Page 203 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 183 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 183

Equation (4.16) is valid only at short times or in infinite bodies. This relationship neglects
reflected stress waves, which can have a significant effect on the local crack-tip fields. Since the
crack speed is proportional to the wave speed, Equation (4.16) is valid as long as the length of
crack propagation (a − a ) is small compared to the specimen dimensions, because reflecting stress
o
waves will not have had time to reach the crack tip (Example 4.1). In finite specimens where stress
waves reflect back to the propagating crack tip, the dynamic stress intensity must be determined
experimentally or numerically on a case-by-case basis.

EXAMPLE 4.1

Rapid crack propagation initiates in a deeply notched specimen with an initial ligament b o (Figure 4.9).
Assuming the average crack speed = 0.2 c 1 , estimate how far the crack will propagate before it encounters
a reflected longitudinal wave.

Solution: At the moment the crack encounters the first reflected wave, the crack has traveled a distance
∆a, while the wave has traveled 2b o − ∆a. Equating the travel times gives
∆a 2 b − ∆ a
= = o
02 c c
.
1 1
thus,
b
∆a = o
3
Equation (4.16) is valid in this case as long as the crack extension is less than b o /3 and the plastic zone
is small compared to b o .

For an infinite body or short times, Freund [10] showed that the dynamic energy release rate
could be expressed in the following form:
V
t
G() = g( ) G( ) (4.19)
0
where g is a universal function of crack speed that can be approximated by
V
V
g() ≈− c r (4.20)
1

FIGURE 4.9 Propagating crack encountering a reflected
stress wave.