Page 164 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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144 Fracture Mechanics: Fundamentals and Applications

3.6.2.2 Effect of Failure Mechanism on the J-Q Locus

The J-Q approach is descriptive but not predictive. That is, Q quantifies the crack-tip constraint,
but it gives no indication as to the effect of constraint on fracture toughness. A J-Q failure locus,
such as Figure 3.37, can be inferred from a series of experiments on a range of geometries. Alter-
natively, a micromechanical failure criterion can be invoked.
Consider, for example, the Ritchie-Knott-Rice (RKR) [32] model for cleavage fracture, which
states that fracture occurs when a critical fracture stress s is exceeded over a characteristic
f
distance r . As an approximation, let us replace the T = 0 reference solution with the HRR field in
c
Equation (3.69):
δQ
σ ij σ ≈ ij HRR σ + ( o ij (3.72)
)
Setting the stress normal to the crack plane equal to s and r = r , and relating the resulting equation
c
f
to the Q = 0 limit leads to
1 1
σ f  J c n  +  J o n  +1 1
n
n 0)
yy
σ o =  α o o Ir   σ ˜ (, +0) Q =   αεσ Ir   σ ˜ (, (3.73)
yy
ε σ 
n
c
o
n
c
o
where J is the critical J value for the Q = 0 small-scale yielding limit. Rearranging gives
o
J c =  Q  σ o    n+1 (3.74)

J o   − 1   σ f  
which is a prediction of the J-Q toughness locus. Equation (3.74) predicts that toughness is highly
sensitive to Q, since the quantity in brackets is raised to the n + 1 power.
The shape of the J-Q locus depends on the failure mechanism. Equation (3.74) refers to stress-
controlled fracture, such as cleavage in metals, but strain-controlled fracture is less sensitive to the
crack-tip constraint. A simple parametric study illustrates the influence of the local failure criterion.
Suppose that fracture occurs when a damage parameter Φ reaches a critical value r ahead of
c
the crack tip, where Φ is given by
 σ  γ
Φ=   σ m o   ε pl 1 −γ (0 ≤ ≤ γ ) 1 (3.75)

where s is the mean (hydrostatic) stress and ε pl is the equivalent plastic strain. When γ = 1,
m
Equation (3.75) corresponds to stress-controlled fracture, similar to the RKR model. The other
limit g = 0 corresponds to strain-controlled failure. By varying g and applying Equation (3.75) to
the finite element results of O’Dowd and Shih [29, 30], we obtain a family of J-Q toughness loci,
which are plotted in Figure 3.39. The J-Q locus for stress-controlled fracture is highly sensitive
to constraint, as expected. For strain-controlled fracture, the locus has a slight negative slope,
indicating that toughness decreases as constraint relaxes. As Q decreases (i.e., becomes more
negative), crack-tip stresses relax, but the plastic strain fields at a given J value increase with
constraint loss. Thus as constraint relaxes, a smaller J is required for failure for a purely strain-
c
controlled mechanism. The predicted J is nearly constant for g = 0.5. Microvoid growth in metals
c
is governed by a combination of plastic strain and hydrostatic stress (see Chapter 5). Consequently,
critical J values for the initiation of ductile crack growth are relatively insensitive to geometry,
as Figure 3.30 indicates.