An Elastic Strip Pressed Against an Elastic Half Plane by a Steadily Moving Force

1978 ◽  
Vol 45 (1) ◽  
pp. 89-94 ◽  
Author(s):  
G. G. Adams
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Mixed Boundary ◽  
Contact Region ◽  
Theory Of Elasticity ◽  
Strain Theory ◽  
Fredholm Integral Equations ◽  
Elastic Strip ◽  
Concentrated Force ◽  
Moving Force

An infinite elastic strip is pressed against an elastic half plane of a different material by a steadily moving concentrated force. Using the plane strain theory of elasticity, it is shown that the problem can be decomposed into its symmetric and antisymmetric parts. These mixed boundary-value problems are then solved by reduction to Fredholm integral equations subject to certain other conditions. For various material combinations, and a range of speed, the extent and location of the contact region as well as the contact pressure will be computed and illustrated graphically.

Cavitation at the Ends of an Elliptic Inclusion Inside a Plate Under Tension

1968 ◽  
Vol 35 (3) ◽  
pp. 505-509 ◽  
Author(s):  
M. A. Hussain ◽  
S. L. Pu ◽  
M. A. Sadowsky
Keyword(s):  
Mixed Boundary ◽  
Contact Region ◽  
Infinite Plate ◽  
Major Axis ◽  
Uniaxial Stress ◽  
Theory Of Elasticity ◽  
Transcendental Equation ◽  
Elliptic Inclusion ◽  
Unstressed State ◽  
The Matrix

An oblong elliptic inclusion is perfectly filled in a hole in an infinite plate in the unstressed state. Cavities at the ends of the inclusion will appear as a result of the application of uniaxial stress at infinity in the direction of the major axis of the ellipse. Analytical formulation of the problem leads to a mixed boundary-value problem of the mathematical theory of elasticity. A Fredholm integral equation of the first kind is derived for the normal stress with the range of integration being unknown (corresponding to the unknown region of contact). Applying the theorem which has recently been established based on a variational principle, a transcendental equation is obtained for determining the contact region. Numerical results are given for various values of the elastic constants of both the matrix and the inclusion. Application of the results to fiber-reinforced composite materials is discussed.

Frictionless Contact of Layered Half-Planes, Part II: Numerical Results

1993 ◽  
Vol 60 (3) ◽  
pp. 640-645 ◽  
Author(s):  
M.-J. Pindera ◽  
M. S. Lane
Keyword(s):  
Stress Distribution ◽  
Half Plane ◽  
Contact Stress ◽  
Mixed Boundary ◽  
Contact Region ◽  
Contact Problems ◽  
Contact Length ◽  
Unidirectional Composite ◽  
Frictionless Contact ◽  
Contact Stress Distribution

In Part I of this paper, analytical development of a method was presented for the solution of frictionless contact problems of multilayered half-planes consisting of an arbitrary number of isotropic, orthotropic, or monoclinic layers arranged in any sequence. The local/global stiffness matrix approach similar to the one proposed by Bufler (1971) was employed in formulating the surface mixed boundary condition for the unknown stress in the contact region. This approach naturally facilitates decomposition of the integral equation for the contact stress distribution on the top surface of an arbitrarily laminated half-plane into singular and regular parts that, in turn, can be solved using a numerical collocation technique. In Part II of this paper, a number of numerical examples is presented addressing the effect of off-axis plies on contact stress distribution and load versus contact length in layered half-planes laminated with unidirectionally reinforced composite plies. The results indicate that for the considered unidirectional composite, the load versus contact length response is significantly influenced by the orientation of the surface layer and the underlying half-plane, while the corresponding contact stress profiles are considerably less affected.

Edge Crack in an Elastic Strip on a Liquid Foundation

1989 ◽  
Vol 33 (03) ◽  
pp. 214-220
Author(s):  
Paul C. Xirouchakis ◽  
George N. Makrakis
Keyword(s):  
Stress Intensity Factor ◽  
Integral Equation ◽  
Fourier Transform ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Edge Crack ◽  
Applied Pressure ◽  
Theory Of Elasticity ◽  
Elastic Strip ◽  
Pressure Loading

The behavior of a long elastic strip with an edge crack resting on a liquid foundation is investigated. The faces of the crack are opened by an applied pressure loading. The deformation of the strip is considered within the framework of the linear theory of elasticity assuming plane-stress conditions. Fourier transform techniques are employed to obtain integral expressions for the stresses and displacements. The boundary-value problem is reduced to the solution of a Fredholm integral equation of the second kind. For the particular case of linear pressure loading, the stress-intensity factor is calculated and its dependence is shown on the depth of the crack relative to the thickness of the strip. Application of the present results to the problem of flexure of floating ice strips is discussed.

Exact determination of gravity stresses in finite elastic slopes: Part I. Theoretical considerations

1983 ◽  
Vol 20 (1) ◽  
pp. 47-54 ◽  
Author(s):  
V. Silvestri ◽  
C. Tabib
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Inclination Angle ◽  
Complex Variable ◽  
Earth Pressure ◽  
Exact Distributions ◽  
Exact Determination ◽  
Upper Half Plane ◽  
Theoretical Considerations

The exact distributions of gravity stresses are obtained within slopes of finite height inclined at various angles, −β (β = π/2, π/3, π/4, π/6, and π/8), to the horizontal. The solutions are obtained by application of the theory of a complex variable. In homogeneous, isotropic, and linearly elastic slopes under plane strain conditions, the gravity stresses are independent of Young's modulus and are a function of (a) the coordinates, (b) the height, (c) the inclination angle, (d) Poisson's ratio or the coefficient of earth pressure at rest, and (e) the volumetric weight. Conformal applications that transform the planes of the various slopes studied onto the upper half-plane are analytically obtained. These solutions are also represented graphically.

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.

scholarly journals A mixed boundary value problem of a cracked elastic medium under torsion

2021 ◽  
pp. 10-10
Author(s):  
Belkacem Kebli ◽  
Fateh Madani
Keyword(s):  
Boundary Value Problem ◽  
Elastic Medium ◽  
Integral Equations ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problem ◽  
Penny Shaped Crack

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

Sign in / Sign up

Export Citation Format

Share Document