Using complex potentials to determine the reaction of a prestressed two-layer elastic half-space to a moving load

Complex potentials in common form for compressible and incompressible elastic bodies are used to formulate and solve the problem of stationary motion of a prestressed two-layer elastic half-space under a moving surface load. The results presented are similar to those obtained earlier using the Fourier transform __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 3–15, May 2008.     

The general solution of second order effects in elastic half-space acted upon by a non-uniform shear load

I.IntroductionInfiniteelasticitytheorythemathematicalequationsgoverningthedeformationofanisotropiccompressibleelasticmaterialarehighlynonlinear.Asaresult,theexactsolutionsoftheboundaryvalueproblemshavebeenpossibleonlyinsomerestrictedcasesand,oftenapproxim…     

The axisymmetric lamb’s problem for a finite prestrained half-space covered with a finite prestretched layer

The piecewise-homogeneous body model and the three-dimensional linearized theory of elastic waves in prestressed bodies are used to solve the axisymmetric time-harmonic Lamb’s problem for a finite prestrained half-space covered with a finite prestretched layer. It is assumed that the half-space and layer are incompressible and their deformation is described by the Treloar potential. The normal stress at the interface is calculated Published in Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 132–143, March 2007.     

Dynamics of a prestressed incompressible layered half-space under moving load

The paper addresses a plane problem: a concentrated force acts on a plate resting on an elastic half-space with homogeneous prestrain. The equations of motion of the plate incorporate shear and rotary inertia. The half-space is assumed to be incompressible and isotropic in the natural state. The elastic potential is given in general form and is only specified for numerical purposes. The dependence of the critical velocity of the load and the stress-strain state on the prestresses is analyzed for different ratios between the stiffnesses of the layer and half-space and different contact conditions. The calculations are carried out for a half-space with Bartenev-Khazanovich potential __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 36–54, March 2008.     

Effect of the Initial Stresses on Waves in the System Consisting of a Viscous Fluid Layer and a Compressible Elastic Half-Space

International Applied Mechanics - The problem of acoustic wave propagation in a prestrained compressible elastic half-space that interacts with a layer of viscous compressible fluid is solved using...     

On the stability of a plate under longitudinal compression on a two-layer half-space with lower layer prestressed by gravity forces

In the plane (plane strain) and axially symmetric statements, we study the problem of stability, under the action of longitudinal compressing forces, of an infinite elastic plate in two-sided contact with an elastic half-space. The upper layer of finite depth is described by the usual equations of linear theory of elasticity; the lower layer, which is geometrically nonlinear, incompressible, and infinite in depth, is prestressed by gravity forces. The total adhesion between the layer of finite depth and the lower half-space is realized. It is also assumed that the same adhesion takes place between the upper layer of the half-space and the plate with the contact tangential stresses taken into account.The results can be used to calculate the working capacity of coated bodies and layered composites and in problems of geophysics.The problem of stability of an infinite elastic plate under longitudinal compression under conditions of two-sided contact with an elastic base was studied earlier in the monograph [1] (Fuss-Winkler base) and in [2–4].     

Propagation of Waves in an Elastic Half-Space Interacting with a Viscous Liquid Layer

International Applied Mechanics - The problem of the propagation of acoustic waves in a layer of a compressible viscous fluid that interacts with an elastic half-space is solved using the...     

Influence of fluid viscosity on waves in an initially deformed,compressible, elastic layer interacting with a fluid medium

The propagation of harmonic waves along the interface of an initially stressed, compressible layer and a viscous, compressible fluid half-space is investigated. A dispersion relation that does not depend on the form of the elastic potential is derived on the basis of the three-dimensional linearized elasticity equations for elastic bodies with uniform initial deformations and on the linearized Navier-Stokes equations for a viscous Newtonian fluid at rest. The phase velocities and attenuation coefficients of the elastic modes are determined numerically as functions of the thickness of the elastic layer using a Murnaghan-type three-invariant elastic potential.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 9, pp. 3–9, September, 1994.     

Spatial contact problems for a prestressed incompressible elastic layer

Two problems are considered on frictionless indentation of a stamp into the upper face of a layer with a homogeneous field of initial stresses present in the layer. The model of an isotropic incompressible nonlinearly-elastic material determined by the Mooney potential is used. The following two cases are studied: the lower face of the prestressed layer is rigidly fixed, and the lower face of a prestressed layer is supported by a rigid foundation without friction. It is assumed that the additional stresses due to the action of the stamp on the layer are small as compared with the initial stresses. This assumption makes it possible to linearize the problems of determining the additional stresses. In what follows, the problems are reduced to solving two-dimensional integral equations (IE) of the first kind with symmetric irregular kernels with respect to the pressure in the contact region. As an example, the case of an elliptic (in plan) stamp acting on a layer is considered. The spatial contact problem for a prestressed elastic half-space was first considered in [1].     

Finite-amplitude Love waves in a pre-stressed compressible elastic half-space with a double surface layer

The dispersive behavior of finite-amplitude time-harmonic Love waves propagating in a pre-stressed compressible elastic half-space overlaid with two compressible elastic surface layers of finite thickness is investigated. The half-space and layers are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. Results for the energy density and energy flux of the waves are also presented. The special case where the interfaces between the layers and the half-space are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the Love wave speed with the pre-stress and the propagation angle.     

