ABSTRACT
The mapping function of a crack into outside of a unit circle can found by substituting c a=
2 and m=1 into 6.5.14, that is,
z a= +
2
1 ζ
ζ (7.1.3)
Exercise 7.1.1: for a crack find ζ versus z
Solution: by squaring both sides of Eq. (7.1.3) we have, z a2 2 2
24 1 2= + +
ζ ζ
z a2 2
2 24 1 2= + +
ζ ζ , and then by deducting a2 from both sides we have: z a a a2 2
4 1 2
4 1
− = + −
= −
ζ ζ
ζ ζ
z a a a2 2 2
4 1 2
4 1
− = + −
= −
ζ ζ
ζ ζ
and taking square root of this provides:
z a a2 2 2
1 − = −
ζ ζ
Then from Eqs. (7.1.3) and (7.1.4) ζ can be found versus z, that is,
ζ = − +z a z a
ζ =
− −z z a a
By substituting Eqs. 7.1.5) into (7.1.1) we can change φ ζ( ) into φ z( ), that is,
φ σz a z a z
a z z a
a ( ) = − + − − −
8
which can be simplified to:
φ σz z a z( ) = − −( )4 2 2 2 (7.1.6)
Exercise 7.1.2: Convert ψ ζI ( ) into ψ I z( ) Solution: Similarly we substitute Eq. (7.1.5) into Eq. (7.1.2) and we have:
ψ σ
I z a
a z a z a z a z
a z z a z
a z a z ( ) =
− +( ) − − +( ) − − − +( )
− +4
1 4
2 2 2( ) −
2 1
and can be simplified to,
ψ σI z a a z a z a z a z a z z a
z a z a z ( ) =
− +( ) − − +( ) − − −( ) − − +(4
1 4
Further simplification ψ σI z z a
z a z a a z z a z a z
( ) = −
− +( ) − − − − − +
8
ψ σ
I z z a
z a z a a z z a z a z
( ) = −
− +( ) − − − − − +
8
2 2 this gives, ψ σI z
z a z a z a z z a( ) =
− − +( ) − − − −( )
8
ψ σ
I z z a
z a z a z z a( ) = −
− +( ) − − − −( )
8
the expression inside square bracket can be simplified as
well, ψ σI z z a
z z a a( ) = −
− − 8
4 4 2 2
ψ σI z z a
z a ( ) = −
−
Now that we have Eqs. (7.1.6) and (7.1.7), we can rewrite Eq. (6.4.27) in terms ψ I z( ):
2µ κφ φ ψu i v z z z zI+( )= ( ) − ( ) − ( )' (7.1.8)
Differentiation of Eq. (7.1.6) versus z provides φ ' z( ), that is,
φ σ σ' z z z a
z z a
( ) = −
−
=
− −
4
2 1 2
φ σ' z z z a
( ) = −
−
2
1 22 2
From Eq. (7.1.7) we need to find:
ψ σI z z a
z a ( ) = −
−
The objective is finding the displacement v of the face of the crack. It is obvious that we need to change z into x and the crack region can be designated by − ≤ ≤a x a.
