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196 Fracture Mechanics: Fundamentals and Applications

where C* is determined from Equation (4.38) using the total displacement rate. In the limit of
long-time behavior, C*/C = 1.0, but this ratio is less than unity for small-scale creep and transient
t
behavior.
Bassani et al. [53] applied the C parameter to experimental data with various C*/C ratios and
t
t
found that C characterized crack growth rates much better than C* or K . They state that C , when
t
I
t
defined by Equation (4.50) and Equation (4.51), characterizes experimental data better than C(t),
as defined by Riedel’s approximation (Equation (4.45)).
Although C was originally intended as an approximation of C(t), it has become clear that these
t
two parameters are distinct from one another. The C(t) parameter characterizes the stresses ahead
of a stationary crack, while C is related to the rate of expansion of the creep zone. The latter
t
quantity appears to be better suited to materials that experience relatively rapid creep crack growth.
Both parameters approach C* in the limit of steady-state creep.
4.2.2.2 Primary Creep
The analyses introduced so far do not consider primary creep. Referring to Figure 4.17, which depicts
the most general case, the outer ring of the creep zone is in the primary stage of creep. Primary creep
may have an appreciable effect on the crack growth behavior if the size of the primary zone is
significant.
Recently, researchers have begun to develop crack growth analyses that include the effects of
primary creep. One such approach [54] considers a strain-hardening model for the primary creep
deformation, resulting in the following expression for total strain rate:

˙ σ
˙ ε = σ + n A σ + A ( + m ) −p ε p 1 (4.51)
E 1 2
Riedel [54] introduced a new parameter C * which is the primary creep analog to C*. The
h
characteristic time that defines the transition from primary to secondary creep is defined as
+
p 1
 C *  p
t =  h  (4.52)
2 pC*
 ( 1+ ) 
The stresses within the steady-state creep zone are still defined by Equation (4.41), but the inter-
polation scheme for C(t) is modified when primary creep strains are present [54]:
 t p+1 
Ct() ≈ t  +  1  t   p +  1  C* (4.53)
2
t 
 
Equation (4.53) has been applied to experimental data in a limited number of cases. This relationship
appears to give a better description of experimental data than Equation (4.45), where the primary
term is omitted.
Chun-Pok and McDowell [55] have incorporated the effects of primary creep into the estimation
of the C parameter.
t
4.3 VISCOELASTIC FRACTURE MECHANICS

Polymeric materials have seen increasing service in structural applications in recent years. Conse-
quently, the fracture resistance of these materials has become an important consideration. Much of
the fracture mechanics methodology that was developed for metals is not directly transferable to
polymers, however, because the latter behave in a viscoelastic manner.