Nachrichten 54 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0024 GfT-Förderpreis 2023 Numerical Study on Anisotropic EHL Contacts with Short Fiber Reinforced Polymers Moritz Lengmüller, Enzo Maier, Thomas Lohner, Karsten Stahl* Das Thema wurde für den GfT-Förderpreis 2023 in der Kategorie „Bachelor- oder ähnliche Arbeiten“ eingereicht, die Auszeichnung findet im Rahmen der GfT-Tagung im September statt. * B.Sc. Moritz Lengmüller, M.Sc. Enzo Maier, Dr.-Ing. Thomas Lohner, Prof. Dr.-Ing. Karsten Stahl Gear Research Center (FZG), Department of Mechanical Engineering, School of Engineering and Design, Technical University of Munich, Boltzmannstrasse 15, 85748 Garching near Munich, Germany 1 Introduction Machine elements made from plain technical polymers allow for highly efficient operation under lubricated conditions due to low stiffness and contact pressures. However, low strength and wear resistance limit the application to low loads. Short fibers are used to specifically reinforce technical polymers, and thus, increase power density. For machine elements, short fiber reinforced polymers (FRPs) can enable tribological contacts with increased load capacity, whereby cost-efficient production is possible by means of injection-molding. The fiber volume content, its distribution, and its orientation in machine elements are essential design criteria. Fiber orientation and distribution are determined during the manufacturing process. This can result in potential processrelated, non-isotropic material properties like stiffness and thermal conductivity and, thus, affect the tribological and structural behavior. Tribological contacts are characterized by film thickness, contact pressure, and subsurface stress. In the engineering of parts with non-isotropic stiffness, analytical formulas are commonly used, which are based on isotropic approximations. FRPs can be considered as transversely isotropic, with a favored direction in fiber orientation representing the axis of symmetry and a perpendicular plane with different isotropic material properties. The effective contact stiffness of non-isotropic materials determines the contact shape and deformation. In elastohydrodynamically lubricated (EHL) contacts, this affects the hydrodynamics, and thus, the general contact behavior and contact regime. To the author’s knowledge, a systematic analysis of EHL contacts with short fiber reinforced polymers has not been performed. For the use of short fiber reinforced plastics in high performance applications, a detailed understanding of the tribological contact conditions is necessary. Therefore, numerical calculations of the oil-lubricated rolling contact between short-fiber-reinforced polyamide and steel were carried out and influences of the fiber properties were investigated. Fiber-reinforced materials feature transversely isotropic elasticity. Although its influence on pressures, shapes, and sizes has been studied extensively for dry contacts, the transferability to lubricated contacts is fragmented. This numerical study investigates how the content and orientation of short fibers in fiberreinforced polymers (FRP) affect elastohydrodynamic lubrication (EHL) of point contacts. Material properties are modeled with Tandon-Weng homogenization. For EHL modeling, a fully coupled approach based on finite element discretization is used. Results on hydrodynamic pressure and film thickness as well as material stress distribution are analyzed. It is shown that the combination of fiber content and orientation defines the effective contact stiffness that determines the contact shape, size, and film thickness. The following extended abstract is an excerpt from the publication “Effect of Transversely Isotropic Elasticity on Elastohydrodynamic Lubrication of Point Contacts” (https: / / doi.org/ 10.3390/ polym14173507) that is based on the awarded thesis “Numerical Study on Anisotropic TEHL Contacts with Short Fiber Reinforced Polymers”) Keywords short fiber reinforced polyamide (PA66); transversely isotropic elasticity; elastohydrodynamic lubrication (EHL) Summary Abstract TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 54 Nachrichten 55 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 This numerical study investigates how a preferred stiffness orientation of a FRP affects the elastohydrodynamic lubrication of ball-on-flat rolling contacts in steady-state conditions. A systematic variation of the fiber content (degree of anisotropy), its orientation, and distribution is performed. Film thickness, contact pressure and stress distribution are investigated in detail. This is significant to identify optimal fiber orientations for the engineering of efficient machine elements. 