Page 390 - Mechanical Behavior of Materials

Section 8.9 Extensions of Fracture Mechanics Beyond Linear Elasticity 391

fracture
P K = f (a , P )
P c c c
c
K = f (a , P )
P Q i Q
P Q
Δa
c
a i a i a c
0 v 0 a 0 v 0 a
(a) (b)

Figure 8.48 Load vs. displacement and load vs. crack extension behavior during fracture
toughness tests under plane strain (a) and plane stress (b).

8.8.2 Effect of Thickness on Fracture Behavior

Fracture under the highly constrained conditions of plane strain generally occurs rather suddenly,
with little crack growth prior to final fracture. Also, the fracture surface is quite flat. In contrast,
◦
plane stress fractures tend to have sloping or V -shaped surfaces inclined at about 45 on planes
of maximum shear stress, as already illustrated by Figs. 8.44 and 8.47. The final fracture in
plane stress is usually preceded by considerable slow-stable crack growth, as shown in Fig. 8.48.
These behaviors correlate with the thickness effect on toughness, as in Fig. 8.31. Flat plane-strain
fractures occur where the thickness is sufficient to reach the lower plateau of the curve—that is,
the minimum toughness K Ic . Inclined or V-shaped plane-stress fractures occur for relatively thin
members, for which the toughness may be well above K Ic . Fracture toughness values K Ic meeting
the requirements for plane strain are expected to be minimum values that can be safely used in
design for any thickness.
Where a thickness less than that required for plane strain fracture in a given material is used in
an engineering application, K Ic may involve an undesirably large degree of conservatism. It may
then be useful to use K Q data for the particular thickness of interest. Also, a toughness K c can be
defined that corresponds to the point of final fracture, as illustrated in Fig. 8.48(b). Since the amount
of slow-stable crack growth may be considerable, the crack extension a c from the initial length a i
to the final length a c needs to be measured. The corresponding K can then be calculated from the
load P c at the point of final fracture.

8.9 EXTENSIONS OF FRACTURE MECHANICS BEYOND LINEAR ELASTICITY

If Eq. 8.39 is not satisfied, so that LEFM does not apply due to excessive yielding, several methods
still exist for analyzing cracked members. Excessive yielding causes K to no longer correctly
characterize the magnitude of the stress field around the crack tip—specifically, K underestimates
the severity of the crack. An introduction to various approaches for extending fracture mechanics
beyond linear elasticity follows.