Sliding and indentation by a rigid half-wedge with friction and displacement coupling effects

Dynamic rigid indentation induced by sliding contact in the presence of Coulomb friction is studied by considering a rigid half-wedge which simultaneously translates tangentially and normally at constant speeds with respect to the surface of an elastic half-plane. The tangential displacements of half-plane points in the contact zone are not a priori assumed to be negligible, and are thus coupled with the normal displacements in imposing the contact zone boundary conditions. The contact zone itself, as defined on the undeformed half-plane, extends at a constant rate. A complete solution to the problem shows that the coupling effects may be small for low indentation speeds and relatively flat half-wedges. The coupling effect decreases the singularity order at the half-wedge apex, while the friction increases it. Indeed, the two effects may cancel out in this and other aspects of the solution. It is also found that apex particle velocity singularities occur only due to the tangential motion of the half-wedge.     

Torsional impact of a layer or a cylinder bonded to an elastic half-space

In this paper two torsional impact problems are considered. The first problem deals with the solution of a layer bonded to an elastic half-space when the layer is driven by the torsional impact over a bonded rigid circular disc. In the second problem sudden torsion by a rigid disc attached over the plane face of a circular cylinder is considered and the rest of the plane surface of the cylinder is stress free. The cylinder is bonded to the half-space, making use of Laplace and Hankel transforms the solution of each problem is reduced into Fredholm integral equations of the second kind. A numerical Laplace inversion technique is then used to recover the time depencence of the solution. The numerical values for the applied torque at the surface of rigid disc are calculated for each problem and then are displayed graphically.     

Epicenter motion of a transversely isotropic elastic half-space due to a suddenly applied buried point source

A linear, transversely isotropic, elastic half-space is excited by a suddenly applied buried point source. The stress free surface of the half-space is normal to the axis of material symmetry. An integral transform solution is found for a restricted class of materials. Explicit results are given for the normal component of the epicenter displacement. Numerical results are tabulated and discussed for nine hexagonal crystals.     

Direct boundary integral equations for elastodynamics in 3-D half-spaces

Vector boundary integral equations (BIE's) based on Somigliana's integral formula are presented with Stokes' (full-space) and Lamb's (half-space) fundamental tensors (Green's functions), for radiation and scattering of time-harmonic elastic waves by bodies embedded in or laying on the surface of a three-dimensional homogeneous, isotropic, linear-elastic half-space. Numerical work is based on BIEs free from principal-value integrals. Whereas the Stokes'-tensor BIE requires discretization of the infinite half-space surface, all discretization is confined to (finite) surfaces of the body when Lamb's tensors are used. The nonuniqueness of the integral equation solution at fictitious eigenfrequencies is addressed. Numerical results are presented for a rigid circular footing, a rigid hemispherical foundation and a fully embedded spherical cavity.     

A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space

The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.     

The indentation of a half-space of hexagonal elastic material by a circular punch of arbitrary end-profile

Summary The problem is considered of the indentation by a smooth rigid punch of a half-space composed of linear elastic material of hexagonal symmetry whose plane boundary is parallel to the basal planes. The case is considered in which the area of contact between the punch and the half-space is circular, the end of the punch with is in contact with the half-space having an arbitrary profile. An integral equation is formulated and solved for the boundary value of the normal displacement in the half-space, and an expression is derived for the distribution of pressure under the punch.     

Application of a random-walk method for stresses in polycrystalline metals

A probabilistic method is developed for the calculation of some stresses in a half-space of polycrystalline metal subject to a single normal surface point force as in the classical Boussinesq problem. A random-walk model is used, leading to a diffusion equation and an analogy is made between the time in the diffusion problem and the depth in the Boussinesq problem. The resulting diffusion analogy had earlier been developed for granular media, but here the fairly rigid crystalline arrangement of atoms and the random orientation of crystal grains are used to calculate the average diffusion analogy constant for polycrystalline metals. The normal and shear stresses are calculated, and the stresses due to a uniform circular surface pressure are compared with the corresponding results from the theory of elasticity.     

