Page 202 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 182 Thursday, April 21, 2005 5:38 PM

182 Fracture Mechanics: Fundamentals and Applications

According to Equation (4.13) and the Roberts and Wells analysis, the crack speed reaches a
limiting value of 0.38c when a >> a . This estimate compares favorably with measured crack speeds
o
o
in metals, which typically range from 0.2 to 0.4c [30].
o
Freund [2–4] performed a more detailed numerical analysis of a dynamically propagating crack
in an infinite body and obtained the following relationship:
V c = r   1 − a  (4.14)
o
a 

where c is the Raleigh (surface) wave speed. For Poisson’s ratio = 0.3, the c /c ratio = 0.57. Thus
r
o
r
the Freund analysis predicts a larger limiting crack speed than the Roberts and Wells analysis. The
limiting crack speed in Equation (4.14) can be argued on physical grounds [26]. For the special
case where w = 0, a propagating crack is merely a disturbance on a free surface that must move
f
at the Raleigh wave velocity. In both Equation (4.13) and Equation (4.14), the limiting velocity is
independent of the fracture energy; thus the maximum crack speed should be c for all w .
f
r
Experimentally observed crack speeds do not usually reach c . Both the simple analysis that
r
resulted in Equation (4.13) and Freund’s more detailed dynamic analysis assumed that the fracture
energy does not depend on crack length or crack speed. The material resistance actually increases
with crack speed, as discussed below. The good agreement between experimental crack velocities
and the Roberts and Wells estimate of 0.38c is largely coincidental.
o
4.1.2.2 Elastodynamic Crack-Tip Parameters
The governing equation for Mode I crack propagation under elastodynamic conditions can be
written as
Kt() = K ( V) (4.15)
I
I
D
where K is the instantaneous stress intensity and K is the material resistance to crack propagation,
I
ID
which depends on the crack velocity. In general, K (t) is not equal to the static stress-intensity
I
factor, as defined in Chapter 2. A number of researchers [8–10, 31–33] have obtained a relationship
for the dynamic stress intensity of the form
Kt() = k ( V K ( ) (4.16)
)
0
I
I
where k is a universal function of crack speed and K (0) is the static stress-intensity factor. The
I
function k(V) = 1.0 when V = 0, and decreases to zero as V approaches the Raleigh wave velocity.
An approximate expression for k was obtained by Rose [34]:
 V 
k() ≈ V   − 1 c   1 − hV (4.17)
r

where h is a function of the elastic wave speeds and can be approximated by

2
h ≈ 2   c   −   c   2 (4.18)
 
2
2
 1
c  c    c  
1 r  1 
where c and c are the longitudinal and shear wave speeds, respectively.
1
2