Page 181 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 161 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 161

FIGURE A3.5 Two circular contours around the crack tip. r 1 is in the zone dominated by the elastic singularity,
while r 2 is in the plastic zone where the leading term of the Hutchinson asymptotic expansion dominates.

Solving for the integrand in the J integral at r leads to
2
n
w = ασεκ n r ( n+ +1 s − 2) σ ˜ e n+ 1)( 1 (A3.32a)
o o
n +1
and

u ∂
′ −
T i x ∂ i = oo n r n+ s +1 ( − 1)( 2) {sin θ rr u θ u ˜ ) rθ uσ [ ˜ (˜ r +σ ˜ (˜ u − ασεκ θ ′ ˜ )]
r
+ − cos [ ( ns 2 +θ ][ ˜ ˜ u + ) 1 σ ˜ ˜ ]}u (A3.32b)
σ
θθ θ
rr r
where ˜ u r and ˜ u θ are dimensionless displacements, defined by
u nn s−+( ) 21 u ˜ θ ()
r = αεκ
r
r
o
(A3.33)
u θ αεκ nn s−+( ) 21 u ˜ θ θ ()
r =
o
u and u can be derived from the strain-displacement relationships. Evaluating the J integral at r 2
r
q
gives
J o o n r = αεσκ 2 n+ s +1 ( − 1)( 2 1 I n (A3.34)
+
)
where I is an integration constant, given by
n
n
+π
I = ∫ −π  n +1 ˜ σ n+1 cos θ − σ rr u [ ˜ (˜ θ u −{sinθ ˜ ) rθ u ˜ (˜ r u +σ θ ′ ˜ )]
′ −

e
n

+ − cos [ ( ns 2 +θ ]( ˜ ˜ + ) 1 σ ˜ ˜ )}u  d (A3.35)
σ
 θ u
θθ θ
rr r
