Page 116 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 96 Thursday, April 14, 2005 6:28 PM

96 Fracture Mechanics: Fundamentals and Applications

the Airy stress function as follows:

y
Φ= Re Z + Im Z (A2.30)

where Re and Im denote real and imaginary parts of the function, respectively, and the bars over
Z represent integrations with respect to z, i.e.,

dZ dZ
Z = and Z =
dz dz

Applying Equation (A2.6) gives
σ = xx Z Re y − Z Im ′ (A2.31a)

σ = yy Z Re y + Z Im ′ (A2.31b)

τ xy y=− Re Z ′ (A2.31c)

Note that the imaginary part of the stresses vanishes when y = 0. In addition, the shear stress
vanishes when y = 0, implying that the crack plane is a principal plane. Thus, the stresses are
symmetric about θ = 0 and Equation (A2.31) implies Mode I loading.
The Westergaard stress function, in its original form, is suitable for solving a limited range of
Mode I crack problems. Subsequent modifications [38–41] generalized the Westergaard approach
to be applicable to a wider range of cracked configurations.
Consider a through crack in an infinite plate subject to biaxial remote tension (Figure A2.2).
If the origin is defined at the center of the crack, the Westergaard stress function is given by

z σ
Zz() = (A2.32)
z 2 a − 2

where σ is the remote stress and a is the half-crack length, as defined in Figure A2.2. Consider the
crack plane where y = 0. For −a < x < a , Z is pure imaginary, while Z is real for |x| > |a|. The
normal stresses on the crack plane are given by
σx
σ = σ = Z = Re (A2.33)
yy
xx
x 2 a − 2
*
Let us now consider the horizontal distance from each crack tip, x = x − a, Equation (A2.33)
becomes
σ a
σ xx σ = yy = (A2.34)
2 x *
*
for x << a. Thus, the Westergaard approach leads to the expected inverse square-root singularity.
One advantage of this analysis is that it relates the local stresses to the global stress and crack size.
From Equation (A2.28), the stresses on the crack plane (θ = 0) are given by

K
σ rr σ = θθ σ = xx σ = yy = I (A2.35)
2 πx *