Page 250 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 230 Monday, May 23, 2005 5:47 PM

230 Fracture Mechanics: Fundamentals and Applications

2
Values of q = 1.5, q = 1.0, and q = q are typically used for metals. Setting f = 0 recovers
2
1
3
1
the von Mises yield surface for an incompressible material.
In the most recent formulation of the GT model [17], the void growth rate has the following
form:
f ˙ ( − 1 f = ) ˙ +ε kk p Λ ˙ ε eq p (5.13)
where the first term defines the growth rate of the preexisting voids and the second term quantifies
the contribution of new voids that are nucleated with plastic strain. The scaling coefficient Λ, which
is applied to the plastic strain rate of the cell matrix material, is given by

f  1  ε p − ε  2 
Λ= N exp −   eq N   (5.14)
s N 2π  2  s N   

In the above expression, the plastic strain range at nucleation of new voids follows a normal
distribution with a mean value ε , a standard deviation s , and a volume fraction of void nucleating
N
N
particles f . Introducing void nucleation into the GT model results in additional fitting parameters
N
(ε , s , and f ). Moreover, this void nucleation expression is not consistent with the models presented
N
N
N
in Section 5.1.1 because the former implies that nucleation does not depend on hydrostatic stress.
Further research is obviously necessary to sort out these inconsistencies.
In the absence of the void nucleation term in Equation (5.13), the key input parameters to the
GT model are the initial void fraction f and the critical void fraction f . In real materials, voids
o
c
grow very rapidly when the void fraction exceeds 10 to 20%. Equation (5.12) does not adequately
capture the final stage of rapid void growth and coalescence. Consequently, failure is often assumed
when the void fracture reaches a critical value f . This is a reasonable assumption because failure
c
in real materials is very abrupt with little additional macroscopic strain once the void fraction
exceeds f . A typical assumption for carbon steels is f = 0.15. The initial void fraction f is normally
c
o
c
used as a fitting parameter to experimental data.
Tvergaard and Needleman [18] have attempted to model void coalescence by replacing f with
*
an effective void volume fraction f :
f f  ≤ f for c


∗
f =  f f ∗ − (5.15)
 f − u c f ( f − for f f > )
 c f F f − c c c

*
where f , f , and f are fitting parameters. The effect of hydrostatic stress is amplified when f > f ,
u
c
F
c
which accelerates the onset of a plastic instability. As a practical matter, it is probably sufficient
to assume failure when f exceeds f . The marginal benefit of applying Equation (5.15) is offset by
c
*
the need to define the additional fitting parameters f and f . Consequently, Equation (5.15) has
F
u
fallen out of favor in recent years.
Thomason [11] developed a simple limit load model for internal necking between microvoids.
This model states that failure occurs when the net section stress between voids reaches a critical
value σ n(c) . Figure 5.12 illustrates a two-dimensional case, where cylindrical voids are growing in
a material subject to plane strain loading (ε = 0). If the in-plane dimensions of the voids are 2a
3
and 2b, and the spacing between voids is 2d, the row of voids illustrated in Figure 5.12 is stable if
d
σ > σ (5.16a)
+
nc() db 1