A Theoretical Packing Density Model (TPDM) for ordered and disordered packings

In previous papers, the 4-parameter Compressible Packing Model (CPM) was theoretically developed and validated for the packing density of binary mixtures of spheres and crushed aggregate particles. The 4-parameter CPM is now called the Theoretical Packing Density Model (TPDM). The TPDM is a self-consistent packing density model which is now validated for both disordered and ordered packings. Thanks to the compaction index K which is representative of the packing process efficiency and thanks to the critical cavity size ratio from which the loosening effect appears, the TPDM is successfully applied to densest binary crystalline structures (K = 100) and to disordered ternary mixtures of spherical, round aggregate and crushed aggregate particles (from K = 4.7 to K = 15). The critical cavity size ratio can be selected in the range between 0 for irregular particles with strong surface roughness and 0.2 for frictionless spherical particles. The TPDM predictions are excellent for the peak packing density obtained for each of the densest binary crystalline structures identified. The TPDM also proves its efficiency for disordered ternary mixtures from the analysis of 408 results published in the literature when compared with the PAL (Prior, Almeida, Loureiro) model and with the 3-parameter Particle Packing Model (3PPM, Kwan et al.). The question of optimization of ternary mixtures is finally addressed. A simplified model is proposed for the purpose of building the densest ternary mixture. The analytical expressions for the highest packing density and for the corresponding volume fractions of each granular class are established. Thanks to its theoretical wall effect and loosening effect functions, the TPDM appears to be a useful tool for optimizing the granular skeleton in all circumstances: whatever its packing process (simple pouring or highly sophisticated packing process) and whatever the material (crushed aggregate particles with strong surface roughness or frictionless spherical particles).