Axisymmetric Contact Problems for the Elastic Half-Space

1980 ◽  
pp. 457-527
Author(s):  
G. M. L. Gladwell
Keyword(s):  
Half Space ◽  
Contact Problems ◽  
Elastic Half Space

scholarly journals Two Contact Problems for Annular Rigid Stamp on an Elastic Half Space

2018 ◽  
Vol 17 (6) ◽  
pp. 458-464
Author(s):  
S. V. Bosakov
Keyword(s):  
Functional Equation ◽  
Half Space ◽  
Bessel Functions ◽  
Infinite System ◽  
Annular Plate ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

A Rigid Punch on an Elastic Half-Space Under the Effect of Friction: A Finite Difference Solution

2008 ◽  
Author(s):  
Avraham Dorogoy ◽  
Leslie Banks-Sills
Keyword(s):  
Finite Difference ◽  
Contact Problem ◽  
Half Space ◽  
Contact Zone ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Rigid Punch ◽  
Normal Loading ◽  
Receding Contact ◽  
Stationary Contact

The accuracy of the finite difference technique in solving frictionless and frictional advancing contact problems is investigated by solving the problem of a rigid punch on an elastic halfspace subjected to normal loading. Stick and slip conditions between the elastic and the rigid materials are added to an existing numerical algorithm which was previously used for solving frictionless and frictional stationary and receding contact problems. The numerical additions are first tested by applying them in the solution of receding and stationary contact problems and comparing them to known solutions. The receding contact problem is that of an elastic slab on a rigid half-plane; the stationary contact problem is that of a flat rigid punch on an elastic half-space. In both cases the influence of friction is examined. The results are compared to those of other investigations with very good agreement observed. Once more it is verified that for both receding and stationary contact, load steps are not required for obtaining a solution if the loads are applied monotonically, whether or not there is friction. Next, an advancing contact problem of a round rigid punch on an elastic half-space subjected to normal loading, with and without the influence of friction is investigated. The results for frictionless advancing contact, which are obtained without load steps, are compared to analytical results, namely the Hertz problem; excellent agreement is observed. When friction is present, load steps and iterations for determining the contact area within each load step, are required. Hence, the existing code, in which only iterations to determine the contact zone were employed, was modified to include load steps, together with the above mentioned iterations for each load step. The effect of friction on the stress distribution and contact length is studied. It is found that when stick conditions appear in the contact zone, an increase in the friction coefficient results in an increase in the stick zone size within the contact zone. These results agree well with semianalytical results of another investigation, illustrating the accuracy and capabilities of the finite difference technique for advancing contact.

Inclined Flat Punch of Arbitrary Shape on an Elastic Half-Space

1986 ◽  
Vol 53 (4) ◽  
pp. 798-806 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Integral Representation ◽  
Half Space ◽  
Arbitrary Shape ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Elastic Contact ◽  
Circular Sector ◽  
Flat Punch ◽  
Circular Segment ◽  
Arbitrary Planform

A new method is proposed for the analysis of elastic contact problems for a flat inclined punch of arbitrary planform under the action of a normal noncentrally applied force. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some simple yet accurate relationships are established between the tilting moments and the angles of inclination of an arbitrary flat punch. Specific formulae are derived for a punch whose planform has a shape of a polygon, a triangle, a rectangle, a rhombus, a circular sector and a circular segment. All the formulae are checked against the solutions known in the literature, and their accuracy is confirmed.

Self similar solutions to adhesive contact problems with incremental loading

1968 ◽  
Vol 305 (1480) ◽  
pp. 55-80 ◽  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Half Space ◽  
Boundary Value ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
The Body ◽  
Similar Solutions ◽  
Dimensional Argument

The boundary-value problem for axisymmetric distortion of an elastic half space by a rigid indentor is formulated. A dimensional argument is used to infer the form of the distribution of radial displacement within the contact circle in terms of the shape of the body, assuming the load to be applied progressively, with interfacial friction sufficient to prevent any slip taking place between the indentor and the half space. This obviates the need for solving a preliminary integral equation for the boundary conditions, as proposed by Goodman (1962) and Mossakovski (1963). The resulting boundary-value problem is cast in the form of an integral equation of Wiener-Hopf type, which has been solved in a separate paper (Spence 1968, referred to as II). The solution is used to calculate stresses, displacements and contact radii for adhesive indentation by (i) a flat faced cylinder, (ii) an almost flat conical indentor and (iii) a sphere. The results are compared with those for frictionless indentation, for a range of values of Poisson’s ratio (iv). Adhesive indentation of a half space by a sphere of radius R rolling with angular velocity ω and linear velocity V (excluding dynamical effects) is also treated, and a value found for the creep 1 ( V / R ω in the absence of torsional or tractive forces.

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