Buoyancy-induced deformations in a compressible elastic solid

Given a rectangular slab of a purely elastic material (neo-Hookean) with different temperatures imposed on the lateral surfaces, it is shown, within the context of a linearized theory, that the resulting deformation field is, under suitable boundary conditions, virtually identical to its fluid-mechanical counterpart given by a rectangular trench filled with a Newtonian liquid. A second, more general analogy between problems involving buoyancy induced deformations and the classical theory of thin plates is also presented.     

Constitutive branching in compressible elastic materials

Summary Branching analysis for the homogeneous deformations of a compressible elastic unit cube under dead loading is performed. Critical conditions for branching of the equilibrium paths are derived and the post-critical equilibrium paths are described. Special attention is given to the compound branching.     

Wave reflection in slightly compressible, finitely deformed elastic media

Summary  In this paper, the reflection of a plane wave at an incrementally traction-free boundary of a half-space composed of nearly incompressible elastic material is considered. It is shown that two distinct cases exist, these being dependent on the underlying primary deformation. In the first case, the appropriate slowness sections are each approximately elliptical, and the corresponding reflection phenomena closely mirrors that associated with the corresponding linear isotropic theory. Specifically, an angular range of direction of incident wave exists, for which both a quasi-longitudinal and quasi-shear wave are reflected, the former being replaced by a surface wave outside this angular range. In the second case, the outer slowness section is re-enrant and, in addition to the scenarios previously mentioned, it is possible for two quasi-shear waves to be reflected. Numerical illustrations of reflection coefficients are presented in respect of a modified Varga material and the case of increasing bulk modulus is investigated. Received 17 January 2000; accepted for publication 22 February 2000     

The growth of plane discontinuities propagating into a homogeneously deformed elastic material

It is shown that, in general, plane acceleration discontinuities propagating into an isotropic elastic material in a state of homogeneous deformation either become infinite in a finite time or decay to zero in an infinite time. Exceptions to this result are transverse discontinuities which propagate along a principal axis of strain without change in strength. Conditions governing the growth of acceleration discontinuities travelling into undeformed material are found to be identical with the thermodynamic conditions derived by Bland [2] for shock propagation. Plane discontinuities of order higher than the second are shown to propagate with constant strength.     

Universal deformations in incompressible isotropic elastic materials

One is concerned with the problem of determining the static deformations which can be produced in every isotropic, homogeneous, incompressible elastic body by the action of surface tractions alone. It is shown that any new solution cannot have more than one of the proper vectors of the deformation tensorc determining a vector field of constant non-zero abnormality.     

Antiplane deformations of inhomogeneous elastic materials

The system of equations governing antiplane deformations of inhomogeneous elastic media is examined with a view to achieving its reduction to a canonical form associated with the Cauchy-Riemann system.     

Equilibrium shocks in plane deformations of incompressible elastic materials

This paper is concerned with piecewise smooth plane deformations in an isotropic, incompressible elastic material. An explicit necessary and sufficient condition for the existence of piecewise homogeneous equilibrium states is established, and the set of all such states is precisely characterized. A particularly simple expression is derived for the driving traction on a surface of discontinuity in the deformation gradient.     

Moderate deformations in extension-torsion of incompressible isotropic elastic materials

It has been previously shown by anand (1979) that the classical strain energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain energy function is replaced by the Hencky or logarithmic measure of finite strain. The basis in Anand's paper for relating Hencky's strain energy function to experiment was data from experiments on metals and rubbers in uniaxial strain, simple tension and compression, and pure shear. Here, to test further the validity of this strain energy function for moderate deformations, its predictions for the twisting moment and the axial force in simple torsion and combined extension-torsion of solid cylinders of incompressible materials are calculated and shown to be in good agreement with data from the classical experiments of Rivlin and Saunders (1951) on vulcanized natural rubber. Indeed, the predictions from Hencky's strain energy function are in better accord with experiment than the predictions from the widely used Mooney (or Mooney-Rivlin) strain energy function.     

