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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, recalculated using an elementary volume approach which transforms spherical inclusions to cubic inclusions. The EV approach led to expression for the composites moduli that allow for introducing an adhesion factor kadh ranging from 0 and 1 to take into account none perfect 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 kadh for PA66, PBT, PP, PE-LD and BR which are in line with their surface energies. Further analysis showed that stiffening only occurs if kadh exceeds <span class="math-tex">\( { \ \sqrt{E^M/E^F} \ }\) 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 EM, filler modulus EF, filler volume content vF and adhesion factor kadh. 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.
Introduction of Matrix-Filler Adhesion to Modelling of Elastic Moduli of Particulate Composites
(2022)
Cube in cube elementary volume (EV) concept serves to predict a filler-content dependent Young´s moduli of particle filled composites using moduli of a matrix EM and a filler EF. Paul and Ishai-Cohen derived formulas for composites moduli considering different load transfer boundaries in the EV assuming a complete filler-matrix adhesion. In this paper it is confirmed that their models represent the upper and lower bounds, respectively, with the respect to the experimental data. However, in vast majority of composites a filler-matrix adhesion is not complete. Therefore, an adhesion factor kadh gaining values between 0 and 1 was introduced into Paul´s model to consider the reduced adhesion as the reduction of the filler-matrix contact area for glass beads filled in polar and unpolar thermoplastic matrices as well as elastomer. The evaluation of these composite systems provides reasonable adhesion coefficients of PA66 > PBT > PP > PE-LD >> BR. It was also found that stiffening only occurs if kadh exceeds the minimum value adhesion of root square of E(M) divided by E(F). The determined kadh correspond to scanning electron microscopy observations of the composites fracture surfaces. Additionally, finite element analysis of the cubic and hexagonal arrangements of the EV show that the stress distributions are different, but they affect the calculated moduli only for the filler volume contents exceeding 20 %. The introduction of the filler-matrix adhesion provides more reliable predictions of Young´s moduli of particulate composites.
Process-dependent thermo-mechanical viscoelastic properties and the corresponding morphology of HDPE extrusion blow molded (EBM) parts were investigated. Evaluation of bulk data showed that flow direction, draw ratio, and mold temperature influence the viscoelastic behavior significantly in certain temperature ranges. Flow induced orientations due to higher draw ratio and higher mold temperature lead to higher crystallinities. To determine the local viscoelastic properties, a new microindentation system was developed by merging indentation with dynamic mechanical analysis. The local process-structure-property relationship of EBM parts showed that the cross-sectional temperature distribution is clearly reflected by local crystallinities and local complex moduli. Additionally, a model to calculate three-dimensional anisotropic coefficients of thermal expansion as a function of the process dependent crystallinity was developed based on an elementary volume unit cell with stacked layers of amorphous phase and crystalline lamellae. Good agreement of the predicted thermal expansion coefficients with measured ones was found up to a temperature of 70 °C.
Influence of design of extrusion blow molding (EBM) in terms of extrusion direction set-up and draw ratio as well as process conditions (mold temperature) on storage modulus of high density polyethylene EBM containers was analyzed with dynamic mechanical analysis. All three parameters - mold temperature, flow direction and draw ratio - are statistically significant and lead to relative and absolute evaluation of storage modulus. Furthermore, flow induced changes in crystallinity was analyzed by differential scanning calorimetry. Obtained data on deformation properties can be employed for more sophisticated finite element simulations with the aim to reach more sustainable extrusion blow molding production.