The Normal Impact of a Rod-Mass System on a Viscoelastic Layered Half Space

1988 ◽  
Vol 55 (4) ◽  
pp. 879-886
Author(s):  
H. A. Downey ◽  
D. B. Bogy
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Elastic Half Space ◽  
Substrate Material ◽  
Viscoelastic Layer ◽  
Lumped Mass ◽  
Mass System ◽  
Laplace Transform Inversion ◽  
Numerical Laplace Transform ◽  
Layered Half Space

A rod with a lumped mass attached to its trailing end travels axially with a uniform velocity and strikes an elastic half space that is covered with an adhering viscoelastic layer. The problem is reduced to integral equations for the average contact stress and the displacement of the rod tip into the contact surface. The kernels of these integral equations are composed of temporal Green’s functions for the rod and the layered half space, which represent the response of each to an impulsive uniform normal traction. The Green’s function for the rod is obtained in closed form, while that for the layered half space is obtained through a numerical Laplace transform inversion. The integral equations are solved numerically with a second-order stable scheme. Solutions are computed for a wide variety of materials and configurations, providing the stress and displacement history, as well as the stress-displacement response. The results show the effects of changes in rod material and length, lumped mass, layer material, substrate material, and viscoelastic material parameters.

The Normal Impact on an Elastic Half Space of an Elastic Rod-Mass System

1987 ◽  
Vol 54 (2) ◽  
pp. 359-366 ◽  
Author(s):  
H. A. Downey ◽  
D. B. Bogy
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Half Space ◽  
Contact Stress ◽  
Elastic Half Space ◽  
Elastic Rod ◽  
Lumped Mass ◽  
Normal Impact ◽  
Mass System ◽  
The Impact

The normal impact problem of a one-dimensional elastic rod with a lumped mass on the trailing end onto an elastic half space is solved for the time-dependent interface displacement and stress. This problem is reduced to an integral equation, whose kernel is the solution of a simpler auxiliary problem, which is solved in closed form. After examining the graph of the kernel it is found that a simple linear expression adequately represents its half space contribution. This approximation allows the integral equation to be solved in closed form and provides insight into its solution. Numerical results are presented, which display the impact and rebound of the rod, and illustrate the presence of major effects from the Rayleigh wave in the half space and the reflected wave from the trailing end of the rod. Results are presented for various half-space materials, rod lengths, and masses. It is found that in the absence of the mass the maximum contact stress depends entirely on the rod material, but with the lumped mass added the contact stress can become much greater and depends on the rod, the half space, and the mass.

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.

On the Torsion of Elastic Half Spaces With Embedded Penny-Shaped Flaws

1972 ◽  
Vol 39 (3) ◽  
pp. 786-790 ◽  
Author(s):  
R. D. Low
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Rigid Inclusion ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Fredholm Integral Equations ◽  
Graphical Displays ◽  
Penny Shaped Crack ◽  
Shaped Crack

The investigation is concerned with some of the effects of embedded flaws in an elastic half space subjected to torsional deformations. Specifically two types of flaws are considered: (a) a penny-shaped rigid inclusion, and (b) a penny-shaped crack. In each case the problem is reduced to a system of Fredholm integral equations. Graphical displays of the numerical results are included.

THREE-DIMENSIONAL SURFACE CRACK ANALYSIS OF ELASTIC HALF-SPACE UNDER ROLLING CONTACT LOAD

2009 ◽  
Vol 06 (02) ◽  
pp. 317-332 ◽  
Author(s):  
MENG-CHENG CHEN ◽  
HUI-QIN YU
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Half Space ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Rolling Contact ◽  
Contact Load ◽  
Hypersingular Integral ◽  
Space Body ◽  
Planar Crack

In this work a three-dimensional planar crack on the surface of elastic half-space was analyzed under rolling contact load. The stresses interior to an elastic half-space body under rolling contact load and those produced by an infinitesimal displacement jump loop for the elastic half-space body were used to reduce the planar crack problem to the solution of a system of two-dimensional hypersingular integral equations with unknown displacement jump. The ideas of finite element discretization were employed to construct numerical solution schemes for solving the integral equations. An appropriate treatment of the associated hypersingular integral in the numerical solution to the integral equations was proposed in Hadamard's finite-part integral sense. The numerical results showed that the present procedure yields solutions with high accuracies. The stress intensity factors near the crack front edge under rolling contact load were indicated in graphical form with varying the crack shape, the radius of rolling contact zone and the friction coefficients, respectively. In addition, the influence of the lubricant infiltrating the crack surfaces on the crack propagation was also discussed in the paper.

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.

