Page 271 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 251 Monday, May 23, 2005 5:47 PM

Fracture Mechanisms in Metals 251

FIGURE A5.1 Definition of r, θ, and area for a prin-
cipal stress contour.

microcrack density (i.e., the number of critical microcracks per unit volume) should depend
only on the maximum principal stress. 5
ρ ρ = () (A5.1)
σ
1
This quantity must be integrated over the volume ahead of the crack tip. In order to perform this
integration, it is necessary to relate the crack-tip stresses to the volume sampled at each stress level.
Recall Section 3.5, where dimensional analysis indicated that the stresses ahead of the crack tip
in the limit of small-scale yielding are given by

σ  J 
1 = f θ , (A5.2)
σ o  r  σ o  

assuming Young’s modulus is fixed in the material and thus does not need to be included in the
dimensional analysis. Equation (A5.2) can be inverted to solve for the distance ahead of the crack
tip (at a given angle) which corresponds to a particular stress value:

J
θ
θ
r(/σσ , ) = g(/σσ , ) (A5.3)
1 o σ o 1 o
By fixing σ and varying θ from −π to +π, we can construct a contour of constant principal stress,
1
as illustrated in Figure A5.1. The area inside this contour is given by
J 2
A(/σσ ) = h(/σσ ) (A5.4)
1 o σ 2 o 1 o

where h is a dimensionless integration constant:

θ
h σσ ) = 1 ∫ + π g(/ (/σσ , ) θ (A5.5)
d
1 o 1 o
2 − π

5 Although this derivation assumes that the maximum principal stress at a point controls the incremental cleavage probability,
the same basic result can be obtained by inserting any stress component into Equation (A5.1). For example, one might
assume that the tangential stress σ θθ governs cleavage.