ζ T hk) −1, where h and k are respectively the initial amplitudes of the shock and the accompanying second order discontinuity, ζ T being a dimensionless material constant. This conclusion is in contrast with that which we obtained before for dilatational shock waves where the effects of nonlinearity are stronger, becoming most pronounced for the shorter distances of travel of order (ζ D k) −1 where ζ D for dilatational shocks corresponds to ζ T for transverse shocks.

Acceleration waves and shock waves in transversely isotropic elastic non-conductors

We consider the propagation of plane acceleration waves and weak nonlinear shock waves in an unstrained transversely isotropic elastic non-conductor which is incompressible but extensible along the preferred direction denoted by the unit vector e. It is shown that such a material may transmit two different shock waves in the direction of any unit vector n which is neither perpendicular nor parallel to the preferred direction e. One of these shock waves is polarised in the plane Π spanned by e and n and carries an entropy jump of third order in the shock amplitude, whilst the other is polarised perpendicularly to Π and carries merely a fourth order entropy jump. In the degenerate cases when the angle θ between n and e is either π/2 or 0, the two different shock waves which can propagate both carry fourth order entropy jumps. In all of these cases, an asymptotic evolution law describing the decay of the shock amplitude is obtained from a reduced first order governing partial differential equation which is itself obtained by differentiating a certain Riemann invariant that is approximately constant under the isentropic assumption adopted here (in common with all studies of non-conductors). We consider also the limiting case in which the material is inextensible along the preferred direction whilst remaining incompressible. It is shown that when θ = π 2 , two shock waves can propagate as before, but when θ = 0, no shock wave can propagate, and when θ ≠ π 2, 0 , only one shock wave can propagate. Shock waves which can propagate in such a constrained material are all of the second type. The shock wave travelling in the direction of inextensibility is allowed by the purely linear theory of such constrained materials but is forbidden by the weak nonlinear theory.

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 ε 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 ε, 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.

Measurement of Bond Strength at Metal/Ceramic Interfaces

AbstractWe report on a novel method for measuring the bond strength of metal/ceramic interfaces. Test specimens are created by vapor depositing a metal film on a ceramic substrate. The specimen is impacted with a thin metal flyer sending a short planar shock pulse into the ceramic. If the shape and amplitude of the wave is properly controlled the interface will spontaneously debond creating new free surfaces.Measurements indicate the debonding process occurs in less than 1.0 ns, which we believe is too short for crack propagation along existing flaws. Therefore, we conclude that simultanious breaking of atomic bonds rather than propagation and coalescence of cracks is the means by which the film and substrate are separated. The free surface velocity of the metal overlayer is monitored during spall by laser interferometry. The data constitute a direct measurement of the bond strength. The measured bond strengths are reproducible and do not show a dependence on shock amplitude for identically prepared specimens.

On methods of calculating successive positions of a shock front

In this paper we have discussed limits of the validity of Whitham's characteristic rule for finding successive positions of a shock in one space dimension. We start with an example for which the exact solution is known and show that the characteristic rule gives correct result only if the state behind the shock is uniform. Then we take the gas dynamic equations in two cases: one of a shock propagating through a stratified layer and other down a nonuniform tube and derive exact equations for the evolution of the shock amplitude along a shock path. These exact results are then compared with the results obtained by the characteristic rule. The characteristic rule not only incorrectly accounts for the deviation of the state behind the shock from a uniform state but also gives a coefficient in the equation which differ significantly from the exact coefficients for a wide range of values of the shock strength.

Shear strength of shock-loaded alumina as determined with longitudinal and transverse manganin gauges

The shear strength of shock-loaded commercial alumina (AD-85 manufactured by Coors) is determined in the 0–140-kbar range of shock stresses. Longitudinal and transverse manganin gauges were used to determine the principal stresses in the shocked specimens. Shear strengths were determined from the difference between the longitudinal and lateral stresses. It was found that the shear strength remains essentially constant at about 27 kbar for shock stresses between 60 kbar (the Hugoniot elastic limit) and the maximum shock amplitude tested in this series (142 kbar). The source for the high shear strength is attributed to the confining pressures that strengthen the comminuted ceramic. Evidence for this interpretation is obtained by considering the release profiles as recorded by the longitudinal gauges when the free-surface rarefactions reach gauge location.

On induced discontinuities in a class of linear materials with internal state parameters

The governing differential equations of induced discontinuities behind longitudinal and transverse shock waves are derived for a class of linear materials with internal state parameters. These equations indicate that the behavior of the induced discontinuities depends, in particular, on the behavior of the shock amplitudes and non-linearly on the wave surface geometries. Solutions for the case of plane waves with initially flat profiles are obtained, and they indicate that the global behavior of the induced discontinuities need not be monotone depending on the interpretation of the material responses.

Hippocampal lesions and trace conditioning in the rabbit.

