Shear of two half planes pressed to each other and containing a surface groove. Part 2. Incomplete contact
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SHEAR OF TWO HALF PLANES PRESSED TO EACH OTHER AND CONTAINING A SURFACE GROOVE. PART 2. INCOMPLETE CONTACT R. M. Martynyak, N. I. Malanchuk, and B. E. Monastyrs’kyi
UDC 539.3
We consider the elastic problem of contact of two half planes made of identical materials with regard for the presence of an intercontact gap caused by a surface groove in the case of successive loading with normal and shear forces. By using the Kolosov – Muskhelishvili complex potentials, the problem is reduced to a sequence of singular integral equations the first of which is solved analytically and the second numerically. According to the procedure of numerical analysis, we separate the root singularity and represent the regular part in the form of expansion in Chebyshev polynomials of the first kind with unknown coefficients. The coefficients are determined by the collocation method. We analyze the dependences of the length of the gap and the size of the slip region on the external load and the distributions of tangential and normal stresses over the contact surface.
In [1], we studied the case of complete frictional contact of isotropic semiinfinite half planes with small surface groove in one of these planes in the case of successive application of normal and shear forces. The phenomenon of local slip of the edges of the bodies in the vicinity of the surface inhomogeneity was analyzed. In the present work, we consider the case of incomplete contact, i.e., the bodies are in contact over the entire interface except a certain local region where the intercontact gap is formed. We investigate the influence of the gap on the parameters of contact interaction and, in particular, on the mutual slip of the surfaces caused by the shear forces. Statement of the Problem We consider two elastic half planes D1 and D2 made of identical materials in a Cartesian coordinate system x O y (Fig. 1). The boundary of the body D2 is rectilinear and the boundary of the body D1 contains a shallow gently sloping groove located in a segment x ∈ [ – b, b ]. The contour of the groove is described by a smooth function 2
⎛ x ⎞ 3/ 2 r ( x ) = − r0 ⎜1 − 2 ⎟ , ⎝ b ⎠
0 < r0 0,
− 1/ 2 ⎛ ⎛ x x 2 exp ⎜ m ln ⎜ = π ⎛ ⎛⎝ ⎞⎠ − 1⎞ − ⎝ ⎠ 2 ⎝ c c ⎝ 1 − (t / c ) ( x − t )
Tm (t / c) dt
x ≤ c,
2 ⎛ x ⎞ − 1⎞ ⎞ , ⎟⎟ ⎝ c⎠ ⎠⎠
x > c.
(24)
S HEAR OF TWO HALF PLANES P RESSED TO EACH OTHER
AND
CONTAINING A SURFACE GROOVE. PART 2.
557
Fig. 2. Dependences of the half length of the slip region (c = c / b ) on the shear forces at infinity (T ∞ / f , T ∞ = T ∞ b (1 + κ) /3Gr0 ) for the following half lengths of the gap: (1) a = 0.2, (2) 0.4, (3) 0.6, (4) 0.8.
Thus, for the solution of the problem, it is necessary to determine L the unknown coefficients d1 , d2 , … , dL and the half length of the slip region (i.e., the parameter c ) obtained from the condition of continuity of the tangential stresses τxy for x = c. By virtue of relation (24), it is easy to see that the stresses τ xy( x, 0 ) are continuous whenever L
∑ dm
= 0.
(25)
m =1
To determine the unknown coefficients d1
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