Thixoforming : Semi-solid Metal Processing

and liquid are both assumed to be isotropic and incompressible so that their flow
behaviour can be expressed by means of a single modulus, namely the viscosity m,by
the following equations:
s l ij ¼ 2ml _eij with m l ¼ K l
s s ij ¼ 2ms _eeq _eij with m s ¼ K s _eeq
for the Newtonian liquid ð7:8Þ
m 1
for the viscoplastic solid ð7:9Þ
where the superscripts l and s refer to the liquid and solid phase, respectively, sij is the
deviatoric stress tensor, _eij is the deviatoric strain rate tensor, K is the consistency and
m is the strain rate sensitivity parameter.
Actually, one may distinguish the constitutive equation for the entrapped and nonentrapped
liquid and for the solid particles and the solid bonds by using different
values for the consistency and the strain rate sensitivity parameter. In the following
sections, as a first approximation, we assume that the entrapped and non-entrapped
liquids have the same properties. Concerning the solid phase, we assume that the
consistencies of the solid bonds and the solid particles are different. This choice
implies that the strength of the solid bonds may be adjusted to account for the effect
of rest time or temperature on the semi-solid microstructure and mechanical
features.
7.3.2
Homogenized Estimate of the Semi-solid Viscosity: Concentration
and Homogenization Steps
As illustrated in Figure 7.2 and discussed previously, the semi-solid is represented by
a spherical coated inclusion . The inclusion and the coating, also called the active
zone, are both composed of liquid and solid. Therefore, the determination of the
effective properties of the semi-solid requires first the evaluation of the overall
behaviour of the inclusion and of the coating from the behaviour of the solid and
liquid phases. The effective viscosity mSS of the coated inclusion, representing the
semi-solid material, is then determined from the previous results. To do so, we use
the self-consistent approximation. This approach implies that each particle of liquid
(solid or coated inclusion) feels the surroundings (i.e. the neighbouring particles)
through a fictitious homogeneous medium having the effective properties of the
heterogeneous material (Figure 7.4).
7.3.2.1 Step a
The effective viscosities of the inclusion and the active zone are defined by
Equation 7.10 using the same formalism as for local behaviours (Equation 7.9):
SB ¼ 2m B _ EB
7.3 Modelling Semi-solid Behaviourj227
ð7:10Þ
The suffix B is equal to I when it refers to the inclusion and to A for the active zone.
S B and _ EB are the overall stress and strain rate tensors of the medium B. We assume
that the medium B is submitted to boundary conditions _ EB. It is now required to