Page 206 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 186 Thursday, April 21, 2005 5:38 PM

186 Fracture Mechanics: Fundamentals and Applications

where V is the limiting crack speed in the material and m is an experimentally determined constant.
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As Figure 4.11(b) illustrates, K increases and V decreases with increasing material toughness.
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The trends in Figure 4.11(a) and Figure 4.11(b) have not only been observed experimentally, but
have also been obtained by numerical simulation [40, 41]. The upturn in propagation toughness at
high crack speeds is apparently caused by local inertia forces in the plastic zone.
4.1.2.4 Crack Arrest

Equation (4.15) defines the conditions for rapid crack advance. If, however, K (t) falls below the
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minimum K value for a finite length of time, propagation cannot continue, and the crack arrests.
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There are a number of situations that might lead to crack arrest. Figure 4.8 illustrates one possibility:
If the driving force decreases with crack extension, it may eventually be less than the material
resistance. Arrest is also possible when the material resistance increases with crack extension. For
example, a crack that initiates in a brittle region of a structure, such as a weld, may arrest when it
reaches a material with higher toughness. A temperature gradient in a material that exhibits a
ductile-brittle transition is another case where the toughness can increase with position: A crack
may initiate in a cold region of the structure and arrest when it encounters warmer material with
a higher toughness. An example of this latter scenario is a pressurized thermal shock event in a
nuclear pressure vessel [42].
In many instances, it is not possible to guarantee with absolute certainty that an unstable fracture
will not initiate in a structure. Transient loads, for example, may occur unexpectedly. In such
instances crack arrest can be the second line of defense. Thus, the crack arrest toughness K is an
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important material property.
Based on Equation (4.16), one can argue that K (t) at arrest is equivalent to the quasistatic
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value, since V = 0. Thus it should be possible to infer K from a quasistatic calculation based on
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the load and crack length at arrest. This quasistatic approach to arrest is actually quite common,
and is acceptable in many practical situations. Chapter 7 describes a standardized test method for
measuring crack-arrest toughness that is based on quasistatic assumptions.
However, the quasistatic arrest approach must be used with caution. Recall that Equation (4.16) is
valid only for infinite structures or short crack jumps, where reflected stress waves do not have sufficient
time to return to the crack tip. When reflected stress wave effects are significant, Equation (4.16) is no
longer valid, and a quasistatic analysis tends to give misleading estimates of the arrest toughness.
Quasistatic estimates of arrest toughness are sometimes given the designation K ; for short crack jumps
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K = K .
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The effect of stress waves on the apparent arrest toughness K was demonstrated dramatically
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by Kalthoff et al. [43], who performed dynamic propagation and arrest experiments on wedge-loaded
double cantilever beam (DCB) specimens. Recall from Example 2.3 that the DCB specimen exhibits
a falling driving-force curve in displacement control. Kalthoff et al. varied the K at initiation by
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varying the notch-root radius. When the crack was sharp, fracture initiated slightly above K and
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arrested after a short crack jump; the length of crack jump increased with the notch-tip radius.
Figure 4.13 is a plot of the Kalthoff et al. results. For the shortest crack jump, the true arrest
toughness and the apparent quasistatic value coincide, as expected. As the length of crack jump
increases, the discrepancy between the true arrest and the quasistatic estimate increases, with K > K .
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Note that K appears to be a material constant but K varies with the length of crack propagation.
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Also note that the dynamic stress intensity during crack growth is considerably different from the
quasistatic estimate of K . Kobayashi et al. [44] obtained similar results.
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A short time after arrest, the applied stress intensity reaches K , the quasistatic value.
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Figure 4.14 shows the variation of K after arrest in one of the Kalthoff et al. experiments. When
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the crack arrests, K = K , which is greater than K . Figure 4.14 shows that the specimen “rings down”
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to K after ~2000 µs. The quasistatic value, however, is not indicative of the true material-arrest
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properties.