Trace conditioning of the nictitating membrane response (NMR) was examined in rabbits with lesions of the dorsal hippocampus and fimbria-fornix. Using a white noise conditional stimulus and an electrical shock unconditional stimulus, the number and amplitude of conditional responses (CRs) was similar in hippocampus-lesioned and control subjects. At some stages of conditioning, the latencies of CRs from hippocampus-lesioned subjects were slightly shorter than those of the controls. We suggest that the hippocampus is not essential for trace conditioning but may exert a modulatory influence on the timing of the CR.

Comparative Study of Tiapride and Neuroleptics with Anti-Dopamine Activity on Convulsive Seizure in Mice

The effects of tiapride on the convulsive seizures induced by pentylenetetrazole, strychnine, picrotoxin and bemegride, and on electric seizure are reported and compared with those of sulpiride, chlorpromazine, haloperidol and reserpine. The number of deaths and intensity of convulsion increased dose-dependently and also with the increase in amplitude of electric shock. Tiapride and a similar compound, sulpiride, did not affect these seizures, whereas chlorpromazine potentiated strychnine-induced and electric seizure. Haloperidol and reserpine potentiated electric seizure, and chlorpromazine and reserpine tended to potentiate bemegride-induced seizure. Reserpine also tended to potentiate pentylenetetrazole-induced seizure. These results suggest that tiapride would be clinically safer than other drugs with anti-dopamine activity, except for sulpiride.

Shocks and induced discontinuities in materials with internal state variables ? Applications to finite linear viscoelasticity

In this paper, we derive the coupled governing differential equations of shock amplitudes and induced discontinuities in materials with internal state variables. For the special case of finite linear viscoelasticity and weak shocks and under physically reasonable conditions, two examples of the solutions of the coupled equations are given. In the first example, the solutions of the shock and the induced discontinuity approach finite nonzero values in an oscillatory manner, very much akin to the forced vibrations of springmass-damper systems. In the second example, both solutions decay monotonically to zero.

Attenuation of Shocks by Viscoelastic Support

The method of singular surfaces was used in a recent paper to develop a numerical procedure for calculating the growth and decay of the amplitude of shock waves propagating into a one‐dimensional nonlinearly elastic body. Here we extend this approach to estimate the influence of such external effects as elastic support and viscous friction on the propagation of shocks. We conclude that even in the case of a homogeneous linearly elastic material these external effects account for some attenuation of the amplitude of the shock and/or the secondary waves. Further analysis of a nonlinear elastic material readily shows that an increase in the viscosity produces, among other things, a slower growth of the shock amplitude and that for some critical value it will even start with a decay. Numerical examples illustrate the applicability of the technique in the case of a homogeneous nonlinear elastic body subjected to external viscous friction.

The deflagration-to-detonation transition process for high-energy propellants - A review

Introduction T HE deflagration-to-detonation transition (DDT) in solid energetic materials has been of interest to researchers for many decades since it has a variety of areas of applicability, ranging from industrial to military. A recent example of the former is the detonation that occurred during the production of propelling charges for hunting ammunition. In the military area, the applicability extends from gun systems and projectile impact hazards to rocket motors. Hence, the DDT process has been investigated for both voidless (cast) and porous systems. We shall use the rocket motor environment to assess the DDT hazards associated with high-energy propellants. In the area of solid propellant rocket motors, an oft-asked question is Will a cast, well-manufactured rocket propellant grain undergo a transition to detonation from the burning mode? The answer is no, to the best of our knowledge. Although there are few journal articles directly providing this assessment, our knowledge of the DDT mechanism in gases, liquids, and solids provides a rationale for reaching this conclusion. The rationale is based on the thesis that the deflagration process must ultimately produce a shock wave to drive the system to detonation.' That is, the shock-to-detonation transition (SDT) is the final stage in any DDT process. Consequently, in the solid-propellant rocket motor situation, the confinement provided by the motor case must be sufficient to allow the pressure from deflagration to build up to a sufficiently high shock pressure to initiate the cast propellant. As will be shown below, these shock amplitudes cannot be reached for cast propellant systems confined in rocket motor cases.

Thermodynamic influences on one dimensional shock waves and induced discontinuities in thermoelastic bodies and second order effects

The behavior of one dimensional shock waves and induced discontinuities propagating in thermoelastic bodies is examined. We consider specific representations of the constitutive relations and show that to within second order of the shock strength differences between the predictions of thermoelastic and elastic theories are manifested. In particular, the shock speed is greater, the evolutionary behavior of the shock amplitude and that of the induced discontinuity are less pronounced in the thermoelastic theory. We also exhibit certain features of the solution of the coupled equations governing the behavior of shocks and induced discontinuities.

Hydrodynamic attenuation of weak shock waves

A pair of ordinary differential equations is derived, giving the shock amplitude and pressure gradient at the front as functions of range, for a weak shock propagating in a homogeneous medium. These equations enable us to complete a discussion originally given by G. I. Taylor in his work on the decay of blast waves. It is shown that the equations can be solved asymptotically at large ranges to give the well‐known results for amplitude decay and pulse spreading for plane, cylindrical, and spherical week shocks. [This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W‐7405‐Eng‐48.]