2 Methods This study considers a finite element (FE) approach to calculate the isothermal EHL contact considering FRPs in a ball-on-flat configuration. In Figure 2 the investigated configuration is visualized schematically, including the global and local coordinate systems used throughout this study. The steel ball (R x = R y = 20 mm; 100Cr6) is pressed onto the FRP (PA66 - GF) flat space by a normal force F N . Both surfaces move in the same direction with v 1 = v 2 (SRR = 0 %). Surfaces are assumed ideally smooth and are thermally stable at bulk temperature ϑ B = 40 °C. The mechanical properties are shown in Table 1.The contact is fully flooded with mineral oil (ISO VG 100). The FRP’s local (123-) coordinate system is orthonormal with the (11-) axis pointing in the fiber direction. The FRP compound is heterogeneous because of the different mechanical properties of matrix and fiber. The material properties can be calculated from either representative volume elements or theoretical models [1]. The fiber orientation can be obtained from either numerical mold simulations or derived experimentally from tomography or microscopy [2, 3]. As the injection-molded fibers are typically shorter than 1 mm, the Tandon-Weng model [4] can be applied to derive the transversely isotropic elastic stiffness tensor. Table 2 shows the calculated elastic constants for an increasing fiber content. The aspect ratio of the fiber is assumed to be constant (a f = l f ⁄ d f = 100). The different fiber orientations used in this study are shown in Figure 1. All considered lubricant properties and model coefficients are documented in [5, 6]. For numerical simulation of the EHL contact, the Navier- Stokes equations are simplified for lubricant flow with unidirectional entrainment in gap length resulting in the Reynolds equation to calculate the hydrodynamics. A penalty term to prevent cavitation is used. The elastic deformations of the surfaces are calculated separately to evaluate the subsurface stress states and are obtained by solving Hooke’s law for transversely isotropic materials. Detailed information regarding the governing equation are found in [5]. The computational domain is shown in [7]. Within the domain, the equations are solved fully coupled using COMSOL Multiphysics. DOI 10.24053/ TuS-2023-0024 Table 1: Mechanical properties of the considered steel ball, matrix, and fiber Table 2: Elastic constants and degree of anisotropy with varying fiber content, acc. to [4, 8] Figure 1: Visualization of the fiber angles Figure 2: Schematic representation of the investigated contact configuration Material Elastic modulus Poisson’s ratio Density Steel 100Cr6 210000 MPa 0.30 7800 kg/ m³ Matrix PA66 2350 MPa 0.35 1140 kg/ m³ Fiber E-Glass 72500 MPa 0.20 2550 kg/ m³ Fiber Content E 11 in MPa E 22 in MPa G 12 in MPa G 23 in MPa 12 Anisotropy 0 wt.% 2350 2350 870.4 870.4 0.35 0 10 wt.% 5576 2359 951.4 937.3 0.34 0.699 20 wt.% 9214 2507 1053.0 1020.4 0.33 1.672 30 wt.% 13347 2724 1182.1 1126.7 0.32 2.614 TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 55 Nachrichten 56 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0024 Figure 3: Hydrodynamic pressure p and film thickness h for increasing fiber content in y direction Figure 4: von Mises stresses in the xz-plane at y = 0 and the respective principal stresses in depth direction z 3 Results This section presents the studies’ results and discusses hydrodynamic pressure, film thickness, deformation, and subsurface stress. 3.1 Increase in Fiber Content Figure 3 shows the hydrodynamic pressure p and film thickness h along the central gap length x (left) and along the central width y (right) for increasing fiber content ϕ in gap width direction y. The general pressure and film thickness distribution follow isotropic EHL steel polymer contacts, which are explained in detail in [5]. In point contacts, the entrainment flow in the x-direction diverges. Thus, the minimum film thickness is found at an offset from the x-axis in the contact center. An increase in fiber content ϕ y leads to a higher pressure p and a lower film thickness h. This is due a decreased contact area resulting from an increase in the effective stiffness of the solid. A maximum pressure of p max = 100 MPa in the contact center and a minimum film thickness of h m = 320 nm at an offset from the x-axis sets in for ϕ y = 30 wt.%. TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 56 Nachrichten 57 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 Figure 4 visualizes the von Mises stresses σ mieses in the xzplane for ϕ y = 0 wt.% and 30 wt.%, respectively and the principal stresses σ 1,2,3 . The maximum von Mises stress is higher in the FRP (66 MPa) than in the isotropic polymer (55 MPa). This is due to the increased stiffness and the resulting smaller contact area. As stresses are strongly multiaxial in the proximity of the surface, neither the shape modification hypothesis (von Mises stresses) nor the maximum principal stress hypothesis applies to the contacts. Other failure criteria are needed for FRPs [20]. In contrast to the isotropic case, the first and second principal stresses σ 1,2 show large differences in depth direction for the transversally isotropic configuration. This is due to the different stiffnesses (and Poisson ratios) in fiber (11-) direction and in the orthogonal (22-) direction. The σ 3 stress corresponds to the applied hydrodynamic pressure boundary condition. Thus, these stresses behave similar for both. 3.2 Fiber Rotation Besides the fiber content, also the fiber orientation affects the EHL contact. Based on Figure 1, the fiber rotations in α,β,γ from 0 to 90° are varied with the fiber content set constant at ϕ = 30 wt.