Page 397 - Mechanical Behavior of Materials

398 Chapter 8 Fracture of Cracked Members

J σ o + σ u
= 2σ Y , where σ Y = (8.48)
a 2
The quantities σ o and σ u are the tensile yield and ultimate strengths of the material.
Lines are then drawn on the J- a curve that are parallel to this blunting line and which intersect
the a axis at 0.15 mm and 1.5 mm. A curve is fitted through the data between these lines, and J Q
is defined as the intersection of this curve and a third parallel line that intersects the a axis at
0.2 mm. Hence, J Q corresponds to a crack extension by tearing of 0.2 mm (0.008 in), which value
does not include the apparent extension due to plastic blunting. Finally, J Q is qualified as a fracture
toughness, J Ic , if the test meets the size requirement

J Q
t,(b − a)> 10 (8.49)
σ Y
An equivalent K Ic can then be calculated from J Ic by Eq. 8.44.
Note that the crack extension at J Q is quite small, so that the material may have considerable
reserve toughness beyond J Ic due to an ability to withstand tearing crack growth. It is thus
sometimes useful to employ the full R-curve and consider this behavior in engineering applications.

8.9.4 Crack-Tip Opening Displacement (CTOD)
The elastic stress field analysis employed for K can also be used to estimate the displacements
separating the crack faces. Then, pursuing logic that partially parallels that for plastic zone sizes,
it is possible to make an estimate of the separation of the crack faces near the tip—that is, of the
CTOD, which is denoted δ. For ductile materials, this estimate is
K 2 J
δ ≈ ≈ (8.50)
Eσ o σ o
where δ is illustrated in Fig. 8.4. In this equation, the yield strength σ o is sometimes replaced by
the higher stress σ Y of Eq. 8.48. Experimentally determined values of δ are also used as the basis
of fracture toughness tests, so that the toughness is expressed as a critical value δ c ,asinASTM
Standard Nos. E1290 and E1820.
Values of δ may be determined for situations of engineering interest, such as a flaw in a pressure
vessel wall, and then compared with δ c . Hence, the CTOD concept also provides an engineering
approach to fracture beyond linear elasticity.

8.10 SUMMARY

Using linear-elastic fracture mechanics (LEFM), the severity of a crack in a component can be
√
characterized by the value of a special variable called the stress intensity factor, K = FS πa,
where S is stress and a is crack length, both consistently defined relative to the dimensionless
quantity F.Use of K depends on the behavior being dominated by linear-elastic deformation, so
that the zone of yielding (plasticity) at the crack tip must be relatively small. Simple equations and
handbooks provide values of F for a wide range of cases of cracked bodies, some examples of
which are given in Figs. 8.12 to 8.14 and also Figs. 8.17 to 8.20. The value of F depends on the