An analytical solution of an axisymmetric problem of deforming an isotropic half-space with an elastic fixed boundary under the action of a distributed load

An analytical solution to the axisymmetric problem on the action of a distributed load on an isotropic half-space when the load is given by a function dependent on the radial coordinate is obtained. The surface of the half-space is elastically fixed outside the circular domain of load application, the shear stresses are absent along the entire boundary, and the stresses vanish at infinity. At the boundary and inside the elastic half-space, the solutions are represented by the formulas for the stress tensor components and for the displacement vector components.     

Solution of a multidimensional impact deformation problem for an elastic half-space with curved boundary on the basis of a modified ray method

A generalization of the method for constructing approximate solutions of boundary value problems of impact deformation dynamics in the form of ray expansions for two-dimensional plane deformation problems is presented. For each shock wave, the solution near its front is determined on the basis of ray coordinates consistent with this wave. The nonlinear divergence of curvilinear rays is taken into account. A mechanism of transformation from one ray coordinate system to another, which is crucially important in the ray method, is described. The developed technique is illustrated by solving the impact deformation problem for a half-space with boundary of nonzero curvature.     

Pressure pulse on the boundary of an elastic inhomogeneous half-plane

The problem about the motion of a pressure pulse at constant velocity along the boundary of an elastic homogeneous half-plane has been examined in [1–3]. The problem was considered as stationary in [1, 2], while in [3] it was solved by using a Laplace time transformation. An analogous problem is considered in this paper for an elastic half-plane with variable Lamé parameters and density of the medium.     

Method of homogeneous solutions in a mixed problem for an elastic halfstrip

Institute of Hydromechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 2, pp. 98–108, February, 1990.     

Second-order torsion problem of a homogeneous isotropic compressible multiply-connected elastic cylinder

For simply-connected regions, some solutions are available for the second-order torsion problem of homogeneous isotropic compressible elastic cylinders based on the theory given by Green and others. In the present paper, these theories are extended to cover the second-order torsion problem for multiply-connected regions. As an example, results for torsion of a confocal elliptical ring are given.     

Uniform motion of a plane punch on the boundary of an elastic half-plane

The dynamic contact problem of a plane punch motion on the boundary of an elastic half-plane is considered. The punch velocity is constant and does not exceed the Rayleigh wave velocity. The moving punch deforms the elastic half-plane penetrating into it so that the punch base remains parallel to itself at all times. The contact problem is reduced to solving a two-dimensional integral equation for the contact stresses whose two-dimensional kernel depends on the difference of arguments in each variable. A special approximation to the kernel is used to obtain effective solutions of the integral equation. All basic characteristics of the problem including the force of the punch elastic action on the elastic half-plane and the moment stabilizing the punch in the horizontal position in the process of penetration are obtained. A similar problem was considered in [1] and earlier in the “mode of steady-state motions” in [2, 3] and in other publications.     

Exact solution of external tangential contact problem for a transversely isotropic elastic half-space

Summary  The following mixed boundary-value problem for a transversely isotropic elastic half-space is considered. Arbitrary tangential displacements are prescribed at the exterior of a circle, while the interior of the circle is free of tangential stress, and the normal stress vanishes all over the boundary. The governing integral equation is solved exactly, in closed form, and in terms of elementary functions. The method of continuation of solutions previously published by the author has been used here. Several examples are considered. No similar results has been reported before, even in the case of an isotropic body. Received 8 May 2000; accepted for publication 20 July 2000     

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.     

The dynamic indentation of an elastic half-space

A method is given for finding the distribution of traction beneath a conical or a wedge-shaped indentor, moving into an elastic half-space, for the two extreme cases of perfect adhesion between indentor and half-space, and perfectly lubricated indentors. Relations between indenting force, velocity of indentor and area of contact are investigated and found to be fairly insensitive to the friction, although in applications in which the shear traction is required a model with friction would be essential. Mathematically, the paper provides the first accurate solutions of any dynamic mixed boundary value problems with essentially coupled mixed conditions.

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Zusammenfassung Es wird eine Methode zur Bestimmung der Belastung eines keil-oder kegelförmigen Körpers angegeben, der in einen elastischen Halbraum eindringt, wobei die beiden Grenzfäll-vollständige Adhäsion zwischen eindringendem Körper und Halbraum sowie perfekte Schmierung-betrachtet werden. Beziehungen zwischen Eindrückkraft, Eindringgeschwindigkeit und Kontaktfläche werden untersucht. Dabei ergibt sich eine gewisse Unempfindlichkeit gegenüber Reibung, obschon ein Modell unter Berücksichtigung der Reibung dann erforderlich ist, wenn die Schubbeanspruchung benötigt wird. Mathematisch gesehen liefert die vorliegende Arbeit die erste exakte Lösung eines dynamischen gemischten Randwertproblems mit gekoppelten gemischten Bedingungen.

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School of Mathematics, Bath University     

Dynamics of a linearly inhomogeneous incompressible isotropic elastic half-space

The study presented in this paper treats the harmonic and transient wave motion of an incompressible isotropic semi-infinite elastic medium with a shear modulus increasing linearly with depth. The medium has a constant mass density and an initial hydrostatic stress distribution due to a constant gravity. In particular, attention is given to the case of a vanishing top rigidity. For this case it is shown that the governing equations resemble the equations governing the deep water motion, and that under normal loading the behaviour of the upper surface resembles that of the upper surface of (deep) water.