Universal solutions for fiber-reinforced compressible isotropic elastic materials

Universal solutions for fiber-reinforced compressible isotropic elastic materials under large elastic deformations are obtained, by using inverse methods.The following deformations are investigated: bending, stretching and shearing of a rectangular block; straightening, stretching and shearing of a sector of a circular tube; inflation, eversion, extension, torsion, bending and shearing of a sector of a circular tube; inflation and eversion of a spherical shell.The significance of the reinforcement and the deformed configuration of the fibers is duscussed.

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Zusammenfassung Unter Benützung inverser Methoden werden universelle Lösungen für faser-vorgespanntes, Kompressibles, isotropes, elastisches Material angegeben.Die folgenden Deformationen werden untersucht: Biegung, Zug und Schub eines rechteckigen Blockes; Ausgradung, Zug und Schub eines Sektors eines Kreisrohrausschnitts; isotrope Dehnung, Umstülpung, Erweiterung, Tordierung, Biegung und Schub eines Sektors eines Kreisrohres; isotrope Dehnung und Umstülpung einer Kugelschale.Die Bedeutung der Vorspannung und der verformten Konfiguration der Fasern wird diskutiert.

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Graduate Student.     

Response of moving load due to irregularity in slightly compressible,finitely deformed elastic media

The present paper has been framed to study the stresses produced on the rough surface of a slightly compressible, finitely deformed half space due to a normal moving load. The surface of the medium is irregular with parabolic type of irregularity. The perturbation method is applied to find the displacement field. The normal and shear stresses have been obtained in closed form and discussed numerically by means of figures. It has been observed that the shear stress developed at different depths below the surface depends on the irregularity depth, frictional coefficient and irregularity factor of the rough surface of the medium. Also, surface plots have been drawn to analyze the combined variation of non-dimensional stresses and irregularity factor against depth.     

Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium

This investigation is concerned with the plane strain deformation of an infinite slab, containing a circular cavity, within the theory of finite elastostatics for a particular homogeneous isotropic compressible material, the so-called Blatz-Ko material. The body is subjected to uniform pressure, either internal or external. Exact closed-form solutions for the axisymmetric deformation and stress fields are obtained. In the case of internal pressure, it is found that the applied pressure may not exceed a certain maximum value p _max. At a value of pressure p _e (<p _max), the governing equations lose ellipticity at the cavity wall. For greater values of pressure this solution remains smooth, though involving both elliptic and non-elliptic regions. Non-existence of axisymmetric solutions with discontinuous strain fields is established. The possibility of bifurication into a surface mode is considered and it is shown that this occurs at a value of pressure slightly smaller than p _e. Such surface wrinkling leads to a periodic distribution of points of stress concentration, from which shear bands may initiate.This work was supported by the U.S. Army Research Office under Grant DAAG29-83-K-0145 (R.A. & C.O.H.) and by the U.S. National Science Foundation under Grant MEA 78-26071 (C.O.H.).     

Propagation,reflection, and transmission of SH-waves in slightly compressible,finitely deformed elastic media

The propagation, reflection, and transmission of SH waves in slightly compressible, finitely deformed elastic media are considered in this paper. The dispersion relation for SH-wave propagation in slightly compressible, finitely deformed layer overlying a slightly compressible, finitely deformed half-space is derived. The present paper also deals with the reflection and refraction (transmission) phenomena due to the SH wave incident at the plane interface between two distinct slightly compressible, finitely deformed elastic media. The closed form expressions for the amplitude ratios of reflection and refraction coefficients of the reflected and refracted SH waves are obtained from suitable boundary conditions. For the numerical discussions, we consider the Neo-Hookean form of a strain energy function. The phase speed curves, the variations of reflection, and transmission coefficients with the angle of incidence, and the plots of the slowness sections are presented by means of graphs.     

Plane infinitesimal waves in homogeneous elastic plates

Using the theory of elastic Cosserat plates, we obtain conditions for reality of the frequencies of vibration.

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Résumé En utilisant la théorie des plaques élastiques de Cosserat, on obtient les conditions pour réalité des fréquences de vibration.