Page 118 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 98 Thursday, April 14, 2005 6:28 PM

98 Fracture Mechanics: Fundamentals and Applications

which leads to

σ = I cos  θ K  1 − sin θ   sin  3 θ      (A2.38a)
xx
2 πr       2  
2
2
σ yy = I cos  θ K  1 + sin θ   sin  3 θ      (A2.38b)
2
2
2 πr       2  
K  θ  
θ 
τ = I cos sin (A2.38c)
xy
2
2
2 πr    
assuming r >> a. Equation (A2.38) is equivalent to Equation (A2.28), except that the latter is
expressed in terms of polar coordinates.
Westergaard published the following stress function for an array of collinear cracks in a plate
in biaxial tension (Figure 2.21):
σ
Zz() = (A2.39)
   π a   2  / 12
  sin  2 W   


 1 −  z   
  sin  π  
    2 W   



where a is the half-crack length and 2W is the spacing between the crack centers. The stress intensity
for this case is given in Equation (2.45); early investigators used this solution to approximate the
behavior of a center-cracked tensile panel with finite width.
Irwin [9] published stress functions for several additional configurations, including a pair of
crack-opening forces located at a distance X from the crack center (Figure 2.32):
Pa 1 − (/ 2
Xa)
Zz() = (A2.40)
pz − X z 1 − (/ 2
az)
)
(
where P is the applied force. When there are matching forces at ±X , the appropriate stress function
can be obtained by superposition:
Xa)
Zz() = 2 Pa 1 − (/ 2 (A2.41)
z π 2 − X ( 2 ) 1 − (/ 2
az)
In each case, the stress function can be expressed in the form of Equation (A2.37) and the near-
tip stresses are given by Equation (A2.38). This is not surprising, since all of the above cases are
pure Mode I and the Williams analysis showed that the inverse square root singularity is universal.
For plane strain conditions, the in-plane displacements are related to the Westergaard stress
function as follows:

1
Z −
u x 2µ [( 12ν) Z = − Re y Im ] (A2.42a)
1
u y 2µ [( − Z = − y Re ] (A2.42b)
Z 21 ν)Im
For the plate in Figure A2.2, the crack-opening displacement is given by
1−ν 2 1−ν ( 2 4 1−ν ) ( 2 ) σ
2 u y µ Im Z = = E Im Z = E a 2 x − 2 (A2.43a)