Page 138 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 118 Monday, May 23, 2005 5:42 PM

118 Fracture Mechanics: Fundamentals and Applications

FIGURE 3.15 Edge-cracked plate in pure bending.

Thus the total angular displacement can be written as
Ω Ω = nc Ω + c (3.33)

If the crack is deep, Ω >> Ω . The energy absorbed by the plate is given by
nc
c
U ∫ Ω M = d Ω (3.34)
0
When we differentiate U with respect to the crack area in order to determine J, only Ω contributes
c
to the energy release rate because Ω is not a function of crack size, by definition. By analogy
nc
with Equation (3.16), J for the cracked plate in bending can be written as
M  ∂Ω  M  ∂Ω 
J =  ∫  c  dM =−  ∫  c  dM (3.35)
0 a ∂  M 0 b ∂  M

If the material properties are fixed, dimensional analysis leads to

M 
Ω = F   b  (3.36)
c
2
assuming the ligament length is the only relevant length dimension, which is reasonable if the crack
is deep. When Equation (3.36) is differentiated with respect to b and inserted into Equation (3.35),
the resulting expression for J is as follows:

J = 2 ∫ Ω c MdΩ (3.37)
b 0 c

The decision to separate Ω into ‘‘crack’’ and ‘‘no-crack’’ components was somewhat arbitrary.
The angular displacement could have been divided into elastic and plastic components as in the
previous example. If the crack is relatively deep, Ω should be entirely elastic, while Ω may
nc
c
contain both elastic and plastic contributions. Therefore, Equation (3.37) can be written as
J = 2   ∫ Ω c( el) Md Ω cel + ∫ Ω p Md Ω p 
b  0 () 0 