Spherical wave propagation in a viscoplastic medium

Summary A study is presented of the problem of spherical wave propagation in an infinite viscoplastic medium. The surface of a spherical cavity is subjected to a uniform impact load which is continuously maintained. The material is elastic/viscoplastic, satisfying Mises condition, isotropic hardening, and viscoplastic incompressibility. A generalized form of Malvern's constitutive relation is used with a bilinear static shear stressshear strain curve, the method of characteristics, and an IBM 7040–7094 digital computer. Particular emphasis is placed on the influence of strain hardening and of the viscosity coefficient. It is shown that changes in the strain hardening coefficient and in the viscosity coefficient have considerable influence on the results, which is to some extent unexpected in view of previous work on this subject.

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Übersicht Eine konkave innere Oberfläche wird durch eine konstante gleichmäßig verteilte Kraft belastet. Das Material wird als elastisch-viscoplastisch angenommen mit isotroper Verfestigung und viscoplastischer Inkompressibilität; es genügt der von Mises'schen Fließbedingung. Es werden eine sinngemäße Erweiterung der Malvernschen Materialgleichung und numerische Lösungen für ein Material mit bilinearer Schubspannungs- and Scherungskennlinie angegeben. Dabei zeigt sich überraschenderweise, daß Änderungen des Verfestigungskoeffizienten und der Viscosität die spherische Wellenausbreitung erheblich beeinflussen.

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This investigation was supported by the National Science Foundation under a grant to Yale University.     

Effect of a shock wave on heat propagation

The nature of the propagation of a thermal wave produced by a powerful explosion was described in a number of papers, for example, [1–6]. It was shown by a numerical method [4] that a shock wave is present together with the thermal wave. In this paper, the effect of a homothermal shock wave on heat propagation is evaluated by an approximate method.     

Axisymmetric wave propagation in a layered transversely isotropic medium

Summary In this paper, the method of numerical integration along bicharacteristics is generalized to the case of layered transversely isotropic medium for analysing the axisymmetric stress wave propagation. The stability of the present scheme is studied. The advantages and limitations of the method are discussed. Received 12 June 1996; accepted for publication 6 May 1997     

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovski to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.     

Uniaxial wave propagation in a nonlinear thermoviscoelastic medium with temperature dependent properties

The problem of finite wave propagation in a nonlinearly thermoviscoelastic thin rod whose viscoelastic properties are temperature dependent is considered. The rod is subjected to mechanical or thermal time-dependent loading. The coupled equations of motion and heat conduction are based on a constitutive theory of nonisothermal nonlinear viscoelasticity which is described by single-integral terms only. This theory is reformulated here for the uniaxial motion of a compressible rubbery material. The solution of the field equations is obtained by a numerical procedure which is developed for the present case and is able to handle successfully shock waves whenever they built up in the nonlinear material.     

On the theory of transient wave propagation in a dispersive medium

Summary Choosing a simple parallel-plate waveguide, an exact expression is obtained for the transient response when a step-modulated carrier signal is applied. The analysis constitutes a modest extension of the early work of Sommerfeld and the more recent investigations of Rubinowicz and Knop. Numerical calculations are presented for a range of the parameters which have some relevance to propagation in the earth-ionosphere waveguide. The exact form of the transient envelope function is compared with an approximate version which is in the form of a Fresnel integral. It is shown that the approximate method gives a good qualitative estimate for the transient response characteristics. Thus, confidence is gained in applying it to other situations where an exact solution is not available.The research reported here was supported by the Advanced Research Projects Agency, Washington, D.C., under ARPA Order No. 183-62.Formerly the Central Radio Propagation Laboratory of the National Bureau of Standards.     

Shock wave propagation through a model one dimensional heterogeneous medium

We study the problem of impact-induced shock wave propagation through a model one-dimensional heterogeneous medium. This medium is made of a model material with spatially varying parameters such that it is heterogeneous to shock waves but homogeneous to elastic waves. Using the jump conditions and maximal dissipation criteria, we obtain the exact solution to the shock propagation problem. We use it to study how the nature of the heterogeneity changes material response, the structure of the shock front and the dissipation.     

Unsteady one-dimensional flow in a saturated porous medium with a volume heat source

The one-dimensional process of the heating of a saturated porous medium by a volume heat source as a result of the absorption of the energy of a high-frequency (frequency R~ 10¹–10³ MHz) electromagnetic wave is investigated. It is assumed that in the initial state the saturating (second) component is in the high-viscosity liquid or solid state. Under the action of the heat it is heated, melts, expands, becomes less viscous and under the pressure head created may flow relative to the stationary rock skeleton (first component). On the basis of the mathematical model proposed the basic laws of the process are analyzed and numerically investigated in the case of one-dimensional axisymmetric motion. It is shown that under actual conditions the dimensions of the thermal influence zone may be very considerable. Thus, by varying certain external factors it is possible to modify the dynamics of the process and the distributions of the temperature, pressure and phase velocity fields.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–124, July–August, 1991.     

Mean wave propagation in a slab of one-dimensional discrete random medium

A study has been made of the propagation of time harmonic waves through a one-dimensional medium of discrete scatterers randomly positioned over a finite interval L. The random medium is modeled by a Poisson impulse process with density λ. The invariant imbedding procedure is employed to obtain a set of initial value stochastic differential equations for the field inside the medium and the reflection coefficient of the layer. By using the Markov properties of the Poisson impulse process. exact integro-differential equations of the Kolmogorov-Feller type are derived for the probability density function of the reflection coefficient and the field. When the concentration of the scatterers is low, a two variable perturbation method in small λ is used to obtain an approximate solution for the mean field. It is shown that this solution, which varies exponentially with respect to λL, agrees exactly with the mean field obtained by Feldy's approximate method.     

