Page 225 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 205 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 205

where f(t) is an as yet unspecified aging function, and the complementary strain energy density is
defined by

w c ∫ eL d = ε σ ij
i
j

For uniaxial deformation, Equation (4.89) reduces to

 σ  n σ
ε eL = f   + (4.90)
 σ ref  E R

Comparing Equation (4.85) and Equation (4.90), it can be seen that

f = Dt() if ε eL ≡ ε t() for constant stress creep.
Dt() ED t()
L
R
L
The latter relationship for pseudo-strain agrees with the conventional definition in the limit of linear
behavior.
Let us now consider the case where the inner and outer singularities, Equation (4.87) and
Equation (4.88), exist simultaneously. For the outer singularity, the second term in Equation (4.90)
dominates, the stresses are given by Equation (4.88), and J is related to K as follows:
I
L
2
K 1( −ν 2 )
J = I E R (4.91)
L
Closer to the crack tip, the stresses are characterized by J through Equation (4.87), but J is not
v
L
necessarily equal to J , because f appears in the first term of the modified constitutive relationship
v
(Equation (4.90)), but not in Equation (4.86). These two definitions of J coincide if σ in
ref
Equation (4.90) is replaced with σ ref f 1 n . Thus, the near-tip singularity in terms of J is given by
L
1
 J n  +1
σ ij σ = ref  L  σ ˜ ij θ n (, ) (4.92)
I
r
 σ f ref n 
therefore,
J
J = f L (4.93)
v

Schapery showed that f = 1 in the limit of purely linear behavior; thus J is the limiting value of
L
J when the nonlinear zone is negligible. The function f is indicative of the extent of nonlinearity.
v
In most cases, f increases with time, until J reaches J , the limiting value when the specimen is
v
n
dominated by nonlinear viscoelasticity. Schapery also confirmed that
f = Dt() (4.94)
Dt()
L