Thixoforming : Semi-solid Metal Processing

. Introduction of a scalar parameter (e.g. k) to describe the degree of agglomeration
of solid particles. The limits are k ¼ 0 for a completely deagglomerated structure
and k ¼ 1 for a completely agglomerated structure.
. Direct description of the temporal change of the microstructure by taking into
account the number of bonds between solid particles [17].
. Use viscosity–time data to base a theory on Ref. [18].
In the present work, the chosen constitutive equation is a combination of the
Herschel–Bulkley law and a thixotropic model, and belongs to the first group in the
above list. It is based on an approach initially suggested by Moore [19]. The model,
which includes a finite yield stress, is composed for semi-solid alloys at different solid
fractions and is fitted to experimental data (see below and e.g. Ref. [11]). The shear
stress is defined as a function of shear rate and a time-dependent structural
parameter, describing the structural influence on the flow behaviour, such as the
current state of agglomeration. Furthermore, the shear stress is assumed to grow
exponentially with increasing solid fraction. The equation of state is assumed to be
h ¼ t
g_ ¼ t0ð f SÞ g_
þ k ð f S Þ _
g mð f S Þ 1 k þ h L ð6:1Þ
In the strict sense, hL does not represent the liquid viscosity. If k equals zero, the
suspension consists of non-interacting particles suspended in the liquid phase. As a
consequence, hL rather describes the viscosity of this suspension, although its
contribution to the total stress is negligible.
The yield stress t0, the consistency factor k and the exponent m are dependent on
the solid fraction. The generalized shear rate g_ is defined on the basis of the second
invariant of the deformation rate tensor:
g_ ¼ ffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2
2IID ¼ 2tr D_ with D_ ¼ 1
2 r_ u T
_ þ r_ u_ ð6:2Þ
The structural changes in time follow the deformation history of the material. In
the special case of the step change of shear rate experiment (see Section 6.1.5.1), the
structural parameter will approach an equilibrium value ke corresponding to the
current shear rate. It is assumed that the equilibrium flow curve is also a Herschel–
Bulkley curve but with another flow exponent n:
h e ¼ t0ð f S Þ
_
g
þ k ð f SÞg_ mð f SÞ 1
6.1 Empirical Analysis of the Flow Behaviourj173
ke ¼ t0ð f S Þ
_
g
þ kð f SÞg_ n 1
ð6:3Þ
By rearranging Equation 6.3, we obtain the equilibrium structural parameter in the
form
keðg_ Þ¼ t0ð f SÞþkð f SÞg_ n
t0ð f SÞþk ð f SÞg_ mð f SÞ ; 0