Thixoforming : Semi-solid Metal Processing

198j 6 Modelling the Flow Behaviour of Semi-solid Metal Alloys
or three-dimensional, etc.) [34]. Here, the one- and two-phase models based on the
Herschel–Bulkley approach will be presented.
One-phase Model The thixotropic model applied is based on an approach initially
suggested by Moore [19] and described in Section 6.1.2.1. The model, which includes
a finite yield stress, is composed of semi-solid alloys at different solid fractions and is
fitted to experimental data [35]. 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.
Two-phase Model This numerical approach is based on a two-phase modified
Herschel–Bulkley model, considering the yield stress, shear thinning and thixotropic
behaviour (see Section 6.1.2.1). It describes the metal alloy as a mixture of a
continuous matrix liquid phase and dispersed solid particles regarded as a pseudo-continuum.
The mathematical model uses the two-phase approach, with two sets
of conservation equations for the bulk and liquid phases, where the relation between
solid and liquid phase is modelled by Darcy s law [9]. The mass and momentum
conservation equations are solved for the bulk and liquid phases. The mass
conservation equation of the bulk phase is
r _ u _ ¼ 0 ð6:16Þ
The velocity u is defined as the sum of the velocities of individual phases (uS, solid
phase velocity; uL, liquid phase velocity) weighted by the corresponding fractions of
solid, fS, and liquid, fL:
u_ ¼ f u S _ S þ f Lu_ L ð6:17Þ
The bulk flow is governed by the general momentum conservation equation in the
classical form:
Du_ r
Dt ¼ r_ p þ r S
_ _ þ rg
_ ð6:18Þ
with the solid-phase density rS, the liquid phase density rL and the isotropic pressure
p; g is the acceleration due to gravity. The non-Newtonian flow behaviour is expressed
by the extra stress tensor S:
S _ ¼ 2h B D _ ð6:19Þ
where D is the deformation rate tensor and hB describes the viscosity of the bulk
phase (Equation 6.1). For the liquid phase, the momentum conservation equation has
the following form:
r L
D f u L _ L
Dt
¼ f L r _ p þ r _ f L S _ L þ r L f L g _ þ q _
ð6:20Þ