Fiberoripy

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

For example, the model comparison notebook reproduces Figure 2 in Favaloro, A.J., Tucker III, C.L., Composites Part A, 126 (2019):

Installation

pip install fiberoripy              # library
pip install "fiberoripy[examples]"  # plus interactive plotting for the notebooks

Quickstart

import numpy as np
from scipy.integrate import solve_ivp

from fiberoripy.closures import IBOF_closure, compute_closure
from fiberoripy.orientation import folgar_tucker_ode, integrate_ori_ode

# Close a second-order orientation tensor to fourth order.
A = compute_closure(np.diag([0.7, 0.2, 0.1]), "IBOF")


# Evolve an orientation state in simple shear.
def L(t):
    return np.array([[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, 0.0]])


t = np.linspace(0.0, 100.0, 500)
solution = solve_ivp(
    integrate_ori_ode,
    (t.min(), t.max()),
    (np.eye(3) / 3.0).ravel(),
    t_eval=t,
    args=(L, IBOF_closure, folgar_tucker_ode, {"xi": 1.0, "Ci": 0.01}),
)
a = solution.y.T.reshape(-1, 3, 3)

Closures also accept stacked input of shape (N, 3, 3) or (N, 3, 3, 3, 3). See the examples and the API reference in the documentation.

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

Closures are selected by name via compute_closure(a, closure=...).

Second- to fourth-order ((N,)3x3 -> (N,)3x3x3x3)

Accepted names: IBOF (default), LINEAR, QUADRATIC, HYBRID, ORF, ORW, ORW3, SQC, SIQ, SIHYB.

• 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)
• SIQ, 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)

Fourth- to sixth-order ((N,)3x3x3x3 -> (N,)3x3x3x3x3x3)

Accepted names: LINEAR, QUADRATIC, HYBRID.

• Linear, Quadratic, Hybrid:
  Suresh G. Advani and Charles L. Tucker III,
  'The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites',
  Journal of Rheology 31(8), 751-784, 1987.
  (https://doi.org/10.1122/1.549945)

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)

Citing

Please cite the archived release, 10.5281/zenodo.4679755; full metadata is in CITATION.cff.