Problems Involving a Receding Contact Between a Layer and a Half Space

1972 ◽  
Vol 39 (4) ◽  
pp. 1115-1120 ◽  
Author(s):  
L. M. Keer ◽  
J. Dundurs ◽  
K. C. Tsai
Keyword(s):  
Integral Equation ◽  
Half Space ◽  
Contact Pressure ◽  
Auxiliary Function ◽  
Unexpected Result ◽  
Wide Range ◽  
Receding Contact ◽  
Axisymmetric Problems ◽  
Uniformly Distributed Loads ◽  
Two Bodies

The work reconsiders the smooth receding contact between an elastic layer and a half space when the two bodies are pressed together. The analysis leads to a Fredholm integral equation of the second kind for an auxiliary function that is directly related to the contact pressure. An unexpected result is that the integral equation is homogeneous, and that finding the extent of contact can be viewed as an eigenvalue problem. The integral equation can be solved numerically to any required degree of accuracy, and the extent of contact and the contact pressure are computed for concentrated and uniformly distributed loads in both plane and axisymmetric problems. The present analysis confirms the results of Weitsman rather than Pu and Hussain over a wide range of mismatch in the elastic constants.

Elastic Layer Pressed Against a Half Space

1974 ◽  
Vol 41 (3) ◽  
pp. 703-707 ◽  
Author(s):  
K. C. Tsai ◽  
J. Dundurs ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Integral Equations ◽  
Half Space ◽  
Elastic Layer ◽  
Numerical Results ◽  
Fredholm Integral Equations ◽  
Auxiliary Functions ◽  
Receding Contact ◽  
Two Bodies

The paper considers the elastic layer which is pressed against a half space by loads that are not necessarily symmetric about the center of the loaded region. It is shown that the receding contact between the two bodies can be treated by means of superposition, leading to two homogeneous Fredholm integral equations for auxiliary functions that are directly related to the contact tractions. The determination of the extent of contact and the shift between the load and contact intervals can be viewed as an eigenvalue problem of the homogeneous integral equations. Specific numerical results are given for two types of triangular loads, and a comparison is made with certain symmetric loads.

Contact Stresses Between an Elastic Cylinder and a Layered Elastic Solid

1974 ◽  
Vol 96 (2) ◽  
pp. 250-257 ◽  
Author(s):  
P. K. Gupta ◽  
J. A. Walowit
Keyword(s):  
Integral Equation ◽  
Layer Thickness ◽  
Contact Pressure ◽  
Contact Zone ◽  
Numerical Solutions ◽  
Elastic Solid ◽  
Elastic Cylinder ◽  
Elastic Solids ◽  
Actual Contact ◽  
Wide Range

The generalized plane strain problem of the contact of layered elastic solids is reduced to an integral equation using Green’s function approach. Approximate numerical solutions are obtained by replacing the integral equation by a matrix inversion when the trapezoidal rule is used to represent the integral. Results for determining the actual contact pressure at the center of the contact zone and size of contact zone for a wide range of layer thicknesses are presented for two most practical cases, (i) when the indenter is rigid, and (ii) when the indenter is elastic having a modulus of elasticity equal to that of the substrate of the indented body. When the layer is softer than the substrate it is found that the actual contact pressure distribution is very closely determined by a weighted sum of elliptic and parabolic functions. For a substrate softer than the layer the pressures substantially deviate from an elliptical or parabolic behavior, for the cases when the layer thickness is finite. The analysis checks with the Hertzian solution in the extreme cases when the layer thickness either tends to zero or approaches infinity.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

scholarly journals The first boundary value problem of the dynamics of an anisotropic elastic half-space under the action of subsonic transport loads

2020 ◽  
Vol 4 (78) ◽  
pp. 45-52
Author(s):  
G. K. ZAKIR’YANOVA ◽  
◽  
L. A. ALEXEYEVA ◽  
Keyword(s):  
Rock Mass ◽  
Boundary Value Problem ◽  
Half Space ◽  
Wide Class ◽  
Boundary Value ◽  
Operator Method ◽  
Elastic Half Space ◽  
Theory Of Elasticity ◽  
Wide Range ◽  
Anisotropic Elastic

The first boundary value problem of the theory of elasticity for an anisotropic elastic half-space is solved when a transport load moves along its surface. The subsonic Raleigh case is considered, when the velocity of motion is less than the velocity of propagation of bulk and surface elastic waves. The Green’s tensor of the transport boundary value problem is constructed and on its basis the solution of boundary value problems for a wide class of distributed traffic loads is given. To solve the problem, the methods of tensor and linear algebra, integral Fourier transform, and operator method for solving systems of differential equations were used. The obtained solution makes it possible to investigate the dynamics of the rock mass for a wide class of transport loads, in a wide range of velocities, both low velocities and high velocities, and to evaluate the strength properties of the rock mass under the influence of road transport. In particular, determine the permissible velocities of its movement and carrying capacity. In addition, a investigation on its basis of the movement of the day surface along the route will make it possible to establish criteria for the seismic resistance of ground structures and the permissible distances of their location from the route.

A Test Facility for the Measurement of Torques at the Shaft to Seal Interface in Brush Seals

1999 ◽  
Vol 121 (1) ◽  
pp. 160-166 ◽  
Author(s):  
P. E. Wood ◽  
T. V. Jones
Keyword(s):  
Contact Pressure ◽  
Indirect Measurement ◽  
Test Facility ◽  
Axial Loads ◽  
Rotating Shaft ◽  
Brush Seal ◽  
Wide Range ◽  
Bearing Assembly ◽  
Brush Seals ◽  
Wide Speed Range

An important factor in the performance of brush seals for a wide range of gas turbine applications is the rate of wear at the seal to shaft interface, which is dependent on the contact pressure that exists between the bristles and rubbing surface. This is dependent on a variety of effects. Principally, these are the aerodynamic forces bending the bristles onto the rubbing surface, frictional effects within the bristle pack and at the backing ring that arise with the application of pressure differential, geometrical changes due to centrifugal and thermal growths, and transient differential movements of the rotor that develop in flight manoeuvres. In order to investigate the effect of these phenomena on contact pressure, a test facility has been devised in which the torque exerted by a brush seal on a rotating shaft is used as an indirect measurement of contact pressure. This has necessitated the design of a test facility in which all system torques can be fully calibrated. Consequently, a pressure balanced design has been adopted in which applied seal differential and pressure levels have a minimal effect on axial loads at the rotor bearing assembly. The primary method of torque measurement is the instantaneous deceleration of the rotor. Thus, measurements over a wide speed range are acquired with high frequency instrumentation. The means whereby small parasitic torques are evaluated and corrected is given. Results demonstrating the dependence of contact pressure on seal differential and pressure levels are presented.

Sign in / Sign up

Export Citation Format

Share Document