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.     

The impact of a plane punch on an elastic half-plane

The transient dynamic contact problem of the impact of a plane absolutely rigid punch on an elastic half-plane is considered. The solution of the integral equation of this problem in terms of the unknown Laplace transform of the contact stresses at the punch base is constructed by a special method of successive approximations. The solution of the transient dynamic contact problem is obtained after applying an inverse Laplace transformation to the solution of the integral equation over the whole time range of the impact process, and the law of the penetration of the punch into the elastic medium is determined from a Volterra-type integrodifferential equation. The conditions for the punch to begin to separate from the elastic half-plane are formulated from the solution obtained, and all the stages of the separation process are investigated in detail. The law of the punch motion on the elastic half-plane and the width of the contact area, which varies during the separation, are then determined from the solution of the Volterra-type integrodifferential equation when an additional condition is satisfied.     

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.     

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 indentation of elastic and viscoelastic strips by rigid or elastic cylinders

Résumé Nous examinons les effets produits par la compression d'une bande élastique infinie entre deux cylindres élastiques à surface lisse d'une part, et par celle d'une bande visco-élastique entre deux cylindres rigides à surface lisse d'autre part. Nous déterminons en particulier la distribution de la pression sur la surface de contact. La pression de contact est donnée par une équation intégrale singulière dont le noyau peut être ramené avec une très bonne approximation à un logarithme plus un polynôme, cette dernière équation intégrale pouvant alors être résolue sous forme fermée. Il apparaît ainsi un paramètri géométrique, définissant le rapport entre l'épaisseur de la demi-bande et le rayon du cylindre, et nous obtenons des solutions valables, dans le cadre de l'approximation infinitésimale des contraintes, pour des valeurs de allant jusqu'à l'ordre de 10^–3, cas correspondant à une bande mince. D'autre part,on obtient un résultat très voisin de la solution du problème du demi-plan lorsque est de l'ordre de 10, cas correspondant à une bande épaisse. Ces deux solutions sont déduites pour la bande élastique, tandis que dans le cas visco-élastique, on a présenté la solution de la bande mince.     

Impact indentation of a rigid body into an elastic layer. Axisymmetric problem

An axisymmetric contact-impact problem is considered for an elastic layer subjected to normal indentation of a rigid body. An exact analytical solution is obtained in the case of a blunt shape of the indenter having a given velocity, and the stress pattern under multiple reflections is analyzed depending on the layer thickness. A numerical solution of the problem with arbitrary indenter shape is obtained on the basis of the simplified model of the theory of elasticity having a single displacement coincident with the impact direction. The explicit finite difference algorithm is designed on the basis of the mesh dispersion minimization technique. A parametric analysis is presented of the stress pattern developed with time with respect to variations of irregular shapes of the indenter and its masses.     

Harmonic oscillations of a rigid punch on a porous elastic layer

The plane dynamic contact problem of the harmonic oscillations of a rigid punch on the free surface of an elastic layer of porous isotropic material with linear properties is considered. The Fourier transformation of the problem is reduced to a Fredholm integral equation of the first kind in the contact pressure. The properties of the kernel of the fundamental integral equation are investigated and a numerical method of solving it is constructed. Numerical results are compared with existing results in classical limiting cases.     

The indentation of two rigid stamps into an elastic sphere with a spheroidal inclusion

On the basis of the expansion formulas of the vector solutions of the Lamé equations in spherical coordinates with respect to the solutions of the Lamé equations in oblate spheroidal coordinates and on the basis of their inverse formulas, one solves the problem of the compression of an elastic ball with an absolutely rigid inclusion in the form of an oblate spheroid. The problem is reduced to an infinite system of linear algebraic equations of the second kind with a completely continuous operator in ₂. Results of the numerical solution of the infinite system are given and the obtained results are analyzed.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 9–13, 1989.     

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.     

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.     

The influence of frictional forces on the interaction between an anisotropic half-plane and an absolutely rigid body with a cavity in its surface

By applying functions of a complex variable we reduce the problem of incomplete mechanical contact with friction between an anisotropic half-plane and a rigid body to solving a singular integral equation of second kind with respect to the height of the gap between the interacting bodies. We give the results of numerical analysis of the geometric contact characteristics of the gap for a particular cavity shape. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Hertz problem for a rigid punch moving across the surface of a semi-infinite elastic solid

The elastodynamic problem of a rigid punch moving at a constant sub-Rayleigh speed across the surface of an elastic half-space is investigated in the present paper. The unknown contact region is determined as part of solution from the unilateral or Signorini conditions. Numerical results are plotted showing how the eccentricity of the contact ellipse changes with the punch speed. Some asymptotic properties of the solution for the case where the punch speed is comparable with the Rayleigh wave speed are explored in details.     

A simple method for spherical indentation of an elastic thin layer with surface tension bonded to a rigid substrate

An alternative method is proposed to solve the spherical indentation problem of an elastic thin layer with surface tension bonded to a rigid substrate. Based on the Kerr model, we establish a simple modified governing equation incorporating the surface tension effects for describing the relationship between the pressure and downward deflection of the impressed surface of the layer. This modified governing equation holds both inside and outside the contact zone, making it possible to analyze the whole layer by a unified differential equation. Numerical results are presented for the contact pressure inside the contact zone, the surface deflection of the elastic layer and the load-contact zone width relation to illustrate the present method. The validity and accuracy of the present method are demonstrated by comparing our results with those available in the existing literature.     

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.     

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.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

Optimization of the stress tensor in an elastic anisotropic half-plane

We consider the problem of optimizing the components of the stress tensor and their integral characteristics. The normal and tangential forces prescribed on the boundary of the elastic anisotropic half-plane y≥0 are chosen from certain function classes of curvilinear strip type. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 110–116.     

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.