On computational modeling of sintering based on homogenization
Liquid-phase sintering is the process where a precompacted powder, “green body”, is heated
to the point where (a part of) the solid material melts, and the specimen shrinks while keeping
(almost) net shape. In the case of hardmetal, the microstructure is defined by WC-Co-particles
with large pores, whereby molten Co represents the liquid phase. In the ideal case, a fully
dense material is achieved when the sintering is completed. The “driving force” of the sintering
procedure is surface tension along the free surfaces, i.e. Co-pore interfaces. In this thesis, the
intrinsic deformation of both the solid phase and the melt phase is modeled as the creeping
flow of the Stokes’ type, whereby elastic deformation is ignored.
The macroscopic properties are derived via computational homogenization that utilizes a
highly idealized mesostructure within each Representative Volume Element (RVE). 2D RVE’s
are used predominantly; however, 3D-mesostructures are also analyzed. Within the FE2
algorithmic setting, the homogenization is carried out at the Gaussian integration points in
the macroscale FE-mesh. This allow for the investigation of properties that are not easily
captured with traditional macroscopic constitutive models, which inevitably would become
highly complex with many material parameters that lack physical interpretation.
The finite element mesh of the RVE becomes heavily deformed as the surface tension pulls
the particles closer; hence, it was necessary to develop a surface tracking method with remeshing.
As an element in the mesh reaches a certain deformed state, defined by the condition number
of the Jacobian, a new mesh is created.
The FE2 algorithm has been implemented in the open source FE-code OOFEM (written in
C++) where the code is parallelized w.r.t. the elements in the macroscale mesh.
A number of (more generic or less generic) issues related to the homogenization theory
and algorithm are discussed in the thesis: (i) The implications of Variationally Consistent
Homogenization (VCH) and the consequent satisfaction of the “macrohomogeneity condition”.
One issue is how to homogenize the stress and volumetric rate-of-deformation when pores are
present. (ii) How to establish a variational framework on both scales, based on a suitable
mixture of fields, that allows for a seamless transition from macroscopically compressible to
incompressible response. Such a transition is of utmost importance for the practical use of the
FE2 algorithm in view of eventual macroscopic incompressibility of each individual RVE (as the
porosity vanishes locally). In particular, the corresponding RVE-problems are designed in such
a fashion that they are “fed” by the deviatoric part of the macroscopic rate-of-deformation
and the macroscopic pressure. (iii) The role of boundary conditions on RVE, in particular how
bounds on the “macroscale energy density” can be established via the use of Dirichlet and
Neumann boundary conditions. Numerical examples are shown for different loading scenarios,
where the macroscopic behavior is studied.
Mixed variational formulations
Liquid phase sintering