Page 211 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 191 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 191

Most analytical treatments of creep crack growth assume limiting cases, where one or more
of these regimes are not present or are confined to a small portion of the component. If, for
example, the component is predominantly elastic, and the creep zone is confined to a small region
near the crack tip, the crack growth can be characterized by the stress-intensity factor. In the other
extreme, when the component deforms globally in steady-state creep, elastic strains and tertiary
creep can be disregarded. A parameter that applies to the latter case is described below, followed
by a brief discussion of approaches that consider the transition from elastic to steady-state creep
behavior.
4.2.1 THE C * INTEGRAL

A formal fracture mechanics approach to creep crack growth was developed soon after the J
integral was established as an elastic-plastic fracture parameter. Landes and Begley [45], Ohji
et al. [46], and Nikbin et al. [47] independently proposed what became known as the C* integral
to characterize crack growth in a material undergoing steady-state creep. They applied Hoff’s
analogy [48], which states that if there exists a nonlinear elastic body that obeys the relationship
ε = f(σ ) and a viscous body that is characterized by ˙ ε ij = f (σ ), where the function of stress is
ij
ij
ij
the same for both, then both bodies develop identical stress distributions when the same load is
applied. Hoff’s analogy can be applied to steady-state creep, since the creep rate is a function
only of the applied stress.
The C* integral is defined by replacing strains with strain rates, and displacements with displacement
rates in the J contour integral:

 u ∂ ˙ 
n
C* =  ∫  ˙ σ ij j i ds (4.30)
wdy −
Γ x ∂ 
where ˙ w is the stress work rate (power) density, defined as

∫ σε ij (4.31)
˙ w ˙ ε kl ˙ d =
ij
0

Hoff’s analogy implies that the C * integral is path independent, because J is path independent.
Also, if secondary creep follows a power law:

˙ ε A σ = n (4.32)
ij ij

where A and n are material constants, then it is possible to define an HRR-type singularity for
stresses and strain rates near the crack tip:

1
 C * n  +1
σ   σ = ˜ n (, ) (4.33a)
θ
ij
n 
 AI r ij
and
n
 C* n  +1
˙ ε ε = ˜ (, ) θ n (4.33b)
ij  AI r  ij
 n 