Page 229 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 209 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 209

where the subscripts + and − correspond to the upper and lower crack surfaces, respectively. The
following functions satisfy the boundary conditions and lead to integrable strain energy density
and finite displacement at the crack tip:
C −2β C
Fz = Gz = 2 (A4.11)
′′()
′′( )
1 z 2 + ( 2 )1 β z
1 2 2
where C is a constant. Making the substitution z r e= i 1 θ and z r e= i 2 θ leads to the following
1 1 2 2
expressions for the Mode I crack-tip stress fields:
Kt() 1 + β 2   θ 4 β  β r  θ  r 
σ = I 2 β + ( 2 β − 2 )  12 cos 1 − 1 2 cos 2  (A4.12a)
xx
2 πrD t()   1 2  2  r 1 1 + β 2 2  2  r 2  
Kt() 1 + β 2   θ 4 β  β r  θ  r 
σ = I 2  −+ ( β )1 2 cos 1 + 1 2 cos 2  (A4.12b)
yy
2 πrD t()   2  2  r 1 1 + β 2 2  2  r 2  

Kt() β β ( 2 2   θ + )1 r  θ   r 
τ = I 1 2 sin 1 − sin 2 (A4.12c)
t
xy
πr 2 D ()    2  r 1  2  r 2 
where
Dt() = − 4ββ ( 12 + 2 2 )1 β 2

Equation (A4.12) reduces to the quasistatic relationship (Table 2.1) when V = 0.
Craggs [25] and Freund [10] obtained the following relationship between K (t) and the energy
I
release rate for crack propagation at a constant speed:

2
K (1 −ν 2 )
I
G = AV() (A4.13)
E
for plane strain, where

V β
2
AV() = 1
( −ν ) cD t)
2
1
(
2
It can be shown that lim A = 1 , and Equation (A4.13) reduces to the quasistatic result. Equation
V→0
(A4.13) can be derived by substituting the dynamic crack-tip solution (Equation (A4.12) and the
corresponding relationships for strain and displacement) into the generalized contour integral given
by Equation (4.26).
The derivation that led to Equation (A4.12) implies that Equation (A4.13) is a general rela-
tionship that applies to accelerating cracks as well as constant speed cracks.
A4.2 DERIVATION OF THE GENERALIZED ENERGY RELEASE RATE

Equation (4.26) will now be derived. The approach closely follows that of Moran and Shih [11],
who applied a general balance law to derive a variety of contour integrals, including the energy
release rate. Other authors [8–10] have derived equivalent expressions using slightly different
approaches.