Page 137 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 117 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 117

Substituting Equation (3.28) into Equation (3.26) leads to a relationship for the elastic and plastic
components of J:

J =− 1 ∫ P     ∂∆ el   +   ∂∆ p   dP
 
b ∂ 
0    b2 P  ∂  P 

K 2 1 P ∂∆  (3.29)
= I − ∫  p  dP
E 2 0  ∂ 
b′
P
2
where E′ = E for plane stress and E′ = E/(1 − n ) for plane strain, as defined in Chapter 2. Thus
we need only be concerned about plastic displacements because a solution for the elastic component
of J is already available (Table 2.4). If plastic deformation is confined to the ligament between the
crack tips (Figure 3.14(b)), we can assume that b is the only length dimension that influences ∆ .
p
That is a reasonable assumption, provided the panel is deeply notched so that the average stress in
the ligament is substantially higher than the remote stress in the gross cross section. We can define
a new function for ∆ :
p
∆ = bH   P  (3.30)
b 
p
Note that the net-section yielding assumption has eliminated the dependence on the a/b ratio.
Taking a partial derivative with respect to the ligament length gives

P
 ∂∆ p      P
P
H
  = H −′
b
b
b
 ∂  P     b
where H′ denotes the first derivative of the function H. We can solve for H′ by taking a partial
derivative of Equation (3.30) with respect to load:
 ∂∆ p   
P
  = H ′
 P b  
b
∂ 
Therefore
 ∂∆ p  1   ∂∆ p  
  = ∆ p − P    (3.31)
b
∂ 
 ∂  P b    P b  
Substituting Equation (3.31) into Equation (3.29) and integrating by parts gives
K 2 1  ∆ p 
J = I +  2 ∫ Pd∆ − P∆  (3.32)
E 2 b′  0 p p 

Recall that we assumed a unit thickness at the beginning of this derivation. In general, the plastic
term must be divided by the plate thickness; the term in square brackets, which depends on the
load displacement curve, is normalized by the net cross-sectional area of the panel. The J integral
has units of energy/area.
Another example from the Rice et al. article [13] is an edge-cracked plate in bending
(Figure 3.15). In this case they chose to separate displacements along somewhat different lines
from the previous problem. If the plate is subject to a bending moment M, it would displace by
an angle Ω if no crack were present, and an additional amount Ω when the plate is cracked.
nc
c