Page 338 - Mechanical Behavior of Materials

Section 8.2 Preliminary Discussion 339

exists near the crack tip, and the stresses somewhat farther away are higher than they would be for
an ideal crack.

8.2.3 Effects of Cracks on Strength
If the load applied to a member containing a crack is too high, the crack may suddenly grow and
cause the member to fail by fracturing in a brittle manner—that is, with little plastic deformation.
From the theory of fracture mechanics, a useful quantity called the stress intensity factor, K, can be
defined. Specifically, K is a measure of the severity of a crack situation as affected by crack size,
stress, and geometry. In defining K, the material is assumed to behave in a linear-elastic manner,
according to Hooke’s law, Eq. 5.26, so that the approach being used is called linear-elastic fracture
mechanics (LEFM).
A given material can resist a crack without brittle fracture occurring as long as this K is below
a critical value K c , called the fracture toughness. Values of K c vary widely for different materials
and are affected by temperature and loading rate, and secondarily by the thickness of the member.
Thicker members have lower K c values until a worst-case value is reached, which is denoted K Ic
and called the plane strain fracture toughness. Hence, K Ic is a measure of a given material’s ability
to resist fracture in the presence of a crack. Some values of this property are given for various
materials in Tables 8.1 and 8.2.
For example, consider a crack in the center of a wide plate of stressed material, as illustrated in
Fig. 8.5. In this case, K depends on the remotely applied stress S and the crack length a, measured
from the centerline as shown:
√
K = S πa (a
b) (8.2)
This equation is accurate only if a is small compared with the half-width b of the member. For a
given material and thickness with fracture toughness K c , the critical value of remote stress necessary
to cause fracture is thus
K c
S c = √ (8.3)
πa
Hence, longer cracks have a more severe effect on strength than do shorter ones, as might be
expected.
Some test data illustrating the effect of different crack lengths on strength are shown in
Fig. 8.5. These particular data correspond to 2014-T6 aluminum plates of thickness t = 1.5 mm,
◦
tested at −195 C. Note that the failure data fall far below the material’s yield strength σ o .This
behavior cannot be explained merely by yielding and the loss of cross-sectional area due to the
crack, which is indicated by the dotted line. (See Fig. A.16(a), which gives the dotted line as
S = P/(2bt) = σ o (1 − a/b).) Substituting the K c value for this case into Eq. 8.3 gives the solid
curve, which agrees quite well with most of the data, indicating a degree of success for LEFM.
However, as the stress S approaches the material’s yield strength σ o , the data fall below Eq. 8.3, as
shown by the dashed line. This deviation occurs because Eq. 8.2 assumes that linear-elastic behavior
is exhibited and so is accurate only if the plastic zone is small, which is not the case for the failures
at high stresses for short cracks.