Thixoforming : Semi-solid Metal Processing

3. Macroscopic averages or homogenization: This step aims at determining the overall,
also called effective, behaviour. The average stresses and strains over the RVE are
defined as
s ðÞ¼ r
1
ð
s ðrÞdV
V V
e ð rÞ ¼ 1
ð
e ðrÞdV ð7:1Þ
V V
They are equal to the overall stresses and strains when homogeneous boundary
conditions are applied at the RVE boundaries.
In this chapter, this methodology is applied to semi-solid behaviour.
7.3
Modelling Semi-solid Behaviour
The behaviour and properties of materials at each length scale are controlled by the
observable microstructure at the corresponding length scale. Therefore, whether a
material is heterogeneous or not depends on the length scale used in the observation.
In this section, we first analyse semi-solid microstructure to define the RVE. Then, we
determine strain rate concentration tensors. Finally, macroscopic averages are
calculated for the determination of the effective properties.
7.3.1
Definition of the Representative Volume Element
7.3 Modelling Semi-solid Behaviourj223
7.3.1.1 Morphological Pattern
Figure 7.1 displays a typical microstructure of a semi-solid Sn–15wt%Pb alloy. The
material is a two-phase system with a solid volume fraction equals to f s . The system
exhibits a particular morphology consisting of a more or less globular solid phase
bonded by surrounding liquid. A more accurate observation of the picture reveals
(i) the presence of liquid entrapped within the globular solid particles and (ii) the
presence of connections between the solid particles. These two relevant scales are
taken into account in the representation of the microstructure: agglomerates
constituted of both solid and liquid are embedded in a contiguous zone composed
also of both liquid and solid. In a statistical representation of the microstructure, this
complex system can be assumed to be equivalent to one inclusion gathering all the
agglomerates surrounded by a coating gathering the contiguous phase (Figure 7.2, A).
This representation is all the more relevant where the evolution of the system is
concerned. Indeed, it is commonly admitted that the deformation takes place in local
sites such as the bonds between the solid grains and the liquid that is not entrapped in
the agglomerated solid particles [13–15]. Therefore, the parts of solid and liquid
contained in the coating contribute to the deformation and are called the active zone,
whereas the inclusions associated with solid grains and entrapped liquid hardly do so.
To individualize the mechanical role of the non-entrapped and entrapped liquid or
of the solid bonds and the solid particles in the deformation mechanisms and to give