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.     

Stress wave propagation in plates subjected to a transient line source

A method of treating the response of plates to a transient load, which was recently proposed, is here elaborated for the particular case of a plate subjected to a transient line source. Expressions presenting the various stresses as functions of time and position are derived. A numerical scheme, by which these stresses are calculated, is described. Some results concerning the behavior of stresses in the course of time are illustrated and discussed.     

Transient elastic wave propagation in a spherically symmetric bimaterial medium modeling the thorax

The transient response resulting from an impact wave on an elastic bimaterial, made out of a “hard” medium and a “soft” medium, welded at a spherical interface, have been investigated by using an integral transform technique. This technique permits isolation of the pressure and shear waves contributions to the wave field. The method of solution makes use of the generalized ray/Cagniard-de Hoop (GR/CdH) method associated with a “flattening approximation” (FA) technique, similar to the Earth flattening transformation used in geophysics. The GR/CdH method and the FA technique are briefly presented, together with their numerical implementations. The FA has proved to be useful in geophysical application, however, as far as the authors know, it has never been investigated for other applications. For the purpose of this paper, numerous tests of the method have been performed in order to check that the FA is appropriate to compute transient responses in the special case presented here. We could determine appropriate values for some parameters involved in the FA. This paper follows Grimal et al. [Int. J. Solid Struct. 39 (2002) 5345] in which we investigated the same bimaterial with a plane––instead of spherical––interface. Numerical examples are concerned with the propagation of an impact wave in the thorax modeled as a bimaterial (thoracic wall-lung). In addition to the effects of the weak coupling of the two media already observed in our previous study, we found that, for interface curvatures characteristic of those measured in the thorax, focalization of energy is manifest.     

Forced convection from an isolated heat source in a channel with porous medium

A numerical study is made of flow and heat transfer characteristics of forced convection in a channel that is partially filled with a porous medium. The flow geometry models convective cooling process in a printed circuit board system with a porous insert.The channel walls are assumed to be adiabatic. Comprehensive numerical solutions are acquired to the governing Navier-Stokes equations, using the Brinkman-Forchheimer-extended Darcy model for the regions of porous media. Details of flow and thermal fields are examined over ranges of the principal parameters; i.e., the Reynolds number Re, the Darcy number Da (≡K/H²), the thickness of the porous substrate S, and the ratio of thermal conductivities R_k (≡k_eff/k). Two types of the location of the porous block are considered. The maximum temperature at the heat source and the associated pressure drop are presented for varying Re, Da, S, and R_k. The results illustrate that as S increases or Da decreases, the fluid flow rate increases. Also, as R_k increases for fixed Da, heat transfer rates are augmented. Explicit influences of Re on the flow and heat transport characteristics are also scrutinized. Assessment is made of the utility of using a porous insert by cross comparing the gain in heat transport against the increase in pressure drop.     

非均匀吸收介质中地震波的广义传播矩阵解法

本文在复频域内,通过应用混合变量粘弹性波方程和线性常微分方程组的指数矩阵解法,给出了一种计算非均匀吸收介质中地震波传播的广义传播矩阵解法。该方法适用于各种粘弹性模型,可模拟任意震源及所产生的各种体波、面波,数值结果表明具有很高的计算精度。     

Impact of the volume of rooms on shock wave propagation within a multi-chamber system

The behavior of a shock wave generated by a hemispherical gaseous charge and propagating within a confined multi-chamber system is analyzed through the evolution of some of the shock parameters (maximum overpressure and positive impulse). The influence of a variation in the volume of the rooms on the pressure history inside the building is also studied. Several small-scale experiments have been carried out using an adjustable model representative of a pyrotechnic workshop. The experimental results show that the pressure histories are very complex. Yet, using a global approach, we were able to link the evolution of the arrival time of the shock wave within the building with the reference obtained in the free field. New parameters were developed to best fit the experimental maximal overpressure in the cells and in the corridor leading to two predictive laws used to estimate the maximal overpressure in the model.     

Thermal stresses in an isotropic trimaterial interacted with a pair of point heat source and heat sink

A general analytical solution for an isotropic trimaterial interacted with a point heat source is provided in this paper. Based on the method of analytical continuation in conjunction with the alternating technique, the solutions to heat conduction and thermoelasticity problems for three dissimilar media are first derived. A rapidly convergent series solution for both the temperature and stress functions, which is expressed in terms of an explicit general term of the complex potential of the corresponding homogeneous problem, is obtained in an elegant form. As a numerical illustration, the distributions of thermal stresses along the interface are presented for various material combinations and for different positions of the applied heat source and heat sink.     

Three-dimensional formulation of time-dependent wave propagation in a gyroanisotropic medium by Bergeron''s method

This paper discusses a numerical vector analysis method for wave propagation and scattering in three-dimensional space and the time domain. The approach is based on both the equivalent-circuit representation of Maxwell's equations and the formulation by Bergeron's method in the time domain. As an example of the application of this technique to a complicated medium, we consider time-dependent electromagnetic wave propagation in a gyroanisotropic medium, specifically the magnetized ferrite and the magnetized plasma. We demonstrate that the numerical approach accurately models relevant physical phenomena, including the Faraday rotation effect.     

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.     

Shear wave propagation in periodic phononic/photonic piezoelectric medium

Coupled electro-elastic SH waves propagating oblique to the lamination of a one dimensional piezoelectric periodic structure are considered in the framework of the full system of Maxwell’s electrodynamic equations. The dispersion equation has been obtained and numerical analyses carried out for two kinds of composites both consisting of two different piezoelectric materials. The results demonstrate the significant effect of piezoelectricity on the widths of band gaps at acoustic frequencies and confirm that it does not affect the band gaps at optical frequencies.