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 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 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.

     

Asymptotic analysis of an elastic rod with rounded ends

We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m∈(0,1). Error estimates are obtained in the case m∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4.     

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 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 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 periodic contact problem for an elastic half-space

A method of solving the periodic contact problem for a system of indentors of arbitrary shape and an elastic half-space is proposed. Different versions of the arrangement of the indentors, at one and at several levels, are considered. The results are used to analyse the effect of the parameters of the microgeometry of the characteristics of a discrete contact and the stressed state of solids possessing regular microrelief.     

Vertical oscillations of a circular die on an elastic half-space with a cylindrical cavity

The problem on vertical oscillations of a circular die with a plane base that freely lies on an elastic half-space and contains a cylindrical cavity, whose axis is perpendicular to the plane of the base of the die and passes through its center, is considered. The problem is formulated in the form of paired integral equations that are related to integral Weber transforms. The paired equations are reduced to an equivalent Fredholm equation of the second kind. Some results of numerical calculations of amplitude-frequency characteristics of oscillations of the die are given. A comparative analysis with a known solution of an analogous problem for a continuous elastic half-space is given.Translated from Dinamicheskie Sistemy, No. 8, pp. 30–36, 1989.     

The Reissner-Sagoci type problem for a non-homogeneous elastic cylinder embedded in an elastic non-homogeneous half-space

The problem considered is that of the torsion of a non-homogeneouselastic cylinder, which is embedded in a non-homogeneous elastichalf-space (matrix) of different rigidity modulus. A rigid discis bonded to the flat surface of the cylinder and torque isapplied to the cylinder through a rigid disc. It is assumedthat there is perfect bonding at the common cylindrical surface.Using integral transformation techniques the solution of theproblem is reduced to dual integral equations. Later on thesolution of the dual integral equations is transformed intothe solution of a Fredholm integral equation of the second kind.Solving the Fredholm integral equation numerically the numericalresults for torque and shear stress inside the cylinder areobtained and displayed graphically to demonstrate the effectof non-homogeneity of the elastic material on the torque andshear stress.     

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.     

Contact problem for elastic half-space with an ellipsoidal cavity

We consider the axisymmetric problem of elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on its surface. The method of p-analytical functions is applied to reduce the solution of the problem to an infinite quasi-completely regular system of linear algebraic equations with upper bounded free terms that tend to zero as the index increases. The behavior of the normal stress near the contact line of the different boundary conditions is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 94–103, 1988.     

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.     

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.     

Optimization of the contact pressure in the problem of the interaction of a punch and an elastic medium

The problem of optimizing the pressure distribution under a rigid punch, which interacts without friction with an elastic medium filling a half-space, is investigated. The shape of the punch is taken as the initial variable of the design, while the root mean square deviation of the pressure distribution, which occurs under the punch, from a certain specified distribution, plays the role of the minimized functional. The values of the total forces and moments, applied to the punch, are assumed to be given, which leads to limitations imposed on the pressure distribution by the equilibrium conditions. It is shown that the optimization problem allows of decomposition into two successively solvable problems. The first problem consists of finding the pressure distribution which makes the optimized quality functional a minimum. The second problem is reduced to the problem of obtaining directly the optimum shape of the punch that yields the pressure distribution found. The optimization problem is investigated analytically for punches of different shape in plan. The optimum shapes are given in explicit form for punches with rectangular bases.     

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.     

A normal crack in an elastic half-space with stress-free surface

The problem of an elastic half-space with stress-free surface and a crack of arbitrary shape with prescribed displacements or tractions is reduced to an equivalent system of integral equations on the crack. For a pressurized crack in a plane perpendicular to the free surface, a scalar integral equation is derived. In properly chosen function spaces, unique solvability of the integral equation and regularity of solutions for regular data are proven.     

The indentation of a flat punch into an ideal rigid-plastic half-space under the action of shear contact stresses

A general plane problem of the impression of a flat punch into a rigid-plastic half-space under the action of transverse and longitudinal shear contact stresses is considered. The condition of complete plasticity and the hyperbolic equations of the general plane problem of the theory of ideal plasticity [1] are used. The reduction of the limit pressure on the punch is determined as a function of the shear contact stresses.     

The displacements in an elastic half-space when a load moves along a beam lying on its surface

The motion, with constant velocity, of a normal load along an elastic beam lying on an elastic isotropic homogeneous half-space is considered. A method for the approximate calculation of the normal displacements of the surface of the half-space for subsonic velocities of motion is developed. An estimate is given of the expressions obtained and a comparison is made with existing results for the problem of the motion of a point load along a half-space.     

The deformation of a half-space with a gradient elastic coating under arbitrary axisymmetric loading

An analytical-numerical method of solving the Neumann boundary-value problem for an elastic half-space with a gradient elastic coating is proposed. The problem is formulated and the construction of the fundamental solution (Green’s function) is described. The method enables a solution of the problem to be obtained for a fairly wide class of types of non-uniformity of the medium, and effects related to the non-uniformity are investigated analytically. A procedure for calculating the displacement, stress and strain fields is described. Particular attention is devoted to analysing the mechanical characteristics in the transition region from the coating to the elastic substrate.