Numerical simulations of shock wave propagation in condensed multiphase materials

We propose to find out numerical solutions of a travelling shock wave in condensed mixtures by using a direct numerical simulation. Condensed multiphase materials under shock wave conditions are mechanically characterized by a unique pressure and a unique velocity. In this study, the mixture is considered as a collection of grains separated by interface between each material: this problem of interfaces is solved by a diffuse interface method. The results will be compared to existing one-dimensional numerical models, analytical solutions and also experimental data. The volume fraction (or the phase temperature) is not measured in experiments and it is then important to verify the behaviour of a phase quantity through various methods. A non-monotonous evolution of the volume fraction is obtained with analytical solution as well as numerical simulation.      

Modelling shock waves in composite materials using generalised orthotropic pressure

Continuum Mechanics and Thermodynamics - Excellent mechanical properties of composite materials have numerous engineering applications, especially in aerospace structures. The main characteristics...     

DDT and detonation waves in dust-air mixtures

This paper summarizes the studies of DDT and stable detonation waves in dust-air mixtures at the Stosswellenlabor of RWTH Aachen. The DDT process and propagation mechanism for stable heterogeneous dust detonations in air are essentially the same as in the oxygen environment studied previously. The dust DDT process in tubes is composed of a reaction compression stage followed by a reaction shock stage as the pre-detonation process. The transverse waves that couple the shock wave and the chemical energy release are responsible for the propagation of a stable dust-air detonation. However, the transverse wave spacing of dust-air mixtures is much larger. Therefore, DDT and propagation of a stable detonation in most industrial and agricultural, combustible dust-air mixtures require a tube that has a large diameter between 0.1 m and 1 m and a sufficient length-diameter ratio beyond 100, when an appropriately strong initiation energy is used. Two dust detonation tubes, 0.14 m and 0.3 m in diameter, were used for observation of the above-mentioned results in cornstarch, anthraquinone and aluminum dust suspended in air. Smoked-foil technique was also used to measure the cellular structure of dust detonations in the 0.3 m detonation tube. Received 11 February 2000 / Accepted 1 August 2000     

Propagation of shock and detonation waves in dust-laden gases

This article examines the flows of a two-phase mixture of a gas with solid particles arising as a result of the propagation of shock waves or detonation waves through a homogeneous medium at rest. It is assumed that the basic assumptions of the mechanics of mutually penetrating continua hold [1], whereby it is possible to describe the flow of each phase of the mixture within the framework of the mechanics of a continuous medium. We assume that the solid phase consists of identical, incompressible, and nondeformable particles of spherical shape. It is assumed that the temperature inside the particles is homogeneous. Collisions between particles and their Brownian motion are ignored. It is assumed that the carrier phase is an ideal gas (the viscosity is only allowed for in the interaction forces between phases). The contribution of the volume of the particles is not considered. On the basis of these assumptions, the following problems are considered: the propagation of a detonation wave in a mixture of a detonating gas and chemically inert particles and the motion of a dust-gas mixture in a shock tube in the presence of combustion of the particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 93–99, November–December, 1984.     

Reflection of shock waves from a solid boundary in a mixture of condensed materials 2. Nonequilibrium approximation

The process of reflection of shock waves from a solid wall in a two-component mixture of condensed materials is numerically studied with account of the difference in velocities and pressures of the components within the framework of mechanics of heterogeneous media. It is shown that a shock wave (SW) of the dispersed type with monotonic velocity profiles. A dispersed SW with a nonmonotonic velocity profile in the light component and a monotonic velocity profile in the heavy component is reflected by an SW of the dispersed-frozen type. When a frozen-dispersed SW is reflected, its type is either preserved, or changed to the dispersed-frozen structure depending on the initial parameters of the mixture. A dispersed-frozen SW is reflected by an SW of the same type with slight changes in the velocity and pressure profiles. A frozen SW of the two-front configuration can be reflected as an SW of the dispersed-frozen type or a frozen SW of the two-wave configuration. It is shown that a boundary layer is formed near the wall, where the volume concentration and the density of the light component exceed the corresponding values behind the reflected SW. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 3–10, November–December, 1999.     

