Page 120 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 100 Thursday, April 14, 2005 6:28 PM

100 Fracture Mechanics: Fundamentals and Applications

where A is a real constant. Equation (A2.47) is equivalent to the Irwin modification of the
Westergaard approach if

z
2 ′() =φ Z − () z A (A2.48)
Substituting Equation (A2.48) into Equation (A2.47) gives

σ xx = Z Re y − Z Im A ′ − 2 (A2.49a)
σ yy = Z Re y + Z Im ′ (A2.49b)
τ xy y=− Re Z (A2.49c)

Comparing Equation (A2.49) with Equation (A2.31) and Equation (A2.46), it is obvious that the
Sih and Irwin modifications are equivalent, and 2A = σ .
oxx
Sanford [41] showed that the Irwin-Sih approach is still too restrictive, and he proposed
replacing A with a complex function η(z):

z
z
2 ′() =φ Z − () z η () (A2.50)
The modified stresses are given by
σ xx = y − Z Im y ′ + Im η ′ − Re2 η Z Re (A2.51a)

σ yy Z Re y + Z Im y ′ + Im η = ′ (A2.51b)
τ xy Z y + Re η ′ + Im η y=− Re (A2.51c)

Equation (A2.51) represents the most general form of Westergaard-type stress functions. When
η(z) = a real constant for all z, Equation (A2.51) reduces to the Irwin-Sih approach, while Equation
(A2.51) reduces to the original Westergaard solution when η(z) = 0 for all z.
The function η can be represented as a polynomial of the form
M
η ∑ α() z m z = m /2 (A2.52)
m=0
Combining Equation (A2.37), Equation (A2.50), and Equation (A2.52) and defining the origin at
the crack tip gives

K M
2 ′ = φ I − ∑ α z m/ 2 (A2.53)
2πz m
m = 0
which is consistent with the Williams [11, 38] asymptotic expansion.

A2.4 ELLIPTICAL INTEGRAL OF THE SECOND KIND

The solution of stresses in the vicinity of elliptical and semielliptical cracks in elastic solids [10, 44]
involves an elliptic integral of the second kind:
2 / π c 2 − a 2
d
Ψ= ∫ 1 − sin φφ (A2.54)
2
0 c 2