Pulse propagation in heat-conducting elastic materials

The influence of second-order effects on the propagation of a weak dilatational stress pulse in a heat-conducting elastic material is investigated. As a first approximation, the problem is studied using the linear theory of thermoelasticity. It is found that thermal diffusion dominates the wave motion, and a time is reached when second-order terms must be considered. The wave motion is then found to be isentropic and the shock structure is governed by Burgers's equation. Solutions to this equation are obtained and the influence of heat-conduction on pulse propagation in elastic materials is discussed, with some numerical results being presented for copper. Also, previous work in linear thermoelasticity theory is clarified and related to known results in linear and nonlinear elastodynamics.     

On the propagation of elastic waves through composite media

Summary With the aid of an ultrasonic pulse technique, the propagation of elastic waves (longitudinal as well as transverse) through polyurethane rubbers filled with different amounts of sodium chloride particles was studied. The velocity of both longitudinal and transverse waves was found to increase with filler content. From the measured wave velocities, the effective modulus for longitudinal waves,L, bulk modulus,K, and shear modulus,G, were calculated according to the relations for a homogeneous isotropic material. All three moduli appear to be monotonously increasing functions of the filler content over the whole experimentally accessible temperature range (–70 °C to + 70 °C forL andK;}-70 °C to about –20 °C forG) and they, moreover, reflect the glassrubber transition of the binder.Poisson's ratio,, was found to decrease with increasing filler content and show a rise at the high temperature side of the experimentally accessible temperature range (about –20 °C) as a result of the approach of the glass-rubber transition.In addition to the velocities, the attenuation of both longitudinal and transverse waves was measured in the temperature ranges mentioned. It was found that in the hard region tan _L as well as tan _G are independent of the filler content within the accuracy of the measurements. In the rubbery region, however, tan _L, increases with increasing filler content.Finally, the experimental data are compared with a simple macroscopic theory on the elastic properties of composite media.     

On the propagation of elastic waves through composite media II

Summary The propagation of elastic waves (both longitudinal and transverse) through polyurethane rubbers filled with different amounts of sodium chloride particles was studied at 0.8 MHz and 5 MHz. At a constant filler concentration (∼10% by volume), the velocity of these waves appeared to be independent of filler size. On the other hand, both velocities were found to increase with filler content. From the wave velocities, the effective modulus for longitudinal waves, L, bulk modulus, K, and shear modulus, G, were calculated according to the relations for a homogeneous isotropic material. All three moduli appear to be monotonically increasing functions of filler content, c, over the whole experimentally accessible temperature range (−80°C to +80°C for L and K; −80°C to about −30°C for G) and they, moreover, reflect the glass-rubber transition of the binder. Poissons ratio, μ, was found to decrease with increasing filler content and shows a rise at about −30°C as a result of the approach of the glass-rubber transition. The attenuation of the elastic waves was also measured in the temperature ranges mentioned. For filler particles beyond a critical size both tan δ_L and tan δ_G in the hard region are independent of the filler content within the accuracy of the measurements. The critical size depends on the type of wave and on its frequency. In the rubbery region, however, tan δ_L increases with particle size (at a constant content of 10% by volume) and even shows an enhancement with the smallest particles (1–5 μ) at 0.8 MHz. Moreover, it is found that for the same filler size tan δ_L increases with filler content. In some cases an anomalous damping behaviour was found, such that in the rubbery region the attenuation rises indefinitely with temperature. For filler particles larger than the above-mentioned critical size, tan δ_G and tan δ_L increase in the hard region as well. Finally, the experimental results are compared with existing theories on the elastic properties of and wave propagation through composite media.     

On the propagation of elastic waves in a multiperiodically reinforced medium

By a multiperiodically reinforced medium (multiperiodic composite) we mean a composite in which the matrix material is reinforced by two or more families of periodically spaced fibres. Moreover, at least along one direction the periods corresponding to different families are different. An example of this composite is shown in Fig. 1, where along the x ¹-axis we deal with two different periods . The aim of the contribution is twofold. First, we propose a macroscopic (averaged) model of a multiperiodic composite, describing the effect of period lengths on the overall dynamic behaviour of the medium, in contrast to the known homogenized models. Second, we apply this model to the analysis of elastic waves propagating across a composite reinforced by two pairs of families of parallel periodically spaced fibres with different periods along certain direction.     

On the propagation of thermoelastic waves in temperature rate-dependent materials

The one-dimensional propagation of thermoelastic waves in isotropic homogeneous half spaces within the theory of Green and Lindsay [1] is studied. Padé-extended ray methods are employed to obtain the desired information. Comparisons between the predictions of the Green and Lindsay theory and the theory of Lord and Shulman [2] are made. Our ray series solutions show that for discontinuous thermal disturbances the displacement according to the Green and Lindsay theory is also discontinuous. This violates the fundamental continuum hypothesis that matter is impenetrable. For a simple numerical example we show also that a compressive behaviour in the displacement may be associated with a tensile behaviour in the stress and vice versa. This prediction of the Green and Lindsay theory is also unrealistic from the physical point of view.     

