Page 215 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 195 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 195

Note the similarity to the transition time concept in dynamic fracture (Section 4.1.1). In both
instances, a transition time characterizes the interaction between two competing phenomena.
4.2.2.1 The C Parameter
t
Unlike K and C*, a direct experimental measurement of C(t) under transient conditions is usually not
I
possible. Consequently, Saxena [52] defined an alternate parameter C which was originally intended
t
as an approximation of C(t). The advantage of C is that it can be measured relatively easily.
t
Saxena began by separating global displacement into instantaneous elastic and time-dependent
creep components:
∆ ∆ = e ∆ + t (4.46)

The creep displacement ∆ increases with time as the creep zone grows. Also, if load is fixed,
t
∆ ˙ t ∆ = ˙ t . The C parameter is defined as the creep component of the power release rate:
t
C =− 1   ∂ a ∂ ∫ ˙ t ∆ Pd∆ ˙ t   (4.47)
t
B


0
˙ t ∆
Note the similarity between Equation (4.36) and Equation (4.47).
For small-scale creep conditions, Saxena defined an effective crack length, analogous to the
Irwin plastic zone correction described in Chapter 2:
a eff =+ β r c (4.48)
a

1
where β ≈ and r is defined at θ = 90°. The displacement due to the creep zone is given by
c
3
dC
∆ t ∆ = ∆ − e = P da r β c (4.49)

where C is the elastic compliance, defined in Chapter 2. Saxena showed that the small-scale creep
limit for C can be expressed as follows:
t
 f ′   a  P∆ ˙


W 

C
() =     a  BW t (4.50)

t ssc

 f   W 

where fa W( / ) is the geometry correction factor for Mode I stress intensity (see Table 2.4):
f  a  = KB W
I
 W  P
and f' is the first derivative of f. Equation (4.50) predicts that (C ) is proportional to K ; thus C t
4
t ssc
I
does not coincide with C(t) in the limit of small-scale creep (Equation (4.42)).
Saxena proposed the following interpolation between small-scale creep and extensive creep:
 ∆ 
˙
C C = t ssc  1 − ()  C + * (4.51)
˙
t
 ∆ t 