%. A maximum rotation of α = 90° results in a minimum film thickness of h m = 324 nm and a maximum hydrodynamic pressure of p max = 100 MPa. Thereby, the pressure p and film thickness h in gap length direction x and gap width direction y follow the same characteristic profiles as shown in Figure 3. It confirms that a rotation around any axis in the isotropic plane of transversely isotropic FRP barley affects the hydrodynamic pressure and film thickness. This ensues from the isotropic stiffness in the (22-) direction. The deviatoric stress in zdirection remains constant, where upon a rotation of 90°, the x-stress component transitions to become a y-stress component, and vice versa. The von Mises stresses distributions stretch in fiber direction and are lower in magnitude. If the fibers are aligned with the normal z-direction (parallel to load direction, β = 90°), the high stiffness leads to small contact areas and corresponding high pressures. A maximum pressure of p max = 175 MPa in the contact center and a minimum film thickness of h m = 257 nm is reached. A rotation of the fibers from gap length direction x to contact normal direction z (γ = 90°) shows no significant differences. Corresponding plots and more detailed analysis can be found in [7]. 3.3 Fiber Content and Orientation Distribution Injection-molded technical FRP parts can feature an amorphous boundary layer with little to no fibers. In addition, a variation of fiber orientation in depth direction due to the manufacturing process occurs. Based on a sample injection-molded part , the boundary layer thickness is assumed to be z = 0.1 mm with ϕ = 0 wt.%, before it ramps to ϕ = 30 wt.% at a depth of z = 0.5 mm. The fiber orientation rotates 90° around the isotropic plane. As fiber content increases closely below the surface, the effect on the stiffness is small. Further, on, the effective stiffness is also not significantly affected due to the fiber orientation . The stress distribution shows an almost isotropic behavior. The increase in fiber content leads to a small dent in the second principal stress profile σ 2 and consequently in the von Mises stress. The second rotation shows no significant influence due to the distance to the contact zone. 4 Conclusion The short fiber content and orientation affect the degree of non-isotropy, the effective contact stiffness, and the shape of the contact region of EHL contacts. Any increase of stiffness leads to less surface normal deformation and thus lower film thickness, smaller contact size, higher pressure, and higher stresses. A fiber orientation parallel to the contact plane results in a higher contact area and higher film thickness compared to an orientation perpendicular to the contact plane. Subsurface stresses increase with fiber content. Fiber orientations in the contact plane can cause high von Mises stress in the proximity of the surface, while fiber orientations in the contact normal direction result in maximum von Mises stress at higher depths. Typical fiber orientation distribution after injection molding may effectively lead to isotropic-like contact behavior. Future research needs to focus on the critical review of the Tandon-Weng homogenization approach including an experimental validation. References [1] V. Müller, M. Kabel, H. Andrä, and T. Böhlke, “Homogenization of linear elastic properties of short-fiber reinforced composites - A comparison of mean field and voxel-based methods,” International Journal of Solids and Structures, vol. 67-68, pp. 56-70, 2015, doi: 10.1016/ j.ijsolstr.2015.02.030. [2] M. Gupta and K. K. Wang, “Fiber orientation and mechanical properties of short-fiber-reinforced injectionmolded composites: Simulated and experimental results,” Polym. Compos., vol. 14, no. 5, pp. 367-382, 1993, doi: 10.1002/ pc.750140503. [3] R. Żurawik, J. Volke, J.-C. Zarges, and H.-P. Heim, “Comparison of Real and Simulated Fiber Orientations in Injection Molded Short Glass Fiber Reinforced Polyamide by X-ray Microtomography,” Polymers, early access. doi: 10.3390/ polym14010029. [4] G. P. Tandon and G. J. Weng, “Average stress in the matrix and effective moduli of randomly oriented composites,” Composites Science and Technology, vol. 27, no. 2, pp. 111-132, 1986, doi: 10.1016/ 0266-3538(86)90067-9. [5] A. Ziegltrum, E. Maier, T. Lohner, and K. Stahl, “A Numerical Study on Thermal Elastohydrodynamic Lubrication of Coated Polymers,” Tribol Lett, vol. 68, no. 2, 2020, doi: 10.1007/ s11249-020-01309-6. [6] M. Tošić, R. Larsson, and T. Lohner, “Thermal Effects in Slender EHL Contacts,” Lubricants, vol. 10, no. 5, p. 89, 2022, doi: 10.3390/ lubricants10050089. [7] E. Maier, M. Lengmüller, and T. Lohner, “Effect of Transversely Isotropic Elasticity on Elastohydrodynamic Lubrication of Point Contacts,” Polymers, early access. doi: 10.3390/ polym14173507. [8] S. I. Ranganathan and M. Ostoja-Starzewski, “Universal elastic anisotropy index,” Physical review letters, early access. doi: 10.1103/ PhysRevLett.101.055504. DOI 10.24053/ TuS-2023-0024 TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 57