Page 213 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 193 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 193

where P is the applied load and ∆ is the load-line displacement. Similarly, C* can be defined in
terms of a power release rate:

∂
˙
∆
C* =− 1   ∫ ∆ Pd   (4.36)

B a ∂ 0  ˙ ∆
The J integral can be related to the energy absorbed by a laboratory specimen, divided by the
ligament area: 2

η ∆
J = ∫ Pd∆ (4.37)
Bb 0

where η is a dimensionless constant that depends on geometry. Therefore, C* is given by

η ˙ ∆
C* = ∫ Pd∆ (4.38)
˙
Bb 0
For a material that creeps according to a power law (Equation (4.32)), the displacement rate is
n
proportional to P , assuming global creep in the specimen. In this case, Equation (4.38) reduces to
n η
C* = P∆ ˙ (4.39)
n B + 1 b

The geometry factor η has been determined for a variety of test specimens. For example, η = 2.0
for a deeply notched bend specimen (Equation (3.37) and Equation (4.6)).

4.2.2 SHORT-TIME VS. LONG-TIME BEHAVIOR

The C* parameter applies only to crack growth in the presence of global steady-state creep. Stated
another way, C* applies to long-time behavior, as discussed below.
Consider a stationary crack in a material that is susceptible to creep deformation. If a remote
load is applied to the cracked body, the material responds almost immediately with the corresponding
elastic strain distribution. Assuming the loading is pure Mode I, the stresses and strains exhibit a
1/ r singularity near the crack tip and are uniquely defined by K . However, large-scale creep defor-
I
mation does not occur immediately. Soon after the load is applied, a small creep zone, analogous to
a plastic zone, forms at the crack tip. The crack-tip conditions can be characterized by K as long as
I
the creep zone is embedded within the singularity dominated zone. The creep zone grows with time,
eventually invalidating K as a crack-tip parameter. At long times, the creep zone spreads throughout
I
the entire structure.
When the crack grows with time, the behavior of the structure depends on the crack growth
rate relative to the creep rate. In brittle materials, the crack growth rate is so fast that it overtakes
the creep zone; crack growth can be characterized by K because the creep zone at the tip of the
I

2 The load-line displacement ∆ in Equation (4.37)–(4.39) corresponds to the portion of the displacement due to the presence
of the crack, as discussed in Section 3.2.5. This distinction is not necessary in Equation (4.35) and Equation (4.36), because
the displacement component attributed to the uncracked configuration vanishes when differentiated with respect to a.