Abstract The two-scale asymptotic homogenization method is used to find closed-form formulas for effective properties of periodic multi-phase fiber-reinforced composites where constituents have complex-valued transport properties and parallelogram unit cells. An antiplane problem relevant to linear elasticity is formulated in the frame of transport properties. The application of the method leads to the need of solving some local problems whose solution is found using potential theory and shear effective coefficients are explicitly obtained for n - phase fiber-reinforced composites. Simple formulae are explicitly given for three- and four-phase fiber-reinforced composites. The broad applicability, accuracy and generality of this model is determined through comparison with other methods reported in the literature in relation to shear elastic moduli and several transport problems of multi-phase fiber-reinforced composites in their realm, such as conductivity in a biological context and permittivity leading to gain and loss enhancement of dielectrics. Also, the example of gain enhancement of inertial mass density is looked into. Good agreement with other theoretical approaches is obtained. The formulas may be useful as benchmarks for checking experimental and numerical results.

中文翻译：

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具有复杂成分和平行四边形晶胞的周期性多相纤维增强复合材料的有效输运性能

摘要 使用两尺度渐近均质化方法寻找周期性多相纤维增强复合材料有效性能的闭式公式，其中成分具有复值输运特性和平行四边形晶胞。与线弹性相关的反平面问题在输运性质的框架中被公式化。该方法的应用导致需要解决一些局部问题，这些问题的解决方案是使用势理论找到的，并明确获得 n 相纤维增强复合材料的剪切有效系数。明确给出了三相和四相纤维增强复合材料的简单公式。适用范围广，该模型的准确性和通用性是通过与文献中报道的其他方法进行比较来确定的，这些方法涉及剪切弹性模量和多相纤维增强复合材料在其领域中的几个传输问题，例如生物环境中的电导率和介电常数导致电介质的增益和损耗增强。此外，还研究了惯性质量密度增益增强的例子。获得了与其他理论方法的良好一致性。这些公式可用作检查实验和数值结果的基准。研究了惯性质量密度增益增强的例子。获得了与其他理论方法的良好一致性。这些公式可用作检查实验和数值结果的基准。研究了惯性质量密度增益增强的例子。获得了与其他理论方法的良好一致性。这些公式可用作检查实验和数值结果的基准。