Rigid Stamp Research Articles

Waer of composite thin-walled structural elements with consideration of thermal effects

The contact problem on interaction of rigid stamps with anisotropic plates with regard to wear and the corresponding frictional heating is considered. The procedure developed is based on the reduction of the governing equation to a system of Volterra integral equations of the second kind. The numerical analysis allows us to study the effects of both the anisotropic thermoelastic characteristics of materials and the nature of the interaction of the contact-pair elements on the wear process.

Some dynamic properties of the nonhomogeneous thermoelastic prestressed media

The problem of steady-state harmonic oscillations is considered for a nonhomogeneous thermoelastic prestressed medium. The mechanical parameters of this medium are considered to be arbitrary, sufficiently smooth functions of one of the coordinates (depth). The oscillations are realized under the action of the load or the rigid stamp oscillated on the medium’s surface. Two models of a nonhomogeneous thermoelastic medium are investigated: a nonhomogeneous layer of finite thickness on the un-deformed foundation and a nonhomogeneous layer on the surface of the homogeneous half-space. This investigation is based on the numeric construction of the Green’s function of the medium and the analysis of this function’s properties. The different types of changing with the depth of the numerical parameters of the layer’s material have been considered. The maximal changing of the parameters is located near the upper or lower bound of the layer. [Work supported by RFFI.]

Stress Distribution at the Contact of Absolutely Rigid Stamp and Elastic Orthotropic Strip

A plane elastic problem for an orthotropic infinite strip with mixed boundary conditions is investigated. A model of the strip has been built by using the method of integral Fourier transforms. We obtain relationships which allow us to formulate singular integral equations for the various types of boundary conditions on one of its edges. The problems in the case of both smooth stamp-strip contact and rigid stamp-strip adhesion have been considered. It is shown that the effect of anisotropy on the contact stress distribution is minor. The stress intensity factors at the stamp corners, which are the main parameters of fracture, are evaluated. The quasi-invariance of a certain combination of the stress intensity factors is confirmed.

Rigid Stamp Indentation on a Curvilinear Hole Boundary of an Anisotropic Elastic Body

A general solution for the problems of rigid stamp indentation on a curvilinear hole boundary of an anisotropic elastic body is obtained by employing the Stroh formalism, the method of analytical continuation, and the technique of mapping a closed curve to a unit circle. With this general solution, two typical curvilinear holes are studied. One is an ellipse, the other is a polygon. Since the transformation functions used in our solutions are not always one to one, some of the solutions are not exact but only approximate. For example, the solutions to the problems of anisotropic plates containing an elliptic hole and isotropic plates containing a polygonal hole are exact, but the solutions to the problems of anisotropic plates containing a polygonal hole are only approximate. Because the solutions presented in this paper are new and no other analytical solution has been found in the literature, the correctness of the present results can only be checked analytically by their reduced forms such as those for isotropic media or those for the stress boundary value problems. To show the generality of our solutions and to see clearly the physical behavior of the indentation problems two numerical examples are given, and their related hoop stress and stress contour are also plotted.

Mixed boundary–value problems of two–dimensional anisotropic elasticity with perturbed boundaries

The mixed boundary–value problems of two–dimensional anisotropic elasticity are considered in this paper. The boundary treated may be the one perturbed by a straight line or an ellipse. A general solution up to the first–order perturbation has been found by using Stroh9s formalism, analytical continuation method, conformal mapping function, and perturbation technique. As to higher–order perturbation solutions, general procedure is depicted in this paper. In order to illustrate the use of general solution, two typical examples are solved completely. One is a cosine wave–shaped surface indented by a rigid flat–ended punch, the other is a triangular hole boundary indented by a rigid stamp.

Unsymmetrical elastic stamp on a nonlocal elastic half-plane

The statical problem of an elastic stamp on a nonlocal elastic half-plane is solved. The starting point is the solution of the problem of a rigid stamp on an elastic half-plane. This solution is given in the appendix. The classical solution of the elastic stamp is obtained by making appropriate changes in the classical solutions of the rigid stamp. The classical and nonlocal problem of elastic stamp has not yet been solved in the literature. The nonlocal solution of the problem is carried out by using the classical solution. The classical solution gives infinite stresses at the tips of the edges, but the nonlocal solution produces infinite stresses nowhere. This situation is shown by several diagrams.

