Anisotropic fiber-reinforced composites are used in lightweight construction, which is of great industrial relevance. During mold filling of fiber suspensions, the microstructural evolution of the local fiber arrangement and orientation distribution is determined by the local velocity gradient. Based on the Folgar–Tucker equation, which describes the evolution of the second-order fiber orientation tensor in terms of the velocity gradient, the present study addresses selected states of deformation rates that can locally occur in complex flow fields. For such homogeneous flows, exact solutions for the asymptotic fiber orientation states are derived and discussed based on the quadratic closure. In contrast to the existing literature, the derived exact solutions take into account the fiber-fiber interaction. The analysis of the asymptotic solutions relying upon the common quadratic closure shows disadvantages with respect to the predicted material symmetry, namely, the anisotropy is overestimated for strong fiber-fiber interaction. This motivates us to suggest a novel normalized fully symmetric quadratic closure. Two versions of this new closure are derived regarding the prediction of anisotropic properties and the fiber orientation evolution. The fiber orientation states determined with the new closure approach show an improved prediction of anisotropy in both effective viscous and elastic composite behaviors. In addition, the symmetrized quadratic closure has a simple structure that reduces the effort in numerical implementation compared to more elaborated closure schemes.

中文翻译：

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二次闭合 Folgar-Tucker 方程的渐近纤维取向状态和随后的闭合改进

各向异性纤维增强复合材料用于轻质结构，具有重要的工业意义。在纤维悬浮液的模具填充过程中，局部纤维排列和取向分布的微观结构演变由局部速度梯度决定。基于 Folgar-Tucker 方程，该方程描述了二阶纤维取向张量在速度梯度方面的演变，本研究解决了可能在复杂流场中局部发生的变形率的选定状态。对于此类均匀流动，基于二次闭包推导出和讨论渐近纤维取向状态的精确解。与现有文献相比，导出的精确解考虑了纤维-纤维相互作用。依赖于共同二次闭包的渐近解的分析显示出相对于预测的材料对称性的缺点，即对于强纤维 - 纤维相互作用高估了各向异性。这促使我们提出一种新颖的归一化完全对称二次闭包。关于各向异性特性和纤维取向演化的预测，推导出了这种新闭合的两个版本。使用新的闭合方法确定的纤维取向状态显示了对有效粘性和弹性复合材料行为的各向异性的改进预测。此外，对称二次闭包具有简单的结构，与更详细的闭包方案相比，可以减少数值实现的工作量。