PRESSURE OF A NARROW BAND-LIKE PUNCH UPON AN ELASTIC HALF-SPACE
I. I. Argatov UDC 539.3
An asymptotic solution is obtained to the contact problem of a band-like punch acting upon an elastic
half-space. The method of joined asymptotic expansions is used. The results of numerical calculations are
presented. The efficiency of the approach is tested by comparing it with another method.
The method of joined asymptotic expansions is used to construct the principal terms of the asymptotics of the solution
of a contact problem for a narrow band-like punch held at the level of the unperturbed boundary of an elastic half-space
x
2
≤ 0, upon which a concentrated force Q e
2
acts. The ratio of the punch halfwidth h to the distance L from the punch to the
force axis is assumed small. The equation for the density of linear pressure is written out and is solved by the method of
integral Fourier transformation. Results of numerical calculations are presented. The distribution of the overturning moment
is found. A problem for a narrow band-like punch with a corrugated base is considered. A comparison is made with results
obtained earlier.
1. Formulation of the Problem. Let a band-like punch with a flat horizontal base | x
1
| ≤ h rest on the boundary of
an elastic half-space x
2
≤ 0 loaded by a concentrated force Q e
2
at the point (L, 0, 0) (Fig. 1). The friction between the contacting
bodies is neglected. The ratio
ε =
h
L
(1.1)
is assumed a small parameter.
The exact solution of the contact problem for a band-like punch was derived by V. L. Rvachev [14]. Aleksandrov
[1] found an asymptotic solution for a narrow band-like punch with a corrugated base by the method of large λ. In the
present paper, the method of joined asymptotic expansions is used to construct the principal terms of the asymptotics of the
displacement field and the contact pressure [5, 8, etc.].
2. External Displacement Field. The vector of displacements of points of the elastic half-space at a distance from
the punch can be presented as [15, 10, 12]
v (P, x ) =
∫
- ∞
+ ∞
P ( s ) T ( x
1
, x
2
, x
3
- s ) d s - Q T (x
1
- L, x
2
, x
3
). (2.1)
Here, P ( s ) are contact pressures per unit length of the punch and T ( x ) is the solution of the Boussinesq problem [13] on a
half-space x
2
≤ 0 subjected to the action of a unit concentrated force opposite to the axis O x
2
,
2 π E (1 + ν)
-1
T
2
( x ) = - 2 (1 - ν) | x |
-1
- x
2
2
| x |
-3
, | x | = (x
1
2
+ x
2
2
+ x
3
2
)
1 ∕ 2
,
2 π E (1 + ν)
-1
T
i
( x ) = - x
i
| x |
-1
x
2
| x |
-2
+ (1 - 2 ν) ( | x | - x
2
)
-1
, i = 1, 3.
State Marine Academy, St. Petersburg, Russia. Translated from Prikladnaya Mekhanika, Vol. 36, No. 10, pp. 109–114,
October, 2000. Original article submitted September 5, 1999.
1063-7095/00/3610-1363$25.00 ©2001 Plenum Publishing Corporation 1363
International Applied Mechanics, Vol. 36, No. 10, 2000