Page 226 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 206 Thursday, April 21, 2005 5:38 PM

206 Fracture Mechanics: Fundamentals and Applications

for small-scale nonlinearity. Equation (4.93) and Equation (4.94) provide a reasonable description
of the transition to nonlinear behavior. Schapery defined a transition time by setting J = J in
n
v
Equation (4.93):
J
J = L (4.95a)
n
ft()
τ
or

 J 
t f = −1  L  (4.95b)
τ
 J 
n
For the special case of linear behavior followed by viscous creep, Equation (4.95b) becomes

J
t = L (4.96)
τ
( n C + ) 1 *

which is identical to the transition time defined by Riedel and Rice [50].

APPENDIX 4: DYNAMIC FRACTURE ANALYSIS
A4.1 ELASTODYNAMIC CRACK TIP FIELDS

Rice [31], Sih [35], and Irwin [62] each derived expressions for the stresses ahead of a crack
propagating at a constant speed. They found that the moving crack retained the 1/ r singularity,
but that the angular dependence of the stresses, strains, and displacements depends on crack speed.
Freund and Clifton [32] and Nilsson [33] later showed that the solution for a constant speed crack
was valid in general; the near-tip quantities depend only on instantaneous crack speed. The following
derivation presents the more general case, where the crack speed is allowed to vary.
For dynamic problems, the equations of equilibrium are replaced by the equations of motion,
which, in the absence of body forces, are given by

∂σ ji
∂x j = ρu ˙˙ i (A4.1)

where x denotes the orthogonal coordinates and each dot indicates a time derivative. For quasistatic
j
problems, the term on the right side of Equation (A4.1) vanishes. For a linear elastic material, it
is possible to write the equations of motion in terms of displacements and elastic constants by
invoking the strain-displacement and stress-strain relationships:

2
∂ u ∂ u
2
µ i λ + ( µ + j = u ˙˙ (A4.2)
ρ )
∂x 2 j ∂∂xx j i
i
where µ and λ are the Lame′ constants; µ is the shear modulus and

λ = 2 µν
−
12 ν