Abstract
Sheet Molding Compounds (SMC) offer a cost efficient way to enhance mechanical properties of a polymer with long discontinuous fibers, while maintaining formability to integrate functions, such as ribs, beads or other structural reinforcements. During SMC manufacturing, fibers remain often in a bundled configuration and the resulting fiber architecture determines part properties. Accurate prediction of this architecture by simulation of flow under consideration of the transient rheology and transient fiber orientations can speed up the development process. In particular, the interaction of bundles is of significance to predict molding pressures correctly in a direct simulation approach, which resolves individual fiber bundles. Thus, this work investigates the tangential short-range lubrication forces between fiber bundles with analytical and numerical techniques. A relation between the effective sheared gap between bundles and the bundle separation distance at the contact point is found and compared to experimental results from literature. The result is implemented in an ABAQUS contact subroutine to incorporate short-range interactions in a direct bundle simulation framework.
References
Abrams, L. M., Castro, J. M., “Predicting Molding Forces during Sheet Molding Compound (SMC) Compression Molding. I: Model Development”, Polym. Compos., 24, 291–303 (2003), DOI:10.1002/pc.1002910.1002/pc.10029Search in Google Scholar
Alnersson, G., Tahir, M. W., Ljung, A.-L. and Lundström, T. S., “Review of the Numerical Modeling of Compression Molding of Sheet Molding Compound”, Processes, 8, 179–190 (2020), DOI:10.3390/pr802017910.3390/pr8020179Search in Google Scholar
Barone, M. R., Caulk, D. A., “A Model for the Flow of a Chopped Fiber Reinforced Polymer Compound in Compression Molding”. J. Appl. Mech., 53, 361–371 (1986), DOI:10.1115/1.317176510.1115/1.3171765Search in Google Scholar
Batchelor, G. K., “Slender-Body Theory for Particles of Arbitrary Cross-Section in Stokes Flow”, J. Fluid Mech., 44, 419–440 (1970), DOI:10.1017/S002211207000191X10.1017/S002211207000191XSearch in Google Scholar
Batchelor, G. K., “The Stress Generated in a Non-Dilute Suspension of Elongated Particles by Pure Straining Motion”, J. Fluid Mech., 46, 813–829 (1971), DOI:10.1017/S002211207100087910.1017/S0022112071000879Search in Google Scholar
Bertóti, R., Böhlke, T., “Flow-Induced Anisotropic Viscosity in Short FRPs”, Mechancis of Advancd Materials and Modern Processes, 3, 1–12 (2017), DOI:10.1186/s40759-016-0016-710.1186/s40759-016-0016-7Search in Google Scholar
Butler, J. E., Snook, B., “Microstructural Dynamics and Rheology of Suspensions of Rigid Fibers”, Annu. Rev. Fluid Mech., 50, 299 – 318 (2018), DOI:10.1146/annurev-fluid-122316-04514410.1146/annurev-fluid-122316-045144Search in Google Scholar
Dinh, S. M., Armstrong, R. C., “A Rheological Equation of State for Semiconcentrated Fiber Suspensions”, J. Rheol., 28, 207–227 (1984), DOI:10.1122/1.54974810.1122/1.549748Search in Google Scholar
Djalili-Moghaddam, M., Toll, S., “A Model for Short-Range Interactions in Fibre Suspensions”, J. Non-Newtonian Fluid Mech., 132, 73–83 (2005), DOI:10.1016/j.jnnfm.2005.08.01410.1016/j.jnnfm.2005.08.014Search in Google Scholar
Dumont, P. J. J., Orgéas, L., Le Corre, S. and Favier, D., “Anisotropic Viscous Behavior of Sheet Molding Compounds (SMC) during Compression Molding”, Int. J. Plast., 19, 625–646 (2003), DOI:10.1016/S0749-6419(01)00077-810.1016/S0749-6419(01)00077-8Search in Google Scholar
