Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

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.     

Extreme elastic deformations

Communicated by M. E. Gurtin     

Thermally generated stress waves in a dispersive elastic rod

The propagation of thermally generated stress waves in a dispersive elastic rod was investigated both experimentally and analytically. In the experimental investigation, the end of a circular colored-glass rod was heated very rapidly by the deposition of luminous energy from a Q-switched ruby laser. The light from the laser was directed parallel to the axis of the rod and deposited on the polished end of the rod. The depth of deposition was of the same order as the radius of the rod. The length of the energy pulse from the laser was 20 nsec. This results in heating at such a rate that it can be considered as instantaneous when compared to the mechanical response of the material used. The resulting stress wave was measured using a thin quartz crystal in a Hopkinson pressure-bar arrangement. Radial inertia precluded the use of the simple wave equation; Love's modified wave equation was used to describe the motion. The thermoelastic problem was reduced to a homogeneous partial differential equation with appropriate initial and boundary conditions which is solved by the separation of variables technique. The experimental results are in good agreement with Love's theory. The amplitude of the stress waves was found to be directly proportional to the total energy deposited. The very short stress pulses generated by Q-switched laser deposition on the end of the thin rod gave rise to the higher modes of longitudinal wave propagation. The existence of wave propagation in a thin rod at near dilatational velocities was experimentally confirmed. It is concluded that the experimental techniques developed can be used to model stress-wave generation due to electromagnetic-energy depositions. Also, laser deposition provides an efficient means for generating the higher modes of longitudinal wave propagation in thin rods. Paper was presented at 1968 SESA Spring Meeting held in Albany, N. Y., on May 7–10. This work was supported by the U. S. Atomic Energy Commission at University of California, Lawrence Radiation Laboratory, Livermore, Calif.     

Large elastic deformations in thin rods

Summary In this paper an appropriate analytical treatment for the determination, through exact formulae, of large elastic deformations in thin skew-curved rods is presented. This problem is associated with a system of fifteen nonlinear, ordinary, differential equations of the first order; the unknowns of the system are the final curvature and torsion functions, as well as the generalized internal forces and displacements of the rod. Subsequently, the problem of a thin cantilever circular rod subjected to terminal co-planar forces is examined and closed formulae determining its generalized displacements are obtained. Finally, the effectiveness and the potentialities of the method are demonstrated by several numerical applications.

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Übersicht In diesem Artikel wird eine analytische Methode zur Bestimmung von großen elastischen Verformungen eines schief gekrümmten Stabes durch exakte Formeln entwickelt. Dieses Problem wird durch ein System von fünfzehn nichtlinearen, gewöhnlichen Differentialgleichungen erster Ordnung beschrieben; die Unbekannten des Systems sind sowohl die endlichen Krümmungs- und Torsionsfunktionen als auch die verallgemeinerten inneren Kräfte und Verschiebungen des Stabes. Ferner wird das Problem des dünnen beidseitig gelagerten zylindrischen Stabes, welcher koplanaren Endlasten unterliegt, untersucht, und geschlossene Formeln werden erhalten. Schließlich werden die Effektivität und die Möglichkeiten der Methode durch mehrere numerische Anwendungen dargestellt.

     

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.     

Thin-plate theory for large elastic deformations

Non-linear plate theory for thin prismatic elastic bodies is obtained by estimating the total three-dimensional strain energy generated in response to a given deformation in terms of the small plate thickness. The Euler equations for the estimate of the energy are regarded as the equilibrium equations for the thin plate. Included among them are algebraic formulae connecting the gradients of the midsurface deformation to the through-thickness derivatives of the three-dimensional deformation. These are solvable provided that the three-dimensional strain energy is strongly elliptic at equilibrium. This framework yields restrictions of the Kirchhoff-Love type that are usually imposed as constraints in alternative formulations. In the present approach they emerge as consequences of the stationarity of the energy without the need for any a priori restrictions on the three-dimensional deformation apart from a certain degree of differentiability in the direction normal to the plate.     

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.     

Thermally induced natural-frequency variations in a thin disk

A number of studies has shown that natural frequencies and, accordingly, the minimum critical speed for the formation of a standing wave in thin, rotating, circular disks can be beneficially altered by purposely induced initial membrane stresses. The possibility of controlling natural frequencies by induced thermal membrane stresses, rather than initial stresses, has received some previous theoretical attention and is experimentally examined here for a stationary, constant-thickness, centrally clamped, circular disk. The primary advantages of thermal membrane stresses are manifested in the inherent flexibility in adjustment of the thermal as opposed to the initial stresses. Increases in the minimum critical speed, which is proportional here to the zero nodal circle—two nodal-diameter natural frequency, of 20 percent were determined with moderate heating. This can be considered a relatively small critical-speed increase when compared with variations expected in many common rotating disk environments. A thermal model, which utilizes as input the peripheral disk heat flux and the controlled disk temperature at some known radius, is shown to predict the temperature distribution and natural frequencies with reasonable accuracy. The applicability of this model enhances the potential practicality of the induced thermal-membrane-stress method of natural frequency and/or critical speed control.     

Finite deformations of an elastic anisotropic body

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 166–174, May–June, 1989.     

One-dimensional deformations of nonlinearly elastic micropolar bodies

We find families of finite deformations of a Cosserat elastic continuum on which the system of equilibrium equations is reduced to a system of ordinary differential equations. These families can be used to describe the expansion, tension, and torsion of a hollow circular cylinder, cylindrical bending of a rectangular slab, straightening of a circular arch, reversing of a cylindrical tube, formation of screw and wedge dislocations in a hollow cylinder, and other types of deformations. In the case of a physically nonlinear material model, the above-listed families of deformations can be used to construct exact solutions of several problems of strong bending of micropolar bodies.     

Coupling of elastic and thermal deformations I

Flow, Turbulence and Combustion - The general solution for the stress and temperature distribution in an isotropic thermoelastic solid occupying the half space has been given when the classical...