Page 60 - Tribology in Machine Design

Basic principles of tribology 47

and also varies slightly with the coefficient of friction (0* varies from 0° to
15° as tana goes from 0 to 1.37). The stress intensity associated with the
crack is obtained from a weighted average of the stresses calculated
previously. The stress intensity corresponding to a combined uniform
shear-compression stress on the crack can be expressed as

where tan/? is the coefficient of friction between the opposing faces of the
crack and la is the crack length. According to eqn (2.110) the crack is driven
by the shear stress and retarded by the friction forces arising from the
compressive stresses. It is suggested that eqn (2.110) can be adopted to a
non-uniform stress field by evaluating the quantity a xy — tan f}a yy along the
crack and integrating according to the procedure described below.

2.12.1. Estimation of stress intensity under non-uniform applied loads
There are situations where one needs to know the stress intensity associated
with cracks in non-uniform stress fields, for example when there is a
delamination type of wear.
The approximation is derived for a semi-infinite plate containing a crack
of length la. The applied stress <r(x) can be either tensile or shear so that
Mode I, II or III stress intensities can be approximated. If <r(x) is the stress
that would be acting along the crack plane if the crack were not there

where K is the stress intensity and 2a is the crack length. Equation (2.111)
evaluates the stress intensity at x = a. When x = acos0, eqn (2.111)
becomes
n

If a term 0- eff given by

is introduced, eqn (2.112) becomes

Equation (2.114) can be evaluated by the Simpson rule. If the crack length is
divided into two intervals, the Simpson rule approximation is