Elastic Wave Scattering From an Interface Crack in a Layered Half Space

1985 ◽  
Vol 52 (1) ◽  
pp. 42-50 ◽  
Author(s):  
H. J. Yang ◽  
D. B. Bogy
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Interface Crack ◽  
Scattered Field ◽  
Crack Tip ◽  
Wave Number ◽  
Graphical Form ◽  
Integral Theorem ◽  
Special Cases ◽  
Layered Half Space

Many applications in industry utilize a layered elastic structure in which a relatively thin layer of one material is bonded to a much thicker substrate. Often the fabrication process is imperfect and cracks occur at the interface. This paper is concerned with the plane strain, time-harmonic problem of a single elastic layer of one material on a half space of a different material with a single crack at the interface. Green’s functions for the uncracked medium are used with the appropriate form of Green’s integral theorem to derive the scattered field potentials for arbitrary incident fields in the cracked layered half space. These potentials are used in turn to reduce the problem to a system of singular integral equations for determining the gradients of the crack opening displacements in the scattered field. The integral equations are analyzed to determine the crack tip singularity, which is found, in general, to be oscillatory, as it is in the corresponding static problem of an interface crack. For many material combinations of interest, however, the crack tip singularity in the stress field is one-half power, as in the case of homogeneous materials. In the numerical work presented here attention is restricted to this class of composites and the integral equations are solved numerically to determine the Mode I and Mode II stress intensity factors as a function of a dimensionless wave number for various ratios of crack length to layer depth. The results are presented in graphical form and are compared with previously published analyses for the special cases where such results are available.

Elastic Wave Scattering From an Interface Crack in a Layered Half Space Submerged in Water: Part I: Applied Tractions at the Liquid-Solid Interface

1986 ◽  
Vol 53 (2) ◽  
pp. 326-332 ◽  
Author(s):  
S. M. Gracewski ◽  
D. B. Bogy
Keyword(s):  
Elastic Wave ◽  
Integral Equations ◽  
Half Space ◽  
Interface Crack ◽  
Wave Scattering ◽  
Plane Waves ◽  
Crack Tip ◽  
Solid Interface ◽  
Elastic Wave Scattering ◽  
Layered Half Space

In Part I of this two-part paper, the analytical solution of time harmonic elastic wave scattering by an interface crack in a layered half space submerged in water is presented. The solution of the problem leads to a set of coupled singular integral equations for the jump in displacements across the crack. The kernels of these integrals are represented in terms of the Green’s functions for the structure without a crack. Analysis of the integral equations yields the form of the singularities of the unknown functions at the crack tip. These singularities are taken into account to arrive at an algebraic approximation for the integral equations that can then be solved numerically. Numerical results in the form of crack tip stress intensity factors are presented for the cases in which the incident disturbance is a harmonic uniform normal or shearing traction applied at the liquid-solid interface. These results are compared with a previously published solution for this problem in the absence of the liquid. In Part II, which immediately follows Part I in the same journal issue, the more realistic disturbances of plane waves and bounded beams incident from the liquid are considered.

The In-Plane Loading of a Rigid Disk Inclusion Embedded in an Elastic Half-Space

1991 ◽  
Vol 58 (2) ◽  
pp. 362-369 ◽  
Author(s):  
A. P. S. Selvadurai ◽  
B. M. Singh ◽  
M. C. Au
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Space Region ◽  
Rigid Disk ◽  
Lateral Translation

The paper examines the problem of the in-plane loading of a rigid disk inclusion which is embedded in bonded contact with an isotropic elastic half-space region. The governing coupled integral equations, derived via a Hankel transform technique, are evaluated numerically to generate results for the in-plane stiffness of the rigid disk inclusion and the rotation which accompanies the lateral translation.

The Vibration of Pile Groups Embedded in a Layered Poroelastic Half Space Subjected to Harmonic Axial Loads by Using Integral Equations Method

2012 ◽  
Vol 204-208 ◽  
pp. 1170-1173
Author(s):  
Chun Bo Cheng ◽  
Man Qing Xu ◽  
Bin Xu
Keyword(s):  
Dynamic Response ◽  
Layer Thickness ◽  
Integral Equations ◽  
Half Space ◽  
Pile Group ◽  
Dynamic Interaction ◽  
Fredholm Integral Equations ◽  
Pile Groups ◽  
Axial Loads ◽  
Layered Half Space

The dynamic response of a pile group embedded in a layered poroelastic half space subjected to axial harmonic loads is investigated in this study. Based on Biot's theory and utilizing Muki's method, the second kind of Fredholm integral equations describing the dynamic interaction between the layered half space and the pile group is constructed. Numerical results show that in a two-layered half space, for the closely populated pile group with a rigid cap, the upper softer layer thickness has considerably different influence on the center pile and the corner piles, while for sparsely populated pile group, it has almost the same influence on all the piles.

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