Shock wave propagation into a surface depression

An aircraft travelling at supersonic speeds close to the ground generates a bow wave which is reflected off the ground surface. If a valley is traversed a complex reflection pattern will be generated. Similar patterns will evolve with a plane wave traversing a depression on a surface or structure. To simulate the process a planar shock wave, generated in a shock tube, is moved over several notched wedge configurations. Schlieren photographs were produced to assist in identifying the resulting complex three-dimensional wave structures and then verified and extended by three- dimensional computations. The valley geometries investigated are rectangular, triangular, parabolic and conical with a number of valley floor inclinations. The main features are extracted in surface models to demonstrate the complexity of the flow, and in particular in the case where the reflection is regular on the ground plane and Mach reflection in the valley.      

Shock wave propagation in a branched duct

The propagation of a planar shock wave in a 90° branched duct is studied experimentally and numerically. It is shown that the interaction of the transmitted shock wave with the branching segment results in a complex, two-dimensional unsteady flow. Multiple shock wave reflections from the duct's walls cause weakening of transmitted waves and, at late times, an approach to an equilibrium, one-dimensional flow. While at most places along the branched duct walls calculated pressures are lower than that existing behind the original incident shock wave, at the branching segment's right corner, where a head on-collision between the transmitted wave and the corner is experienced, pressures that are significantly higher than those existing behind the original incident shock wave are encountered. The numerically evaluated pressures can be accepted with confidence, due to the very good agreement found between experimental and numerical results with respect to the geometry of the complex wave pattern observed inside the branched duct. Received 15 July 1996 / Accepted 20 February 1997     

Second-gradient homogenized model for wave propagation in heterogeneous periodic media

The paper is focused on a homogenization procedure for the analysis of wave propagation in materials with periodic microstructure. By a reformulation of the variational-asymptotic homogenization technique recently proposed by Bacigalupo and Gambarotta (2012a), a second-gradient continuum model is derived, which provides a sufficiently accurate approximation of the lowest (acoustic) branch of the dispersion curves obtained by the Floquet–Bloch theory and may be a useful tool for the wave propagation analysis in bounded domains. The multi-scale kinematics is described through micro-fluctuation functions of the displacement field, which are derived by the solution of a recurrent sequence of cell BVPs and obtained as the superposition of a static and dynamic contribution. The latters are proportional to the even powers of the phase velocity and consequently the micro-fluctuation functions also depend on the direction of propagation. Therefore, both the higher order elastic moduli and the inertial terms result to depend by the dynamic correctors. This approach is applied to the study of wave propagation in layered bi-materials with orthotropic phases, having an axis of orthotropy parallel to the direction of layering, in which case, the overall elastic and inertial constants can be determined analytically. The reliability of the proposed procedure is analysed by comparing the obtained dispersion functions with those derived by the Floquet–Bloch theory.     

Special case of a traveling wave in one model of a multicomponent medium

The Kuropatenko model for a multicomponent medium whose components are polytropic gases is considered. It is assumed that, as x → ±∞, the multicomponent medium is in a homogeneous state with constant gas-dynamic parameters — velocity, pressure, and temperature. For the traveling wave flows, conditions similar to the Hugoniot conditions are obtained and used to uniquely determine the flow parameters for x → −∞ from the flow parameters x → +∞ and traveling wave velocity. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 39–47, July–August, 2009.     

Love wave propagation in heterogeneous micropolar media

The present paper framed to study the impact of heterogeneity on propagation of Love wave in a heterogeneous micropolar layer over an elastic inhomogeneous stratum, when both rigidity and density are assumed to vary linearly with depth. The equations of motion have been formulated separately for layer and half-space under suitable boundary conditions. Analytical solution for the dispersion equation has been obtained using method of separation of variables by means of the Airy function and Whittaker function. Some particular cases have also been investigated. Further, as a special case the velocity equation for isotropic layer over a homogeneous half-space coincides with the standard result of Love wave. Numerical calculations of frequency relation have been performed and depicted by means of graphs to exhibit the substantial impact of heterogeneity, micropolar parameters and wave number on the phase velocity of Love wave. The wave velocity is strongly influenced by these parameters.     

