On one-dimensional shock waves in composite materials modelled as interpenetrating solid continua

A one dimensional version of a theory of composite materials modelled as interpenetrating solid continua is used to study the propagation of shock waves in composites with two identifiable constituents. It is found that two distinct types of shock waves may propagate except when one of the constituents is a chopped fiber. The speeds at which the shock waves propagate are determined as are the differential equations which govern the evolutionary behaviour of the amplitudes of the waves. The implications of these results are studied in detail in a number of particular situations. Finally, the special results which hold when the amplitudes of the shock waves are infinitesimal are also presented.     

A one-dimensional meshfree particle formulation for simulating shock waves

In this paper, a one-dimensional meshfree particle formulation is proposed for simulating shock waves, which are associated with discontinuous phenomena. This new formulation is based on Taylor series expansion in the piecewise continuous regions on both sides of a discontinuity. The new formulation inherits the meshfree Lagrangian and particle nature of SPH, and is a natural extension and improvement on the traditional SPH method and the recently proposed corrective smoothed particle method (CSPM). The formulation is consistent even in the discontinuous regions. The resultant kernel and particle approximations consist of a primary part similar to that in CSPM, and a corrective part derived from the discontinuity. A numerical study is carried out to examine the performance of the formulation. The results show that the new formulation not only remedies the boundary deficiency problem but also simulates the discontinuity well. The formulation is applied to simulate the shock tube problem and a 1-D TNT slab detonation. It is found that the proposed formulation captures the shock wave at comparatively lower particle resolution. These preliminary numerical tests suggest that the new meshfree particle formulation is attractive in simulating hydrodynamic problems with discontinuities such as shocks waves.Received: 8 October 2002, Accepted: 10 May 2003, Published online: 15 August 2003     

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.     

Some remarks on the growth behavior of one-dimensional shock waves in non-linear materials

The growth behavior of both compressive and expansive one-dimensional shock waves which propagate into an unstrained region of a non-linear material exhibiting anelastic response, in the sense of Eckart, is analyzed. In each case, a differential equation governing the growth of the amplitude of the shock is derived and it is shown that a critical strain gradient may be defined. The growth behavior of the waves closely resembles the growth behavior of compressive and expansive shock waves propagating in sufficiently smooth non-linear materials with fading memory, i.e., in materials which can be approximated by linear viscoelastic materials for small relative strains.     

On the behaviour of one-dimensional waves in thermo-viscoelastic fluids

Summary We investigate the behaviour of one-dimensional acceleration waves propagating into a thermo-viscoelastic fluid through a model characterized by constitutive equations with both thermal and viscous relaxation times. The differential equation governing the amplitude of thermomechanical longitudinal waves is shown to be a Bernoulli equation. Waves entering a region at rest in thermal equilibrium are precisely discussed: our results confirm that longitudinal waves are not exceptional. Finally, attention is confined to purely mechanical transverse waves: it is proved that the amplitude of such waves satisfies a linear equation, hence transverse waves propagating into a region at equilibrium are exceptional.

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Sommario Si analizza il comportamento delle onde di accelerazione unidimensionali che si propagano in un fluido termo-viscoelastico caratterizzato da equazioni costitutive con tempi di rilassamento sia termico che viscoso. Si dimostra che l'ampiezza delle onde longitudinali termomeccaniche soddisfa un'equazione differenziale del tipo di Bernoulli. Si esaminano più in dettaglio le onde che entrano in regioni in equilibrio, termico e meccanico: i risultati ottenuti confermano che le onde longitudinali ammesse dalla teoria non sono eccezionali. Infine, si concentra l'attenzione sulle onde trasversali puramente meccaniche: si prova che l'ampiezza di tali onde soddisfa un'equazione lineare, quindi le onde trasversali propagantisi in regioni in equilibrio sono eccezionali.

     

On the propagation of one-dimensional acoustic waves in liquids

Within the framework of one-dimensional unsteady compressible flow in deformable pipes, the present work proposes expressions able to represent the stress state of a “fluid-pipe” continuous system. The coefficients needed to define the friction forces are calibrated against experimental results. The proposed expressions highlight the crucial role played by the fluid compressibility on the wave resistance.     

