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.     

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.     

Finite amplitude one-dimensional wave propagation in a thermoelastic half-space

The problem of finite wave propagation in a nonlinearly thermoelastic half-space is considered. The surface of the half-space is subjected to a time-dependent thermal and normal mechanical loading. The solution is obtained by a numerical procedure, which is shown to furnish accurate results, and linear dynamic thermoelastic problems are obtained as special cases. The accuracy of the results is checked by comparison with some known analytical solutions which can be obtained in some special cases of both the linear and the nonlinear problems. In those cases where the solution contains shocks, it is shown that the numerical results satisfy the necessary jumps conditions which need to hold across such discontinuities.     

Transient stress wave propagation in one-dimensional micropolar bodies

Certain types of structures and materials, such as engineered multi-scale systems and comminuted zones in failed ceramics, may be modeled using continuum theories incorporating additional kinematic degrees of freedom beyond the scope of classical continuum theories. If such material systems are to be subjected to high strain rate loads, such as those resulting from ballistic impact or blast, it will be necessary to develop models capable of describing transient stress wave propagation through these media. Such a model is formulated, solved, and applied to the impact between two bodies and to a two-layer bar or strip subjected to an instantaneously applied stress. Results from these examples suggest that the model parameters, and therefore constitutive properties and geometries, may be tuned to reduce and control the transmission of stress through these bodies.     

Nonlinear wave effects in a fluid-saturated porous medium

Some one-dimensional nonlinear effects associated with wave propagation in weakly permeable fluid-saturated porous media are investigated. The effect of nonlinearity on the damping of monoharmonic waves is estimated and, moreover, the characteristics of the nonlinear parametric interaction of two waves excited in the medium by two monoharmonic sources of different frequencies are established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 74–77, January–February, 1992.     

Nonlinear wave propagation in optical fibres

A generalised mathematical theory leading to the evolution equation of nonlinear pulses propagating in an optical fibre waveguide is presented. The magnitudes of the optical self-steepening and third-order dispersion coefficients are calculated and their relative importance is assessed. The possibility of self-steepening is carefully examined and represented on a pulse-displacement-distance parameter diagram. It is concluded, for specimen single mode fibre data, that, even for a 1% deviation from zero group dispersion, the shock term interacts with the group dispersion to produce a pulse distortion in the form of a velocity change limited by the group dispersion rather than selfsteepening. Also, for such a fibre, self-steepening is so small that vast runs down the fibre are required for it to be seen. Any observation of self-steepening in fibres will require much closer tuning to zero group dispersion and will need the suppression of the third-order term.     

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     

On wave propagation and uniqueness in one-dimensional elastic bodies

A notion referred to as the Wave Propagation Property is analyzed in the context of the nonlinear theory of one-dimensional elastic bodies. Roughly speaking, a body possesses this property if mechanical disturbances propagate with bounded speed. A uniqueness theorem is proven with the aid of the results on wave propagation.

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Zusammenfassung Ein Begriff, bezeichnet als Wellenfortpflanzungseigenschaft, wird in Zusammenhang mit der nicht-linearen Theorie des eindimensionalen elastischen Körpers untersucht. Ein Körper besitzt, gross gesprochen, diese Eigenschaft, wenn sich mechanische Störungen mit beschränkter Geschwindigkeit fortpflanzen. Mit Hilfe des Ergebnisses für Wellenfortpflanzung wird ein Eindeutigkeitsatz bewiesen.

     

Symplectic analysis for regulating wave propagation in a one-dimensional nonlinear graded metamaterial

An analytical method, called the symplectic mathematical method, is proposed to study the wave propagation in a spring-mass chain with gradient arranged local resonators and nonlinear ground springs. Combined with the linearized perturbation approach, the symplectic transform matrix for a unit cell of the weakly nonlinear graded metamaterial is derived, which only relies on the state vector. The results of the dispersion relation obtained with the symplectic mathematical method agree well with t...     

Deformation trapping due to thermoplastic instability in one-dimensional wave propagation

One-dimensional shear wave propagation in a half-space of a nonlinear material is considered. The surface of the half-space is subjected to a time dependent but spatially uniform tangential velocity. The half-space material exhibits strain hardening, thermal softening and strain rate sensitivity of the flow stress. For this system, a well-defined band of intense shear deformation can develop adjacent to the loaded surface, even though the material has no imperfections or other natural length scale. Representative particle velocity and strain profiles, which have been obtained numerically, are described for several different models.     

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.     

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 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 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.     

FFT-based spectral analysis methodology for one-dimensional wave propagation in poroelastic media

The general one-dimensional equilibrium equations describing the dynamic behaviour of a porous medium form a system of coupled hyperbolic partial differential equations. A transition from the time to the frequency domain is made by spectral decomposition of the displacements. The equations simplify to a set of coupled ordinary differential equations. A solution can be obtained by solving a frequency-dependent eigenvalue problem. The characteristic equation clarifies the double wave-pattern and the attenuation of each wave. A spectrally formulated element uses the frequency-dependent eivenvectors as shape functions. The mass distribution is treated exactly without the need of subdividing a member into smaller elements and therefore wave propagation within an element is also treated exactly.     

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.     

Nonlinear wave propagation in binary mixtures of Euler fluids

In the present paper, we study the propagation of acceleration and shock waves in a binary mixture of ideal Euler fluids, assuming that the difference between the atomic masses of the constituents is negligible. We evaluate the characteristic speeds, proving that they can be separated into two groups: one is related to the case of a single Euler fluid, provided that an average ratio of specific heats is introduced; the other is new and related to the propagation speed due to diffusion. We evaluate the critical time for sound acceleration waves and compare its value to that of a single fluid. We then study shock waves, showing that three types of shock waves appear: sonic and contact shocks, which have counterparts in the single fluid case, and the diffusive shock, which is peculiar to the mixture. We discuss the admissibility of the shock waves using the Lax-Liu conditions and the entropy growth criterion. It is proved that the sonic and the characteristic shock obey the same properties as in the single fluid case, while for the diffusive shock there exists a locally exceptional case that is determined by a particular value of the concentration of the constituents, for which the genuine nonlinearity is lost and no shocks are admissible. For other values of the unperturbed concentration, the diffusive shock is stable in a bounded interval of admissibility.Received: 15 December 2002, Accepted: 28 June 2003 Correspondence to: T. RuggeriS. Simi: On leave from the Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Serbia