Three-dimensional propagation of the transmitted shock wave in a square cross-sectional chamber

An investigation into the three-dimensional propagation of the transmitted shock wave in a square cross-section chamber was described in this paper, and the work was carried out numerically by solving the Euler equations with a dispersion-controlled scheme. Computational images were constructed from the density distribution of the transmitted shock wave discharging from the open end of the square shock tube and compared directly with holographic interferograms available for CFD validation. Two cases of the transmitted shock wave propagating at different Mach numbers in the same geometry were simulated. A special shock reflection system near the corner of the square cross-section chamber was observed, consisting of four shock waves: the transmitted shock wave, two reflection shock waves and a Mach stem. A contact surface may appear in the four-shock system when the transmitted shock wave becomes stronger. Both the secondary shock wave and the primary vortex loop are three-dimensional in the present case due to the non-uniform flow expansion behind the transmitted shock.PACS: 43.40.Nm     

Shock wave propagation and admissibility criteria in a nonlinear dielectric medium

We study the propagation of electromagnetic shock waves in an isotropic nonlinear dielectric medium. In order to select the physical shocks among all the mathematical solutions the usualLax conditions are applied. However, here they do not appear sufficient since strong shocks are present and the differential system is not strictly hyperbolic. So, two additional criteria are studied, theentropy growth condition and thereflection and transmission criterion, and a comparative analysis is developed. Finally, some experimental checks are suggested considering in particular the possible shape changes of an initial shock wave during its propagation.     

Shock propagation along a two-layer interface in confined geometry

The paper presents direct numerical simulations of the propagation of a strong planar shock wave along a two-layer interface confined between two walls. Transient stage and steady state of the propagation are described. In the steady state regime, a stationary irregular refraction of the shock and a vorticity sheet are characterized. Physics of the stationary irregular refraction state is explained. Shock velocity is studied. Quantities are shown to verify a set of generalized Rankine–Hugoniot relations. The unknown coupling mechanism between shear layer and pressure step is predicted by a neural network which leads to a predictive, gray-box type model.     

Dynamics of stress wave propagation in a chain of photoelastic discs impacted by a planar shock wave: Part II, numerical investigation

The propagation of stress waves through a chain of discs has been studied experimentally in Part I (Glam et al. [1]) and is completed here with numerical investigation using the standard package ABAQUS. A fair agreement is found between experimental findings and their simulations. Based on this agreement, parametric study of wave propagation through disc-chains was conducted. Specifically, effects associated with changes in the disc diameter, material density, stiffness/rigidity and the number of discs in the chain on the stressed chain have been studied. It was found that the propagation velocity of the evolved waves increases with improving contacts between the chain’s discs by exposing the chain to a static load before its dynamic loading. The wave- propagation velocity decreases with increase in the discs material density and it increases when its diameter increases. In case of a chain composed of small diameter discs and/or small material density, the transmitted stress wave is first strengthened and only at discs further down the chain it starts decaying. When checking the influence of the dynamic-loading duration it was found that long dynamic-load duration dissolves quickly into short pulses. It was also found that there is a ‘characteristic’ wave for a given chain. This wave propagates with minimal dispersion. Dynamic loads having shorter time duration than the ‘characteristic’ one experiences significant attenuation.     

Dynamics of stress wave propagation in a chain of photoelastic discs impacted by a planar shock wave; Part I, experimental investigation

The propagation of stress waves through a chain of discs has been studied experimentally. Optically transparent 20-mm diameter discs, made of epoxy, were loaded dynamically by head-on collision with an incident planar shock wave. The loading was done in a vertical shock tube. The head-on collision between the punch-plate, placed on top of the chain of discs, and the incident shock wave resulted in a head-on reflected shock wave inducing behind it a fairly uniform step-wise pressure pulse having duration of about 6 ms. The recorded fringe patterns of the stress field, in the discs-chain, show that the input pressure pulse was broken into several oscillating cycles. The back and forth bouncing of stress waves gave rise to two different modes of the contact stress oscillations, which continued until the overall stress reaches equilibrium with the input conditions. The registered propagation velocity of the stress wave was significantly lower than the appropriate speed of sound in the material from which the discs were made.      

Influence of a rigid skeleton pore structure on wave propagation in a fluid-filling porous medium

This work concerns an analysis of the influence of a rigid skeleton pore structure on wave propagation in a fluid-filling porous medium. The analysis is based on the continuum theory of a deformable porous medium in which the pore structure is described by two macroparameters. Considerations comprise two questions: the influence of the pore structure on wave-propagation velocity analysed for the quasilinear case and the role of structure in the reflection-refraction wave phenomenon in fluid at the contact surface of two porous media. It has been shown that the pore structure reduces the velocity of wave and together with the angle of incidence it defines the reflection-refraction wave phenomenon.     

