Page 249 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Fracture Mechanisms in Metals 229

FIGURE 5.11 The continuum assumption for modeling a porous medium. The material is assumed to be
homogeneous, and the effect of the voids is averaged through the solid.

where R R = R + ( R + )/3 and ε p is the equivalent (von Mises) plastic strain. Subsequent inves-
1 2 3 eq
tigators found that Equation (5.11) could be approximately modified for strain hardening by
replacing the yield strength with σ , the effective stress [20].
e
Since the Rice and Tracey model is based on a single void, it does not take account of
interactions between voids, nor does it predict ultimate failure. A separate failure criterion must be
applied to characterize microvoid coalescence. For example, one might assume that fracture occurs
when the nominal void radius reaches a critical value.
A model originally developed by Gurson [13] and later modified by Tvergaard [15, 16] analyzes
the plastic flow in a porous medium by assuming that the material behaves as a continuum. Voids
appear in the model indirectly through their influence on the global flow behavior. The effect of
the voids is averaged through the material, which is assumed to be continuous and homogeneous
(Figure 5.11). The main difference between the Gurson-Tvergaard (GT) model and classical plas-
ticity is that the yield surface in the former exhibits a hydrostatic stress dependence, while classical
plasticity assumes that yielding is independent of hydrostatic stress. This modification to conven-
tional plasticity theory has the effect of introducing a strain-softening term.
Unlike the Rice and Tracey model, the GT model contains a failure criterion. Ductile fracture
is assumed to occur as the result of a plastic instability that produces a band of localized deformation.
Such an instability occurs more readily in a GT material because of the strain softening induced
by hydrostatic stress. However, because the model does not consider discrete voids, it is unable to
predict the necking instability between voids.
The model derives from a rigid-plastic limit analysis of a solid having a smeared volume fraction
f of voids approximated by a homogeneous spherical body containing a spherical void. The yield
surface and flow potential g is given by

g e , m , , f(σσ   σ e  2 + 2 q ) = 1 f cosh  3 q σ  − ( + 1 qf ) = 2 0 (5.12)
m
σ
2
σ 
σ 
3
 2
where
σ = macroscopic von Mises stress
e
σ = macroscopic mean stress
m
σ = flow stress for the matrix material of the cell
f = current void fraction