Analytical Solutions of Quadratically-Closed Folgar–Tucker Equations for Benchmark Flows: A Computational App

  • Ranjbar, Maedeh (Hochschule für Technik und Wirtschaft Berlin)
  • Papenfuss, Christina (Hochschule für Technik und Wirtschaft Berlin)

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Predicting fiber orientation evolution is fundamental for modeling manufacturing processes of fiber-reinforced composites, as the resulting mechanical response strongly depends on the dominant fiber orientation. In engineering workflows, fiber-orientation fields are typically computed using commercial software such as Autodesk Moldflow [1], which is indispensable for industrial flow and geometry conditions. However, these tools provide limited physical insight compared to analytical solutions and remain constrained by the specific assumptions implemented for the Folgar–Tucker equation (FTE) and its closure relations. [2] In our recent work [2, 3], we derived exact three-dimensional analytical solutions of the quadratically closed FTE for several benchmark planar flows, including Couette shear, elongational, and combined three-dimensional flow configurations. These closed-form solutions reveal deeper insight into the intrinsic structure and behavior of the FTE, such as the oscillatory evolution of orientation tensor components, and provide reliable reference data for validating numerical orientation models. Their applicability, however, remains restricted to the classical FTE formulation with quadratic closure, and their direct use is nontrivial, as they require solving coupled linear differential equations together with an eigenvalue problem. To make these analytical solutions accessible and practically useful, we present a computational application that consolidates all benchmark results into a user-friendly platform. The tool enables users to visualize the evolution of orientation tensor components and examine the corresponding orientation distribution functions (ODFs). Such an open-access app can support engineers and researchers in exploring model behavior and investigating reverse-engineering strategies in composite processing. As an outlook, our goal is to develop a numerical solution framework that is not restricted to a specific FTE formulation or closure relation, but achieves predictive capabilities comparable to commercial tools—using our analytical solutions as a robust foundation for algorithm validation.