Discrete element simulation of shock wave propagation in polycrystalline copper

The implementation of the characteristic of compressive plasticity into the Discrete Element Code, DM2, while maintaining its quasi-molecular scheme, is described. The code is used to simulate the shock compression of polycrystalline copper at 3.35 and 11.0 GPa. The model polycrystal has a normal distribution of grain sizes, with mean diameter 14 μm, and three distinct grain orientations are permitted with respect to the shock direction; 〈1 0 0〉, 〈1 1 0〉, and 〈1 1 1〉. Particle velocity dispersion (PVD) is present in the shock-induced flow, attaining its maximum magnitude at the plastic wave rise. PVD normalised to the average particle velocity of and are yielded for the 3.35 and 11.0 GPa shocks, respectively, and are of the same order as those seen in the experiment. Non-planar elastic and plastic wave fronts are present, the distribution in shock front position increasing with propagation distance. The rate of increase of the spread in shock front positions is found to be significantly smaller than that seen in probabilistic calculations on nickel polycrystals, and this difference is attributed, in the main, to grain interaction. Reflections at free surfaces yield a region of tension near to the target free surface. Due to the dispersive nature of the shock particle velocity and the non-planarity of the shock front, the tensile pressure is distributed. This may have implications for the spall strength, which are discussed. Simulations reveal a transient shear stress distribution behind the shock front. Such a distribution agrees with that put forward by Lipkin and Asay to explain the quasi-elastic reloading phenomenon. Simulation of reloading shocks show that the shear stress distribution can give rise to quasi-elastic reloading on the grain scale.     

Shock enhancement of cellular structures under impact loading: Part I Experiments

This paper aims at showing experimental proof of the existence of a shock front in cellular structures under impact loading, especially at low critical impact velocities around 50 m/s. First, an original testing procedure using a large diameter Nylon Hopkinson bar is introduced. With this large diameter soft Hopkinson bar, tests under two different configurations (pressure bar behind/ahead of the supposed shock front) at the same impact speed are used to obtain the force/time histories behind and ahead of the assumed shock front within the cellular material specimen.Stress jumps (up to 60% of initial stress level) as well as shock front speed are measured for tests at 55 m/s on Alporas foams and nickel hollow sphere agglomerates, whereas no significant shock enhancement is observed for Cymat foams and 5056 aluminium honeycombs. The corresponding rate sensitivity of the studied cellular structures is also measured and it is proven that it is not responsible for the sharp strength enhancement.A photomechanical measurement of the shock front speed is also proposed to obtain a direct experimental proof. The displacement and strain fields during the test are obtained by correlating images shot with a high speed camera. The strain field measurements at different times show that the shock front discontinuity propagates and allows for the measurement of the propagation velocity.All the experimental evidences enable us to confirm the existence of a shock front enhancement even at quite low impact velocities for a number of studied materials.     

Shock enhancement of cellular structures under impact loading: Part II analysis

Numerical simulations of two distinct testing configurations using a Hopkinson bar (pressure bar behind/ahead of the shock front) are performed with an explicit finite element code. It allows us to confirm the observed test data such as velocity and force time histories at the measurement surface. A comparison of the simulated local strain fields during shock front propagation with those measured by image correlation provides an additional proof of the validity of such simulations.Very simple rate insensitive phenomenological constitutive model are used in such simulations. It shows that the shock effect is captured numerically with a basic densification feature. It means that strength enhancement due to shock should not be integrated in the constitutive model of foam-like materials used in industrial FE codes.In order to separate shock enhancement from entire strength enhancement, an improvement of an existing model with easily identifiable parameters for shock enhancement prediction is proposed. For a quick estimate of the shock enhancement level, a simple power law densification model is proposed instead of the classical RPPL model proposed by Reid and co-workers [Tan et al., 2005. Dynamic compressive strength properties of aluminium foams. Part I—experimental data and observations. J. Mech. Phys. Solids 53, 2174-2205]. It is aimed at eliminating the parameter identification uncertainty of the RPPL model. Such an improved model is easily identifiable and gives a good prediction of the shock enhancement level.     

