Page 258 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 238 Monday, May 23, 2005 5:47 PM

238 Fracture Mechanics: Fundamentals and Applications

where C , in this case, is the carbide thickness, and τ and k are the friction stress and pile-up
y
o
i
constant, respectively, as defined in the Hall-Petch equation:
τ y τ = i + kd −12 (5.19)
/
y
where τ is the yield strength in shear. The second term on the left side of Equation (5.18) contains
y
the dislocation contribution to cleavage initiation. If this term is removed, Equation (5.18) reduces
to the Griffith relationship for a grain boundary microcrack.
Figure 5.22 shows SEM fractographs that give examples of cleavage initiation from a grain
boundary carbide (a) and an inclusion at the interior of a grain (b). In both cases, the fracture origin
was located by following river patterns on the fracture surface.
Susceptibility to cleavage fracture is enhanced by almost any factor that increases the yield
strength, such as low temperature, a triaxial stress state, radiation damage, high strain rate, and
strain aging. Grain size refinement not only increases the yield strength but also increases σ . There
f.
are a number of reasons for the grain size effect. In mild steels, a decrease in grain size implies
an increase in the grain boundary area, which leads to smaller grain boundary carbides and an
increase in σ . In fine-grained steels, the critical event may be the propagation of the microcrack
f
across the first grain boundary it encounters. In such cases, the Griffith model implies the following
expression for fracture stress:
E
 πγ gb  / 12
σ =   (5.20)
f
2
)
 (1 − ν d
where γ is the plastic work per unit area required to propagate into the adjoining grains. Since
gb
there tends to be a high degree of mismatch between grains in a polycrystalline material, γ > γ .
gb
p
Equation (5.20) assumes an equiaxed grain structure. For martensitic and bainitic microstructures,
Dolby and Knott [28] derived a modified expression for σ based on the packet diameter.
f
In some cases cleavage nucleates, but total fracture of the specimen or structure does not occur.
Figure 5.23 illustrates three examples of unsuccessful cleavage events. Part (a) shows a microcrack
that has arrested at the particle–matrix interface. The particle cracks due to strain in the matrix,
but the crack is unable to propagate because the applied stress is less than the required fracture
stress. This microcrack does not reinitiate because subsequent deformation and dislocation motion
in the matrix causes the crack to blunt. Microcracks must remain sharp in order for the stress on
the atomic level to exceed the cohesive strength of the material. If a microcrack in a particle
propagates into the ferrite matrix, it may arrest at the grain boundary, as illustrated in Figure 5.23(b).
This corresponds to a case where Equation (5.20) governs cleavage. Even if a crack successfully
propagates into the surrounding grains, it may still arrest if there is a steep stress gradient ahead
of the macroscopic crack (Figure 5.23(c)). This tends to occur at low applied K values. Locally,
I
the stress is sufficient to satisfy Equation (5.18) and Equation (5.20) but there is insufficient global
driving force to continue the crack propagation. Figure 5.24 shows an example of arrested cleavage
cracks in front of a macroscopic crack in a spherodized 1008 steel [29].

5.2.3 MATHEMATICAL MODELS OF CLEAVAGE FRACTURE TOUGHNESS

A difficulty emerges when trying to predict fracture toughness from Equation (5.18) to Equation (5.20).
The maximum stress ahead of a macroscopic crack occurs at approximately 2δ from the crack tip,
but the absolute value of this stress is constant in small-scale yielding (Figure 5.14); the distance
from the crack tip at which this stress occurs increases with increasing K, J, and δ. Thus, if attaining
a critical fracture stress were a sufficient condition for cleavage fracture, the material might fail
upon the application of an infinitesimal load, because the stresses would be high near the crack
tip. Since ferritic materials have finite toughness, the attainment of a critical stress ahead of the
crack tip is apparently necessary but not sufficient.