Fiberoripy

This python package provides basic functionality and tools for fiber orientations and closure models.

For example, the Jupyter Notebook

example/orientation/comparison_perfectshear.ipynb

should reproduce Figure 2 in Favaloro, A.J., Tucker III, C.L., Composites Part A, 126 (2019):

Orientation models

Following models have been implemented:

• Jeffery:
  G.B. Jeffery,
  'The motion of ellipsoidal particles immersed in a viscous fluid',
  Proceedings of the Royal Society A, 1922.
  (https://doi.org/10.1098/rspa.1922.0078)
• Folgar-Tucker:
  F. Folgar, C.L. Tucker,
  'Orientation behavior of fibers in concentrated suspensions',
  Journal of Reinforced Plastic Composites 3, 98-119, 1984.
  (https://doi.org/10.1177%2F073168448400300201)
• FTMS:
  A. Latz, U. Strautins, D. Niedziela,
  'Comparative numerical study of two concentrated fiber suspension models',
  Journal of Non-Newtonian Fluid Mechanics 165, 764-781, 2010.
  (https://doi.org/10.1016/j.jnnfm.2010.04.001)
• iARD(-RPR):
  H.C. Tseng, R.Y. Chang, C.H. Hsu,
  'An objective tensor to predict anisotropic fiber orientation in concentrated susp ensions',
  Journal of Rheology 60, 215, 2016.
  (https://doi.org/10.1122/1.4939098)
• pARD(-RPR):
  H.C. Tseng, R.Y. Chang, C.H. Hsu,
  'The use of principal spatial tensor to predict anisotropic fiber orientation in concentrated fiber suspensions',
  Journal of Rheology 62, 313, 2017.
  (https://doi.org/10.1122/1.4998520)
• MRD:
  A. Bakharev, H. Yu, R. Speight and J. Wang,
  “Using New Anisotropic Rotational Diffusion Model To Improve Prediction Of Short Fibers in Thermoplastic Injection Molding",
  ANTEC, Orlando, 2018.
• RSC:
  J. Wang, J.F. O'Gara, and C.L. Tucker,
  'An objective model for slow orientation kinetics in concentrated fiber suspensions: Theory and rheological evidence',
  Journal of Rheology 52, 1179, 2008.
  (https://doi.org/10.1122/1.2946437)
• ARD-RSC:
  J. H. Phelps, C. L. Tucker,
  'An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics',
  Journal of Non-Newtonian Fluid Mechanics 156, 165-176, 2009.
  (https://doi.org/10.1016/j.jnnfm.2008.08.002)
• Mori-Tanaka:
  T. Karl, T. Böhlke,
  'Generalized Micromechanical Formulation of Fiber Orientation Tensor Evolution Equations',
  International Journal of Mechanical Sciences, 2023.
  (https://doi.org/10.1016/j.ijmecsci.2023.108771)

Closures

• Linear, Quadratic, Hybrid:
  Kyeong-Hee Han and Yong-Taek Im,
  'Modified hybrid closure approximation for prediction of flow-induced fiber orientation',
  Journal of Rheology 43, 569, 1999.
  (https://doi.org/10.1122/1.551002)
• IBOF:
  Du Hwan Chung and Tai Hun Kwon,
  'Invariant-based optimal fitting closure approximation for the numerical prediction of flow-induced fiber orientation',
  Journal of Rheology 46(1), 169-194, 2002.
  (https://doi.org/10.1122/1.1423312)
• ORF, ORW, ORW3:
  Joaquim S. Cintra and Charles L. Tucker III,
  'Orthotropic closure approximations for flow-induced fiber orientation',
  Journal of Rheology, 39(6), 1095-1122, 1995. (https://doi.org/10.1122/1.550630)
  Du Hwan Chung and Tai Hun Kwon,
  'Improved model of orthotropic closure approximation for flow induced fiber orientation',
  Polymer Composites, 22(5), 636-649, 2001.
  (https://doi.org/10.1002/pc.10566)
• SQC:
  Tobias Karl, Davide Gatti, Bettina Frohnapfel and Thomas Böhlke,
  'Asymptotic fiber orientation states of the quadratically closed Folgar-Tucker equation and a subsequent closure improvement',
  Journal of Rheology 65(5) : 999-1022, 2021
  (https://doi.org/10.1122/8.0000245)
• SIC, SIHYB:
  Tobias Karl, Matti Schneider and Thomas Böhlke,
  'On fully symmetric implicit closure approximations for fiber orientation tensors',
  Journal of Non-Newtonian Fluid Mechanics 318 : 105049, 2023.
  (https://doi.org/10.1016/j.jnnfm.2023.105049)

Approximations for equivalent aspect ratios

• Cox:
  R.G. Cox,
  'The motion of long slender bodies in a viscous fluid. Part 2. Shear flow.',
  J. Fluid Mech. 1971, 45, 625–657.
  (http://doi.org/10.1017/S0022112071000259)
• Zhang:
  D. Zhang, D.E. Smith, D.A. Jack, S. Montgomery-Smith,
  'Numerical Evaluation of Single Fiber Motion for Short-Fiber-Reinforced Composite Materials Processing',
  J. Manuf. Sci. Eng. 2011, 133, 51002.
  (http://doi.org/10.1115/1.4004831)