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.

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 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.

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.

Transient fundamental solutions for a transversely isotropic elastic half space

1993 ◽  
Vol 442 (1916) ◽  
pp. 505-531 ◽  
Keyword(s):  
Half Space ◽  
Transient Response ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Integral Transforms ◽  
Fundamental Solutions ◽  
Elastic Half Space ◽  
Kernel Functions ◽  
Two Dimensional

This paper is concerned with the study of transient response of a transversely isotropic elastic half-space under internal loadings and displacement discontinuities. Governing equations corresponding to two-dimensional and three-dimensional transient wave propagation problems are solved by using Laplace–Fourier integral transforms and Laplace−Hankel integral transforms, respectively. Explicit general solutions for displacements and stresses are presented. Thereafter boundary-value problems corresponding to internal transient loadings and transient displacement discontinuities are solved for both two-dimensional and three-dimensional problems. Explicit analytical solutions for displacements and stresses corresponding to internal loadings and displacement discontinuities are presented. Solutions corresponding to arbitrary loadings and displacement discontinuities can be obtained through the application of standard analytical procedures such as integration and Fourier expansion to the fundamental solutions presented in this article. It is shown that the transient response of a medium can be accurately computed by using a combination of numerical quadrature and a numerical Laplace inversion technique for the evaluation of integrals appearing in the analytical solutions. Comparisons with existing transient solutions for isotropic materials are presented to confirm the accuracy of the present solutions. Selected numerical results for displacements and stresses due to a buried circular patch load are presented to portray some features of the response of a transversely isotropic elastic half-space. The fundamental solutions presented in this paper can be used in the analysis of a variety of transient problems encountered in disciplines such as seismology, earthquake engineering, etc. In addition these fundamental solutions appear as the kernel functions in the boundary integral equation method and in the displacement discontinuity method.

The Singularity at the Apex of a Bonded Wedge-Shaped Stamp

1979 ◽  
Vol 46 (3) ◽  
pp. 577-580 ◽  
Author(s):  
K. S. Parihar ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Solution ◽  
Integral Equations ◽  
Half Space ◽  
Singular Integral ◽  
Green's Functions ◽  
Green’S Functions ◽  
Singular Integral Equations ◽  
Elastic Half Space

The problem of determining the singularity at the apex of a rigid wedge bonded to an elastic half space is formulated by considerations of Green’s functions for the loaded half space. The eigenvalue problem is reduced to finding the solution of a coupled pair of singular integral equations. A numerical solution for small wedge angles is given.

Thermoelastic Contact Between a Rolling Rigid Indenter and a Damaged Elastic Body

1990 ◽  
Vol 112 (2) ◽  
pp. 382-390 ◽  
Author(s):  
T. Goshima ◽  
L. M. Keer
Keyword(s):  
Heat Transfer ◽  
Stress Intensity ◽  
Half Space ◽  
Singular Integral Equations ◽  
Elastic Half Space ◽  
Two Dimensional ◽  
Intensity Factors ◽  
Thermoelastic Contact ◽  
Surface Breaking Crack ◽  
Thermoelastic Contact Problem

The two-dimensional thermoelastic contact problem of a rolling, rigid cylinder on an elastic half space containing a surface-breaking crack is solved using complex variable techniques. The effects of heat generation and friction between the cylinder and half space and of friction and heat transfer on the faces of the crack are considered. The problem is reduced to a pair of singular integral equations which are solved numerically. Numerical results are obtained when the loading is a Hertzian distributed heat input. By consideration of combinations of parameters, stress intensity factors for which the crack is likely to grow are shown.

Rapid Contact on a Prestressed Highly Elastic Half-Space: The Sliding Ellipsoid and Rolling Sphere

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
L. M. Brock
Keyword(s):  
Half Space ◽  
Contact Zone ◽  
Compressive Load ◽  
Reference Value ◽  
Singular Integral Equations ◽  
Elastic Half Space ◽  
Polar Coordinates ◽  
Rigid Sphere ◽  
Cauchy Stress ◽  
Specific Contact

A neo-Hookean half-space, in equilibrium under uniform Cauchy stress, undergoes contact by a sliding rigid ellipsoid or a rolling rigid sphere. Sliding is resisted by friction, and sliding or rolling speed is subcritical. It is assumed that a dynamic steady state is achieved and that deformation induced by contact is infinitesimal. Transform methods, modified by introduction of quasi-polar coordinates, are used to obtain classical singular integral equations for this deformation. Assumptions of specific contact zone shape are not required. Signorini conditions and the requirement that resultant compressive load is stationary with respect to contact zone stress give an equation for any contact zone span in terms of a reference value and an algebraic formula for the latter. Calculations show that prestress can significantly alter the ratio of spans parallel and normal to the direction of die travel, an effect enhanced by increasing die speed.

A Theoretical Study on the Indentation of Viscoelastic Materials

2003 ◽  
Vol 795 ◽  
Author(s):  
Guanghui Fu
Keyword(s):  
Mechanical Properties ◽  
Half Space ◽  
Theoretical Study ◽  
Contact Region ◽  
Indentation Load ◽  
Fundamental Relation ◽  
Polymeric Thin Films ◽  
Complete Contact ◽  
Simply Connected ◽  
Rigid Smooth

ABSTRACTNanoindentation experiments have been used to investigate mechanical properties of polymeric thin films. In this paper, the problem is modeled as normal indentation of a viscoelastic half-space by a rigid smooth frictionless axisymmetric polynomial indenter. An analytical solution, which relates the indentation load to the penetration depth, is presented. The solution is valid as long as the contact region is simply connected and the indenter is in complete contact with the half-space. It shows that a similar fundamental relation exists in Laplace space.

Solution of contact problems for a transversely isotropic elastic layer bonded to an elastic half-space

2009 ◽  
Vol 223 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
V I Fabrikant
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Integral Transform ◽  
Transversely Isotropic ◽  
Contact Problems ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Electrostatic Problem ◽  
Crack Problems ◽  
New Interpretation

The idea, first used by the author for the case of crack problems, is applied here to solve a contact problem for a transversely isotropic elastic layer bonded to an elastic halfspace, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the layer's free surface. The governing integral equation is derived; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged discs in the shape of the domain of contact. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.

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