Thixoforming : Semi-solid Metal Processing

_g ¼ ffiffi p
3 _Eeq, where _ 226j 7 A Physical and Micromechanical Model for Semi-solid Behaviour
Eeq is the macroscopic von Mises equivalent strain rate.
As a consequence, the steady-state solid fraction in the active zone is calculated
by [25]
f s
A ¼
f s
f s þ Kdg s
ð1fÞðÞ _g
n
Kag
ð7:3Þ
It is worth noting that Equation 7.3 provides a macroscopic description of the
strain-ratedependenceoftheinternalstructure.
To improve the description of relationships between local deformation mechanisms
and microstructure evolution, we incorporate the fact that the relevant shear rate
is not the overall one but the local one within the solid bonds. We postulate that the
solid bonds break as soon as the local shear reaches a critical value g c. To do so,
Equation 7.2 is rewritten as
_f A
s ¼ Kag f s 1 f A
s 1 f s
_g bonds
f
gc A
s
ð7:4Þ
The shear rate within the solid bonds is naturally given by the micro–macro
modelling (see Equation 7.11 where the medium B is the active zone A) and is
equal to
_g bonds ¼ ffiffiffi p
s
3 _e ð7:5Þ
A eq
During deformation, it is admitted that the kinetics of the agglomeration process are
much lower than the kinetics of the deagglomeration process (Martin et al., 1994).
Consequently, for isothermal thixoforming, we can neglect the agglomeration term
in comparison with the deagglomeration term, so that
_f A
s ¼ 1 f s
_g bonds
f
gc A
s
ð7:6Þ
For non-isothermal processing when the dies are colder than the slug, an increase in
solid fraction related to solidification due to thermal exchanges at the tool–slug
interface is observed. As a result, the solid particles are more agglomerated. In order
to incorporate this agglomeration–solidification phenomenon, we introduce a
phenomenological term in Equation 7.4 accounting for the increase in the solid
fraction within the active zone when temperature decreases. It is written as
_f A
s
A
¼ a 1 f s f sðTÞj_ Tj b
1 f s
_g bonds
f
gc A
s
where a and b are two positive material parameters.
ð7:7Þ
7.3.1.3 Local Behaviours
The description of the RVE requires liquid and solid behaviour representation. The
liquid phase is regarded as a Newtonian fluid. The solid phase is regarded as a
viscoplastic material, the consistency of which may depend on temperature. Solid