Partially stiffened elastic half-plane with an edge crack

In the present paper two contact problems referring to the partially stiffened elastic half-plane along its edge with a reinforcement (or stringer) either in the form of a rigid plate or a smooth rigid stamp, are considered. The half-plane contains an edge crack which is arbitrarily pressurized, and is perpendicular to the bounding edge of the half-space. It is assumed that the mid-point of the stringer is located in the axis of the crack. Each of the above two half-plane contact problems is first reduced to a system of two singular integral equations with fixed singularities. Then by employing the generalized method of integral transforms, this system is further reduced to a system of Wiener–Hopf equations that is equivalent to the Riemann matrix boundary value problem. Exact analytical solutions of the two problems are presented in series form. Asymptotic approximations for the stress intensity factor and the energy release rate at the crack tip are also given. Finally, numerical results for the contact stresses, crack opening displacements, stress intensity factor and crack energy are displayed.

Rectangular rigid stamp on a nonlocal elastic half-plane

First, the solution of the problem of a rectangular rigid stamp uniformly moving on an elastic half plane is given. Secondly, the solution is obtained for a motionless rectangular stamp on a nonlocal elastic half space. The finite distribution of the stress under the stamp has been compared with the unbounded stresses of local case.

Unsymmetrical rigid stamp on a nonlocal elastic half-plane

The problem of an unsymmetrical rigid stamp on a nonlocal elastic half plane is solved. Since the problem has not been solved even for the classical case in its full generality and because the latter is needed for comparison, the solution of a moving unsymmetrical stamp for the classical case is given in the Appendix which, of course, gives the case of the motionless stamp in the special case of the zero velocity. The solution of the problem of the unsymmetrical stamp on the nonlocal half-plane does not give any infinite stresses; hence it does not have the defect of the classical solution and its importance is mainly due to this property. In fact, the classical solution gives infinite stresses at the sharp edges of the stamp embedded in the half plane, but in the case of a nonlocal medium the maximum stresses are finite and what is more astounding, they are not at the tips of the edges. The results are given by several diagrams which, whenever necessary, include also the classical values.

Green's Functions for Axisymmetric Surface Force Problems of an Elastic Half-Space with Transverse Isotropy.

Green's functions are derived for axisymmetric surface force problems of an elastic half-space with transverse isotropy. Green's functions are defined as solutions to the problem of a transversely isotropic elastic half-space subjected axisymmetric surface forces acting on a circle. The surface forces acting on a circle are classified as (i) a radial force, (ii) a torsional force and (iii) an axial force in cylindrical coordinates. Green's functions for an isotropic elastic half-space were derived in a previous paper by applying Love's stress function. In this paper we apply Lekhnitsky's stress function in order to obtain Green's functions for an elastic half space with transverse isotropy. As an example of application of these Green's functions, we consider a problem of a transversely isotropic elastic half-space twisted by a flat annular rigid stamp.

The dynamical problem of a rectangular stamp moving on an elastic half plane

In this work, the stresses under a rectangular rigid stamp moving on an elastic half plane are calculated. The boundary value problem has been formulated in the form of a singular integral equation whose unknown function is the stress distribution under the stamp. The solutions of the equation have been compared for the cases of absence or presence of friction and for the cases of motion or rest. The work is an extension of Muskhelishvili's results and differs numerically from these ones only when the speed of the stamp is comparable with the shear wave speed.

Convex multilevel decomposition algorithms for non‐monotone problems

AbstractA convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non‐monotone, possibly multivalued laws. The theory is developed here for a model structure with non‐monotone interface or boundary conditions. First the non‐monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non‐linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non‐convex min‐min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,1 is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non‐monotone stick‐slip frictional effects illustrate the theory.

Pressure of a rigid stamp with a flat base heated to a constant temperature on an elastic half-space

Pressure of a rigid stamp with a flat base heated to a constant temperature on an elastic half-space

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 Lame equations in spherical coordinates with respect to the solutions of the Lame 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 l2. Results of the numerical solution of the infinite system are given and the obtained results are analyzed.

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.

