Page 425 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C009.fm Page 405 Monday, May 23, 2005 3:58 PM

Application to Structures 405

He proposed the following approximation.
hn() ≈ h 1) (9.44)
(
1 1
where h (n) is the geometry constant for a material with a strain hardening exponent of n and h (1)
1
1
is the corresponding constant for a linear material. By substituting h (1) into Equation (9.29) or
1
Equation (9.32), Ainsworth was able to relate the plastic J to the linear elastic stress-intensity factor:
µ K 2  Eε 
ref
J = pl E I   σ ref − 1   (9.45)

where m = 0.75 for plane strain and m = 1.0 for plane stress.
Ainsworth’s work has important ramifications. When applying the EPRI approach, one must
obtain a stress-intensity factor solution to compute the elastic J, and a separate solution for h in
1
order to compute the plastic term. The h constant is a plastic geometry correction factor. However,
1
Equation (9.45) makes it possible to estimate J from an elastic geometry correction factor. The
pl
original EPRI handbook [23] and subsequent additions [24–26] contain h solutions for a relatively
1
small number of configurations, but there are thousands of stress-intensity factor solutions in
handbooks and the literature. Thus Equation (9.45) is not only simpler than Equation (9.29), but
also more widely applicable.
Ainsworth made additional simplifications and modifications to the reference stress model in
order to express it in terms of a FAD. The expression for the FAD based on the reference stress
concept is given in Section 9.4.3.

9.3.3 DUCTILE INSTABILITY ANALYSIS

Section 3.4.1 outlined the theory of stability of J-controlled crack growth. Crack growth is stable as
long as the rate of change in the driving force (J) is less than or equal to the rate of change of the
material resistance (J ). Equation (3.49) and Equation (3.50) defined the tearing modulus, which is
R
a nondimensional representation of the derivatives of both the driving force and the resistance:
E d  j  E dj
T app = σ o 2  da  ∆ T and T = σ 2 o da R (9.46)
R

where ∆ is the remote displacement:
T
∆ T ∆ = + CP (9.47)
M
where C is the system compliance (Figure 2.12). Recall that the value of C influences the relative
M
M
stability of the structure; C = ∞ corresponds to dead loading, which tends to be unstable, while
M
C = 0 represents the other extreme of displacement control, which tends to be more stable. Crack
M
growth is unstable when
T app > T R (9.48)
The rate of change in driving force at a fixed remote displacement is given by 4

∂  
 dJ   J ∂   J ∂   ∂∆    ∆ −1

  da  ∆ T =   a ∂   P −   ∂ P   ∂   P C m +   P  a    (9.49)
∂ 
a  
 
a
4 The distinction between local and remote displacements (∆ and C M P, respectively) is arbitrary as long as all displacements
due to the crack are included in ∆. The local displacement ∆ can contain any portion of the “no crack” elastic displacements
without affecting the term in square brackets in Equation (9.49).