Contact problems of the theory of elasticity for a transversally isotropic half space with an elliptical line of separation between boundary conditions

1996 ◽  
Vol 32 (6) ◽  
pp. 443-449
Author(s):  
Yu. N. Podil'chuk ◽  
V. F. Tkachenko
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Contact Problems ◽  
Theory Of Elasticity ◽  
Transversally Isotropic

scholarly journals THE OSCILLATIONS ON SURFACE OF THE PRE-STRESSED THERMOELASTIC LAYERED HALF-SPACE

2020 ◽  
Vol 82 (2) ◽  
pp. 123-134
Author(s):  
G.Yu. Levi ◽  
L.A. Igumnov
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Thermal Contact ◽  
Operational Calculus ◽  
Contact Problems ◽  
The Body ◽  
Vertical Displacements ◽  
Deformed State ◽  
The Green Function ◽  
Layered Half Space

The contact problems of the excitation of harmonic oscillations by an oscillating rigid stamp on the thermoelastic layered pre-stressed body surface are considered. The body is represented by a layer lying on half-space. At the interface, two modes of mechanical boundary conditions are assumed: rigid adhesion and the absence of friction along one of the coordinates. In addition thermal contact condition is assumed. The initial deformed state is created due to thermal and mechanical effects. To find a solution of contact problems we introduced a boundary value problems with homogeneous boundary conditions on the surface. The Green function of auxiliary problems is constructed. Using operational calculus, the system of integral equations for the stress distribution function is reduced to a one-dimensional integral equation. Its solution is constructed by using the method of fictitious absorption. As an example, the contact problems on vibrations of a half-space of magnesium oxide coated with cadmium sulfide are considered. The solution to the contact problems, the distribution of vertical displacements over the entire surface of a thermoelastic prestressed body, is calculated and presented graphically. The influence of preheating and conditions at the interface is studied on the distribution of vertical stresses and displacements over the surface of a layered half-space. It is shown that under conditions of preheating and stretching of the layer in the direction of the horizontal coordinate under the action of a hard stamp on the surface the larger vertical stresses arise in the case of hard pinching of the coating with the base than with non-ideal adhesion. On the contrary, preheating and extension of the layer affect vertical displacements outside the contact area more when imperfect coupling.

Periodic Contact and Mixed Problems of the Elasticity Theory (Review)

2021 ◽  
pp. 22-33
Author(s):  
Dmitrii A. Pozharskii
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Half Space ◽  
Three Dimensional ◽  
Contact Problems ◽  
Mixed Problems ◽  
Periodic Problems ◽  
Straight Line ◽  
Periodic Contact ◽  
Space Boundary

Results are reviewed collected in the investigations of periodic contact and mixed problems of the plane, axially symmetric and spatial elasticity theory. Among mixed problems, cut (crack) problems are focused integral equations of which are connected with those for contact problems. The periodic contact problems stimulate research of the discrete contact of rough (wavy) surfaces. Together with classical elastic domains (half-plane, half-space, plane and full space), we consider periodic problems for cylinder, layer, cone and spatial wedge. Most publications including fun-damental ones by Westergaard and Shtaerman deals with plane periodic problems of the elasticity theory. Here, one can mention approaches based on complex variable functions, Fourier series, Green’s functions and potential func-tions. A fracture mechanics approach to the plane periodic contact problem was developed. Methods and approaches are considered which allow us to take friction forces, adhesion and wear into account in the periodic contact. For spatial periodic and doubly periodic contact and properly mixed problems, we describe such methods as the localiza-tion method, the asymptotic methods, the method of nonlinear boundary integral equations, the fast Fourier trans-form. The half-space is the simplest model for elastic solids. But for the simplest straight-line periodic punch system, some three-dimensional contact problems (normal contact or tangential contact for shifted cohesive coatings) turn out to be incorrect because their integral equations contain divergent series. Considering three-dimensional periodic problems, I.G. Goryacheva disposes circular punches in special way (circular orbits, polar coordinated are used for centers of the punches), in this case one can prove convergence of the series in the integral equation (it is important that the punches are circular). For the periodic problems for an elastic layer, V.M. Aleksandrov has shown that the series in integral equations converge but the kernels become more complicated. In the present paper, we demonstrate that for the straight-line periodic punch system of arbitrary form the contact problem for a half-space turns out to be correct in case of more complicated boundary conditions. Namely, it can be sliding support or rigid fixation of a half-plane on the half-space boundary, the half-plane boundary should be parallel to the straight-line (the punch system axis) for arbitrary finite distance between the parallel lines. On this way, for sliding support, the kernel of the period-ic problem integral equation kernel is free of integrals, it consists of single convergent series (normal contact, the kernel is given in two equivalent forms). We consider classical percolation (how neighboring contact domains pene-trate one to another, investigated by K.L. Johnson, V.A. Yastrebov with co-authors) for the three-dimensional periodic contact amplification as well as percolation for the straight-line punch system. A similar approach is suggested for the case of periodic tangential contact (coatings system cohesive with a half-space boundary shifted along its axis or perpendicular to it). Here, one can separate out unique solutions of auxiliary problems because the line of changing boundary conditions on the half-space boundary can provoke non-uniqueness. The method proposed opens possibility to consider more complicated three-dimensional periodic contact problems for straight-line punch systems with changing boundary conditions inside the period.

