Page 123 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 103 Monday, May 23, 2005 5:42 PM

3 Elastic-Plastic Fracture

Mechanics

Linear elastic fracture mechanics (LEFM) is valid only as long as nonlinear material deformation
is confined to a small region surrounding the crack tip. In many materials, it is virtually impossible
to characterize the fracture behavior with LEFM, and an alternative fracture mechanics model is
required.
Elastic-plastic fracture mechanics applies to materials that exhibit time-independent, nonlinear
behavior (i.e., plastic deformation). Two elastic-plastic parameters are introduced in this chapter:
the crack-tip-opening displacement (CTOD) and the J contour integral. Both parameters describe
crack-tip conditions in elastic-plastic materials, and each can be used as a fracture criterion. Critical
values of CTOD or J give nearly size-independent measures of fracture toughness, even for relatively
large amounts of crack-tip plasticity. There are limits to the applicability of J and CTOD (Section 3.5
and Section 3.6), but these limits are much less restrictive than the validity requirements of
LEFM.

3.1 CRACK-TIP-OPENING DISPLACEMENT

When Wells [1] attempted to measure K values in a number of structural steels, he found that
Ic
these materials were too tough to be characterized by LEFM. This discovery brought both good
news and bad news: High toughness is obviously desirable to designers and fabricators, but Wells’
experiments indicated that the existing fracture mechanics theory was not applicable to an important
class of materials. While examining fractured test specimens, Wells noticed that the crack faces
had moved apart prior to fracture; plastic deformation had blunted an initially sharp crack, as
illustrated in Figure 3.1. The degree of crack blunting increased in proportion to the toughness of
the material. This observation led Wells to propose the opening at the crack tip as a measure of
fracture toughness. Today, this parameter is known as CTOD.
In his original paper, Wells [1] performed an approximate analysis that related CTOD to the
stress intensity factor in the limit of small-scale yielding. Consider a crack with a small plastic
zone, as illustrated in Figure 3.2. Irwin [2] postulated that crack-tip plasticity makes the crack
behave as if it were slightly longer (Section 2.8.1). Thus, we can estimate the CTOD by solving
for the displacement at the physical crack tip, assuming an effective crack length of a + r . From
y
Table 2.2, the displacement r behind the effective crack tip is given by
y
κ +1 r 4 r
u y K = I y = K I y (3.1)
2 µ 2 π E′ 2 π

where E′ is the effective Young’s modulus, as defined in Section 2.7. The Irwin plastic zone correction
for plane stress is


K 
1
I
r = 2πσ YS   2 (3.2)

y

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