ABSTRACT
In previous chapter, it was seen that how difficult the calculation of the stresses σ y , σ x and τ xy can be. Those stress formulas were given by Eqs. (7.3.21), (7.3.25) and (7.326). Fortunately, we are interested in stress distribution around the crack tips, since it is only this type of distribution that can lead us to KI or KIC. Therefore, in Eqs. (7.3.21), (7.3.25) and (7.3.26), we transfer the coordinate of the origin to the crack tip (Fig. 8.1), in mathematical term new variable z z a r ei0 = − =
θ will be entered into the discussions, rewriting this provides:
z z a r e z a r ei i0 = − = ⇒ = + θ θ (8.1.1)
Before we start calculations, we should emphasize why the stresses around crack tip and particularly, the Westergaard functions are that important. We can answer to this question by rewriting Eq. (7.3.17) again which is:
Z z z z a
I ( ) = −
−
σ 2
2 1 2 2
(7.3.17)
If we multiply both sides of the above equation into z a− , then we have:
z a Z z z z a z a
z aI− ( ) = −
− − −
σ 2
and then it can be simplified to:
z a Z z z z a
z aI− ( ) = + − −
σ 2
2 , then we will take a limit as z a→
and we have:
lim lim limz a I z a z az a Z z z
z a z a→ → →− ( ) = +
− −
σ 2
2 (8.1.2)
which yields to:
lim z a Iz a Z z a a
a → − ( ) =
=
σ σ
2 2 2 2
(8.1.3)
We saw in Eq. (7.2.13), that SIF in a very simple case is:
K aI =σ pi (8.1.4)
Now we multiply both sides of Eq. (8.1.3) into 2pi and we have:
2pi σ pilim z a Iz a Z z a→ − ( ) = (8.1.5)
By comparing Eqs. (8.1.4) and (8.1.5) we can easily find that:
K z a Z zI z a I= − ( )→2pi lim (8.1.6)
It is Eq. (8.1.6) that says, why the stress distribution around the crack tip and analytical methods for its determination, are so important. Although we have derived Eq. (8.1.6) for a very simple case in mode I, scientists
believe that it can be modified for any cases by a correction factor. This is motivated mathematicians to find the stress distributions near the crack tip and ignore the higher order terms, so that finally it leads them to SIF.