Reflection of shock waves from a solid boundary in a mixture of condensed materials. 1. Equilibrium approximation

The process of reflection of shock waves (SW) from a solid wall in a two-component mixture of condensed materials is studied within the framework of mechanics of heterogeneous media. The velocity of a reflected SW and the values of the parameters behind its front are analytically determined as functions of the velocity of the incident wave and the initial parameters of the mixture. It is shown that the absolute value of the velocity of the reflected SW can be greater than the velocity of the incident SW in mixtures with a small content of the light component and at low velocities of the incident shock wave. The nonmonotonic character of the dependence of pressure in the final equilibrium state behind the incident SW on the initial volume concentration of particles is demonstrated. The velocity of the incident SW is estimated for the case where a similar effect is also observed behind a reflected SW. It is established that, for weak shock waves, the dependence of the amplification factor of the reflected SW on the initial volume concentration of the light component is nonmonotonic and has a local maximum. It is noted that, as the velocity of the incident SW increases, the effect of compacting of the mixture (increase in concentration of the heavy component) behind the reflected SW becomes much less pronounced than in a passing SW. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences. Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 73–78, September–October, 1999.     

Comparison of critical conditions for DDT in regular and irregular cellular detonation systems

Abstract. The results of an experimental study of DDT in mixtures with regular and irregular detonation cellular structures are presented. Experiments were carried out in a tube 174 mm i. d. with obstacles (blockage ratios were 0.1, 0.3, and 0.6). Mixtures used were hydrogen–air and stoichiometric hydrogen–oxygen diluted with , Ar, and He. The critical conditions for DDT are shown to depend on the regularity of the cellular structure of test mixtures. The critical values of the cell sizes in Ar- and He-diluted mixtures are shown to be significantly smaller than those in -diluted mixtures. This means that systems with a highly regular detonation cellular structure have far less capacity for undergoing DDT compared to irregular ones with the same values of detonation cell sizes. Received 18 November 1999 / Accepted 15 May 2000     

On the behavior of stress waves in composite materials—I. A universal set of boundary conditions

In recent years experiments on uniaxially reinforced composites have revealed anomalous behavior in the stress-wave propagation characteristics of these materials. Whenever the exposed ends of both composite constituents were subjected to moderate pressures of a few kilobars the number of stable propagating waves generated within the composite exceeded by one the number of waves calculated through conventional composite models. This effect greatly increased the wave dispersion and rise-time in the experimentally observed stress wave.The key to the origin of this phenomenon is quite elementary. The composite debonds internally. When the bond between the reinforcing and matrix fails, the composite attains an additional degree of freedom which results in an additional stable propagating wave. Since conventional composite models do not allow for this debonding, they cannot account for the resulting wave. However, as was shown in an earlier paper, direct application of the theory of elasticity to this problem results in wave velocities and mode shapes for all of the waves.The solution to the total problem, including the determination of the various wave amplitudes, was previously hampered by an insufficient set of boundary conditions. The usual procedure was to impose continuity of stress and displacement at the boundary between the composite and the adjoining homogeneous material where the volumetric averages of stress and displacement were used for the composite. While these conditions are necessary and sufficient for the bonded composite problem, they are insufficient for the debonded composite problem. The additional degree of freedom in the debonded problem makes the use of an additional boundary condition necessary. This additional boundary condition is the subject of this paper.     

