Thixoforming : Semi-solid Metal Processing

172j 6 Modelling the Flow Behaviour of Semi-solid Metal Alloys
rheological measurement. Koke and Modigell [11, 12] introduced an approach to
predict the influence of convective Ostwald ripening on the viscosity evolution during
material preparation in a rotational rheometer. This approach is used here to
eliminate particle growth from experimental results just to reflect rheological
phenomena.
6.1.2
Mathematical Modelling of the Flow Behaviour
The mathematical approach used to describe experimentally discovered flow phenomena
depends on the solid fraction of the sample under investigation. Semi-solid
material used for thixocasting with solid fraction between 40 and 60% is described
using fluid mechanics whereas material used for thixoforging with higher solid
fraction up to 90% requires approaches from solid-state physics [13].
6.1.2.1 Approach from Fluid Mechanics
It was mentioned earlier that in the semi-solid state the globular solid particles are
dispersed in a liquid Newtonian matrix and consequently modelling approaches
known from classical suspension rheology can be applied. In the equilibrium state,
semi-solid alloys are shear-thinning and the most common descriptions are as
follows:
Ostwald de Waele: t ¼ kg_ n Yh ¼ kg_ n 1
Herschel–Bulkley:
t ¼ t0 þ kg_ n Yh ¼ t0
Cross model:
þ kg_ n 1
g_ t ¼ t¥ þ t0 t¥
1 þ kg_ nYh ¼ h ¥ þ h0 h ¥
1 þ kg_ n
The first two models just differ in the yield stress t0, which is a property thoroughly
discussed in the literature. Barnes and Walters [14] doubt the existence of a yield
stress and claim it to be the stress which cannot be measured due to device-related
limits. Contrary to this, Cheng [15] defines a static and a dynamic yield stress, both of
which are influenced by the internal structure of the material. In Section 6.2, recent
investigations on the low-melting tin–lead alloy using creep tests at very low stresses
show the necessity to distinguish between isostructural, dynamic and static yield
stress. In addition to the yield stress, the models contain the consistency factor k and
the flow exponent n. The stress approaches infinity for very low shear rates and zero
for very high shear rates.
In the Cross model, it is assumed that thixotropic fluids effectively behave like a
Newtonian fluid in the case of very low or very high shear rates. The viscosity tends
towards h0 for shear rates ! 0 and towards h¥ for shear rates ! ¥. Atkinson [2]
fitted several experimental data for tin–lead to the Cross model but stated that
experimental values at the transition to the plateau regions are sparse or even
missing.
To account for thixotropic effects in metallic suspensions, the common theories
can be divided into three groups [2, 16]: