Page 176 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 156 Monday, May 23, 2005 5:42 PM

156 Fracture Mechanics: Fundamentals and Applications

Recall the J-CTOD relationship (Equation (3.44)) derived from the strip-yield model. Let us
define an effective stress intensity for elastic-plastic conditions in terms of the J integral:

K eff J E ≡ (A3.11)

Combining Equation (3.44), Equation (A3.10) and Equation (A3.11) gives

  πσ   12 /
K eff = σ YS π a  8 ln sec    (A3.12)
  π 2  σ YS   
2
which is the strip-yield plastic zone correction given in Equation (2.81) and plotted in Figure 2.33.
Thus the strip-yield correction to K is equivalent to a J-based approach for a nonhardening material
I
in plane stress.

A3.2 THE J CONTOUR INTEGRAL
Rice [4] presented a mathematical proof of the path independence of the J contour integral. He
*
began by evaluating J along a closed contour Γ (Figure A3.2):

J* = *  ∫   wdy − T i u ∂ i ds  (A3.13)
Γ x ∂ 

where the various terms in this expression are defined in Section 3.2.2. Rice then invoked the
divergence theorem to convert Equation (A3.13) into an area integral:

J* = ∫   ∂ w − ∂  σ ij u ∂    dx dy (A3.14)
i
  x A* ∂ ∂ x j  x ∂   

FIGURE A3.2 Closed contour Γ* in a two-dimensional solid.