Indentation of a Rigid Sphere into an Elastic Half-Space in the Direction Orthogonal to the Axis of Material Symmetry

2010 ◽  
Vol 99 (2) ◽  
pp. 147-161 ◽  
Author(s):  
Olesya I. Zhupanska
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Rigid Sphere ◽  
Material Symmetry

Axisymmetric contact with friction of a rigid sphere with an elastic half-space

2009 ◽  
Vol 465 (2108) ◽  
pp. 2565-2588 ◽  
Author(s):  
O. I. Zhupanska
Keyword(s):  
Contact Problem ◽  
Half Space ◽  
Integral Transform ◽  
Elastic Half Space ◽  
Rigid Sphere ◽  
Analytical Treatment ◽  
Normal Contact ◽  
Exact Analytical Solutions ◽  
Single Integral ◽  
Toroidal Coordinates

The problem of normal contact with friction of a rigid sphere with an elastic half-space is considered. An analytical treatment of the problem is presented, with the corresponding boundary-value problem formulated in the toroidal coordinates. A general solution in the form of Papkovich–Neuber functions and the Mehler–Fock integral transform is used to reduce the problem to a single integral equation with respect to the unknown contact pressure in the slip zone. An analysis of contact stresses is carried out, and exact analytical solutions are obtained in limiting cases, including a full stick contact problem and a contact problem for an incompressible half-space.

Fourier integral representation of the Green function for an anisotropic elastic half-space

1993 ◽  
Vol 443 (1918) ◽  
pp. 367-389 ◽  
Keyword(s):  
Green Function ◽  
Integral Representation ◽  
Half Space ◽  
Isotropic Material ◽  
Fourier Integral ◽  
Elastic Half Space ◽  
Material Symmetry ◽  
Vertical Force ◽  
Anisotropic Elastic ◽  
The Green Function

This paper describes the development of a Fourier integral representation of the Green function for an anisotropic elastic half-space. The representation for an isotropic material is integrated in closed form and shown to reduce to Mindlin’s solution. An application of the anisotropic representation is made to deduce the exact displacement caused by a two-dimensional periodic vertical force distribu­tion applied to the interior of a half-space with cubic material symmetry.

Nonlinear oscillations in a constant gravitational field

2021 ◽  
Author(s):  
Stylianos - Vasileios Kontomaris ◽  
Anna Malamou
Keyword(s):  
Gravitational Field ◽  
Numerical Methods ◽  
Differential Equations ◽  
Half Space ◽  
Nonlinear Oscillations ◽  
Elastic Half Space ◽  
Rigid Sphere ◽  
Elementary Functions ◽  
Approximate Expressions ◽  
Special Case

Abstract Exploring non-linear oscillations is a challenging task since the related differential equations cannot be directly solved in terms of elementary functions. Thus, complicated mathematical or numerical methods are usually employed to find accurate or approximate expressions that describe the behavior of the system with respect to time. In this paper, the vertical oscillations of an object under the influence of its weight and an opposite force with magnitude F=cyn, where n>0 are being explored. Accurate and approximate simple solutions regarding the object’s position with respect to time are presented and the dependence of the oscillation’s period from the oscillation’s range of displacements and the exponent n is revealed. In addition, the special case in which n=3/2 (which describes the oscillation of a rigid sphere on an elastic half space) is also highlighted. Lastly, it is shown that similar cases (such as the case of a force with magnitude F=kx+λx2) can be also treated using the same approach.

Impact of a rigid sphere onto an elastic half space

1986 ◽  
Vol 22 (5) ◽  
pp. 403-409 ◽  
Author(s):  
Yu. A. Rossikhin
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Rigid Sphere

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.

scholarly journals Contact Analyses for Anisotropic Half Space With an Anisotropic Coating

2012 ◽  
Author(s):  
C. Bagault ◽  
D. Nélias ◽  
M.-C. Baietto ◽  
T. Ovaert
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Rigid Sphere ◽  
Manufacturing Processes ◽  
Problem Solution ◽  
Numerical Techniques ◽  
General Anisotropy ◽  
Recent Developments ◽  
Anisotropic Elastic ◽  
Wear Coatings

For most composite and mono-crystal materials their compositions or the elaboration and manufacturing processes imply that it exists one or two main directions or even a general anisotropy. Moreover, coatings are often used to prevent or control wear. Coatings do not have, generally, the same properties as the substrate and may have various thicknesses. The influence of the anisotropy orientations (in the coating and in the substrate) have to be taken into account to better predict the distribution of the contact pressure and the subsurface stress-field in order to optimize the service life of industrial components. A contact model using semi analytical methods, relying on elementary analytical solutions, has been developed. It is based on numerical techniques adapted to contact mechanics. Recent developments aim to quantify displacements and stresses of a layered anisotropic elastic half space which is in contact with a rigid sphere. The influence of material properties and layer thickness on the contact problem solution will be more specifically analyzed.

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