Dumont, P. J. J., Vassal, J.-P., Orgéas, L., Michaud, V., Favier, D. and Månson, J.-A. E., “Processing, Characterisation and Rheology of Transparent Concentrated Fibre-Bundle Suspensions”, Rheol. Acta, 46, 639–651 (2007), DOI:10.1007/s00397-006-0153-810.1007/s00397-006-0153-8Search in Google Scholar
Folgar, F., Tucker, C. L., “Orientation Behavior of Fibers in Concentrated Suspensions”, J. Reinf. Plast. Compos., 3, 98–119 (1984), DOI:10.1177/07316844840030020110.1177/073168448400300201Search in Google Scholar
Görthofer, J., Meyer, N., Pallicity, T. D., Schöttl, L., Trauth, A., Schemmann, M., Hohberg, M., Pinter, P., Elsner, P., Henning, F., Hrymak, A. N., Seelig, T., Weidenmann, K., Kärger, L. and Böhlke, T., “Virtual Process Chain of Sheet Molding Compound: Development, Validation and Perspectives”, Composites Part B, 169, 133–147 (2019), DOI:10.1016/j.compositesb.2019.04.00110.1016/j.compositesb.2019.04.001Search in Google Scholar
Guiraud, O., Orgéas, L., Dumont, P. J. J. and Rolland Du Roscoat, S., “Microstructure and Deformation Micromechanisms of Concentrated Fiber Bundle Suspensions: An Analysis Combining X-Ray Microtomography and Pull-Out Tests”, J. Rheol., 56, 593–623 (2012), DOI:10.1122/1.369818510.1122/1.3698185Search in Google Scholar
Hinch, E. J., Leal, L. G., “Time-Dependent Shear Flows of a Suspension of Particles with Weak Brownian Rotations”, J. Fluid Mech., 57, 753–767 (1973), DOI:10.1017/S002211207300199010.1017/S0022112073001990Search in Google Scholar
Hohberg, M., Kärger, L., Henning, F. and Hrymak, A. N., “Rheological Measurements and Rheological Shell Model Considering the Compressible Behavior of Long Fiber Reinforced Sheet Molding Compound (SMC)", Composites Part A, 95, 110–117 (2017), DOI:10.1016/j.compositesa.2017.01.00610.1016/j.compositesa.2017.01.006Search in Google Scholar
Kugler, S. K., Dey, A. P., Saad, S., Cruz, C., Kech, A. and Osswald, T., “A Flow-Dependent Fiber Orientation Model”, J. Compos. Sci., 4, 96–117 (2020), DOI:10.3390/jcs403009610.3390/jcs4030096Search in Google Scholar
Kuhn, C., Walter, I., Täger, O. and Osswald, T., “Simulative Prediction of Fiber-Matrix Separation in Rib Filling During Compression Molding Using a Direct Fiber Simulation”, J. Compos. Sci., 2, 2–12 (2017), DOI:10.3390/jcs201000210.3390/jcs2010002Search in Google Scholar
Le, T.-H., Dumont, P. J. J., Orgéas, L., Favier, D., Salvo, L. and Boller, E., “X-Ray Phase Contrast Microtomography for the Analysis of the Fibrous Microstructure of SMC Composites”, Composites Part A, 39, 91–103 (2008), DOI:10.1016/j.compositesa.2007.08.02710.1016/j.compositesa.2007.08.027Search in Google Scholar
Le Corre, S., Dumont, P. J. J., Orgéas, L. and Favier, D., “Rheology of Highly Concentrated Planar Fiber Suspensions”, J. Rheol., 49, 1029–1058 (2005), DOI:10.1122/1.199359410.1122/1.1993594Search in Google Scholar
Lee, C.-C., Folgar, F. and Tucker, C. L., “Simulation of Compression Molding for Fiber-Reinforced Thermosetting Polymers”, Journal of Engineering for Industry, 106, 114–125 (1984), DOI:10.1115/1.318592110.1115/1.3185921Search in Google Scholar
Londoño-Hurtado, A., “Mechanistic Models for Fiber Flow", Ph. D. Thesis, University of Wisconsin– Madison, USA (2009)Search in Google Scholar
Mackaplow, M. B., Shaqfeh, E. S. G., “A Numerical Study of the Rheological Properties of Suspensions of Rigid, Non-Brownian Fibres”, J. Fluid Mech., 329, 155–186 (1996), DOI:10.1017/S002211209600888910.1017/S0022112096008889Search in Google Scholar
Meyer, N., Saburow, O., Hohberg, M., Hrymak, A. N., Henning, F. and Kärger, L., “Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics”, J. Compos. Sci., 4, 77–96 (2020a), DOI:10.3390/jcs402007710.3390/jcs4020077Search in Google Scholar
