Introduction of an Adhesion Factor to Cube in Cube Models and its Effect on Calculated Moduli of Particulate Composites

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 <em>k</em>_adh 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 <em>k</em>_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 <em>k</em>_adh exceeds \( { \ \sqrt{E^M/E^F} \ }\) 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>E</em>_M, filler modulus <em>E</em>_F, filler volume content <em>v</em>_F and adhesion factor <em>k</em>_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 <em>cube in cube</em> 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.立方包立方（cube in cube）方法由Paul与Ishai-Cohen提出，用于在假设填料与基体完全粘合的前提下，建模并推导颗粒填充复合材料中依赖填料含量的杨氏模量公式。该团队选用的公式因形式简洁，经重新推导时采用了将球形夹杂物转换为立方夹杂物的基本体积（elementary volume, EV）方法。 该基本体积方法推导出的复合材料模量表达式，可引入取值范围为0至1的粘合因子$k_{ ext{adh}}$，用以表征非完全的填料-基体粘合状态。该粘合因子通过缩放立方夹杂物的边长，减小了基体与填料之间的应力传递面积。 利用修正后的Paul模型对实验数据进行拟合，可为PA66、PBT、PP、PE-LD及BR得到合理的粘合因子$k_{ ext{adh}}$，其结果与各材料的表面能相符。进一步分析显示，仅当粘合因子$k_{ ext{adh}}$大于$sqrt{E^M/E^F}$时，复合材料才会出现刚度增强现象，且该现象取决于基体模量与填料模量的比值。 该修正模型可基于已知的基体模量$E_M$、填料模量$E_F$、填料体积分数$v_F$及粘合因子$k_{ ext{adh}}$，快速计算任意颗粒填充复合材料的相关性能。由此可大幅简化任意颗粒填充聚合物部件的有限元分析（finite element analysis, FEA）模拟流程，同时也为材料选型提供便利。 对立方与六方基本体积排布的有限元分析表明，当排布偏离立方结构时，基本体积内部的剪切应力会显著升高。在高填料含量下，“基本体积的性能可代表整个复合材料”这一假设仅适用于填料体积分数不超过15%或20%（对应质量分数30%至40%）的场景。因此，对于绝大多数商业化颗粒填充复合材料，该修正模型均可适用。 此外，该结果也表明立方包立方方法存在两大局限性：其一，当填料含量超过20%时，由于基本体积排布或填料空间分布偏离立方结构（呈现奇异特性），剪切应力会持续升高；其二，颗粒间相互作用不断增强，基体内部会形成颗粒网络，这违背了基本体积方法中填料均匀分散的假设前提。