The paper focuses on the evaluation of the effective properties of linear viscoelastic composites with a periodic structure, containing long cylindrical fibers of circular cross-section and for two different cell arrangements: square and hexagonal unit cells. For this purpose, we use the two-scale asymptotic homogenization method (AHM) and a numerical homogenization method (NHM). Based on the correspondence principle, the local functions and the relaxation overall properties are obtained in explicit form by the AHM using the Prony series. Additionally, the NHM is established for a three-dimensional representative cell, and the problem is solved under appropriate boundary conditions, by using the Finite Element Method. The numerical results obtained by the AHM and NHM are compared and verified with other theoretical approaches. The comparisons show a good agreement and a benchmark for further experimental and theoretical investigations.

中文翻译：

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使用 Prony 级数应用于纤维粘弹性复合材料的渐近和数值均匀化方法

本文重点评估具有周期性结构的线性粘弹性复合材料的有效性能，该复合材料包含圆形横截面的长圆柱形纤维，并具有两种不同的单元格排列：方形和六边形单元格。为此，我们使用两尺度渐近均匀化方法 (AHM) 和数值均匀化方法 (NHM)。基于对应原理，AHM 使用 Prony 级数以显式形式获得局部函数和松弛整体属性。此外，还为三维代表性单元建立了 NHM，并通过使用有限元方法在适当的边界条件下解决了该问题。将 AHM 和 NHM 获得的数值结果与其他理论方法进行了比较和验证。