On computational modeling of sintering based on homogenization

Doctoral thesis, 2014

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.