Page 241 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 221 Monday, May 23, 2005 5:47 PM
Fracture Mechanisms in Metals 221
where σ is the effective stress, given by
e
σ 1 σ = σ [ 2 σ− σ + ( 2 ( σ− ) − ) σ + ( ) 2 ] 12 / (5.2)
e
2 1 2 1 3 3 2
σ is the mean stress, defined as
m
σ σ + σ +
σ = 1 2 3 (5.3)
m
3
and σ , σ , and σ are the principal normal stresses. According to the Argon et al. model, the
2
3
1
nucleation strain decreases as the hydrostatic stress increases. That is, void nucleation occurs more
readily in a triaxial tensile stress field, a result that is consistent with experimental observations.
The Beremin research group in France [7] applied the Argon et al. criterion to experimental
data for a carbon manganese steel, but found that the following semiempirical relationship gave
better predictions of void nucleation at MnS inclusions that were elongated in the rolling direction:

σ c σ = m + σ C( e σ − Y S ) (5.4)

where σ is the yield strength and C is a fitting parameter that is approximately 1.6 for longitudinal
YS
loading and 0.6 for loading transverse to the rolling direction.
Goods and Brown [9] have developed a dislocation model for void nucleation at submicron
particles. They estimated that dislocations near the particle elevate the stress at the interface by the
following amount:

ε b
∆ σ d α = 54 . µ 1 r (5.5)

where
α = constant that ranges from 0.14 to 0.33
µ = shear modulus
ε = maximum remote normal strain
1
b = magnitude of Burger’s vector
r = particle radius

The total maximum interface stress is equal to the maximum principal stress plus ∆σ . Void
d
nucleation occurs when the sum of these stresses reaches a critical value:
σ c σ = d + σ ∆ 1 (5.6)

An alternative but equivalent expression can be obtained by separating σ into deviatoric and
1
hydrostatic components:
σ c σ = d + ∆ 1 σ S + m (5.7)

where S is the maximum deviatoric stress.
1
The Goods and Brown dislocation model indicates that the local stress concentration increases
with decreasing particle size; void nucleation is more difficult with larger particles. The continuum