Page 198 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 178 Thursday, April 21, 2005 5:38 PM

178 Fracture Mechanics: Fundamentals and Applications

According to Figure 4.4, Equation (4.6) provides a good estimate of J in a high-rate test at
times greater than approximately twice the transition time. It follows that if the fracture initiation
occurs after 2t , the critical value of J obtained from Equation (4.6) is a measure of the fracture
τ
toughness for high-rate loading. If small-scale yielding assumptions apply, the critical J can be
converted to an equivalent K through Equation (3.18).
Ic
Given the difficulties associated with defining a fracture parameter in the presence of inertia
forces and reflected stress waves, it is obviously preferable to apply Equation (4.6) whenever
possible. For a three-point bend specimen with W = 50 mm, the transition time is approximately
300 µs [19]. Thus the quasistatic formula can be applied as long as fracture occurs after ~600 µs.
This requirement is relatively easy to meet in impact tests on ductile materials [15, 16]. For more
brittle materials, the transition-time requirement can be met by decreasing the displacement rate
or the width of the specimen.
The transition-time concept can be applied to other configurations by adjusting the geometry
factor in Equation (4.2). Duffy and Shih [17] have applied this approach to dynamic fracture
toughness measurement in notched round bars. Small round bars have proved to be suitable
for the dynamic testing of brittle materials such as ceramics, where the transition time must
be small.
If the effects of inertia and reflected stress waves can be eliminated, one is left with the rate-
dependent material response. The transition-time approach allows material rate effects to be quantified
independent of inertia effects. High strain rates tend to elevate the flow stress of the material. The
effect of flow stress on fracture toughness depends on the failure mechanism. High strain rates tend
to decrease cleavage resistance, which is stress controlled. Materials whose fracture mechanisms
are strain controlled often see an increase in toughness at high loading rates because more energy
is required to reach a given strain value.
Figure 4.5 shows the fracture toughness data for a structural steel at three loading rates [21].
The critical K values were determined from quasistatic relationships. For a given loading rate, the
I
fracture toughness increases rapidly with temperature at the onset of the ductile-brittle transition.
Note that increasing the loading rate has the effect of shifting the transition to higher temperatures.
Thus, at a constant temperature, fracture toughness is highly sensitive to strain rate.
The effect of the loading rate on the fracture behavior of a structural steel on the upper shelf
of toughness is illustrated in Figure 4.6. In this instance, the strain rate has the opposite effect from
Figure 4.5, because ductile fracture of metals is primarily strain controlled. The J integral at a given
amount of crack extension is elevated by high strain rates.

4.1.2 RAPID CRACK PROPAGATION AND ARREST

When the driving force for crack extension exceeds the material resistance, the structure is unstable,
and rapid crack propagation occurs. Figure 4.7 illustrates a simple case, where the (quasistatic)
energy release rate increases linearly with the crack length and the material resistance is constant.
Since the first law of thermodynamics must be obeyed even by an unstable system, the excess
energy, denoted by the shaded area in Figure 4.7, does not simply disappear, but is converted into
kinetic energy. The magnitude of the kinetic energy dictates the crack speed.
In the quasistatic case, a crack is stable if the driving force is less than or equal to the material
resistance. Similarly, if the energy available for an incremental extension of a rapidly propagating
crack falls below the material resistance, the crack arrests. Figure 4.8 illustrates a simplified scenario
for crack arrest. Suppose that cleavage fracture initiates when K = K . The resistance encountered
Ic
I
by a rapidly propagating cleavage crack is less than for cleavage initiation, because plastic
deformation at the moving crack tip is suppressed by the high local strain rates. If the structure
has a falling driving force curve, it eventually crosses the resistance curve. Arrest does not occur at
this point, however, because the structure contains kinetic energy that can be converted to fracture