Contact of elastic half spaces in the presence of an elliptic gap filled with liquid
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CONTACT OF ELASTIC HALF SPACES IN THE PRESENCE OF AN ELLIPTIC GAP FILLED WITH LIQUID R. M. Martynyak1, 2 and B. S. Slobodyan1
UDC 539.3
We study the problem of contact of elastic half spaces in the presence of compressible liquid in an elliptic gap formed between the half spaces. By the method of functions of intercontact gaps, we reduce the posed problem to the integrodifferential equation for the height of the gap. This equation is solved analytically. The dependence of the geometric characteristics of the gap on the external load is analyzed.
The influence of local interface gaps filled with gases or liquids on the contact behavior of bodies with surface grooves under the conditions of plane deformation is studied in [1–4]. An axisymmetric problem of interaction between the bodies in the presence of a circular interface gap filled with a certain material is investigated in [5]. In what follows, we solve a three-dimensional contact problem for half spaces one of which contains a groove with elliptic base in the case where the gap formed between the spaces is filled with a compressible fluid. Statement of the Problem Consider the problem of interaction of two elastic isotropic half spaces D1 (lower) and D2 (upper). The boundary of the half space D2 is formed by a plane Ω identified with the coordinate plane Ox1 x2 of a Cartesian coordinate system Ox1 x2 x3 . The boundary of the lower body D1 in a region S0 bounded by an ellipse L0 with semiaxes a0 and b0 (a0 ≤ b0 ) has a contains an elliptic groove 3/ 2 ⎛ x2 x2 ⎞ r ( x ) = − r0 ⎜ 1 − 12 − 22 ⎟ , ⎝ a0 b0 ⎠
x ∈ S0 ,
(1)
where r0 = ( 0, 0 ) is the maximum height of the groove and x is the point with coordinates ( x1, x2 , x3 ). The groove is shallow ( r0 < < a 0 , r 0 < < b 0 ) and smoothly turns into the plane along its contour (r ( x ) = 0, ∂r ( x ) / ∂x1 = 0, ∂r ( x ) / ∂x2 = 0, x ∈ L0 ). The bodies are in contact under the action of compression stresses P∞ uniformly distributed at infinity. In view of the unevenness of the surface of one of the half spaces, an interface gap of height h ( x ), x ∈ S , is formed between the bodies in a region S ⊂ S 0 (Fig. 1). We assume that the gap is filled with compressible liquid acting with pressure P1 upon the surfaces of the bodies. The region S, the height of the gap, and the pressure of liquid h ( x ) inside the gap are unknown and depend on the load. Beyond the gap, we observe the frictionless contact of the analyzed bodies. 1 Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine. 2 Corresponding author; e-mail: [email protected].
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 45, No. 1, pp. 62–65, January – February, 2009. Original article submitted August 6, 2008. 66
1068–820X/09/4501–0066
© 2009
Springer Science+Business Media, Inc.
CONTACT OF ELASTIC HALF S PACES IN
THE
P RESENCE
OF AN
ELLIPTIC GAP F ILLED
WITH
LIQUID
67
Fig. 1. Schematic diagram of loading. The stress-strain state of the half spaces can be represented in the form of the superpo
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