On the speed of propagation of waves in a deformed elastic material

The secular equation is obtained for small amplitude waves propagated in an arbitrary direction in a body of incompressible isotropic elastic material subjected to a pure homogeneous deformation. Conditions are obtained that the wave speeds be real.

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Zusammenfassung Es wird die Säkulargleichung für Wellen kleiner Amplitude gewonnen, welche sich in beliebiger Richtung in einem inkompressiblen isotropen elastischen Material fortpflanzen, das einer reinen homogenen Deformation unterworfen ist. Es werden Bedingungen dafür aufgestellt, dass die Fortpflanzungsgeschwindigkeiten reell sind.

     

On the propagation of elastic waves in two-phase media

At the present time a number of papers has been already devoted to the dynamics of two-phase media. One may mention the papers by Frenkel' [1], Rakhmatulin [2], Biot [3,4], Zwikker and Kosten [5], and others. However, the basic problem of the setting up of the equations of motion in two-phase media still cannot be considered solved and requires additional study and experimental verification.

This paper is concerned with the study of the simplest case of motion, which is the propagation of elastic waves in a homogeneous isotropic medium consisting of a solid and a fluid phase. The problems of the reflection of plane waves and surface waves at the free boundary of the half-space are solved. It is shown that the stress-strain relations established by Frenkel' are equivalent to the analogous relations proposed by Biot and that the equations of motion of the latter are more general.     

Some stability conditions for a compressible elastic material

Two sets of restrictions on the strain-energy function for a compressible isotropic elastic material are obtained which are necessary conditions for stability of the material. These arise from the following considerations. (i) A rectangular block is subjected to a finite pure homogeneous deformation, and an infinitesimal pure homogeneous deformation with arbitrary principal directions is superposed. The dimensions in two of these principal directions are held constant. Then the incremental modulus associated with the third principal direction must be positive for stability to obtain. (ii) In the initial pure homogeneous deformation one pair of faces of the block is force-free. The superposed inifinitesimal pure homogeneous deformation has one of its principal directions normal to these faces, which remain force-free, and the principal extension ratio corresponding to another is unity. The incremental modulus corresponding to the third principal direction must be positive to obtain stability.

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Zusammenfassung Für ein kompressibles isotropes elastisches Material werden zwei Sätze von Einschränkungen der Funktion der Verformungsenergie als notwendige Bedingungen für die Materialstabilität hergeleitet. Diese entstehen aus den folgenden Überlegungen: (i) Ein rechteckiger Block wird einer endlichen homogenen Verformung, und einer darauf überlagerten infinitesimalen homogenen Verformung mit beliebigen Hauptachsen ausgesetzt. Die Längen in zwei dieser Hauptachsen-Richtungen werden festgehalten. Dann muß der inkrementale Modul in der dritten Hauptrichtung positiv sein, damit die Stabilität gewährleistet ist. (ii) Bei der endlichen homogenen Verformung sind zwei parallele Begrenzungsflächen des Blockes kräftefrei. Die eine Hauptrichtung der überlagerten infinitesimalen homogenen Verformung sei senkrecht zu diesen Flächen, die kräftefrei bleiben, und die Hauptstreckung in einer der anderen Hauptrichtungen sei gleich 1. Der inkrementale Modul in der dritten Hauptrichtung muß zur Gewährleistung der Stabilität positiv sein.

     

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.     

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.     

The propagation of elastic waves in a piezoelectric

Formulas of the ray method are developed in the case of the propagation of high-frequency waves in a piezoelectric. It follows from the equations of the paper that in first approximation the energy of the wave field propagates along rays.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 155–161, 1974.     

The incremental shear modulus in a compressible isotropic elastic material

A compressible isotropic elastic material is subjected to a finite pure homogeneous deformation. An infinitesimal simple shear is superposed on the deformation. Necessary and sufficient conditions are obtained for the shear modulus to be positive for all choices of the direction and plane of shear.

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Zusammenfassung Ein kompressibles isotropes elastisches Material wird einer endlichen homogenen Verformung ausgesetzt. Ein einfacher, infinitesimaler Schub wird dieser Verformung überlagert. Für alle Schubrichtungen und -ebenen werden notwendige und hinreichende Bedingungen für einen positiven Schubmodul hergeleitet.

