Page 174 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 154 Monday, May 23, 2005 5:42 PM

154 Fracture Mechanics: Fundamentals and Applications

and

u = y 1 [( − 2 Z )Im y − ν 1 ( + ν Z )Re ] for plane strain (A3.2b)
21
E
where Z is the integral of Z with respect to z, as discussed in Appendix 2. For a through crack
of length 2a in an infinite plate under biaxial tensile stress s, the Westergaard function is
1
given by

σ z
Z = (A3.3)
z 2 a − 2
1

where the origin is defined at the crack center.
The stress function for a pair of splitting forces P at ± x within a crack of length 2a (see Figure
1
2.32) is given by

2
2 Pz a − x 2
Z = 1 (A3.4)
zπ 2 a − 2 ( z 2 a − 2 )
1 1
For a uniform compressive stress s along the crack surface between a and a (Figure A3.1), the
YS
1
Westergaard stress function is obtained by substituting P = −s dx into Equation (A3.4) and inte-
YS
grating:
2
Z =− ∫ a 1 2σ 2 YS za − x 2 2 ) dx
1
z (
a −
zπ
2
2
a
1 a − 1
(A3.5)

2σ  z  a  1 a z 2 − a 2  
=− YS  cos −   − cot −1  1  
π  z  2 − a 1 2  a 1   z a 1 2 − a 2   
The stress functions of Equation (A3.3) and Equation (A3.5) can be superimposed, resulting in the
strip-yield solution for the through crack. Recall from Section 2.8.2 that the size of the strip-yield

FIGURE A3.1 Strip-yield model for a through crack.