Page 359 - Mechanical Behavior of Materials

360 Chapter 8 Fracture of Cracked Members

In practical applications, cracks having shapes that approximate a circle, half-circle, or quarter-
circle may occur, as illustrated in Fig. 8.17. Half-circular surface cracks as in (b) and (d) are
especially common. Evaluation of stress intensities for these complex three-dimensional cases is
aided by the existence of an exact solution for a circular crack of radius a in an infinite body under
uniform stress S:
2 √
K = S πa (8.20)
π
For embedded (internal) circular cracks, Fig. 8.17(a), this solution is still within 10% for members
of finite size, subject to the limits a/t < 0.5 and a/b < 0.5.
For half-circular surface cracks or quarter-circular corner cracks, and for a values that are
small compared with the other dimensions, the stress intensities are elevated compared with
Eq. 8.20 by a factor around 1.13 or 1.14, giving F values as shown in Fig. 8.17 for cases (b),
(c), and (d). These F values specifically apply for points where the crack front intersects the surface,
where K has its maximum value. They may be applied for either tension or bending, with 10%
accuracy, within the limits indicated. (Note that the factors of 1.13 or 1.14 on K, compared with
the circular crack case, are analogous to the previously discussed free surface factor of 1.12 for
cracks in flat plates.)
More detail is shown for the half-circular surface crack case in Fig. 8.18. The equations given
are based on the paper by Newman (1986), as fitted there to finite element analyses. Note that K is
affected by the proximity of the boundary in two directions and also varies around the periphery of
the crack; that is, K varies with both a/b and a/t and also with θ. Stresses for tension and bending,
S t and S b , respectively, are defined as in Fig. 8.17(b), and three functions f a , f b , and f w , are needed.
Figure 8.18 gives an equation for f w that is accurate for a/b < 0.5, where f w = 1 if either or both
of a/b and a/t are small. Also given are f a and f b for the surface points, θ = 0 and 180 , and
◦
◦
for the deepest point, θ = 90 .The θ variation is rather small, as shown in Fig. 8.18(b) for two
different a/t values. For small a/b and a/t, note that f w = f b = 1, giving f a = 1.144 at the surface
and f a = 1.04 at the deepest point. The former value corresponds to F = 1.144(2/π) = 0.728 in
Fig. 8.17(b).
An exact solution also exists for an elliptical crack in an infinite body under uniform stress.
With reference to Fig. 8.19(a), this solution is

1/4
πa 2 2
!
a 2
K = S f φ , f φ = cos φ + sin φ (a/c ≤ 1) (8.21)
Q c
where the angle φ specifies a particular location P around the elliptical crack front. The quantity Q
is called the flaw shape factor. It is given exactly by

π/2
a 2
2 2 2
Q = E(k) = 1 − k sin β dβ, k = 1 − (8.22)
0 c
E(k) is the standard elliptic integral of the second kind, values of which are given in most books
of mathematical tables and in analogous computer software packages. However, Q maybeclosely