Page 119 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 99 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 99

assuming plane strain, and
4σ
2u = y E a 2 x − 2 (A2.43b)

for plane stress. Equation (A2.43) predicts that a through crack forms an elliptical opening profile
when subjected to tensile loading.
The near-tip displacements can be obtained by inserting Equation (A2.37) into Equation (A2.42):

K r    2 θ
  
θ
u = 2µ I 2π cos     κ −+ sin     (A2.44a)
12
x
2
2
K r    2 θ
  
θ
u = 2µ I 2π sin     κ +− cos     (A2.44b)
12
y
2
2
for r << a, where
κ 34 ν =− for plane strain (A2.45a)
and
k = 3 −ν for plane stress (A2.45b)
1 +ν
Although the original Westergaard approach correctly describes the singular Mode I stresses in
certain configurations, it is not sufficiently general to apply to all Mode I problems; this shortcoming
has prompted various modifications to the Westergaard stress function. Irwin [38] noted that
photoelastic fringe patterns observed by Wells and Post [42] on center-cracked panels did not match
the shear strain contours predicted by the Westergaard solution. Irwin achieved a good agreement
between theory and experiment by subtracting a uniform horizontal stress:

σ xx = Z Re y − Z Im ′− σ oxx (A2.46)
where σ depends on the remote stress. The other two stress components remain the same as in
oxx
Equation (A2.31). Subsequent analyses have revealed that when a center-cracked panel is loaded in
uniaxial tension, a transverse compressive stress develops in the plate. Thus, Irwin’s modification to
5
the Westergaard solution has a physical basis in the case of a center-cracked panel. Equation (A2.46)
has been used to interpret photoelastic fringe patterns in a variety of configurations.
Sih [39] provided a theoretical basis for the Irwin modification. A stress function for Mode I
must lead to zero shear stress on the crack plane. Sih showed that the Westergaard function was
more restrictive than it needed to be, and was thus unable to account for all situations. Sih
generalized the Westergaard approach by applying a complex potential formulation for the Airy
stress function [39]. He imposed the condition τ = 0 at y = 0, and showed that the stresses could
xy
be expressed in terms of a new function φ(z):
σ xx φ = 2Re ′( ) y − 2 Im φ z z − A ′′( ) (A2.47a)
σ yy = 2 φ ′( ) y + 2 Im φ z Re z + A ′′( ) (A2.47b)

τ xy y φ = 2Re ′′( ) (A2.47c)
z

5 Recall that the stress function in Equation (A2.32) is strictly valid only for biaxial loading. Although this restriction was
not imposed in Westergaard’s original work, a transverse tensile stress is necessary in order to cancel with −σ oxx . However,
the transverse stresses, whether compressive or tensile, do not affect the singular term; thus the stress intensity factor is the
same for uniaxial and biaxial tensile loading and is given by Equation (A2.36).