Thixoforming : Semi-solid Metal Processing

displacement boundary conditions or stress boundary conditions, one may envision
that the effects of the applied loads (through the boundary conditions) and the
interaction with other heterogeneities can be accounted for by assuming that the
coating inclusion is placed within a homogeneous matrix having the same properties
as the real material, namely having the effective properties [9]. As a consequence,
the RVE is defined as a volume containing a coated inclusion embedded in the
homogenized medium.
7.3.1.2 Internal Variable
Deformation by shearing breaks the solid bonds because solid bonds mostly carry
the deformation. This leads to a deagglomeration process and a release of some
liquid [16]. Consequently, the resulting bimodal liquid–solid distribution is different
from the initial one (Figure 7.3). To capture the evolution of the microstructure
and the bimodal distribution while the strain rate changes, the solid volume
fraction of the active zone, f s
A , is introduced as an internal variable. This is related
to the volume fraction of solid bonds and decreases with the strain rate. This
internal variable is very similar to the structural parameters introduced first
by Kumar et al. [14, 17]; (Martin et al., 1994) and more recently by many others
[18–22]; (Modigell et al., 2001) [5]. It is equal to one when all particles are
connected to each other and zero when all particles are separated. More accurately,
percolation theories [23, 24] predict that under a critical volume fraction of solid,
the solid phase appears as isolated agglomerates so that f s
A is equal to zero (no
solid bonds). Above a critical volume fraction f c , a macroscopically connected
network of solid is formed and f s
A is given by a differential equation accounting for
competing kinetics for agglomeration and deagglomeration (Equation 7.2). Following
the work of Kumar et al. [14, 17]; (Martin et al., 1994), the evolution law for f s
A is
given by
_f s
A ¼ Kag f s 1 f s
A Kdg 1 f s
ð Þf s
A _g ðÞn
7.3 Modelling Semi-solid Behaviourj225
ð7:2Þ
where Kag, Kdg and n are material parameters describing the agglomeration and
deagglomeration mechanisms, respectively; _g istheoverallshearrategivenby
Figure 7.3 Schematic representation of the evolution of the microstructure with the strain rate.