Page 347 - Mechanical Behavior of Materials

348 Chapter 8 Fracture of Cracked Members

value of the factor K I . On this basis, K I is a measure of the severity of the crack. Its definition in a
formal mathematical sense is
√

K I = lim σ y 2πr (8.8)
r,θ→0
It is generally convenient to express K I as
√
K I = FS πa (8.9)
where the factor F is needed to account for different geometries. For example, if a central crack
in a plate is relatively long, Eq. 8.2 needs to be modified, as the proximity of the specimen edge
causes F to increase above unity. The quantity F is a function of the ratio a/b, as shown in Fig. 8.12,
curve (a). Curves (b) and (c) show the variation of F with a/b for two additional cases of cracked
members under tension, specifically, for double-edge-cracked plates and for single-edge-cracked
plates.

8.3.3 Additional Comments on K and G

For loading in Mode II or III, analogous, but distinct, stress field equations exist, and stress
intensities K II and K III can be defined in a manner analogous to K I . However, most practical
applications involve Mode I. As a convenience, the subscript on K I will be dropped, and K without
such a subscript is understood to denote K I , that is, K = K I .
The quantities G and K can be shown to be related as follows:
K 2
G = (8.10)
E

where E is obtained from the material’s elastic modulus E and Poisson’s ratio ν:
E = E (plane stress; σ z = 0)

E (8.11)

E = (plane strain; ε z = 0)
1 − ν 2
Equation 8.10 and the dependence of G on load versus displacement behavior, Eq. 8.6, can be
exploited to evaluate K. Slopes on P-v curves, as in Fig. 8.9, are employed in a procedure called
the compliance method. See any book on fracture mechanics or Tada (2000) for details.
Since G and K are directly related according to Eq. 8.10, only one of these concepts is
generally needed. We will primarily employ K, which is consistent with most engineering-oriented
publications on fracture mechanics.

8.4 APPLICATION OF K TO DESIGN AND ANALYSIS

For fracture mechanics to be put to practical use, values of stress intensity K must be determined
for crack geometries that may exist in structural components. Extensive analysis work has been
published, and also collected into handbooks, giving equations or plotted curves that enable K
values to be calculated for a wide variety of cases. A special section of the References at the