Modelling of stress softening in elastomeric materials: foundations of simple theories

This work is concerned with formulation of constitutive relations for materials exhibiting the stress softening phenomenon (known as the Mullins effect) typical observed in elastomeric and other amorphous materials during loading–reloading cycles. It is assumed that microstructural changes in such materials during the deformation process can be represented by a single scalar-valued softening variable whose evolution is accompanied by microforces satisfying their own law of balance, besides the classical laws of mechanics underlying macroscopic deformation of a material. The constitutive equations are then derived in consistency with thermodynamics of irreversible processes with the restriction to purely mechanical theory. The general form of the derived constitutive equations is subsequently simplified through introduction of additional assumptions leading to various models of the stress softening phenomenon. As an illustration of the general theory, it is shown that the so-called pseudo-elastic model proposed in the literature may be derived without an ad hoc postulate of the variational principle.     

Explicit expression of polarization vector for surface waves,slip waves,Stoneley waves and interfacial slip waves in anisotropic elastic materials

We present explicit expression of the polarization vector for surface waves and slip waves in an anisotropic elastic half-space, and Stoneley waves and interfacial slip waves in two dissimilar anisotropic elastic half-spaces. An unexpected result is that, in the case of interfacial slip waves, the polarization vector for the material in the half-space _x2≥0$x_2\ge 0$ does not depend explicitly on the material property in the half-space _x2≤0$x_2\le 0$. It depends on the material property in the half-space _x2≤0$x_2\le 0$ implicitly through the interfacial slip wave speed υ$\upsilon$. The same is true for the polarization vector for the material in the half-space _x2≤0$x_2\le 0$.     

Comparative characteristics of strong shock and detonation waves in bubble media by an electrical wire explosion

Strong shock and detonation waves in inert and chemically active bubble media, which are generated by a wire explosion initiated by a capacitor with a stored energy $W_0 =12.3$ –1,600 J, is experimentally studied. The measurements are performed near the wire and far from the wire in a vertical shock tube 4.5 m long with a volume fraction of the gas in the medium $\beta _0 =1$ –4 %. It is shown that in inert bubble medium, a short intensely decaying shock wave (SW) with intense pressure oscillations is formed in the vicinity of wire explosion point; near the explosion point at $\beta _0 \le 2$  % the SW propagates with the velocity of sound in a liquid. In chemically active bubble medium, an unsteady detonation wave generated by a wire explosion is formed. The pressure amplitude and the velocity of this wave are greater and the length is smaller than those of SW in an inert bubble medium in the same range of explosion energy. It is found that in the interval of low energy explosion from ${\sim }12$ to 64 J, the formation of the bubble detonation wave occurs faster than that at high energies ( $3\times 10^{2}$ – $10^{3}$  J).     

High-frequency Rayleigh waves in materials with micro-structure and couple-stress effects

A well-known deficiency of the classical theory of elasticity is that it does not predict dispersive Rayleigh-wave motions at any frequency. This contradicts experimental data and predictions of the discrete particle theory (atomic-lattice approach) for high frequencies. The present work is intended to explore whether the elastic couple-stress theory with micro-structure can overcome the deficiency of the classical theory. Our analysis shows indeed that Rayleigh waves propagating along the surface of a half-space are dispersive at high frequencies, a result that can be useful in applications of high-frequency surface waves where the wavelength is often of the micron order. Provided that certain relations hold between the various micro-structure parameters entering the theory employed here, the dispersion curves of these waves have the same form as that given by previous analyses based on the atomic-lattice theory. In this way, the present analysis gives means to obtain estimates for micro-structure parameters of the couple-stress theory. Besides the Rayleigh-wave results reported here, basic theoretical results for the kinetic energy and momentum balance laws in micro-structured media with couple-stress effects are derived and presented.     