Debonding of a thermoelastic material from a rigid substrate at any constant speed: thermal relaxation effects

Summary A linear isotropic thermoelastic half-space is debonded from a rigid insulated substrate at constant speed by moving shear and normal line loads. A dynamic steady state is examined, and an exact transform solution for the related problem of an insulated half-space subjected to a moving zone of specified surface displacements is obtained. Asymptotic forms are extracted that are valid near the zone edge and for high speeds, and which highlight thermal relaxation effects. They are used to derive analytical results for debonding at any constant speed. In particular, field variables on the debonded surface and the still-bonded interface are given for the sub-Rayleigh, super-Rayleigh/subsonic, lower and upper transonic, and supersonic speed ranges. The degenerate cases that arise at the three body wave speeds and at twice the rotational wave speed are also given. Calculations for the dynamic fracture energy rate and debonding zone temperature change at sub-Rayleigh speeds in 4340 steel indicate that thermal relaxation enhances energy rate, but mutes thermal response. The latter effect, however, itself decreases as the Rayleigh speed is approached.     

A three-dimensional parabolic punch problem in linear elasticity

In this paper an analytic solution for a three-dimensional contact problem, in linear elasticity, is constructed through the separation of Laplace's equation in paraboloidal coordinates. A rigid punch under normal loading is applied to an isotropic elastic medium occupying an infinite half-space where the contact region is parabolic and the punch profile is prescribed. This treatment allows for a general punch profile provided it is physically reasonable so as to ensure the convergence of the solution.     

The axisymmetric boussinesq problem of an initially stressed neo-hookean half-space for a punch of arbitrary profile

A solution of the axisymmetric Boussinesq problem for an initially stressed neo-Hookean half-space is obtained in the closed form, from which are deduced simple formulae for the depth of penetration of the tip of a rigid punch of arbitrary profile and the total load which must be applied to the punch to achieve this penetration. Simple closed form expressions are also deduced for the shape of the deformed surface outside the punch and the distribution of pressure under the punch. The corresponding results are obtained for cylindrical, conical, paraboloidal, ellipsoidal and spherical punch shapes.     

Vibration isolation using open or filled trenches

The problem of isolating structures from surface waves by open or filled trenches under conditions of plane strain is numerically studied. The soil is assumed to be an isotropic, linear elastic or viscoelastic nonhomogeneous (layered) half-space medium. Waves generated by the harmonic motion of a rigid surface machine foundatin are considered. The formulation and solution of the problem are accomplished by the frequency domain boundary element method. The Green's function of Kausel-Peek-Hull for a thin layered half-space is employed and this essentially requires only a discretization of the trench perimeter and the soil-foundation interface. The proposed methodology is used for the solution of a number of vibration isolation problems and the effect of soil inhomogeneity on the wave screening effectiveness of trenches is discussed.     

Steady state thermoelastic contact problem in a functionally graded material

This paper is concerned with the stationary plane contact of a functionally graded heat conducting punch and a rigid insulated half-space. The frictional heat generation inside the contact region due to sliding of the punch over the half-space surface and the heat radiation outside the contact region are taken into account. Elastic coefficient μ, thermal expansion coefficient α_t and coefficient of thermal conductivity k are assumed to vary along the normal to the plane of contact. With the help of Fourier integral transform the problem is reduced to a system of two singular integral equations. The equations are solved numerically. The effects of nonhomogeneity parameters in FGMs and thermal effect are discussed and shown graphically.     