Meyer, N., Schöttl, L., Bretz, L., Hrymak, A. N. and Kärger, L., “Direct Bundle Simulation Approach for the Compression Molding Process of Sheet Molding Compound”, Composites Part A, 132, 105809 (2020b), DOI:10.1016/j.compositesa.2020.10580910.1016/j.compositesa.2020.105809Search in Google Scholar
Motaghi, A., Hrymak, A. N., “Microstructure Characterization in Direct Sheet Molding Compound”, Polym. Compos., 40, E69–E77 (2019), DOI:10.1002/pc.2449510.1002/pc.24495Search in Google Scholar
Orgéas, L., Dumont, P. J. J. and Corre, S. L., “Chapter 5 Rheology of Highly Concentrated Fiber Suspensions”, in Rheology of Non-Spherical Particle Suspensions, Elsevier, Kidlington, p. 119–166 (2015), DOI:10.1016/B978-1-78548-036-2.50005-810.1016/B978-1-78548-036-2.50005-8Search in Google Scholar
Osswald, T. A., Tucker, C. L., “A Boundary Element Simulation of Compression Mold Filling”, Polymer Engineering and Science 28, 413–420 (1988), DOI:10.1002/pen.76028070310.1002/pen.760280703Search in Google Scholar
Osswald, T. A., Tucker, C. L., “Compression Mold Filling Simulation for Non-Planar Parts”, Int. Polym. Proc., 5, 79–87 (1990), DOI:10.3139/217.90007910.3139/217.900079Search in Google Scholar
Phelps, J. H., Tucker, C. L., “An Anisotropic Rotary Diffusion Model for Fiber Orientation in Short and Long-Fiber Thermoplastics”, J. of Non-Newtonian Fluid Mech., 156, 165–176 (2009), DOI:10.1016/j.jnnfm.2008.08.00210.1016/j.jnnfm.2008.08.002Search in Google Scholar
Servais, C., Luciani, A. and Månson, J.-A. E., “Fiber–Fiber Interaction in Concentrated Suspensions: Dispersed Fiber Bundles”, J. Rheol., 43, 1005–1018 (1999a), DOI:10.1122/1.55101510.1122/1.551015Search in Google Scholar
Servais, C., Månson, J.-A. E. and Toll, S., “Fiber–Fiber Interaction in Concentrated Suspensions: Disperse Fibers”, J. Rheol., 43, 991– 1004 (1999b), DOI:10.1122/1.55101410.1122/1.551014Search in Google Scholar
Shaqfeh, E. S. G., Fredrickson, G. H., “The Hydrodynamic Stress in a Suspension of Rods”, Physics of Fluids A: Fluid Dynamics, 2, 7– 24 (1990), DOI:10.1063/1.85768310.1063/1.857683Search in Google Scholar
Silva-Nieto, R. J., Fisher, B. C. and Birley, A. W., “Predicting Mold Flow for Unsaturated Polyester Resin Sheet Molding Compounds”, Polym. Compos., 1, 14–23 (1980), DOI:10.1002/pc.75001010510.1002/pc.750010105Search in Google Scholar
Sommer, D. E., Favaloro, A. J. and Pipes, R. B., “Coupling Anisotropic Viscosity and Fiber Orientation in Applications to Squeeze Flow", J. Rheol., 62, 669–679 (2018), DOI:10.1122/1.501309810.1122/1.5013098Search in Google Scholar
Toll, S., “Packing Mechanics of Fiber Reinforcements”, Polym. Eng. Sci., 38, 1337–1350 (1998), DOI:10.1002/pen.1030410.1002/pen.10304Search in Google Scholar
Tucker, C. L., Folgar, F., “A Model of Compression Mold Filling”, Polym. Eng. Sci., 23, 69–73 (1983), DOI:10.1002/pen.76023020410.1002/pen.760230204Search in Google Scholar
Yamane, Y., Kaneda, Y. and Dio, M., “Numerical Simulation of Semi-Dilute Suspensions of Rodlike Particles in Shear Flow”, J. Non-Newtonian Fluid Mech., 54, 405–421 (1994), DOI:10.1016/0377-0257(94)80033-210.1016/0377-0257(94)80033-2Search in Google Scholar
Acknowledgements
The research documented in this manuscript has been funded by the German Research Foundation (DFG) within the International Research Training Group “Integrated engineering of continuous-discontinuous long fiber-reinforced polymer structures” (GRK 2078). The support by the German Research Foundation (DFG) is gratefully acknowledged.
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