Explicit conditions for the existence of exceptional body waves and subsonic surface waves in anisotropic elastic solids

It is known that a subsonic surface (Rayleigh) wave exists in an anisotropic elastic half-space x₂  0 if the first transonic state is not of Type 1. If the first transonic state is of Type 1 but the limiting wave is not exceptional, a subsonic surface wave exists. If the first transonic state is of Type 1 and the limiting wave is exceptional, a subsonic surface wave exists when . It is shown that an exceptional body wave is necessarily an exceptional transonic wave, and could be an exceptional limiting wave. Only two wave speeds are possible for an exceptional body wave. We present explicit conditions in terms of the reduced elastic compliances for the existence of an exceptional body wave. If an exceptional body wave exists, conditions are given for identifying whether the transonic state is of Type 1. Hence, through the existence of an exceptional body wave we provide explicit conditions for the existence of a subsonic surface wave with the exception when needs to be computed.     

On shock waves in elastic-plastic solids

The behavior of shock waves in an elastic-plastic material is investigated with systematic reference to the theory of shocks in fluids. The classical hydrodynamic theory and the notions of the Hugoniot curve and of the Hugoniot contour are first briefly reviewed. Then, it is shown that continuous adiabatic compression is not isentropic and that, in general, the Hugoniot curve cannot be obtained by the classical rate independent elastic-plastic behavior. Two methods are proposed in order to overcome this difficulty. The second one, which is physically more satisfactory, requires the introduction of rate effects. It is shown that when the shock structure is composed of a purely elastic jump followed by a continuous profile, the Hugoniot curve can be defined independently of the precise formulation of the law for the rate effects.     

On the existence of intermediate magnetohydrodynamic shock waves

We consider the questions related to the structure of shock waves for a system of magnetohydrodynamic equations. Using Conley's connection matrix, we recover and extend earlier results due to C. Conley and J. Smoller. In particular, we give a simpler proof of the existence of fast and slow shocks with structure. We also demonstrate that for some viscosity parameters intermediate shocks occur. Furthermore, under an assumption of transversality, we show that there exist multi-parameter families of these intermediate shocks.This research was done while both authors were visiting the Lefschetz Center for Dynamical Systems at Brown University.Supported in part by the NSF under Grant DMS-8507056.Supported in part by AFOSR 87-0347.     

Propagation of steady shock waves in non-linear thermoviscoelastic solids

This paper is concerned with the dynamic response of a class of thermoviscoelastic solids. In particular a specific one-dimensional model, consistent with the laws of continuum thermodynamics, is proposed and applied to the problem of the propagation of steady shock waves. The governing equations are written in terms of material response functions which can be determined from shock wave, thermophysical, and bulk response data. The results of the analysis are compared with experimental steady wave studies involving the solid polymer, polymethyl methacrylate.     

Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

A set of transport equations for the growth or decay of theamplitudes of shock waves along an arbitrary propagation directionin three-dimensional nonlinear elastic solids is derived using theLagrangian coordinates.The transport equations obtained showthat the time derivative of the amplitude of a shock wave alongany propagation ray depends on (i) an unknown quantity immediatelybehind the shock wave,(ii) the two principal curvatures of theshock surface,(iii) the gradient taken on the shock surface ofthe normal shock wave speed and (iv) the inhomogeneous term.whichis related to the motion ahead of the shock surface.vanisheswhen the motion ahead of the shock surface is uniform.Severalchoices of the propagation vector are given for which the tran-sport equations can be simplified.Some universal relations,which relate the time derivatives of various jump quantities toeach other but which do not depend on the constitutive equationsof the material,are also presented.     

On shock waves in a special class of thermoelastic solids

This paper describes a thermoelastic model for shock waves in uniaxial strain based on a subclass of the so-called materials of Mie–Grüneisen type. We compare the Hugoniot curve with the isotherms and isentropes for this model, and we construct the shock-wave solution to a simple impact problem.     

Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids

The propagation of a three-dimensional shock wave in an elastic solid is studied. The material is assumed to be a simple elastic solid in which the Cauchy stress depends on the deformation gradient only. It is shown that the growth or decay of a discontinuity ψ depends on (i) an unknown quantity φ^? behind the shock wave, (ii) the two principal curvatures of the shock surface, (iii) the gradient on the shock surface of the shock wave speeds and (iv) the inhomogeneous term which depends on the motion ahead of the shock surface and vanishes when the motion ahead of the shock surface is uniform. If a proper choice is made of the propagation vectorb along which the growth or decay of the discontinuity is measured, the dependence on item (iii) can be avoided. However,b assumes different directions depending on the choice of discontinuity ψ with which one is concerned and the unknown quantity φ^? behind the shock wave on which one chooses to depend. As in the case of one-dimensional shock waves, the growth (or decay) of one discontinuity may not be accompanied by the growth (or decay) of other discontinuities. A universal equation relating the growth or decay of discontinuities in the normal stress, normal velocity and specific volume is also presented.     

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane _x3=0$x_3=0$. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.     

Nonequilibrium homogeneous condensation in rarefaction waves

Results of numerical calculations of condensing vapor expanding into a vacuum and in a supersonic conical nozzle are presented. Nonequilibrium homogeneous condensation is compared with the theory of determining parameters. It is proved that equality of the determining parameters ensures the unity of the course of such processes in different devices. The influence of unsteady nucleation on the maximum supercooling was also investigated. All the calculations were made for water and nitrogen vapors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 152–156, March–April, 1986.     

Stability of rarefaction waves in viscous media

We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, Burgers rarefaction wave, for initial perturbations w ₀ with small mass and localized as w ₀(x)= The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error.This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass (log (t)). These diffusion waves have amplitude (t ^-1/2logt) in linear degenerate transversal fields and (t ^-1/2logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.     

Model of the motion and shock waves in two-phase solids with phase transitions

The one-velocity and one-temperature model of the motion of a two-phase solid, in which each phase occupies a certain part of the volume, is considered. The investigation is carried out in Lagrangian variables, which offers certain advantages in solving one-dimensional nonstationary problems. The stress tensor for the mixture is decomposed into two parts -hydrostatic pressure, common to the two phases, associated with the three-term equation of state, and the deviator, which varies elastically up to a certain value and then remains constant. A certain relation, determined by the characteristic reaction time, is proposed for the phase transition kinetics. Then a solution is obtained for the problem of the nonstationary one-dimensional motion of a metal (iron) resulting from the impact of a plate against a target. The phase transitions (FeFe) behind the wave and their characteristic time have an important effect on the damping of the disturbance and on the zone in which these transitions go to completion. A method is proposed for determining the coefficient in the relation for the phase transition rate from the residual effect (hardening) after impact.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 88–95, January–February, 1970.     

Entropy and causality as criteria for the existence of shock waves in low temperature heat conduction

A set of symmetric hyberbolic field equations, describing heat conduction in dielectric solids at low temperatures, is studied with respect to the propagation of temperature shock waves. The field equations have been derived from the Boltzmann-Peierls equation and include the phenomenon of second sound, a special form of wavelike energy transport occuring in some crystals in a temperature range close to absolute zero.Two physical criteria, an entropy shock condition and the Lax condition, which is based on a causality argument, are applied to study the existence of so called hot and cold shocks. These are characterized by a temperature rise or fall across the shock respectively, and it turns out that the only possible solution to the problem is a hot shock, predicted by either one of the criteria.In the recent literature, however, a similar case was treated revealing a partial contradiction between the two criteria. Regarding the fact that there exists a proof of equivalence for small shocks, we were thus led to investigate this equivalence in the general case, which is illustrated by means of a simple example.     

Compression and rarefaction waves in shock-compressed metals

The behavior of duralumin and copper is studied under conditions of specimen loading by two successive shock waves and during unloading after the shock compression. The amplitude of the first shock wave was 150–250 kbar. Direct measurements were performed of the difference in main stresses behind the shock front in duralumin. The results obtained do not agree with existing concepts of the behavior of solids under dynamic loading. Possible causes of this divergence are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 146–153, March–April, 1976.In conclusion, the authors thank G. A. Savel'ev for his aid in the preparation and performance of the experiments.