Dynamics of an elastic half-space with a reinforced cylindrical cavity under moving loads

Partial separation of variables and reexpansion of cylindrical and plane waves are used to find the solution describing the uniform motion of a load along a thin circular cylindrical shell in an elastic half-space with the free surface parallel to the axis of the shell. This is a model problem for studying the dynamics of tunnels and shallow-buried pipelines under transport loads. Dispersion curves for the cases of sliding and tight contact between the shell and the half-space are plotted and analyzed. The effect of the shell parameters on the stress–strain state of the half-space is examined     

The disturbance due to plane and line sources in a pre-stressed semi-infinite elastic solid—I

This paper investigates the disturbance due to harmonic time dependence plane and line load in pre-stressed elastic half-space. A normal force has been considered to act on the surface of a semi-infinite elastic solid. Stresses and displacements due to the load are evaluated. The results obtained here are in agreement with the classical Lamb's problem in elastic wave propagation when the half-space is supposed to be initially stress-free. The numerical results include the stress distribution and the displacement at the surface; further, the results for stresses are shown graphically.     

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.     

Green functions for a compressible linearly non homogeneous half-space

Summary The paper presents a study of time-harmonic vibration of a half-space possessing a shear modulus linearly increasing with depth. Completing the previous paper [1], where the time-harmonic vibration of an incompressible half-space has been considered, the problem is now solved for a compressible as well as an incompressible material. The half-space is subjected to a vertical or horizontal surface load. The solution is represented in terms of Fourier-Bessel integrals containing functions of depth coordinate that are expressed through confluent hypergeometric functions. Numerical results concerning surface displacements due to a point force are given for a wide range of frequency variations and degree of non-homogeneity. The results show that, as compared to the homogeneous case, non-homogeneity can considerably increase vibration amplitudes at large distances from the applied force. Received 19 August 1996; accepted for publication 16, December 1996     

Second order effects in an elastic half-space acted upon by a non-uniform normal load

A closed form solution to the second order elasticity problem, when an isotropic compressible elastic half-space undergoes a deformation owing to a non-uniformly distributed normal load, is presented. The method of integral transform is employed and the case when loading is distributed, in accordance with Hertz's law, is discussed. The limiting solution for incompressible isotropic elastic material is also derived. Numerical calculations for the second order elastic material for the displacement and the normal stress in thez-direction are carried out. It is found that, in comparision to the linear elastic case, the displacement increases and the normal stress decreases in the second order elastic material

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Sommario Si presenta una soluzione del secondo ordine in forma chiusa del problema di un semi-spazio elastico isotropo compressibile soggetto a un carico normale distribuito non uniformemente, ottenuta mediante il metodo delle trasformate integrali. Si discute il caso del carico distribuito secondo la legge di Hertz; si deriva anche la soluzione limite per materiale elastico isotropo incompressibile. Si ottengono numericamente alcuni risultati relativi a spostamenti e tensioni normali nella direzione z: si trova che, a paragone del caso elastico lineare, gli spostamenti aumentano e le tensioni diminuiscono nel materiale elastico del secondo ordine.

     

The contact problem for an elastic half-plane in the case of a transonic load velocity

The stress-strain state of an elastic half-space under the action of a moving load is studied. The problem is solved analytically. The solution obtained is analyzed for the case of a transonic load velocity.     

A Prestressed Elastic Strip with Elastic Reinforcements

A contact problem is studied for a prestressed elastic strip with an elastic reinforcement. The integral Fourier transform is used to construct an influence function for an infinite strip with one face fixed. A unit concentrated force is applied to the strip at an arbitrary angle. The contact problem on force transfer from a thin infinite stringer to the prestressed strip is solved. The problem is mathematically formulated as a system of integro-differential equations for the unknown contact stresses on the assumption that the beam bending model and the uniaxial stress model are valid for the stringer, which is subjected to both vertical and horizontal forces. This system is solved in a closed form using the integral Fourier transform. The contact stresses are expressed in terms of Fourier integrals in a quite simple form. The influence of the initial stresses on the contact stress distribution is analyzed, and effects of concentrated load are revealed     

Propagation of Quasi-Lamb Waves in an Elastic Layer Interacting with a Viscous Liquid Half-Space

International Applied Mechanics - The propagating of quasi-Lamb waves in an elastic layer that interacts with a half-space of a viscous compressible fluid is studied. The three-dimensional...     

Plane problem of filtration consolidation for an elastic half-space with discontinuous initial conditions

The plane consolidation problem for an elastic half-space under an instantaneously applied vertical load is solved in the general case assuming that the Terzaghi hypothesis holds.     

Influence of Prestresses on Quasi-Lamb Modes in Hydroelastic Waveguides

International Applied Mechanics - The propagation of quasi-Lamb waves in a prestrained compressible elastic layer that interacts with a half-space of a compressible ideal fluid and in a prestrained...