## Abstract

This paper pursues a numerical approach to the solution of the contact problem of a rigid punch on an incompressible half-plane, subjected to a shearing force together with a normal force which may be offset from the centerline of the punch. A piecewise-linear representation of the shear tractions is employed and quadratic programming techniques are used to solve the problem. This method enables an arbitrary load history to be followed. Results are presented which show excellent agreement with other solutions to the special cases of monotonic and steady-state cyclic loading. It is shown that the traction distribution reaches a steady-state cycle after only a few cycles of loading. The existence of an interesting stick-slip regime, where a central zone of slip is bordered by two stick regions is highlighted.

## References

1.
,
A.
,
Sackfield
,
A.
, and
Hills
,
D. A.
,
2004
, “
Analysis of a Rocking and Walking Punch—Part I: Initial Transient and Steady State
,”
ASME J. Appl. Mech.
,
71
,
225
233
.
2.
Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
3.
Sackfield
,
A.
,
Truman
,
C. E.
, and
Hills
,
D. A.
,
2001
, “
The Tilted Punch Under Normal and Shear Load (With Application to Fretting Tests)
,”
Int. J. Mech. Sci.
,
43
(
8
), pp.
1881
1892
.
4.
Nowell
,
D.
, and
Dai
,
D. N.
,
1998
, “
Analysis of Surface Tractions in Complex Fretting Fatigue Cycles Using Quadratic Programming
,”
ASME J. Tribol.
,
120
, pp.
744
749
.
5.
Kalker
,
J. J.
,
1971
, “
A Minimum Principle for the Law of Dry Friction With Application to Elastic Cylinders in Rolling Contact
,”
ASME J. Appl. Mech.
,
38
, pp.
875
880
.
6.
Gill, P. E., Murray, W., and Wright, M. H., 1981, Practical Optimization, Academic Press, London.