The stressed state of clayey soil in the contact layer below the base of a rigid strip-type stamp

The stressed state of clayey soil in the contact layer below the base of a rigid strip-type stamp

The torsion of a growing cylinder by a rigid stamp

The contact problem of the torsion of a viscoelastic ageing, growing cylinder by a rigid stamp is considered. Dual series equations reflecting the mathematical content of the problem of different stages of the growing process are derived and investigated. The results of a numerical analysis and the singularities of the qualitative behaviour of the fundamental characteristics are discussed.

A contact problem for an elastic layer indented by multiple rigid annular stamps.

This paper is concerned with a contact problem for an elastic layer on a rigid substrate indented by multiple rigid concentric-annular stamps. This problem is a mixed boundary value problem in which the surface displacement of the layer is specified inside the concentric-annular regions and the traction is zero outside. It is assumed that contact is free from friction so that the shear traction on the whole surface of the layer is zero. The problem is reduced to infinite simultaneous algebraic equations by using Fourier series expansion of the normal traction within the contact region. General solutions for the contact stress and the surface displacement are presented. Sone numerical results are given in graphical form, and the interaction of the contact stresses caused by each annulus is investigated.

Supersonic cleavage of an elastic strip

The problem of the longitudinal cleavage of an infinite elastic strip by a thin smooth rigid wedge is examined. The wedge moves symmetrically with respect to the faces of the strip at a constant supersonic velocity. Formulas are obtained that govern the stresses in the domain of wedge contact with the elastic medium and the displacements of points of the slit edge outside the contact domain for certain relationships between the parameters of the problem. Conditions are set up for which separation of the medium from the wedge surface occurs. Unlike the case of wedge motion at a speed less than the Rayleigh velocity /1, 2/, when a crack is formed ahead of the wedge, no crack is formed when the wedge moves at supersonic speed. The contact problem of the motion of a rigid stamp with a flat smooth base at a supersonic speed over the surface of an elastic strip was investigated /3/ in a similar formulation.

Singularities of the interaction of a vibrating stamp with an inhomogeneous heavy base

A method is developed for studying the fundamental characteristics of the wave process on the surface of an initally isotropic prestressed elastic half-space caused by an oscillating rigid stamp. The following is taken as the model of the inhomogeneous medium: an elastic layer 0 ⩽ x 3 ⩽ h, x 1, x 2, < ∞ whose mechanical characteristics as well as the initial stresses are arbitrary, fairly smooth functions of the coordinate x 3 in the general case, lies on the surface of a homogeneous half-space x 3, ⩾ h, x 1, x 2, < ∞ ( x 1, x 2, x 3, are a rectangular Cartesian coordinate system). The linearized boundary value problem of the dynamic theory of elasticity of vibrations with frequency ω for a rigid stamp on the surface of an inhomogeneous medium reduces to an integral equation or to a system of integral equations of the first kind whose integral operator kernel is constructed numerically. Approximation of the kernel of the integral operator by a special kind of function enables an approximate solution of the integral equation to be constructed by the factorization method /1, 2/. On the basis of the latter, an effective investigation of the influence of the parameters characterising the inhomogeneity of the medium and the initial state of stress on the wave process both under (stress wave) and outside the stamp is possible. The construction of a general linearized theory and the regularities of elastic wave propagation in bodies with homogeneous initial stresses are considered in /3/, where a fairly complete survey is also given of the literature on this question. A systematic exposition of the theory of wave propagation in elastic media with an inhomogeneous initial state was first given in /4/. The contact problem of the vibrations of an inhomogeneous half-space subjected to a rigid stamp oscillating on its surface was examined in /5, 6/ without taking the initial stresses into account. A method of investigating the regularities of electric-wave excitation in semibounded bodies (layers and cylinders) with varying properties and magnitude of the initial stresses was proposed in /7/ on the basis of the solution of the contact problem. A method of investigating the singularities of elàstic wave propagation in an inhomogeneous initially strained half-space caused by an oscillating load distributed in a certain domain on the surface of the medium has been developed (∗Kalinchuk V.V., Lysenko I.V. and Polyakova I.B., Singularities of elastic wave excitation and propagation in an inhomogeneous heavy half-space. Deposited Manuscript, December 10, 1986, 8877-B86, VINITI, Rostov, 1986).