Page 53 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 33 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 33

In an ideally brittle solid, a crack can be formed merely by breaking atomic bonds; γ reflects
s
the total energy of broken bonds in a unit area. When a crack propagates through a metal,
however, a dislocation motion occurs in the vicinity of the crack tip, resulting in additional
energy dissipation.
Although Irwin and Orowan originally derived Equation (2.21) for metals, it is possible to
generalize the Griffith model to account for any type of energy dissipation:

/
 2 Ew f  12
σ =   πa   (2.22)
f
where w is the fracture energy, which could include plastic, viscoelastic, or viscoplastic effects,
f
depending on the material. The fracture energy can also be influenced by crack meandering and
branching, which increase the surface area. Figure 2.6 illustrates various types of material behavior
and the corresponding fracture energy.
A word of caution is necessary when applying Equation (2.22) to materials that exhibit nonlinear
deformation. The Griffith model, in particular Equation (2.16), applies only to linear elastic material
behavior. Thus, the global behavior of the structure must be elastic. Any nonlinear effects, such as
plasticity, must be confined to a small region near the crack tip. In addition, Equation (2.22) assumes
that w is constant; in many ductile materials, the fracture energy increases with crack growth, as
f
discussed in Section 2.5.

EXAMPLE 2.1

A flat plate made from a brittle material contains a macroscopic through-thickness crack with half
length a 1 and notch tip radius ρ. A sharp penny-shaped microcrack with radius a 2 is located near the
tip of the larger flaw, as illustrated in Figure 2.5. Estimate the minimum size of the microcrack required
to cause failure in the plate when the Griffith equation is satisfied by the global stress and a 1 .

Solution: The nominal stress at failure is obtained by substituting a 1 into Equation (2.19). The stress
in the vicinity of the microcrack can be estimated from Equation (2.11), which is set equal to the Griffith
criterion for the penny-shaped microcrack (Equation 2.20):

 2Eγ  12 / a  πγ  12 /
E
2  S  1 =  s 
21 va 
 π a 1  ρ  ( − 2 ) 2
Solving for a 2 gives

πρ
2
a =
2
2 16 1− v )
(
for ν = 0.3, a 2 = 0.68ρ. Thus the nucleating microcrack must be approximately the size of the
macroscopic crack-tip radius.

This derivation contains a number of simplifying assumptions. The notch-tip stress computed from
Equation (2.11) is assumed to act uniformly ahead of the notch, in the region of the microcrack; the
actual stress would decay away from the notch tip. Also, this derivation neglects free boundary effects
from the tip of the macroscopic notch.