Page 166 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 146 Monday, May 23, 2005 5:42 PM

146 Fracture Mechanics: Fundamentals and Applications

where
F = failure probability
s = maximum principal stress at a point
1
V(s ) = cumulative volume sampled where the principal stress is ≥ s
1 1
For a specimen subjected to plane strain conditions along the crack front, V = BA, where B is
the specimen thickness and A is the cumulative area on the x-y plane.

The J Parameter
o
For small-scale yielding, dimensional analysis shows that the principal stress ahead of the crack
tip can be written as

σ  J 
1 = f , θ (3.77)
σ o   σ o r  

Equation (3.77) implies that the crack-tip stress fields depend only on J. When J dominance is lost,
there is a relaxation in triaxiality; the principal stress at a fixed r and q is less than the small-scale
yielding value.
Equation (3.77) can be inverted to solve for the radius corresponding to a given stress and angle:
J
r(/σσ , )θ = g( /σσ , )θ (3.78)
1 o σ o 1 o

Solving for the area inside a specific principal stress contour gives

A(/σσ ) = J 2 h(/σσ ) (3.79)
1 o σ o 2 1 o

where

d
σσ
θ
h σσ ) = 1 ∫ π g(/ 2 (/ , ) θ (3.80)
1 o 1 o
2 − π
2
Thus, for a given stress, the area scales with J in the case of small-scale yielding. Under large-
scale yielding conditions, the test specimen or structure experiences a loss in constraint, and the
area inside a given principal stress contour (at a given J value) is less than predicted from small-
scale yielding:
J 2
A(/σσ ) φ h(/σσ ) (3.81)
=
1 o σ 1 o
0 2
where f is a constraint factor that is ≤1. Let us define an effective J in large-scale yielding that
relates the area inside the principal stress contour to the small-scale yielding case:

J () 2
A(/σσ ) = o h(/σσ ) (3.82)
1 o σ o 2 1 o