Page 345 - Mechanical Behavior of Materials

346 Chapter 8 Fracture of Cracked Members

As listed in the References, Barsom (1987) is a collection of papers reporting some of the early
work on fracture mechanics by Griffith, Irwin, and others.

8.3.2 Stress Intensity Factor, K

The stress intensity factor concept, which has already been introduced, needs to be defined in a more
complete manner. In general terms, K characterizes the magnitude (intensity) of the stresses in the
vicinity of an ideally sharp crack tip in a linear-elastic and isotropic material.
A coordinate system for describing the stresses in the vicinity of a crack is shown in Fig. 8.10.
The polar coordinates r and θ lie in the x-y plane, which is normal to the plane of the crack, and the
z-direction is parallel to the leading edge of the crack. For any case of Mode I loading, the stresses
near the crack tip depend on r and θ as follows:

K I θ θ 3θ
σ x = √ cos 1 − sin sin + ··· (a)
2πr 2 2 2

K I θ θ 3θ
σ y = √ cos 1 + sin sin + ··· (b)
2πr 2 2 2

K I θ θ 3θ
τ xy = √ cos sin cos + ··· (c) (8.7)
2πr 2 2 2
σ z = 0 (plane stress) (d)

σ z = ν σ x + σ y (plane strain; ε z = 0) (e)
τ yz = τ zx = 0 (f)

σ y
τ xy
y τ yz
σ τ zx
x
σ z x
r θ

leading edge
of the crack

z

Figure 8.10 Three-dimensional coordinate system for the region of a crack tip. (Adapted
from [Tada 85]; used with permission.)