Page 129 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 109 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 109

FIGURE 3.8 Nonlinear energy release rate.

Thus if the plate in Figure 3.8 is in load control, J is given by

J =  dU  * (3.14)
 da  P

If the crack advances at a fixed displacement, F = 0, and J is given by

J =−  dU  (3.15)
 da  ∆

According to Figure 3.8, dU* for load control differs from −dU for displacement control by the
amount 1 2 dPd∆, which is vanishingly small compared to dU. Therefore, J for load control is equal
to J for displacement control. Recall that we obtained this same result for G in Section 2.4.
By invoking the definitions for U and U*, we can express J in terms of load and displacement:
 ∂ P 
J =   ∫ 0 ∆ dP   P
a ∂

P  ∆ (3.16)
∂ 
=  ∫   dP
0 a ∂  P
or
 ∂ ∆ 
∆
J =−   ∫ Pd  
a ∂
0
∆
(3.17)
∆  ∂ P
=−  ∫   d∆
0 a ∂  ∆