Page 141 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 121 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 121

where d = 2u (X = r). Since the strip-yield model assumes s = s within the plastic zone, the
YS
yy
y
J-CTOD relationship is given by
J = σδ (3.44)
YS
Note the similarity between Equation (3.44) and Equation (3.7). The latter was derived from the
strip-yield model by neglecting the higher-order terms in a series expansion; no such assumption
was necessary to derive Equation (3.44). Thus the strip-yield model, which assumes plane stress
conditions and a nonhardening material, predicts that m = 1 for both linear elastic and elastic-
plastic conditions.
Shih [14] provided further evidence that a unique J-CTOD relationship applies well beyond
the validity limits of LEFM. He evaluated the displacements at the crack tip implied by the HRR
solution and related the displacement at the crack tip to J and flow properties. According to the
HRR solution, the displacements near the crack tip are as follows:

n
ασ  EJ  n+1
o
u = E  ασ on  ru θ (, n) (3.45)
˜

i
i
Ir 
2
where ˜ u i is a dimensionless function of q and n, analogous to ˜ σ ij and ε ij (Equation (3.24)). Shih
˜
[14] invoked the 90° intercept definition of CTOD, as illustrated in Figure 3.4(b). This 90° intercept
construction is examined further in Figure 3.17. The CTOD is obtained by evaluating u and u at
y
x
r = r* and q = p:
δ = π −
π = ur ( *, )
x
2 y r * u r ( *, ) (3.46)
Substituting Equation (3.46) into Equation (3.45) and solving for r* gives
 ασ o  n / 1 n+1 J
r* = u π n {˜ (, ) + u π n ˜ (, )} n (3.47)
 E  x y σ on
I
Setting d = 2u (r*, π) leads to
y
dJ
δ = n (3.48)
σ o

FIGURE 3.17 Estimation of CTOD from a 90° inter-
cept construction and HRR displacements.