Page 337 - Mechanical Behavior of Materials

338 Chapter 8 Fracture of Cracked Members

A stress concentration factor for the ellipse can be defined as the ratio of the maximum stress to the
remote stress: k t = σ y /S.
Consider a narrow ellipse where the half-height d approaches zero, so that the tip radius ρ also
approaches zero, which corresponds to an ideal slitlike crack. In this case, σ y becomes infinite, as
does k t . Hence, a sharp crack causes a severe concentration of stress and is special in that the stress
is theoretically infinite if the crack is ideally sharp.

8.2.2 Behavior at Crack Tips in Real Materials

An infinite stress cannot, of course, exist in a real material. If the applied load is not too high,
the material can accommodate the presence of an initially sharp crack in such a way that the
theoretically infinite stress is reduced to a finite value. This is illustrated in Fig. 8.4. In ductile
materials, such as many metals, large plastic deformations occur in the vicinity of the crack tip.
The region within which the material yields is called the plastic zone. Intense deformation at the
crack tip results in the sharp tip being blunted to a small, but nevertheless nonzero, radius. Hence,
the stress is no longer infinite, and the crack is open near its tip by a finite amount, δ, called the
crack-tip opening displacement (CTOD).
In other types of material, different behaviors occur that have a similar effect of relieving the
theoretically infinite stress by modifying the sharp crack tip. In some polymers, a region containing
elongated voids develops, with a fibrous structure bridging the crack faces, which is called a craze
zone. In brittle materials such as ceramics, a region containing a high density of tiny cracks may
develop at the crack tip.
In all three cases, the crack tip experiences intense deformation and develops a finite separation
near its tip. The very high stress that would ideally exist near the crack tip is spread over a larger
region and is said to be redistributed. A finite value of stress that can be resisted by the material thus

ideal crack
real crack polymer
σ
y
δ

x

metal ceramic
ρ
δ plastic zone δ

2r o

Figure 8.4 Finite stresses and nonzero radii at tips of cracks in real materials. A region of
intense deformation forms due to plasticity, crazing, or microcracking.