Page 175 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 155 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 155

zone was chosen so that the stresses at the tip would be finite. Thus

a  πσ 
k ≡ = cos   (A3.6)
a  σ YS 
1

When Equation (A3.6) is substituted into Equation (A3.5) and Equation (A3.3) is superimposed,
the first term in Equation (A3.5) cancel with Equation (A3.3), which leads to

2σ  k z 2 a − 2 
Z = YS  1  (A3.7)
π   z 1 − k 2  

Integrating Equation (A3.7) gives

2σ
Z YS z = ω − a [ ω ] (A3.8)
π 1 2

where

 a 1  2
1−
− 1  a 
ω = cot
1
1 −
k 2 1
and

z 2 − a 2
− 1 1
ω = cot
2 1− k 2

On the crack plane, y = 0 and the displacement in the y direction (Equation (A3.2)) reduces to

2
u = E Im Z (A3.9)
y

for plane stress. Solving for the imaginary part of Equation (A3.8) gives

4σ   a 1 2 z − 2   k a 2 z − 2  
u = YS a  coth −  1 2  − zcoth −1 1  1 2  
y
π
E 
  a 1 1 − k   z 1 − k   
for |z| ≤ a . Setting z = a leads to
1

8 σ a   1
δ = y = 2u πE YS ln   k (A3.10)

which is identical to Equation (3.5).