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.

References

1.
Barwell
,
F. T.
,
1979
,
Bearing Systems, Principles and Practice
,
Oxford University Press
,
Oxford, UK
.
2.
Bayer
,
R. G.
,
1994
,
Mechanical Wear Prediction and Prevention
,
Marcel Dekker
,
New York.
3.
Blau
,
P. J.
,
1996
,
Friction Science and Technology
,
Marcel Dekker
,
New York.
4.
Ahmadi
,
N.
,
Keer
,
L. M.
, and
Mura
,
T.
,
1983
, “
Non-Hertzian Contact Stress Analysis for an Elastic Half-Space—Normal and Sliding Contact
,”
Int. J. Solids Struct.
,
19
, pp.
357
373
.10.1016/0020-7683(83)90032-X
5.
Barber
,
J. R.
,
1983
, “
The Solution of Elasticity Problems for the Half-Space by the Method of Green and Collins
,”
Appl. Sci. Res.
,
40
, pp.
135
157
.10.1007/BF00386216
6.
Johnson
,
K. L.
,
1985
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge, UK.
7.
Hills
,
D. A.
,
Nowell
,
D.
, and
Sackfield
,
A.
,
1993
,
Mechanics of Elastic Contacts
,
Butterworth-Heinemann
,
Oxford, UK
.
8.
Kalker
,
J. J.
,
1990
,
Three-Dimensional Bodies in Elastic Contact
,
Kluwer
,
Dordrecht, The Netherlands
.
9.
Craggs
,
J. W.
, and
Roberts
,
A. M.
,
1967
, “
On the Motion of a Heavy Cylinder Over the Surface of an Elastic Half-Space
,”
ASME J. Appl. Mech.
,
34
, pp.
207
209
.10.1115/1.3607626
10.
Brock
,
L. M.
,
1981
, “
Sliding and Indentation by a Rigid Half-Wedge With Friction and Displacement Coupling Effects
,”
Int. J. Eng. Sci.
,
19
, pp.
33
40
.10.1016/0020-7225(81)90047-1
11.
Churilov
,
V. A.
,
1978
, “
Action of an Elliptical Stamp Moving at a Constant Speed on an Elastic Half-Space
,”
J. Appl. Math. Mech.
,
42
, pp.
1176
1182
.10.1016/0021-8928(78)90067-9
12.
Rahman
,
M.
,
1996
, “
Hertz Problem for a Rigid Punch Moving Across the Surface of a Semi-Infinite Elastic Solid
,”
Z. Angew. Math. Mech.
,
47
, pp.
601
615
.10.1007/BF00914874
13.
Brock
,
L. M.
,
2012
, “
Two Cases of Rapid Contact on an Elastic Half-Space: Sliding Ellipsoidal Die, Rolling Sphere
,”
J. Mech. Mater. Struct.
(in press).
14.
Georgiadis
,
H. G.
, and
Barber
,
J. R.
,
1993
, “
On the Super-Rayleigh/Subseismic Elastodynamic Indentation Problem
,”
J. Elast.
,
31
, pp.
141
161
.10.1007/BF00044967
15.
Green
,
A. E.
, and
Zerna
,
W.
,
1968
,
Theoretical Elasticity
, 2nd ed.,
Oxford University Press
,
London
.
16.
Beatty
,
M. F.
, and
Usmani
,
S. A.
,
1975
, “
On the Indentation of a Highly Elastic Half-Space
,”
Q. J. Mech. Appl. Math.
,
28
, pp.
47
62
.10.1093/qjmam/28.1.47
17.
Brock
,
L. M.
,
2001
, “
Rapid Sliding Contact on a Highly Elastic Pre-Stressed Material
,”
Int. J. Non-Linear Mech.
,
36
, pp.
433
442
.10.1016/S0020-7462(00)00044-5
18.
Van der Pol
,
B.
, and
Bremmer
,
H.
,
1950
,
Operations Based on the Two-Sided LaPlace Integral
,
Cambridge University Press
,
Cambridge, UK
.
19.
Sneddon
,
I. N.
,
1972
,
The Use of Integral Transforms
,
McGraw-Hill
,
New York.
20.
Sokolnikoff
,
I. S.
,
1956
,
Mathematical Theory of Elasticity
, 2nd ed.,
McGraw-Hill
,
New York
.
21.
Brock
,
L. M.
,
2004
, “
Coupled Thermoelastic Analysis of Perfect Rolling Contact: Solution Behavior Near Yield Initiation
,”
J. Therm. Stresses
,
27
, pp.
383
403
.10.1080/01495730490442756
22.
Erdogan
,
F.
,
1985
, “
Mixed Boundary Value Problems in Mechanics
,”
Mechanics Today
, Vol.
4
,
S.
Nemat-Nasser
, ed.,
Pergamon
,
New York
, pp.
1
86
.
23.
Truesdell
,
C. T.
, and
Noll
,
W.
,
1965
, “
The Non-Linear Field Theories of Mechanics
,”
Handbuch der Physik
, Vol.
3
,
W.
Flugge
, ed.,
Springer
,
Berlin
.
24.
Peirce
,
B. O.
, and
Foster
,
R. M.
,
1956
,
A Short Table of Integrals
, 4th ed.,
Ginn and Company
,
New York
.
25.
Stakgold
,
I. S.
,
1967
,
Boundary Value Problems of Mathematical Physics
, Vol.
1
,
MacMillan
,
New York
.
You do not currently have access to this content.