Page 349 - Mechanical Behavior of Materials

350 Chapter 8 Fracture of Cracked Members

end of this chapter lists several such handbooks. What will be done in this portion of the chapter is
to give certain fundamental equations for calculating K and also examples of the type of information
available from handbooks.

8.4.1 Mathematical Forms Used to Express K
It has already been noted that K can be related to applied stress and crack length by an equation of
the form
√
K = FS g πa, F = F(geometry, a/b) (8.12)
The quantity F is a dimensionless function that depends on the geometry and loading configuration,
and usually also on the ratio of the crack length to another geometric dimension, such as the member
width or half-width, b, as defined for three cases in Fig. 8.12. Additional examples for F are given in
Figs. 8.13 and 8.14, specifically, for bending of single-edge-cracked plates and for various loadings
on a circumferentially cracked round bar. In these examples, crack length a is measured from either
the surface or the centerline of loading, and the width dimension b is consistently defined as the
maximum possible crack length, so that for a/b = 1, the member is completely cracked. For each
case in Figs. 8.12 to 8.14, polynomials or other mathematical expressions are given that may be
employed to calculate F within a few percent for any α = a/b. Where trigonometric functions
appear, the arguments for these are in units of radians.
Applied forces or bending moments are often characterized by determining a nominal or
average stress. In fracture mechanics, it is conventional to use the gross section nominal stress,
S g , calculated under the assumption that no crack is present. Note that this convention is followed
for each case in Figs. 8.12 to 8.14. The subscript g is added merely to avoid any possibility of
confusion, as net section stresses, S n , based on the remaining uncracked area, could be used. The
use of S g rather than S n is convenient, as the effect of crack length is then confined to the F and
√
a factors. In general, the manner of defining nominal stress S is arbitrary, but consistency with
F is necessary. The function F must be redefined and its values changed if the definition of S is
changed, and also if the definition of a or b is changed.
It is sometimes convenient to work directly with applied loads (forces), with the following
equation being useful for planar geometries:

P
K = F P √ , F P = F P (geometry, a/b) (8.13)
t b
Here, P is force, t is thickness, and b is the same as before. The function F P is a new dimensionless
geometry factor. Examples are given in Figs. 8.15 and 8.16. Equating K from Eqs. 8.12 and 8.13
allows F P to be related to the previously defined F:
√
S g t πab
F P = F (8.14)
P
This relationship can be used to obtain the function F P for any of the cases where F is given in
Figs. 8.12 to 8.14. Expressing K in terms of F P has the advantage that the dependence on crack
length is confined to the dimensionless function F P .