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194 Fracture Mechanics: Fundamentals and Applications

growing crack remains small. At the other extreme, if the crack growth is sufficiently slow that the
creep zone spreads throughout the structure, C* is the appropriate characterizing parameter.
Riedel and Rice [50] analyzed the transition from short-time elastic behavior to long-time
viscous behavior. They assumed a simplified stress-strain rate law that neglects primary creep:

˙ σ
˙ ε = + A σ n (4.40)
E

for uniaxial tension. If a load is suddenly applied and then held constant, a creep zone gradually
develops in an elastic singularity zone, as discussed earlier. Riedel and Rice argued that the stresses
well within the creep zone can be described by

1
 n  +1
σ  Ct()  σ = ˜ n (, ) (4.41)
θ
ij
 AI r ij
n 
where C(t) is a parameter that characterizes the amplitude of the local stress singularity in the creep
zone; C(t) varies with time and is equal to C* in the limit of long-time behavior. If the remote load
is fixed, the stresses in the creep zone relax with time, as creep strain accumulates in the crack-tip
region. For small-scale creep conditions, C(t) decays as 1/t according to the following relationship:
K ( −ν 2 )
2
1
Ct() = I (4.42)
n E ( +1 t )
The approximate size of the creep zone is given by

2
n
2
I
r θ t (, ) = 1  K    n A ( +1 I n E) t  n  −1 r θ n ˜ (, ) (4.43)
( π 
c
2 π E   21 − ν 2 )  c
At θ = 90°, ˜ r c is a maximum and ranges from 0.2 to 0.5, depending on n. As r increases in size,
c
C(t) approaches the steady-state value C*. Riedel and Rice defined a characteristic time for the
transition from short-time to long-time behavior:
2
K 1−( ν 2 )
t = I (4.44a)
1 n C +
( E 1) *
or

J
t = (4.44b)
1 n +
( C 1) *
When significant crack growth occurs over time scales much less than t , the behavior can be
1
characterized by K , while C* is the appropriate parameter when significant crack growth requires
I
times >> t . Based on a finite element analysis, Ehlers and Riedel [51] suggested the following
1
simple formula to interpolate between small-scale creep and extensive creep (short- and long-time
behavior, respectively):

Ct() ≈ C * t  1 + 1  (4.45)
t  