The behavior of induced discontinuities behind curved shocks in isotropic linear elastic materials

In this paper we examine the behavior of the induced discontinuities behind curved longitudinal and transverse shock waves in isotropic linear elastic materials. It is shown that in either case the governing differential equation of the induced discontinuity differs from that of the shock amplitude. The latter depends linearly on the second fundamental form of the shock surface and exhibits purely geometrical effects. The former, however, depends non-linearly on the second fundamental form of the shock surface, and on the shock amplitude. These terms are dominant for a strong shock and their effects diminish as the shock weakens. In particular, the governing differential equation for an acceleration wave is obtained in the limit as the shock amplitude vanishes. The results obtained are quite unexpected, and they demonstrate the complex evolutionary behavior of mechanical waves due to geometrical considerations alone.

Evolutionary behavior of induced discontinuities behind one dimensional shock waves in non-linear elastic materials

The governing differential equation of induced discontinuities behind one dimensinal shock waves in non-linear elastic materials has been derived. This equation depends, in particular, on the shock amplitude itself. Therefore, its solution depends on the solution of the governing equation of the shock amplitudes which, in turn, depend on the induced discontinuities. It is shown in the special case pertaining to a first-order approximation that there exists a critical shock amplitude Sc such that the evolutionary behavior of the induced discontinuities depends on the relative magnitudes of the shock amplitudes and Sc. However, in the special case pertaining to a second-order approximation the evolutionary behavior of the induced discontinuities is monotone.

The use of in-material stress gauges for estimating the dynamic yield strength of shock-loaded solids

We present stress-time history measurements, with in-material manganin gauges, from which we obtain the dynamic yield strength of shock-loaded 2024 Al and Ti-6 Al-4 V specimens. This yield strength is inferred from the cusp in the release curve which results from the elastoplastic nature of the release process. We show that while the strength of the aluminum alloy increases with shock amplitude, in the 0–100 kb range, that of the titanium alloy does not.

On induced discontinuities behind shock waves in isotropic linear viscoelastic materials

In this paper we derive the governing differential equations of induced discontinuities behind longitudinal and transverse shocks propagating in isotropic linear viscoelastic materials. It is shown that, when the shock amplitudes are finite, the evolutionary behavior of the induced discontinuities depends on additional properties of the stress relaxation functions,viz. the initial values of their first and second derivatives, and nonlinearly on the second fundamental form of the shock surfaces. The results indicate that evolutionary behavior of mechanical pulses containing shocks can be quite complex.

Eyeball retraction latency in the conscious rabbit measured with a new photodiode technique

A new technique is described for accurate, reliable measurement of eyeball retraction in the rabbit. A narrow film strip, on which a linear light intensity grating has been exposed, is attached to a contact lens which is placed on the animal's cornea. The other end of the intensity grating slides between a photodiode and an LED. The contact lens-film grating assembly moves freely with eyeball retraction and relaxation, causing changes in photodiode output. This device appears to be well tolerated by the animal. Large amplitude eyeball retractions occur in response to air puff stimulation directed at the upper or lower eyelid and to periorbital shock. Average eye retraction latency to stimulation of the abducens (VI) nerve with a chronically implanted stimulating electrode was 5.3 ms (S.D. = 0.75 ms) in the conscious rabbit as measured with our device. Latency to periorbital electrical shock was 9.3 ms (S.D. = 2.1 ms). Eye retraction latency decreased with increasing shock amplitudes. Rabbits readily acquired classically conditioned eyeball retractions, monitored with this device, when a white noise auditory stimulus was paired with an air puff directed at the eyelid.

Propagation in air of N waves produced by sparks

Weak sparks, of length 0.5–1.0 cm and energy per discharge 0.01–0.1 J, served to produce intense acoustic transients resembling N waves. Amplitude decay and waveform elongation were studied, for propagation distance up to 2 m, through the use of a wideband capacitor microphone with essentially uniform response from dc to 1 MHz. Within the range of propagation distances for which the first (compression) phase of the N wave was completely formed, the duration of this compression phase T and its amplitude ps were found to agree with the theoretical relations T=T0[1+σ0 ln(r/r0)]1/2 and ps =(r0 ps 0/r)[1+σ0 ln(r/r0)]−1/2, where σ0 is a parameter that depends upon the values of ps and T at a reference distance from the source r0. The time required for the amplitude of the head shock to increase from 5% to 95% of peak value was observed to vary from 0.45 μs (imposed by the microphone response) to greater than 2.0 μs as the wave traveled outward and as its amplitude decreased. Finally, the microphone was calibrated through use of the variation with distance of measured values of T; this new method has led to calculation of a free-field sensitivity that agrees within ±1 dB with the results of other calibrations.