Page 173 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 153 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 153

FIGURE 3.47 Single- and two-parameter assump-
tions in terms of the three independent variables in the
elastic-plastic crack-tip field. The loading path initially
lines in the two-parameter surface and then diverges,
as indicated by the dashed line.

single-parameter line. When it deviates from this line, it may remain in the two-parameter surface
for a time before diverging from the surface.
The loading path in J-A -A space depends on geometry [43]. Low-constraint configurations
4
2
such as the center-cracked panel and shallow notched bend specimens diverge from the single-
parameter theory almost immediately, but follow Equation (3.88) to fairly high deformation levels.
Deeply notched bend specimens maintain a high constraint to relatively high J values, but they do
not follow Equation (3.88) when constraint loss eventually occurs. Thus low-constraint geometries
should be treated with the two-parameter theory, and high-constraint geometries can be treated
with the single-parameter theory in many cases. When high-constraint geometries violate the single-
parameter assumption, however, the two-parameter theory is of little value.

APPENDIX 3: MATHEMATICAL FOUNDATIONS OF ELASTIC-PLASTIC
FRACTURE MECHANICS

A3.1 DETERMINING CTOD FROM THE STRIP-YIELD MODEL
Burdekin and Stone [3] applied the Westergaard [44] complex stress function approach to the strip-
yield model. They derived an expression for CTOD by superimposing a stress function for closure
forces on the crack faces in the strip-yield zone. Their result was similar to previous analyses based
on the strip-yield model performed by Bilby et al. [45] and Smith [46].
Recall from Appendix 2.3 that the Westergaard approach expresses the in-plane stresses (in a
limited number of cases) in terms of Z:
σ xx = Z Re y − Z Im ′ (A3.1a)
σ yy = Z Re y + Z Im ′ (A3.1b)

τ xy y=− Re Z ′ (A3.1c)

where Z is an analytic function of the complex variable z = x + iy, and the prime denotes a first
derivative with respect to z. By invoking the equations of elasticity for the plane problem, it can
be shown that the displacement in the y direction is as follows:

u = y 1 Z 2 [Im y − 1 ( + ν Z )Re ] for plane stress (A3.2a)
E