Influence of flow–fiber coupling during mold-filling on the stress field in short-fiber reinforced composites

Computational Mechanics (2023) 71:991–1013 https://doi.org/10.1007/s00466-023-02277-z ORIGINAL PAPER Influence of flow–fiber coupling during mold-filling on the stress field in short-fiber reinforced composites Tobias Karl1,2 · Jan Zartmann1 Thomas Böhlke1 · Simon Dalpke1 · Davide Gatti2 · Bettina Frohnapfel2 · Received: 28 October 2022 / Accepted: 26 January 2023 / Published online: 1 March 2023 © The Author(s) 2023 Abstract The anisotropic elastic properties of injection molded composites are fundamentally coupled to the flow of the fiber suspension during mold-filling. Regarding the modeling of mold-filling processes, both a decoupled and a flow–fiber coupled approach are possible. In the latter, the fiber-induced viscous anisotropy is considered in the computation of the flow field. This in turn influences the evolution of the fiber orientation compared to the decoupled case. This study investigates how flow– fiber coupling in mold-filling simulation affects the stress field in the solid composite under load based on the final elastic properties after fluid–solid transition. Furthermore, the effects of Newtonian and non-Newtonian polymer matrix behavior are investigated and compared. The entire process is modeled micromechanically unified based on mean-field homogenization, both for the fiber suspension and for the solid composite. Different numerical stabilization methods of the mold-filling simulation are discussed in detail. Short glass fibers with a typical aspect ratio of 20 and a volume fraction of 20% are considered, embedded in polypropylene matrix material. The results show that the flow–fiber coupling has a large effect on the fiber orientation tensor in the range of over ± 30% with respect to the decoupled simulation. As a consequence, the flow–fiber coupling affects the stress field in the solid composite under load in the range of over ± 10%. In addition, the predictions based on a non-Newtonian modeling of the matrix fluid differ significantly from the Newtonian setup and thus the necessity to consider the shear-thinning behavior is justified in a quantifiable manner. Keywords Short-fiber reinforced composites · Flow–fiber coupling · Micromechanics · Homogenization · Numerical stabilization 1 Introduction 1.1 Motivation and state of the art Short-fiber reinforced composites are commonly used in lightweight design. The special case of adding fibers into a polymer matrix allows the mass-production of lightweight components with complex shapes while retaining the desired mechanical stiffness and strength. During injection molding— the manufacturing process through which such components B Tobias Karl 1 Institute of Engineering Mechanics, Chair for Continuum Mechanics, Karlsruhe Institute of Technology (KIT), Kaiserstraße 10, 76131 Karlsruhe, Germany 2 Institute of Fluid Mechanics, Karlsruhe Institute of Technology (KIT), Kaiserstraße 10, 76131 Karlsruhe, Germany are produced—the fibers flow within the polymer matrix and thereby change their orientation based on the experienced flow conditions. At the same time, a local change of the fiber orientation affects the overall rheological properties of the fluid, which become inhomogeneous anisotropic, and in turn the flow characteristics. This mutual flow–fiber interaction is subject of recent numerical research efforts addressed in the following, focussing on the effects of flow–fiber coupling on the fluid side. In the context of this work, flow–fiber coupling expresses that the evolution of fiber orientation is influenced by the effective anisotropic viscosity of the fiber suspension. Three main points can be identified in this context: The effect of the flow–fiber coupling on the flow field, the influenced fiber orientation evolution due to the changed flow conditions and finally the change of the mechanical properties of the composite, which depend on the local fiber orientation. How the flow–fiber coupling during mold-filling affects the stress state in the solid composite under load after fluid–solid 123 992 transition is a question that received less attention in literature and is the objective of the present work. Since the correct prediction of the stress field in the manufactured composite is an essential part in engineering applications, the need for a flow–fiber coupled simulation has to be studied. In this context, the question also arises which influence the two-phase simulation of the actual mold filling has both on the fiber orientation and on the local anisotropic properties. The fiber-induced anisotropic viscosity within the flow– fiber coupled approach causes changes of the velocity field during mold-filling. Latz et al. [1] investigated the flow–fiber coupling from a fluid mechanics perspective in a channel flow, a flow around a cylinder and in a contraction flow. Different coupling intensities were considered with the result, that the coupling effects are significant in the channel and in the contraction flow. It is shown that (e.g. for the channel flow) the stronger the coupling, the flatter the velocity profile. This flattening was already reported in the early studies of Altan et al. [2] and Tang and Altan [3]. The importance of accounting for flow–fiber coupling in mold-filling simulations was demonstrated in a low Reynolds number flow through a tapered channel by Krochak et al. [4]. Both isotropic and aligned orientation states were considered at the inlet, showing that the flow changes towards a plug flow because of flow–fiber coupling. Flow–fiber coupled moldfilling simulation based on probability density function was carried out by Mezi et al. [5] in 2D geometries and by Férec et al. [6] in 3D axisymmetric geometries. Both studies conclude that the influence of flow–fiber coupling on the velocity field is significant and the velocity profile flattens, as mentioned before. In addition, the difference between Newtonian and non-Newtonian matrix fluid is addressed by Mezi et al. [5]. Mezi et al. [7] observed a flattening of the velocity profile in the context of flow–fiber coupling for die swell flows. The results show that the swell ratio of the flow is affected by flow fiber coupling. By using a scalar rheologial model, Li and Luyé [8] studied the flow–fiber coupling effects based on mold-filling simulations of a rectangular plate. In addition, three different orientation evolution models were used in their study. They observed, that the coupling effects strongly depend on the chosen orientation evolution model. In this context, the classical Folgar–Tucker model [9,10] showed less coupling effects than the reduced-strain closure model [11]. The aforementioned flattening of the velocity profile also occured. In addition, the study of Li and Luyé [12] shows that results based on flow–fiber coupling improve parameter optimization of fiber orientation models in the core region of the flow. Tseng and Favaloro [13] introduced the informed isotropic viscosity model in o (...truncated)