On the slow evolution of quasi-stationary shock waves

We consider the Burgers equation with a nonhomogeneous drift term, in the limit of small dissipation. A finite-dimensional manifold of slowly varying shock-like solutions is described, and a formal derivation of the dynamics on this manifold, in terms of a system of ordinary differential equations, is given. We also discuss the interpretation of the stationary solutions to the Burgers equation imbedded on the slow manifold.     

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.     

On the structure of shock waves in liquid-bubble mixtures

Asbtract The structure of shock waves in liquids containing gas bubbles is investigated theoretically. The mechanisms taken into account are the steepening of compression waves in the mixture by convection and the effects due to the motion of the bubbles with respect to the surrounding fluid. This relative motion, radial and translational, gives rise to dissipation and to dispersion caused by the inertia of the radial flow associated with an expanding or compressed bubble. For not too thick shocks the dissipation by radial motion around the bubbles dominates over the dissipation by relative translational motion, in mixtures with low gas content. The overall thickness of the shock appears to be determined by the dispersion effect. Dissipation, however, is necessary to permit a steady shock wave. It is shown that, analogous to undular bores, a stationary wave train may exist behind the shock wave.     

On one-dimensional acceleration waves in composite materials modeled as interpenetrating solid continua

A one-dimensional version of a theory of composite materials modeled as interpenetrating solid continua is applied in the analysis of acceleration waves in composites containing two identifiable constituents. As expected, two distinct acceleration waves always propagate except when one of the constituents consists of a chopped fiber. The influence of viscous type damping is included in only the volumetric interaction between the constituents in portions of the treatment. Equations are derived both for the propagation velocities and the varying amplitudes of the disturbance as a function of the state of the material immediately ahead of the wavefront. These rather general results are specialized to the case of a homogeneous steady-state ahead of the fast wave. The various types of behavior possible and the order of the discontinuities occurring across the wavefront are discussed in detail for a number of special cases.     

On flows behind shock waves reflected from a solid wall

The flow fields associated with the interaction of a normal shock wave with a plane wall kept at a constant temperature were studied based on kinetic theory which can describe appropriately the shock structure and its reflection process. With the use of a difference scheme, the time developments of the distributions of the fluid dynamic quantities (velocity, temperature, pressure and number density of the gas) were obtained numerically from the BGK model of the Boltzmann equation subject to the condition of diffusive-reflection at the wall for several cases of incident Mach number:M ₁=1.2, 1.5, 2.0, 3.0, 4.0, 5.0 and 6.0. The reflection process of the shocks is shown explicitly together with the resulting formation of the flow fields as time goes on. The nonzero uniform velocity toward the wall occurring between the viscous boundary layer and the reflected shock wave is found to be fairly large, the magnitude of which is of the order of several percent of the velocity induced behind the incident shock, decreasing as the incident Mach number increases. It is also seen that a region of positive velocity (away from the wall) within the viscous boundary layer manifests itself in the immediate vicinity of the wall, which is distinct for larger incident Mach numbers. Some of the calculated density profiles are compared with available experimental data and also with numerical results based on the Navier-Stokes equations. The agreement between the three results is fairly good except in the region close to the wall, where the difference in the conditions of these studies and the inappropriateness of the Navier-Stokes equations manifest themselves greatly in the gas behavior.This article was processed using Springer-Verlag TEX Shock Waves macro package 1990.     

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.     

Enhancement of shock waves

The flow field developed behind a shock wave propagating inside a constant cross-section conduit is solved numerically for two different cases. First, when the density of the ambient gas into which the shock propagates has a logarithmic change with distance. In the second, and the more practical case, the ambient gas is composed of pairs of air–helium layers having a continually decreasing width. It is shown that in both cases meaningful pressure amplification can be reached behind the transmitted shock wave. It is especially so in the second case. By proper choice of the number of air–helium layers and their width reduction ratio, pressure amplification as high as 7.5 can be obtained.