Introduction of an adhesion factor to cube in cube models and its effect on calculated moduli of particulate composites

Abstract

Abstract The cube in cube approach was used by Paul and Ishai-Cohen to model and derive formulas for filler content dependent Young’s moduli of particle filled composites assuming perfect filler matrix adhesion. Their formulas were chosen because of their simplicity, and recalculated using an elementary volume approach which transforms spherical inclusions to cubic inclusions. The EV approach led to expression of the composites moduli that allows introducing an adhesion factor k adh ranging from 0 and 1 to take into account reduced filler matrix adhesion. This adhesion factor scales the edge length of the cubic inclusions, thus reducing the stress transfer area between matrix and filler. Fitting the experimental data with the modified Paul model provides reasonable k adh for PA66, PBT, PP, PE-LD and BR which are in line with their surface energies. Further analysis showed that stiffening only occurs if k adh exceeds $$\sqrt{{E}_{\mathrm{M}}/{E}_{\mathrm{F}}}$$ E M / E F and depends on the ratio of matrix modulus and filler modulus. The modified model allows for a quick calculation of any particle filled composites for known matrix modulus E M, filler modulus E F, filler volume content v F and adhesion factor k adh. Thus, finite element analysis (FEA) simulations of any particle filled polymer parts as well as materials selection are significantly eased. FEA of cubic and hexagonal EV arrangements show that stress distributions within the EV exhibit more shear stresses if one deviates from the cubic arrangement. At high filler contents the assumption that the property of the EV is representative for the whole composite, holds only for filler volume contents up to 15 or 20% (corresponding to 30 to 40 weight %). Thus, for vast majority of commercially available particulate composites, the modified model can be applied. Furthermore, this indicates that the cube in cube approach reaches two limits: (i) the occurrence of increasing shear stresses at filler contents above 20% due to deviations of EV arrangements or spatial filler distribution from cubic arrangements (singular), and (ii) increasing interaction between particles with the formation of particle network within the matrix violating the EV assumption of their homogeneous dispersion.