Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

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     

Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties

The contact problem for the impression of spherical indenter into a non-homogeneous (both layered and functionally graded) elastic half-space is considered. Analytical methods for solving this problem have been developed. It is assumed that the Lame coefficients vary arbitrarily with the half-space depth. The problem is reduced to dual integral equations. The developed methods make it possible to find the analytical asymptotically exact problem solution, suitable for a PC. The influence of the Lame coefficients variation upon the contact stresses and size of the contact zone with different radius of indenter as well as values of the impressing forces are studied. The effect of the non-homogeneity is examined. The developed method allows to construct analytical solutions with presupposed accuracy and gives the opportunity to do multiparametric and qualitative investigations of the problem with minimal computation time expenditure.     

On the long-time limit of the Garvin–Alterman–Loewenthal solution for a buried blast load

The Garvin–Alterman–Loewenthal solution refers to the problem of a line blast load suddenly applied in the interior of an elastic half-space. It is expected that the long-time asymptotic limit of this solution should be equal to the solution of a related static problem. This expectation is justified here. First, the solution of the static problem is constructed. Then, the asymptotic limit of the transient problem is found, correcting previously published results.     

The contact problem of electroelasticity for a flat punch with parabolic cross-section

The solution of the problem of a rigid punch with a parabolic cross-section and flat base that is forced into an elastic piezoelectric ceramic half-space is derived in explicit form. The punch is somewhat displaced, being parallel to the isotropy plane that coincides with the boundary surface of the half-space. The symmetry axis coincides with the direction of the force lines of the field with the previous polarization. Formulas are derived to determine the stresses on the surface of the half-space under the punch and the components of the conjugate electric field for certain boundary conditions on the contact area. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 20–26, November, 1999.     

Interaction of Several Dies on an Elastic Half-Space

A contact problem is solved for several rigid dies on an elastic half-space. Relationships between the generalized loads and generalized displacements of a large number of spaced dies are established. The interaction between flat-based dies is described in terms of their capacity characteristics. The general solution is constructed on the basis of Mossakovski's theorem. Explicit formulas are derived for a system of elliptic dies     

The Action of an Arbitrary Load on the Surface of an Orthotropic Half-Space

The exact solution to the first boundary-value problem for a half-space is constructed on the basis of the general solution of the equilibrium equations for an orthotropic medium (nine elastic constants). The stress–strain state of an orthotropic half-space whose surface is under an arbitrarily applied concentrated force is described as an example. The well-known solution for the isotropic case is obtained by the same scheme, which confirms the reliability of the result.     

Bonded contact of a flexible elliptical disk with a transversely isotropic half-space

The article concerns the problem of bonded contact of a thin, flexible elliptical disk with a transversely isotropic half-space. Three different cases of loading have been considered: (a) the disk is loaded by a transverse force, whose line of action passes through the center of the disk and lies in the plane of the disk; (b) the disk is subjected to a rotation by a torque, whose axis is perpendicular to the surface of the half-space; (c) the half-space with the bonded disk is under uniform stress field at infinity. The problem corresponding to all three cases is reduced, in a unified manner, to a set of coupled two-dimensional integral equations. Closed-form solutions for these equations have been obtained by using Galins theorem.     

A half-space problem in the theory of generalized thermoelastic diffusion

In this work we consider the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques.The temperature, displacement, stress and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

On deforming of a linear viscoelastic orthotropic half-space under the turning rigid cylindrical roller

In this paper we consider the problem of rigid cylinder turning on a linear viscoelastic orthotropic half-space with Coulomb's friction acting along the contact area. Results for extents of contact area and pressure under the cylinder are obtained using Volterra's principle. The obtained functions of viscoelastic operators are interpreted by a method based on expansion of such functions in operator continued fractions. A solution is given for the general type of resolvent viscoelastic operators expressing rheological properties of half-space material. Algebra of resolvent Volterrian operators is used to facilitate the calculations. An example is given to illustrate the results for real viscoelastic material with the rheological properties expressed by the operators of Yu.N. Rabotnov.     

On the relationship between the solutions of static contact problems of elasticity and electroelasticity for a half-space

The paper establishes the relationship between the static contact problems of elasticity and electroelasticity (in the absence of friction) for a transversely isotropic half-space whose surface is the isotropy plane. This makes it possible to avoid solving the electroelastic problem by finding all the characteristics of electroelastic contact from known cases of purely elastic interaction. Moreover, the electroelastic state of the half-space can be fully described using a known harmonic function, which is a solution of the purely elastic problem. The approach is exemplified by solving contact problems of electroelasticity for flat, elliptic, two circular, conical, and paraboloidal (circular and elliptic in plan) punches __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 69–84, November 2006.     

The problem of a slender die

The paper deals with the contact behaviour of a slender die indenting an elastic half-space. It is shown that the problem of determining the pressure on the elastic half-space may be reduced (with an error exponentially small relative to the elongation) to a single-variable integral equation, whose solution is commonly represented by an asymptotic series in a small parameter. It was shown for a die of oval form that, depending on the type of contact region, either an increase or decrease in the force acting from the elastic half-space on the die upon approaching the end-points of the die are possible.     

Elasticity solutions for constant and linearly varying loads applied to a rectangular surface patch on the elastic half-space

The solution of the point load problem in the half-space is well known in the theory of elasticity. Using direct integration, the point solution can theoretically be used to develop the solution for loading various contact areas with a variety of loading profiles. Unfortunately, anything more complicated than constant pressure loading has previously required numerical integration, and hence, no closed form solution was obtainable. Partial solutions, i.e. solutions valid only on the surface of the half-space have also been available. This paper presents the methodology to generate complete solutions to the integrals for constant and linearly varying loads applied in both the normal and tangential directions everywhere in the half-space.     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.     

Rapid sliding motion of a rigid frictionless indentor with a flat base over a thermoelastic half-space

The three-dimensional, rapid sliding indentation of a deformable half-space by a rigid indentor of a flat elliptical base is treated in this paper. The response of the material that fills the half-space is assumed to be governed by coupled thermoelasticity. The indentor translates without friction on the half-space surface at a constant sub-Rayleigh speed and the problem is treated as a steady-state one. An exact solution is obtained that is based on a Green’s function approach, integral equations, and Galin’s theorem. A closed-form expression for the distributed contact pressure under the elliptical base of the indentor is derived. Representative numerical results are given illustrating the effects of the indentor velocity, indentor geometry, and parameters of the thermoelastic solid on the contact displacement. Since there is an analogy between the steady-state theories of thermoelasticity and poroelasticity, the present results carry over to the latter case directly.     

Slow nonstationary vertical motions of a die on the surface of an elastic half-space

We consider the dynamic contact problem on vertical motions of an absolutely rigid body on an elastic half-space. We assume that the contact region does not vary during the motion and there is no friction under the die bottom. We construct an approximate solution of the problem under the assumption that the variation in the contact pressure under the die bottom on the time interval in which the Rayleigh wave runs the distance equal to the contact area diameter is small. Computational formulas are obtained for the cases of circular and elliptic dies.     

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.     

On the stress field singularity in an incompressible half-space weakened by two near-surface wedge-like cracks

We consider the equilibrium problem for an elastic incompressible half-space weakened by two near-surface wedge-like cracks, whose lie in the same plane perpendicular to the half-space surface and have a common vertex. We use the Papkovich-Neuber representation to reduce the problem to finding two harmonic functions satisfying the mixed boundary conditions. These functions are constructed in spherical coordinates by using a Mehler-Fock type integral representation in Legendre functions. The analytic solution thus obtained permits finding the character of the stress distribution near the common tip of the cracks.