Multiple Region Contact Solutions for a Flat Indenter on a Layered Elastic Half Space: Plane-Strain Case

1989 ◽  
Vol 56 (2) ◽  
pp. 251-262 ◽  
Author(s):  
T. W. Shield ◽  
D. B. Bogy
Keyword(s):  
Plane Strain ◽  
Half Space ◽  
Contact Region ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Geometrical Parameters ◽  
Restricted Range ◽  
Complete Contact ◽  
The One ◽  
Layered Half Space

The plane-strain problem of a smooth, flat rigid indenter contacting a layered elastic half space is examined. It is mathematically formulated using integral transforms to derive a singular integral equation for the contact pressure, which is solved by expansion in orthogonal polynomials. The solution predicts complete contact between the indenter and the surface of the layered half space only for a restricted range of the material and geometrical parameters. Outside of this range, solutions exist with two or three contact regions. The parameter space divisions between the one, two, or three contact region solutions depend on the material and geometrical parameters and they are found for both the one and two layer cases. As the modulus of the substrate decreases to zero, the two contact region solution predicts the expected result that contact occurs only at the corners of the indenter. The three contact region solution provides an explanation for the nonuniform approach to the half space solution as the layer thickness vanishes.

Some Axisymmetric Problems for Layered Elastic Media: Part I—Multiple Region Contact Solutions for Simply-Connected Indenters

1989 ◽  
Vol 56 (4) ◽  
pp. 798-806 ◽  
Author(s):  
T. W. Shield ◽  
D. B. Bogy
Keyword(s):  
Half Space ◽  
Singular Integral Equations ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Geometrical Parameters ◽  
Restricted Range ◽  
Axisymmetric Problems ◽  
Complete Contact ◽  
Simply Connected ◽  
Layered Half Space

The contact of a flat, simply-connected axisymmetric indenter with a layered elastic half space is examined. The problem is mathematically formulated using integral transforms to derive singular integral equations for the contact pressure. The solution of these equations is obtained by expansion in orthogonal polynomials. The solution predicts complete contact between the indenter and the surface of the layered half space only for a restricted range of the material and geometrical parameters. Outside of this range, solutions exist with multiple contact regions. A parameter space is divided into zones for single and multiple contact solutions and comparisons are made with the solutions for the analogous plane-strain problem.

Some Axisymmetric Problems for Layered Elastic Media: Part II—Solutions for Annular Indenters and Cracks

1989 ◽  
Vol 56 (4) ◽  
pp. 807-813 ◽  
Author(s):  
T. W. Shield ◽  
D. B. Bogy
Keyword(s):  
Half Space ◽  
Contact Region ◽  
Singular Integral Equations ◽  
Integral Transforms ◽  
Lower Surface ◽  
Elastic Half Space ◽  
Solution Technique ◽  
Physical Parameters ◽  
Annular Crack ◽  
Layered Half Space

In Part I, the multiple contact region solutions for an axisymmetric indenter were presented. The solution technique utilized integral transforms and singular integral equations. The emphasis there was the study of the conditions of contact as a function of the physical parameters of the indenter and the layered elastic half space. The method and results were similar to those for the analogous plane-strain problem that was studied in Shield and Bogy (1989). However, several differences in detail were required for the analysis of the axisymmetric geometry. In this Part II, the solution of Part I is used to study some related problems that have been considered previously in the literature for homogeneous half spaces. First we solve the problem of the axisymmetric annular indenter for the layered half space. Multiple contact region solutions are studied and the problem of an axisymmetric punch with internal pressure is solved for the layered half space and also for the special case of a layer with a traction-free lower surface. Finally, the problem of an annular crack in a homogeneous or layered structure is solved.

Normal Indentation of Elastic Half-Space With a Rigid Frictionless Axisymmetric Punch

2001 ◽  
Vol 69 (2) ◽  
pp. 142-147 ◽  
Author(s):  
G. Fu
Keyword(s):  
Half Space ◽  
Complex Variable ◽  
Contact Region ◽  
Pressure Profile ◽  
Elastic Half Space ◽  
Final Solution ◽  
Special Cases ◽  
Complete Contact ◽  
Simply Connected ◽  
Variable Analysis

The contact of a simply connected axisymmetric punch with an elastic half-space is examined. The problem is mathematically formulated by using potential theory and complex variable analysis. The final solution of these equations is obtained by assuming a polynomial punch profile. The conditions for complete contact and incomplete contact are also derived. The solutions give the pressure profile at the punch–elastic half-space interface for any polynomial punch profile, even for noninteger power polynomials, as long as the contact region is simply connected. The results show that some classic solutions in linear elasticity are special cases of the derived solution and determine the range of validity for those solutions.

Contact Stress Analysis of a Layered Transversely Isotropic Half-Space

1992 ◽  
Vol 114 (2) ◽  
pp. 253-261 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Contact Region ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Stress Components ◽  
Contact Failure

The three-dimensional problem of contact between a spherical indenter and a multi-layered structure bonded to an elastic half-space is investigated. The layers and half-space are assumed to be composed of transversely isotropic materials. By the use of Hankel transforms, the mixed boundary value problem is reduced to an integral equation, which is solved numerically to determine the contact stresses and contact region. The interior displacement and stress fields in both the layer and half-space can be calculated from the inverse Hankel transform used with the solved contact stresses prescribed over the contact region. The stress components, which may be related to the contact failure of coatings, are discussed for various coating thicknesses.