On the propagation of elastic-viscoplastic waves in damage-softened polycrystalline materials

Theories for uniaxial wave propagation as, for example, along the longitudinal axis of slender rods composed of materials that behave elastically or plastically with hardening, encounter difficulty when confronted with softening material. For such theories, onset of softening causes the value of the wave speed to become complex thereby transforming the governing partial differential equations from hyperbolic to elliptic, implying no further possibility for wave-like motion in the softened material.The purpose of this paper is to show how an elastic-viscoplastic-damage type of constitutive theory together with the equation of motion produce a system of governing partial differential equations that can be shown to be hyperbolic. As an outgrowth of the calculation for the characteristics of the system, an expression relating the elastic dilatational wave speed with material damage and softening can be derived, demonstrating positive value for all phases of the material deformation including material softening that terminates in fracture. The paper also shows how experimental data from plate impact spall fracture tests can illustrate the reality of wave motion through damage-softened polycrystalline material.     

On the interaction of cubically nonlinear transverse plane waves in an elastic material

The paper presents theoretical results on the interaction of cubically nonlinear harmonic elastic plane waves in a nonlinear material described by the Murnaghan potential. The interaction of two harmonic transverse waves is studied using the method of slowly varying amplitude. Reduced and evolution equations and the Manley-Rowe relations are derived. An analysis is made of the mechanism of energy transfer from the strong pumping wave, which has frequency ω, to the weak signal wave, which has frequency 3ω because of this interaction. A switching mechanism for hypersonic waves in a nonlinear elastic material is described, which is similar to the switching mechanism observed in transistors __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 61–70, June 2006.     

Purely transverse waves in elastic anisotropic media

Formulas are obtained for decompositions of the third- and fourth-rank tensors symmetric in the last two and three indices, respectively, into irreducible parts invariant relative to the orthogonal group of coordinate transformation. The corresponding parts of the decompositions are orthogonal to each other. These decompositions are used to obtain a general representation of the displacement vectors of plane transverse waves in elastic isotropic and anisotropic solids. It is shown that the displacement vectors of transverse waves are second-, third-, and fourth-degree homogeneous polynomials of the wave normal. Special orthotropic materials are found that transmit purely transverse waves for any direction of the wave normal. The eigenmoduli, eigenstates, and engineering constants (bulk moduli, Youngs moduli, Poissons ratios, shear moduli, and Lame constants of the closest isotropic materials) are determined for these materials.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 160–172, January–February, 2005     

Simple waves in general elastic materials

The general theory of simple waves in Green-elastic and Cauchy-elastic materials is given. Such waves generate three-dimensional unsteady deformations. Boundary conditions producing such waves are derived together with conditions under which shocks occur. The theory is used to illustrate conditions behind acceleration fronts moving into homogeneously deformed regions and also the modes of propagation of fronts moving into a simple wave. The steady flow of an elastic material past a rigid developable surface is discussed. Simple waves which are principal waves are also discussed.     

Nonlinear progressive waves in elastic materials

Summary Progressive harmonic waves in nonlinear elastic materials are generally considered as small disturbances superposed on a finite deformation. In this first approximation they are governed by essentially the same laws as in linear elasticity. In the present paper a second-order theory is developed which allows for nonlinear effects on progressive waves in a finitely deformed elastic material. The problem is investigated by means of a perturbation procedure using intrinsically two time scales. After providing the kinematical prerequisites a transport equation is derived which governs the distortion of the wave profile. A closedform solution is obtained for the case of a plane wave in a homogeneous medium. The influence of nonlinearity is closely related to the evolution of acceleration waves.

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Zusammenfassung Fortschreitende harmonische Wellen in nichtlinearen elastischen Stoffen werden meist als kleine Störungen betrachtet, die einer endlichen Verformung überlagert sind. In dieser ersten Näherung genügen sie im wesentlichen den gleichen Gesetzen wie in der linearen Elastizitätstheorie. In der vorliegenden Arbeit wird eine Theorie zweiter Ordnung entwickelt, die auch nichtlineare Effekte bei fortschreitenden Wellen in einem endlich verformten elastischen Material berücksichtigt. Das Problem wird mit Hilfe einer Störungsmethode behandelt, welche implizit zwei verschiedene Zeitmaßstäbe zuläßt. Nach Bereitstellung der kinematischen Hilfsmittel wird eine Transportgleichung hergeleitet, welche die Verzerrung des Wellenprofils beschreibt. Für ebene Wellen in einem homogenen Medium wird eine geschlossene Lösung angegeben. Der Einfluß der Nichtlinearität zeigt eine enge Verwandtschaft zur zeitlichen Entwicklung von Beschleunigungswellen.

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Acceleration waves in constrained elastic materials

Well known results on the propagation and growth of acceleration waves in Cauchy elastic materials are extended to materials which suffer one or two internal constraints. It is proved, under certain restrictions, that acceleration waves will not propagate in a material which has three or more internal constraints. The great simplifications deriving from an assumption of hyperelasticity are indicated. The present results could be extended to materials other than simple elastic.     

Limiting velocity of crack propagation in elastic materials

A crack is represented as a continuous set of linear dislocations. Simple analytical expressions are obtained for the potential and kinetic energies of the environment of moving cracks and the attached mass of cracks for an arbitrary form of the stress applied to the crack P(x). It is shown that the indicated analytical expressions are bilinear integrals of the functions P(x) and ∂P(x)/∂x. These integrals are calculated in explicit form for a constant stress over the entire crack length and the stress due to the action of molecular adhesion forces in a narrow region near the crack openings. It is shown that the calculation results does not depend on the form of molecular adhesion forces. The proposed approach to describing cracks and calculations based on it has made it possible for the first time to obtain a complete analytical expression for the limiting crack propagation velocity in elastic materials as a function of the main mechanical characteristics of such materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 158–166, July–August, 2009.