Equivalent inclusion method-based simulation of particle sedimentation toward functionally graded material manufacturing

A novel numerical approach based on Eshelby’s equivalent inclusion method is presented to simulate the Stokes’ flow of many spherical particles moving in a viscous fluid at a small Reynolds number. Many particles fall toward the bottom of the fluid at different velocities and thus form a graded microstructure in terms of the material phase and the particle size. For each particle, an eigenstrain rate, which is given in a polynomial form, is introduced to represent the mismatch between the particle and the rest of the fluid. Rongved’s fundamental solution of a point force in a semi-infinite domain with a fixed boundary (Rongved in J. Appl. Mech. 22, 545–546, 1955 ) is used to calculate the velocity field caused by the body force and eigenstrain rate. Based on Eshelby’s stress equivalent condition, the eigenstrain rate of each particle can be solved and the sedimentation process of a many-particle system can be simulated as a quasi-equilibrium process. If only one or two particles are considered, the results agree with the finite element results very well. Using a suspension of aluminum and high-density polyethylene (HDPE) powders mixed in ethanol, the microstructural evolution is illustrated along with the sedimentation process, which produces a graded mix collected at the fixed boundary for functionally graded material manufacturing.