Integral properties of solutions to some boundary initial-value problems in linear anisotropic elastodynamics

1995 ◽  
Vol 449 (1937) ◽  
pp. 591-596 ◽  
Keyword(s):  
Contact Problem ◽  
Boundary Conditions ◽  
Half Space ◽  
Initial Value Problems ◽  
Contact Region ◽  
Dynamical Theory ◽  
Theory Of Elasticity ◽  
Initial Contact ◽  
Initial Value ◽  
Various Boundary Conditions

We study some integral properties of weak solutions to some boundary initial-value problems in the linearized dynamical theory of elasticity. These problems arise during the indentation of an anisotropic half-space by a convex punch having an arbitrary indenting velocity and shape. In contrast to previous studies of the problem, we consider various boundary conditions, e. g., adhesive or frictional, in the case when this non-frictionless boundary initial-contact problem has two orthogonal planes of symmetry, which are both orthogonal to the boundary of half-space. We show that if the non-frictionless boundary initial-contact problem has such a symmetry, then the problem for integral characteristics of the solutions is equivalent to a problem of plane-wave propagation in the same medium. A proof is given that the instantaneous value of the force required to indent the punch during the first supersonic stage of contact is directly proportional to the product of the velocity of indentation and the area of contact at that instant, and is independent of the boundary conditions in the contact region.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Bounds on the Maximum Contact Stress of an Indented Elastic Layer

1971 ◽  
Vol 38 (3) ◽  
pp. 608-614 ◽  
Author(s):  
Y. C. Pao ◽  
Ting-Shu Wu ◽  
Y. P. Chiu
Keyword(s):  
Half Space ◽  
Contact Stress ◽  
Elastic Layer ◽  
Theory Of Elasticity ◽  
Contact Conditions ◽  
Practical Applications ◽  
Method Of Solution ◽  
Elastic Layers ◽  
Maximum Contact ◽  
Contact Stress Distribution

This paper is concerned with the plane-strain problem of an elastic layer supported on a half-space foundation and indented by a cylinder. A study is presented of the effect of the contact condition at the layer-foundation interface on the contact stresses of the indented layer. For the general problem of elastic indenter or elastic foundation, the integral equations governing the contact stress distribution of the indented layer derived on the basis of two-dimensional theory of elasticity are given and a numerical method of solution is formulated. The limiting contact conditions at the layer-foundation interface are then investigated by considering two extreme cases, one with the indented layer in frictionless contact with the half space and the other with the indented layer rigidly adhered to the half space. Graphs of the bounds on the maximum normal stress occurring in indented elastic layers for the cases of rigid cylindrical indenter and rigid half-space foundation are obtained for possible practical applications. Some results of the elastic indenter problem are also presented and discussed.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

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