End displacements of semi-infinite cylinders due to annular loadings

Surface displacements at the end of a semi-infinite, circular cylinder due to an axisymmetric ring of forces on the end are examined. The solution which has been found may then be used to find surface displacements for general axisymmetric loadings by convolution. The solution, in tabular form, is given as corrections to the counter-part half-space solution. The method of solution involves a three step superposition process. First, the displacement due to a ring of forces on a half-space is found by using the Boussinesq solution. Then, the excess tractions on the half-space, over that of the cylinder, are removed. This is done in two parts. The problem of an infinite cylinder with linearly varying pressure and shear over a short length of the lateral surface is solved by using Fourier integrals. This is used for the removal of the pressure and shear on the lateral surface of the cylinder by convolution. Next, the stresses at the mid-section of the infinite cylinder are removed. This is done by finding a set of boundary conditions for the end which yields zero tractions on the lateral surface. Then a series of these boundary conditions is used to approximate the tractions which must be removed. With the solution thus obtained, two sample problems are shown: 1. an elastic cylinder in contact with a half-space; 2. a rigid punch in contact with an elastic cylinder.     

Frictionless elastic contact problem for a curved rigid punch of arbitrary shape

Summary A new method is proposed for the analysis of elastic contact problems for a curved punch of non-elliptic planform under the action of a normal force. The punch base is assumed to be a quadratic surface. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some general relationships are established between the applied force and the punch settlement. Specific formulae are derived for a punch whose planform has a shape of a polygon, a rectangle, a rhombus and a cross. An example of a finite rigid cylinder lying on its generator and pressed against an elastic half-space is considered in detail. The method allows to have singular stresses at the cylinder edges and zero stresses at the rest of the boundary of the contact domain. The last condition serves for defining the width of the domain of contact. All the formulae are checked against the solutions known in the literature, and a good accuracy is confirmed in a sufficiently wide range of the aspect ratio.With 3 Figures     

Elastic strain of a transversely isotropic half space caused by a rigid punchElastische Berührung eines Stòssels und eines in der Querrichtung isotropen Halbraumes

An analytical investigation of the displacement, strain and stress field in every point of a transversely isotropic solid under the action of a rigid indentor is made.The results are then applied to the case of contact between a flatended cylindrical punch and a transversely isotropic half-space.     

Axi-symmetric loads on an elastic half-space containing an inclusion

The problem of a linearly elastic half-space containing a rigid-lubricated cylindrical inclusion and subjected to a normal static circular line load at the plane surface is treated. The stress and displacement components on the plane boundary yield simple expressions from which quantitative results are readily determined. The singularities existing under the applied load are examined.     

Contact behaviors of a rigid punch and a homogeneous half-space coated with a graded layer

A contact model between a homogeneous half-space with a linearly graded layer and a rigid punch is proposed and studied in the present paper. The governing equation, which describes the relation of the displacements and the normal tractions at the contact interface, is obtained by means of Fourier transform and a transfer matrix method. Appropriate collocation methods are used in order to solve the equation numerically. Singular behaviors at the edge of a flat punch are revealed. Compared to the case with a graded surface varying according to an exponential law, stress concentration is relatively weaker in the case with the graded surface varying according to a linear law. Furthermore, stress distributions in cases of a flat or cylindrical punch are given for different varying graded laws, thickness of graded layer, ratios of stiffness, and frictional coefficients. All the results are helpful for the design of strong and wear resistance coating surfaces.     

Forced vertical vibration of a rigid elliptical disc on an elastic half-space

The forced vertical vibration of a rigid frictionless elliptical disc on the surface of an elastic half-space is considered. This mixed boundary value problem is reduced to a (two-dimensional) integral equation. An approximation is obtained for the displacement of the disc by using a variational procedure.     

Elastic contact problem of a half-space pressed onto a rigid base with a cylindrical protrusion or pit

We analyse the elastic contact problem of a half-space pressed onto a rigid half-space with a cylindrical protrusion or pit; this is a three-part mixed boundary value problem with moving contact border. Considering the condition of the border, we reduce each problem to the solutions of two infinite systems of simultaneous equations, and show the distributions of surface displacement and contact pressure and the relation between contact region and magnitude of protrusion or pit.