Thixoforming : Semi-solid Metal Processing

are the strain rate concentration tensors associated with the liquid and solid particles,
respectively. The overall response, namely the overall stresses, is calculated by
averaging conditions on stresses:
SB ¼ f s
Bss s
B þ 1 f B slB Introducing local constitutive equations and concentrations relations leads to
SB ¼ f s
Bss s
B þ 1 f B sl B
ð7:15Þ
ð7:16Þ
SB ¼ f s
BmsB T s s
B þ 1 f B mlB T l B _ EB ð7:17Þ
The effective viscosity is thus defined by the following implicit equation:
m B ¼ f s
B ms B
5m B
3mB þ 2ms þ 1 f
B
s
B ml B
5m B
3m B þ 2m l B
ð7:18Þ
7.3.2.2 Step b
As done previously, the mechanical interactions between the coated inclusions are
solved by a self-consistent scheme. However, the mathematical form of the strain
rate concentration tensors is different from that given in Equations 7.13 and 7.14
because now we have to account for the specific spatial distribution of the inclusion
and the active zone. In other words, we have to account for the fact that the inclusion
is surrounded by the active zone. Following the work of Christensen and Lo [27] and
Cherkaoui et al. [28], the effective viscosity mSS of the semi-solid is thus determined by
the following equation:
where
mSS ¼ f AmAT A þ f ImIT I ¼ mA þ f IðmImAÞTI ð7:19Þ
T I ¼
3m SS þ 2m I þ 1 f I
f I
5m SS
6
5 mI m ð AÞ
mSS mA 1
ð7:20Þ
where f I ¼ R3 = ðRþDRÞ 3 , R being the radius of the spherical inclusion and DR the
thickness of the coating.
7.4
Results and Discussion
7.4.1
Isothermal Steady-state Behaviour
7.4 Results and Discussionj229
In this section, the micro–macro model is applied to the case of Sn–15wt%Pb alloys
having a globular-like morphology for the solid phase and solid fractions ranging
from 0.4 to 0.7. Calculations are compared with experimental results found in the