Page 209 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 189 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 189

where F is the energy flux into the area bounded by Γ. The generalized energy release rate, including
inertia effects, is given by

J → ∫   w = T lim ) d y ( − σ n + u ∂ i ds   (4.26)
Γ 0 Γ  ij j x ∂ 

where w and T are the stress work and kinetic energy densities defined as

d =
w ∫ ij ε σε ij (4.27)
ij
0
and

T = 1 ρ u ∂ i u ∂ i (4.28)
2 t ∂ t ∂

Equation (4.26) has been published in a variety of forms by several researchers [8–12]. Appendix 4.2
gives a derivation of this relationship.
Equation (4.26) is valid for time-dependent as well as history-dependent material behavior.
When evaluating J for a time-dependent material, it may be convenient to express w in the following
form:

w ∫ t σε ˙ d = t (4.29)
ij ij
t o
where ˙ ε ij is the strain rate.
Unlike the conventional J integral, the contour in Equation (4.26) cannot be chosen arbitrarily.
Consider, for example, a dynamically loaded cracked body with stress waves reflecting off free
surfaces. If the integral in Equation (4.26) were computed at two arbitrary contours a finite distance
from the crack tip and a stress wave passed through one contour but not the other, the values of
these integrals would normally be different for the two contours. Thus, the generalized J integral
is not path independent, except in the immediate vicinity of the crack tip. If, however, T = 0 at all
points in the body, the integrand in Equation (4.26) reduces to the form of the original J integral.
In the latter case, the path-independent property of J is restored if w displays the property of an
elastic potential (see Appendix 4.2).
The form of Equation (4.26) is not very convenient for numerical calculations, since it is
extremely difficult to obtain adequate numerical precision from a contour integration very close to
the crack tip. Fortunately, Equation (4.26) can be expressed in a variety of other forms that are
more conducive to numerical analysis. The energy release rate can also be generalized to three
dimensions. The results in Figure 4.3 and Figure 4.4 are obtained from a finite element analysis
that utilized alternate forms of Equation (4.26). Chapter 12 discusses the numerical calculations of
J for both quasistatic and dynamic loading.

4.2 CREEP CRACK GROWTH

Components that operate at high temperatures relative to the melting point of the material may fail
by the slow and stable extension of a macroscopic crack. Traditional approaches to design in the
creep regime apply only when creep and material damage are uniformly distributed. Time-dependent
fracture mechanics approaches are required when creep failure is controlled by a dominant crack
in the structure.