A Unified Treatment of Axisymmetric Adhesive Contact on a Power-Law Graded Elastic Half-Space

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Fan Jin ◽  
Xu Guo ◽  
Wei Zhang
Keyword(s):  
Half Space ◽  
Power Law ◽  
Contact Region ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
Hertzian Contact ◽  
Contact Radius ◽  
Abel Transformation ◽  
Numerical Solution Methods ◽  
Special Cases

In the present paper, axisymmetric frictionless adhesive contact between a rigid punch and a power-law graded elastic half-space is analytically investigated with use of Betti's reciprocity theorem and the generalized Abel transformation, a set of general closed-form solutions are derived to the Hertzian contact and Johnson–Kendall–Roberts (JKR)-type adhesive contact problems for an arbitrary punch profile within a circular contact region. These solutions provide analytical expressions of the surface stress, deformation fields, and equilibrium relations among the applied load, indentation depth, and contact radius. Based on these results, we then examine the combined effects of material inhomogeneities and punch surface morphologies on the adhesion behaviors of the considered contact system. The analytical results obtained in this paper include the corresponding solutions for homogeneous isotropic materials and the Gibson soil as special cases and, therefore, can also serve as the benchmarks for checking the validity of the numerical solution methods.

On raytracing in an elastic-anelastic medium

1991 ◽  
Vol 81 (2) ◽  
pp. 667-686 ◽  
Author(s):  
E. S. Krebes ◽  
M. A. Slawinski
Keyword(s):  
Travel Time ◽  
Half Space ◽  
Body Wave ◽  
Seismic Wave Propagation ◽  
Elastic Half Space ◽  
Elastic Plane ◽  
Ray Method ◽  
The One ◽  
Ray Parameter ◽  
Method Of Steepest Descent

Abstract In this article, we investigate seismic wave propagation in a medium consisting of a stack of anelastic layers sandwiched between two half-spaces. The upper half-space is perfectly elastic, and the lower half-space is anelastic. The source is in the upper elastic half-space. To compute a ray going from the source to the receiver (which can be anywhere in the medium), we examine two approaches. The first involves an evaluation of the Sommerfeld wavefield integral by the method of steepest descent, and we refer to the resulting ray as the stationary ray. The second involves assuming that the attenuation vector A1 of the initial ray segment emerging from the source in the elastic half-space is zero (an assumption often made in the literature), and we refer to the resulting ray as the conventional ray. We find that the stationary and conventional rays are, in general, not identical, in that the stationary ray has (a) a complex, rather than real, ray parameter; (b) a smaller travel time; (c) an initial ray segment that corresponds to an inhomogeneous elastic plane body wave (A1 ≠ 0); and (d) a substantially different value for the ray amplitude. The stationary ray actually has the smallest travel time of all possible rays, and hence it is the one that satisfies Fermat's principle of least time. Our results suggest that the stationary ray method is the correct method and that the conventional ray method is generally incorrect. The results might also find application in marine seismology, since water is practically a lossless medium.

scholarly journals On a Certain Method of Selection of Domain for Finite Element Modelling of the Layered Elastic Half-Space in the Static Analysis of Flexible Pavement

2012 ◽  
Vol 58 (4) ◽  
pp. 477-501
Author(s):  
M. Nagórska
Keyword(s):  
Half Space ◽  
Mechanistic Model ◽  
Elastic Half Space ◽  
Original Method ◽  
Fe Model ◽  
Linear Elastic ◽  
Road Pavement ◽  
Viscoelastic Layers ◽  
Selection Of ◽  
Layered Half Space

AbstractIn the flexible road pavement design a mechanistic model of a multilayered half-space with linear elastic or viscoelastic layers is usually used for the pavement analysis.This paper describes a domain selection for the purpose of a FE model creating of the linear elastic layered half-space and boundary conditions on borders of that domain. This FE model should guarantee that the key components of displacements, stresses and strains obtained using ABAQUS program would be in particular identical with those ones obtained by analytical method using VEROAD program.It to achieve matching results with both methods is relatively easy for stresses and strains. However, for displacements, using FEM to obtain correct results is (understandably) highly problematic due to infinity of half-space. This paper proposes an original method of overcoming these difficulties.

A concave indenter on a piezoelectromagneto-elastic substrate or a layer elastically supported

2012 ◽  
Vol 47 (6) ◽  
pp. 362-378 ◽  
Author(s):  
Bogdan Rogowski
Keyword(s):  
Half Space ◽  
Elastic Foundation ◽  
Contact Region ◽  
Approximate Solutions ◽  
Elastic Half Space ◽  
Solution Technique ◽  
Elementary Functions ◽  
Full Field ◽  
Full Contact ◽  
Two Parameter

Indentation of piezoelectromagneto-elastic half-space or a layer on a two-parameter elastic foundation by a cylindrical indenter with a slightly concave base is considered. Full-field magnetoelectro-elastic solutions in elementary functions are obtained for the case of full contact and half-space. If the axial load is small, the contact area will be an annulus the outer circumference of which coincides with the edge of the punch. The inner circumference will shrink with increasing load and there will be a critical load above which the stratum makes contact with the entire punch base. The contact problem for high loads can therefore be treated by classical methods. The more interested case in which the load is less the critical value and the contact region is annulus remains. By use the methods of triple integral equations and series solution technique the solution for an indentured substrate over an annular contact region is also given. For parabolic and conical concave punches the exact or approximate solutions are obtained for full contact or annular contact region, respectively. For the layer on two-parameter elastic foundation and concave punch approximate solution is established.

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