     

Plane isentropic large displacement simple waves in a compressible elastic solid

Zusammenfassung Es gibt drei einfache Wellen, die sich in gegebener Richtung in gegebenem Sinn in einem kompressiblen elastischen Körper fortpflanzen können. In einem isotropen Körper ist eine dieser Wellen zirkular, die andern eben polarisiert. Bei kleinen Amplituden kann eine von den eben polarisierten Wellen rein dilatatorisch sein, in der anderen hat die longitudinale Komponente des Verschiebungsgradienten die Grössenordnung des Quadrats der transversalen Komponente. Es wird ein Beispiel einer solchen quasi-transversalen einfachen Welle angeführt. Die Theorie der einfachen Welle wird kurz in einem Anhang behandelt.     

On the propagation of Lamb waves in a sandwich plate made of compressible materials with finite initial strains

The propagation of flexural Lamb waves in a prestrained sandwich plate made from compressible highly elastic materials is investigated within the scope of a piecewise homogeneous body model by utilizing TLTEWISB. The mechanical relations of layer materials are described by a harmonic-type potential, and numerical results are obtained for the first and second vibration modes. According to the results, the influence of problem parameters and of the initial stretching strain along the layers on the wave propagation speed is examined. The asymptotic values of the speed are considered in the cases of short and long wavelengths, and the influence of the initial strains on these asymptotic (limit) values are also analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 231–244, March–April, 2008.     

On the expansion of a deformed cylindrical elastic membrane

In the present note we consider the effect of radially expandingone end of a cylindrical elastic membrane that is also subjectedto longitudinal extension coupled with a twist. In doing thiswe start with the results of Tait, Steigmann and Zhong who studiedthe extension and twist of a cylinder with fixed end radii.We use tension field theory to provide a model of the membranein wrinkled regions giving rise to explicit analytic solutionswithin these regions. Elsewhere we rely on a numerical schemeto provide graphical results which are displayed for variousquantities of interest.     

The propagation of elastic waves in a viscous-fluid-saturated porous medium

Elastic shock waves in a viscous-fluid-saturated porous medium are investigated. The porosity is only taken into account with respect to pores communicating with one another, and isolated pores are considered as elements of the elastic part of the porous skeleton. It is shown, using the theory of discontinuity, that in such a medium there are two types of vortex-free waves and one equivoluminal wave. Differential equations and their solution for determining the change in the wave-front intensity are obtained. The effect of the fluid viscosity and porosity on the propagation of spherical waves is demonstrated using an example.     

On the acceleration waves in second-order elastic,isotropic, compressible,and homogeneous materials

In this paper we propose a Signorini’s perturbation method to investigate the propagation of acceleration waves in second-order elastic, isotropic, compressible, and homogeneous materials. The method is applied when the undisturbed region is subjected to a simple extension or to a simple shear. In these cases, we evaluate the first-order terms of the speeds and the amplitudes of the acceleration waves in any arbitrary direction of propagation.     

Reflection of plane waves from the boundary of a pre-stressed compressible elastic half-space

The effect of pre-stress on the propagation and reflection ofplane waves in an incompressible isotropic elastic half-spacehas been examined recently by the authors (Ogden & Sotiropoulos,1997). In the present paper the corresponding analysis for compressiblematerials is detailed. In the two-dimensional context consideredfor incompressible materials the (homogeneous) plane waves werenecessarily shear waves. By contrast, in the compressible contextpure shear waves can propagate only in specific directions inthe considered principal plane and, in a general direction,a quasi-shear wave may be accompanied by a quasi-longitudinalwave, as is the case in the anisotropic linear theory. The dependenceof the (in-plane) slowness section on the pre-stress (and finitedeformation) and on the choice of constitutive law is elucidated.This information is used to determine the reflection coefficientsfor reflection of either a (quasi-) shear wave or a (quasi-)longitudinal wave from the boundary of the half-space and tocharacterize the different cases which arise depending on thegeometry of the slowness section. The theoretical results are illustrated by numerical calculationsfor the range of possible types of behaviour with referenceforms of strain-energy function and different states of finitedeformation and to the question of stability of the half-space.     

Dead loading of a unit cube of compressible isotropic elastic material

A unit cube of compressible isotropic elastic material undergoes a homogeneous dilatation by dead loading forces applied to its faces. Conditions are obtained for stability of the resulting equilibrium state. The physical nature of these conditions is described and the results are illustrated for a compressible Blatz-Ko foamed rubber material.     

On elastic waves in a wedge

The existence of waves propagating along the edge of an elastic wedge has been established by many authors by physically rigorous arguments on the base of numerical computations. A mathematically rigorous proof for a wedge with aperture angle less than π/2 was presented by I. Kamotskii. We supplement the I. Kamotskii result and prove the existence of fundamental modes for some range of aperture angles greater than π/2. Bibliography: 7 titles.