An experimental investigation of shock wave propagation in periodically layered composites

In heterogeneous media, scattering due to interfaces/microstructure between dissimilar materials could play an important role in shock wave dissipation and dispersion. In this work, the influence of interface scattering on finite-amplitude shock waves was experimentally investigated by impacting flyer plates onto periodically layered polycarbonate/6061 aluminum, polycarbonate/304 stainless steel and polycarbonate/glass composites. Experimental results (obtained using velocity interferometer and stress gage) show that these periodically layered composites can support steady structured shock waves. Due to interface scattering, the effective shock viscosity increases with the increase of interface impedance mismatch, and decreases with the increase of interface density (interface area per unit volume) and loading amplitude. For the composites studied here, the strain rate within the shock front is roughly proportional to the square of the shock stress. This indicates that layered composites have much larger shock viscosity due to the interface/microstructure scattering in comparison with the increase of shock strain rate by the fourth power of the shock stress for homogeneous metals. Experimental results also show that due to the scattering effects, shock propagation in the layered composites is dramatically slowed down and the shock speed in composites can be lower than that either of its components.     

Head on collision of a planar shock wave with a dusty gas layer

The head-on collision of a planar shock wave with a dust-air suspension is studied numerically. In this study the suspension is placed inside a conduit adjacent to its rigid end-wall. It is shown that as a result of this collision two different types of transmitted shock waves are possible, depending on the strength of the incident shock wave and the dust loading ratio in the suspension. One possibility is a partially dispersed shock wave, the other is a compression wave. The flow fields resulting in these two options are investigated. It is shown that in both cases, at late times after the head-on reflection of the transmitted shock wave from the conduit end-wall a negative flow (away from the end-wall) is evident. The observed flow behavior may suggest a kind of dust particle lifting mechanism that could shed new light on the complex phenomenon of dust entrainment behind sliding shock waves.      

On shock wave propagation in a branched channel with particles

The problem of wave propagation in a dust–air mixture inside a branched channel has not been studied widely in literature, even though this topic has many important applications especially in process safety (dust explosions). In this paper, a shock wave interaction with a cloud of solid particles, and the further behaviour of both gas and particulate phases were studied using numerical techniques. The geometry mimicked a real channel where bends or branches are common. Two numerical approaches were used: Eulerian–Eulerian and Eulerian–Lagrangian. Using Eulerian-Lagrangian simulation, it was possible to include the effects of particle–particle and particle–wall collisions in a realistic and direct manner. Results are mainly shown as snap-shots of particle positions during the simulations and statistics for the particle displacement. The results show that collisions significantly influence the process of particle cloud formation. PACS47.40.Nm, 02.60.Cb, 47.55.kf     

Shock wave diffraction by a square cavity filled with dusty gas

A simple two-dimensional square cavity model is used to study shock attenuating effects of dust suspension in air. The GRP scheme for compressible flows was extended to simulate the fluid dynamics of dilute dust suspensions, employing the conventional two-phase approximation. A planar shock of constant intensity propagated in pure air over flat ground and diffracted into a square cavity filled with a dusty quiescent suspension. Shock intensities were and , dust loading ratios were and , and particle diameters were and {\rm \mu}$m. It was found that the diffraction patterns in the cavity were decisively attenuated by the dust suspension, particularly for the higher loading ratio. The particle size has a pronounced effect on the flow and wave pattern developed inside the cavity. Wall pressure histories were recorded for each of the three cavity walls, showing a clear attenuating effect of the dust suspension. Received 15 November 1999 / Accepted 25 October 2000     

Moving shocks through metallic grids: their interaction and potential for blast wave mitigation

Numerical simulations and laboratory measurements have been used to illuminate the interaction of a moving shock wave impacting on metallic grids at various shock strengths and grid solidities. The experimental work was carried out in a large scale shock tube facility while computational work simulated the flow field with a time-dependent inviscid and a time-dependent viscous model. The pressure drop measured across the grids is a result of two phenomena which are associated with the impact of the shock on the metallic grids. First are the reflection and refraction of the incoming shock on the grid itself. This appears to be the main inviscid mechanism associated with the reduction of the strength of the transmitted shock. Second, viscous phenomena are present during the reflection and refraction of the wave as well as during the passage of the induced flow of the air through the grid. The experimental data of pressure drop across the grid obtained in the present investigation are compared with those obtained from computations. The numerical results slightly overpredict the experimental data of relative pressure drop which increases substantially with grid solidity at fixed flow Mach numbers. The processes of shock reflection and refraction are continuous and they can be extended in duration by using thicker grids that will result in lower compression rates of the structural loading and increase the viscous losses associated with these phenomena which will further attenuate the impacting shock. Preliminary theoretical analysis suggests that the use of a graded porosity/solidity material will result in higher pressure drop than a constant porosity/solidity material and thus provide effective blast mitigation.      

Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.