The indentation of a flat punch into a rigid-plastic half-space

The indentation of a flat punch into a rigid-plastic half-space is modelled by a centred field of slip lines with rotation of the rectilinear free boundary about the corner point of the punch. Adjacent to the rectilinear boundary, there is a rigid, stress-free region which is calculated using a velocity hodograph and determines the curvature of the initial horizontal boundary of the half-space during indentation up to the steady-state stage of the motion of the punch in the unbounded rigid-plastic medium.     

The indentation of a spherical punch into an ideally plastic half-space when there is contact friction

Calculations are presented of the indentation of a spherical punch into an ideally plastic half-space under condition of complete plasticity and taking account of contact friction, which is modelled according to Prandtl and Coulomb. Friction leads to the formation of a rigid zone at the centre of the punch when there is slipping of the material on the remaining part of the contact boundary. Limit values of the friction coefficients are obtained for which the rigid zone extends over the whole of the contact boundary. The dependence of the indentation force on the radius of the plastic area is in good agreement with experimental data.     

The indentation of a transversely isotropic half-space by a rigid punch

Zusammenfassung Der Verfasser gibt eine allgemeine Lösung für die Verteilung des Druckes zwischen einem axialsymmetrischen Stempel und einem transversal-isotropen Halbraum. Es wird gezeigt, dass die Verteilung des Druckes für den flachen Stempel mit allgemeiner Belastung unabhängig ist von den elastischen Eigenschaften des Halbraums und auch genau dieselbe, als ob der Halbraum isotrop wäre.     

On the impact deformation of an incompressible half-space under the action of a shear load of variable direction

Using the example of a one-dimensional planar problem for a nonlinear elastic incompressible half-space, the loading process is considered in which the shear action on the boundary plane changes with respect to both intensity and direction. It is shown that, in the regions of the space where the nonlinearity of the medium becomes a significant factor, the solution in the nearfront region of the shock wave is determined by a system of nonlinear evolution equations. The general solution of the evolution system is obtained. As an example, a partial solution of the system is considered for one of the simplest boundary conditions. A parametric method is presented for determining the displacements on the basis of solution of the evolution system.     

The indentation of a punch in the form of an elliptic paraboloid into the plane boundary of an elastic body

The three-dimensional contact problem for an elastic body of arbitrary geometry with a single plane face, into which a punch in the shape of an elliptic paraboloid is indented, is considered. The curvilinear boundary of the body is partially clamped, and the remaining boundary (outside the contact region) is stress-free. It is assumed that the dimensions of the contact area are small compared with the characteristic dimension of the body. Using the method of matched asymptotic expansions a model problem of unilateral contact without friction is derived for the boundary layer, which is solved using the apparatus of Hertz's theory. Asymptotic models of the contact interaction of different degrees of accuracy are constructed, including corrections to the geometry and clamping conditions of the elastic body. The sensitivity of the parameters of the elliptic region of the contact to these factors is investigated.     

The dynamics of an elastic half-space under the action of a moving load

Fundamental solutions of a problem in the theory of elasticity are constructed for a half-space under the action of a load moving at constant velocity which does not change with time in a moving system of coordinates. On the basis of these solutions, the displacements of the medium are determined in the case of a load which moves along a cylindrical surface in the medium itself or over its boundary surface. Subsonic, transonic and supersonic cases are considered.     

Plane contact problem of the indentation of a stamp into an elastic wedge

We consider the contact interaction of a stamp with rectilinear base and an elastic wedge. One of the wedge faces is fixed, and the stamp edge touches the wedge vertex. Using the Wiener–Hopf method, we have obtained an exact solution of this problem. We have also determined the stress distributions in the contact region and on the wedge fixed face as well as the displacements of its free boundary.     

The pressure of a punch with a rounded edge on an elastic half-space

The problem of the unilateral contact without friction for a punch, the face of which is characterized by a rapid change in the neighbourhood of the a priori unknown boundary of the contact area, is investigated. Asymptotic formulae are obtained for the function which describes the variation of the contact area and the contact-pressure density in the boundary-layer region. The problem of the behaviour of the contact pressures in the neighbourhood of a smoothed stress concentrator is considered.     

