Multipolar elastic fields in a layered half space

1968 ◽  
Vol 58 (5) ◽  
pp. 1519-1572 ◽  
Author(s):  
Ari Ben-Menahem ◽  
Sarva Jit Singh
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Strike Slip ◽  
The Other ◽  
Source Depth ◽  
Elastic Fields ◽  
New Type ◽  
Boussinesq Solution ◽  
Horizontal Thrust ◽  
Layered Half Space

Abstract Hansen's expansion is used to derive integral expressions for the displacement field due to a localized buried source of the mth order in a layered half space. The dipolar case (m ≦ 2) is worked out in detail for arbitrary source-depth in the layer and in the substratum. A new type of representation of the source is used which gives the final results in a concise form. Explicit expressions for the displacements at the free surface are obtained for a center of explosion, a vertical strike-slip fault and a vertical dip-slip fault. The results for a horizontal thrust are found to be the same as for a vertical dip-slip fault. The relations between the Galerkin vector and the biharmonic eigenvectors are clarified. It is shown that the Galerkin-Boussinesq solution for the elastic half space cannot be extended to structures of higher complexity, except for a few simple sources. On the other hand, the Hansen Solution is valid for a wide class of sources and structures. Both dynamic and static regimes are considered.

On strike-slip Somigliana dislocations in a layered elastic half-space

1983 ◽  
Vol 6 (3) ◽  
pp. 261-270 ◽  
Author(s):  
M. Dragoni ◽  
D. Golinelli
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Strike Slip

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.

The Wavefield Radiated Into an Elastic Half-Space by a Transducer of Large Aperture

1988 ◽  
Vol 55 (2) ◽  
pp. 398-404 ◽  
Author(s):  
John G. Harris
Keyword(s):  
Power Series ◽  
Plane Wave ◽  
Half Space ◽  
Wave Equations ◽  
Ultrasonic Transducer ◽  
Elastic Half Space ◽  
The Other ◽  
Circular Region ◽  
Asymptotic Power ◽  
Normal Traction

The wavefield radiated into an elastic half-space by an ultrasonic transducer, as well as the radiation admittance of the transducer coupled to the half-space, are studied. Two models for the transducer are used. In one an axisymmetric, Gaussian distribution of normal traction is imposed upon the surface, while in the other a uniform distribution of normal traction is imposed upon a circular region of the surface, leaving the remainder free of traction. To calculate the wavefield, each wave emitted by the transducer is expressed as a plane wave multiplied by an asymptotic power series in inverse powers of the aperture’s (scaled) radius. This reduces the wave equations satisfied by the compressional and shear potentials to their parabolic approximations. The approximations to the radiated waves are accurate at a depth where the wavefield remains well collimated.

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 Numerical Solution for Layered Solid Contact Problems With Application to Bearings

1983 ◽  
Vol 105 (4) ◽  
pp. 585-590 ◽  
Author(s):  
Y. P. Chiu ◽  
M. J. Hartnett
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Contact Problems ◽  
Steel Rolling ◽  
Rolling Element ◽  
Method Of Solution ◽  
Rectangular Area ◽  
Layered Solid ◽  
Boussinesq Solution ◽  
Layered Half Space

Presented herein is a method of solution for three dimensional counterformal contact problems involving layered solids. Based on the generalized Boussinesq solution for a layered half space, displacement and stress coefficients are formulated for a uniformly distributed load applied over a rectangular area on the surface of a layered half space. A precise analytical solution has been developed to find the surface pressure, contact area, approach and subsurface stresses for contact of arbitrary surface shapes. Numerical results have been obtained for the indentation of a second order surface with a layered solid for the case the layer to substrate shear modulus ratio equal to 3, which simulates the contact of a steel rolling element with a steel bearing ring supported by aluminum substrate (or housing) in a transmission system.

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.

scholarly journals To Solution of Contact Problem for Rectangular Plate on Elastic Half-Space

2020 ◽  
Vol 19 (3) ◽  
pp. 224-229
Author(s):  
S. V. Bosakov
Keyword(s):  
Contact Problem ◽  
Rectangular Plate ◽  
Half Space ◽  
Chebyshev Polynomials ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Rectangular Region ◽  
Double Row ◽  
Linear Algebraic Equations ◽  
Boussinesq Solution

Until the present time there is no exact solution to the contact problem for a rectangular plate on an elastic base with distribution properties. Practical analogues of this design are slab foundations widely used in construction. A lot of scientists have solved this problem in various ways. The methods of finite differences, B. N. Zhemochkin and power series do not distinguish a specific feature in contact stresses at the edges of the plate. The author of the paper has obtained an expansion of the Boussinesq solution for determining displacements of the elastic half-space surface in the form of a double series according to the Chebyshev polynomials of the first kind in a rectangular region. For the first time, such a representation for the symmetric part of the Boussinesq solution was obtained by V. I. Seimov and it has been applied to study symmetric vibrations of a rectangular stamp, taking into account inertial properties of the half-space. Using this expansion, the author gives a solution to the problem for a rectangular plate lying on an elastic half-space under the action of an arbitrarily applied concentrated force. In this case, the required displacements are specified in the form of a double row in the Chebyshev polynomials of the first kind. Contact stresses are also specified in the form of a double row according to the Chebyshev polynomials of the first kind with weight. In the integral equation of the contact problem integration over a rectangular region is performed while taking into account the orthogonality of the Chebyshev polynomials. In the resulting expression the coefficients are equal for the same products of the Chebyshev polynomials. The result is an infinite system of linear algebraic equations, which is solved by the amplification method. Thus the sought coefficients are found in the expansion for contact stresses.

The Elastic Field in a Half-Space With a Circular Cylindrical Inclusion

1996 ◽  
Vol 63 (4) ◽  
pp. 925-932 ◽  
Author(s):  
L. Z. Wu ◽  
S. Y. Du
Keyword(s):  
Half Space ◽  
Elastic Field ◽  
Elastic Half Space ◽  
Elliptic Integrals ◽  
Cylindrical Inclusion ◽  
Stress Fields ◽  
Function Technique ◽  
Explicit Solutions ◽  
Elastic Fields ◽  
Complete Elliptic Integrals

The problem of a circular cylindrical inclusion with uniform eigenstrain in an elastic half-space is studied by using the Green’s function technique. Explicit solutions are obtained for the displacement and stress fields. It is shown that the present elastic fields can be expressed as functions of the complete elliptic integrals of the first, second, and third kind. Finally, numerical results are shown for the displacement and stress fields.

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