Global modelling of heat release during initiation and propagation of detonation and deflagration waves in methane-air-particle systems

In the present paper the direct initiation of a self supporting detonation and propagation of a low-speed combustion in methane-air-coal particles mixtures are solved. For particles, a heterogeneous regime of combustion is used, for methane one overall chemical reaction is taken into account: CH + 2O = CO + 2HO. The heat release rate is assumed to be defined as a delay time based on the well-known thermal theory of Frank-Kamenetsky (1967). The proposed model allows one to investigate the influence inert particles or coal dust on the explosion limits of methane-air mixtures. It is shown that the addition of a limited quantity of particles leads to detonation stability. In low speed combustion problems this method allows one to get a good correlation between theoretical and experimental velocities of steady flame propagation in carbon-hydrogen gaseous mixtures. Coal dust influence on gasdynamics of a methane-air mixture combustion is investigated in an unsteady problem by using of the global modelling. It is shown that limited coal dust concentration increases the flame wave intensity in lean methane-air mixtures in contrast to inert particles. In stoichiometric gas mixtures, sand and coal dusts decrease a flame velocity. Far from the ignition point flame, the velocity is largely defined by the dust mass concentration and not by the size of particles. Received 5 July 1997 / Accepted 13 July 1998     

Propagation of detonation waves in electric and magnetic fields appearing as a result of concentrated energy input

The propagation of detonation waves in an electromagnetic field is investigated. The effect of a constant external electromagnetic field on the motion of the detonation wave and the combustion products behind the wave is analyzed for small magnetic Reynolds numbers. The detonation is initiated by a plane point explosion.     

Superposition of finite-amplitude transverse waves in deformed Mooney-Rivlin and neo-Hookean materials

In this paper, waves propagating in Mooney-Rivlin and neo-Hookean non-linear elastic materials subjected to a homogeneous pre-strain are considered. In a previous paper, Boulanger and Hayes [Finite-amplitude waves in deformed Mooney-Rivlin materials, Q. J. Mech. Appl. Math. 45 (1992) 575-593] showed, for deformed Mooney-Rivlin materials, that the superposition of two finite-amplitude shear waves polarized in different directions (orthogonal to each other) and propagating along the same direction is an exact solution of the equations of motion. The two waves do not interact. Here, we are interested in superpositions of waves propagating in different directions. Two types of superpositions are considered: superpositions of waves polarized in the same direction, and also superposition of waves polarized in different directions. It is shown that such superpositions are exact solutions of the equations of motion with appropriate choices of the propagation and polarization directions.     

The propagation and growth of acceleration waves in constrained thermoelastic materials

Propagation conditions are derived for a thermoelastic material subject to arbitrary thermo-kinematic constraints. The results are rendered more concise by rephrasing existing thermodynamic theories for constrained materials in such a way, that the constraint equations may be absorbed into the free energy function. Propagation conditions for homothermal and homentropic waves are shown to be of Fresnel-Hadamard type. Equations governing the growth of acceleration amplitude are also derived for plane homothermal or homentropic waves; these are shown to be similar to the corresponding equations for unconstrained materials.     

On the types and number of plane waves in hypoelastic materials

General principles are formulated for modeling the elastic deformation of materials and analyzing plane waves in nonlinearly elastic materials such as hyperelastic, hypoelastic, and those governed by the general law of elasticity. The results of studying the propagation of plane waves in hypoelastic materials are further outlined. The influence of initial stresses and initial velocities on the types and number of plane waves is studied. Wave effects characteristic of hypoelastic materials are predicted theoretically. One of such effects is blocking of certain types of plane waves by initial stresses __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 96–107, November 2005.     

Finite and symmetric thermomechanical waves in materials with internal state variables

The paper is devoted to the study of wave propagation through materials with internal state variables under two main assumptions: finite speed of propagation and symmetry of acceleration waves. Additional constitutive assumptions are also used: heat flux does not depend on the temperature gradient and the time-derivatives of internal state variables are linear functions of the temperature gradient. Under all these hypotheses, in the neighborhood of a strong equilibrium state, one finds four real and symmetric possible acceleration waves, at least two of them being coupled waves, and heat flux results an internal state variable. All these results are obtained in the general three-dimensional case. As an illustration, the isotropic linear theory is considered, where both acceleration and shock waves are treated.