Axisymmetric indentation of a circular stamp with adhesion into an elastic half-space with a spherical cavity

On the basis of the expansion formulas of the vector solutions of the Lame equations in cylindrical and spherical coordinates, the problem of a circular stamp is formulated in the form of an integro-algebraic system of equations. By the method of orthogonal polynomials, it is reduced to a collection of infinite systems of linear algebraic equations, for which the method of reduction is justified. Formulas for the normal and tangential stresses under the stamp are given.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 14–20, 1987.     

Axisymmetric contact of a punch of polynomial profile with an elastic half-space when there is friction and adhesion

The axisymmetric problem of the contact interaction of a punch of polynomial profile and an elastic half-space when there is friction and partial adhesion in the contact area is considered. Using the Wiener–Hopf method the problem is reduced to an infinite system of algebraic Poincare–Koch equations, the solution of which is obtained in series. The radii of the contact area and of the adhesion zone, the distribution of the contact pressures and the indentation of the punch are obtained.     

The effect of heat flow on the contact area between a continuous rigid punch and a frictionless elastic half-space

A general proof is given of the theorem that, if a continuous rigid punch indenting an elastic half space is heated, the separation of the solids will increase and part of the contact area will be lost. It is also shown that if the punch is convex, the contact area cannot be multiply-connected.

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Résumé On donne une preuve générale du théorème que, si un poinçon continu rigid, pénétrant un demi-espace élastique, est chauffé, la séparation des solides grandira et une partie de la région de contact sera perdue. On montre également que la région de contact ne peut pas être multiplement connexe si le poinçon est convexe.

     

Static frictional indentation of an elastic half-plane by a rigid unsymmetrical punch

Static rigid 2-D indentation of a linearly elastic half-plane in the presence of Coulomb friction which reverses its sign along the contact length is studied. The solution approach lies within the context of the mathematical theory of elastic contact mechanics. A rigid punch, having an unsymmetrical profile with respect to its apex and no concave regions, both slides over and indents slowly the surface of the deformable body. Both a normal and a tangential force may, therefore, be exerted on the punch. In such a situation, depending upon the punch profile and the relative magnitudes of the two external forces, a point in the contact zone may exist at which the surface friction changes direction. Moreover, this point of sign reversal may not coincide, in general, with the indentor's apex. This position and the positions of the contact zone edges can be determined only by first constructing a solution form containing the three problem's unspecified lengths, and then solving numerically a system of non-linear equations containing integrals not available in closed form.The mathematical procedure used to construct the solution deals with the Navier-Cauchy partial differential equations (plane-strain elastostatic field equations) supplied with boundary conditions of a mixed type. We succeed in formulating a second-kind Cauchy singular integral equation and solving it exactly by analytic-function theory methods.Representative numerical results are presented for two indentor profiles of practical interest—the parabola and the wedge.     

The dynamics of the motion of a flat punch on the boundary of an elastic half-plane

The dynamic contact problem of the motion of a flat punch on the boundary of an elastic half-plane is considered. During motion, the punch deforms the elastic half-plane, penetrating it in such a manner that its base remains parallel to the boundary of the half-plane at each instant of time. In movable coordinates connected to the moving punch, the contact problem reduces to solving a two-dimensional integral equation, whose two-dimensional kernel depends on the difference between the arguments for each of the variables. An approximate solution of the integral equation of the problem is constructed in the form of a Neumann series, whose zeroth term is represented in the form of the superposition of the solutions of two-dimensional integral equations on the coordinate semiaxis minus the solution of the integral equation on the entire axis. This approach provides a way to construct the solution of the two-dimensional integral equation of the problem in four velocity ranges of motion of the punch, which cover the entire spectrum of its velocities, as well as to perform a detailed analysis of the special features of the contact stresses and vertical displacements of the free surface on the boundary of the contract area. An approximate method for solving the integral equation, which is based on a special approximation of the integrand of the kernel of the integral equation in the complex plane, is proposed for obtaining effective solutions of the problem that do not contain singular quadratures.     

