Page 146 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 126 Monday, May 23, 2005 5:42 PM

126 Fracture Mechanics: Fundamentals and Applications

Therefore, the relevant partial derivatives are given by

∂ 
 J 2 Pa
2
  =
∂ 
 a P BEI
 J 2 Pa 2
∂ 
  =
∂ 
 P a BEI
 ∂∆  2 Pa 2
  =
∂ 
 a P EI
 ∂∆  2 a 3
  =
∂ 
 P a 3 EI
Substituting the above relationships into Equation (3.50) and Equation (3.52) gives
a 
−1
a  
2
3
3
2
T app = σ o Pa    1 − 2 EI   C + M 2 3 EI    
 
BI
2


As discussed in Section 2.5, the point of instability in a material with a rising R curve depends
on the size and geometry of the cracked structure; a critical value of J at instability is not a material
property if J increases with crack growth. It is usually assumed that the R curve, including the J Ic
value, is a material property, independent of the configuration. This is a reasonable assumption,
within certain limitations.
3.4.2 COMPUTING J FOR A GROWING CRACK
The geometry dependence of a J resistance curve is influenced by the way in which J is calculated.
The equations derived in Section 3.2.5 are based on the pseudo energy release rate definition of J
and are valid only for a stationary crack. There are various ways to compute J for a growing crack,
including the deformation J and the far-field J, which are described below. The former method is
typically used to obtain experimental J resistance curves.
Figure 3.22 illustrates the load-displacement behavior in a specimen with a growing crack.
Recall that the J integral is based on a deformation plasticity (or nonlinear elastic) assumption for
material behavior. Consider point A on the load-displacement curve in Figure 3.22. The crack has
grown to a length a from an initial length a . The cross-hatched area represents the energy that
o
1
would be released if the material were elastic. In an elastic-plastic material, only the elastic portion
of this energy is released; the remainder is dissipated in a plastic wake that forms behind the
growing crack (see Figure 2.6(b) and Figure 3.25).
In an elastic material, all quantities, including strain energy, are independent of the loading
history. The energy absorbed during crack growth in an elastic-plastic material, however, exhibits
a history dependence. The dashed curve in Figure 3.22 represents the load-displacement behavior
when the crack size is fixed at a . The area under this curve is the strain energy in an elastic
1
material; this energy depends only on the current load and crack length:

 ∆ 
U D U = D P a(, ) ∫ P = d  ∆   (3.55)
0
=
aa 1