Page 147 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 127 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 127

FIGURE 3.22 Schematic load-displacement curve for a specimen with a crack that grows to a 1 from an initial
length a o . U D represents the strain energy in a nonlinear elastic material.

where the subscript D refers to the deformation theory. Thus the J integral for a nonlinear elastic
body with a growing crack is given by

D
J =− 1   ∂ U   ∆
a ∂ 
B 
D
ηU (3.56a)
= D
Bb
or

K 2 η U
J = E′ I + p Bb D p () (3.56b)
D

where b is the current ligament length. When the J integral for an elastic-plastic material is defined
by Equation (3.56), the history dependence is removed and the energy release rate interpretation
of J is restored. The deformation J is usually computed from Equation (3.56b) because no correction
is required on the elastic term as long as K is determined from the current load and crack length.
I
The calculation of U D(p) is usually performed incrementally, since the deformation theory load-
displacement curve (Figure 3.22 and Equation (3.55)) depends on the crack size. Specific procedures
for computing the deformation J are outlined in Chapter 7.
One can determine a far-field J from the contour integral definition of Equation (3.19), which
may differ from J . For a deeply cracked bend specimen, Rice et al. [15] showed that the far-field
D
J contour integral in a rigid, perfectly plastic material is given by

J f . o ∫ Ω b = 073σ d Ω (3.57)
0