The contact of a solid with an elastic half-space through a thin coating

More accurate equations of the deformation of thin plates, which are more convenient for solving contact problems for bodies with coatings and containing, as a special case, the equations of all known applied theories, are derived by an asymptotic analysis of the first fundamental problem of the theory of elasticity. The equations of the deformation of thin-walled elastic bodies are classified, their qualitative correspondence to the equations of the theory of elasticity is clarified, and the forms of the features that arise along the shift lines of the boundary conditions in the corresponding contact problems are established. A criterion for selecting approximate models to describe the properties of the coatings depending on the geometrical and mechanical characteristics of the coating and the substrate and also on their degree of adhesion is given.     

The plastic indentation of a semi-infinite solid by a perfectly rough circular punch

Zusammenfassung In dieser Arbeit werden das plastische Spannungsfeld und ein zulässiges Geschwindigkeitsfeld für den eben begrenzten Halbraum gegeben, der unter dem Einfluss eines ideal rauhen, starren Stempels mit kreisförmigem Querschnitt steht. Das Material ist als starr-plastisch vorausgesetzt, ohne Verfestigung, und der Fliessbedingung vonTresca genügend. Es wird gezeigt, dass die Hypothese vonHaar undvon Kármán auf dieses Problem anwendbar ist, wonach zwei von den drei Hauptspannungen gleich sind. Es wird auch eine gültige Fortsetzung des plastischen Spannungsfeldes ins starre Gebiet in der Nähe des Stempels erhalten.     

Thermo-elastic stresses in a half-space having an isolated heat source on the boundary

Zusammenfassung Die Arbeit behandelt das stationäre thermoelastische Spannungsfeld, das in einem Halbraum durch eine Wärmequelle an der Grenzebene erzeugt wird, während für den Rest der Grenzebene eine Strahlungsbedingung gilt. Die Lösung wird mit Hilfe von Hankel-Transformationen erhalten, und es stellt sich heraus, dass der Spannungszustand eben sowie parallel zur Oberfläche ist.     

Electro-mechanical sliding frictional contact of a piezoelectric half-plane under a rigid conducting punch

This paper investigates the two-dimensional sliding frictional contact of a piezoelectric half-plane in the plane strain state under the action of a rigid flat or a triangular punch. It is assumed that the punch is a perfect electrical conductor with a constant electric potential. By using the Fourier integral transform technique and the superposition theorem, the problem is reduced to a pair of coupled Cauchy singular integral equations and then is numerically solved to determine the unknown contact pressure and surface electric charge distribution. The effects of the friction coefficient and electro-mechanical loads on the normal contact stress, normal electric displacement, in-plane stress and in-plane electric displacement are discussed in detail. It is found that the friction coefficient has a significant effect on the electro-mechanical sliding frictional contact behaviors of the piezoelectric materials.     

Solving the problem of contact equilibrium of a punch and an elastic body

The variational problem of contact equilibrium of a punch and an elastic body is considered. An equivalent formulation of the problem is given in variational inequality form. Existence and uniqueness of the solution is investigated in a particular case. A penalty method is proposed for approximate solution of the problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 97–103, 1985.     

Dynamic stresses in a compound body with circular crack under sliding contact on an interface

We have considered the three-dimensional problem of harmonic loading of a circular crack in an elastic composite consisting of two dissimilar half-spaces under sliding contact on the surface of their bonding. The defect is situated in one of the half-spaces perpendicularly to the interface of materials. Using the representations of solutions in the form of Helmholtz potentials, we have reduced the problem to a boundary integral equation for the function of dynamic defect opening. Based on the numerical solution of this equation, we have obtained the frequency dependences of mode I stress intensity factor near the crack for different relations between the elastic moduli of components of the composite and the depths of crack location with respect to the interface.