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Research Papers

Mechanics of Random Discontinuous Long-Fiber Thermoplastics—Part I: Generation and Characterization of Initial Geometry

[+] Author and Article Information
Ahmed I. Abd El-Rahman

Graduate Research Assistant
Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
1206 W. Green Street,
Urbana, IL 61801
e-mail: eabd@illinois.edu

Charles L. Tucker, III

Alexander Rankin Professor
Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
1206 W. Green Street,
Urbana, IL 61801
e-mail: ctucker@illinois.edu

1Current affiliation: Assistant Professor, Department of Physics, School of Science and Engineering, American University in Cairo, P.O. Box 74, New Cairo 11835, Egypt, e-mail: ahmedibrahim@aucegypt.edu.

2Corresponding author.

Manuscript received May 3, 2012; final manuscript received October 21, 2012; accepted manuscript posted January 31, 2013; published online July 12, 2013. Assoc. Editor: Krishna Garikipati.

J. Appl. Mech 80(5), 051007 (Jul 12, 2013) (10 pages) Paper No: JAM-12-1179; doi: 10.1115/1.4023537 History: Received May 03, 2012; Revised October 21, 2012; Accepted January 31, 2013

The deformation mechanics of dry networks of large-aspect-ratio fibers with random orientation controls the processing of long-fiber thermoplastics (LFTs) and greatly affects the mechanical properties of the final composites. Here, we generate initial geometries of fiber networks in a cubic unit cell with a fiber aspect ratio of l/d = 100 and fully periodic boundary conditions for later numerical simulation. The irreversible random sequential adsorption (RSA) process is first used to generate a quasi-random structure due to the excluded-volume requirements. In order to investigate the nonequilibrium character of the RSA, a second method, which is similar to the mechanical contraction method (MCM) (Williams and Philipse, 2003, “Random Packings of Spheres and Spherocylinders Simulated by Mechanical Contraction,” Phys. Rev. E, 67, pp. 1–9) and based on a simplified Metropolis Monte Carlo (MC) simulation is then developed to produce quasi-equilibrium fiber geometries. The RSA packing results (ϕ ≈ 4.423% when using a fiber aspect ratio of 100) are in good agreement with the maximum unforced random packing limits (Evans and Gibson, 1986, “Prediction of the Maximum Packing Fraction Achievable in Randomly Oriented Short-Fibre Composites,” Compos. Sci. Technol., 25, pp. 149–162). The fiber structures were characterized by several distribution functions, including pair-spatial and pair-orientation distributions, based on either the center-to-center distance or the shortest distance between the particles. The results show that the structures generated by the RSA have an easily-detectable long-range spatial correlation but very little orientational correlation. In contrast, the quasi-equilibrium structures have reduced spatial correlation but increased short-range orientational correlation.

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References

Evans, K. E. and Gibson, A. G., 1986, “Prediction of the Maximum Packing Fraction Achievable in Randomly Oriented Short-Fibre Composites,” Compos. Sci. Technol., 25, pp. 149–162. [CrossRef]
Torquato, S., 2002, Random Heterogeneous Materials—Microstructure and Macroscopic Properties, Springer-Verlag, New York.
Evans, K. E. and Ferrar, M. D., 1989, “The Packing of Thick Fibres,” J. Phys. D, 22(2), pp. 354–360. [CrossRef]
Viot, P., Tarjus, G., Ricci, S. M., and Talbot, J., 1992, “Random Sequential Adsorption of Anisotropic Particles. I. Jamming Limit and Asymptotic Behavior,” J. Chem. Phys., 97(7), pp. 5212–5218. [CrossRef]
Ricci, S. M., Talbot, J., Tarjus, G., and Viot, P., 1994, “A Structural Comparison of Random Sequential Adsorption and Equilibrium Configurations of Spherocylinders,” J. Chem. Phys., 101(10), pp. 9164–9180. [CrossRef]
Williams, S. R. and Philipse, A. P., 2003, “Random Packings of Spheres and Spherocylinders Simulated by Mechanical Contraction,” Phys. Rev. E, 67, pp. 1–9. [CrossRef]
Yamane, Y., Kaneda, Y., and Doi, M., 1994, “Numerical Simulation of Semi-Dilute Suspensions of Rodlike Particles in Shear Flow,” J. Non-Newtonian Fluid Mech., 54, pp. 405–421. [CrossRef]
Abd El-Rahman, A. I., 2012, “Mechanics of Random Discontinuous Long-Fiber Thermoplastics—Part II: A Direct Simulation of Unaxial Compression,” J. Rheology. (submitted).
Abd El-Rahman, A. I., 2009, “Mechanics of Random-Fiber Networks: A Direct Simulation,” Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana.
Toll, S., 1998, “Packing Mechanics of Fiber Reinforcements,” Polym. Eng. Sci., 38(8), pp. 1337–1350. [CrossRef]

Figures

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Fig. 1

Coordinate system and definitions of θ, φ, and p for a rigid fiber

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Fig. 2

Geometry of two neighboring fibers, showing the gap distance hij

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Fig. 3

(a) The image of the new parent fiber j intersecting with a pre-existing parent fiber i, and (b) a unit cell containing 272 parent fibers; B = 1.0, r = 100 and ϕ = 2%

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Fig. 4

Maximum packing fractions at the RSA jamming limit in terms of the fiber aspect ratio. Maximum random packing fractions, experimentally found (solid line) ϕ0 = 5.3/r, and theoretically defined (dotted line) ϕ0 = 4.0/r by Evans and Gibson [2].

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Fig. 5

A sketch of the definitions of the shell radius ρ and shell thickness Δρ associated with the reference fiber i and the relative orientation angles η and λ with respect to the fiber j

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Fig. 6

The (a) long-range, and (b) short-range radial pair spatial-distribution function for the RSA and equilibrium configurations

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Fig. 7

The (a) long-range, and (b) short-range center-to-center orientation-distribution function for the RSA and equilibrium configurations

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Fig. 8

A cross section of the spherocylindrical shell surrounding the reference fiber i of length l and a neighboring fiber j. The angle between the two fibers is η. The shell excluded area is obtained by following the center of the fiber j while traveling around fiber i, keeping constant the surface shortest distances between the two fibers.

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Fig. 9

The spatial (top) and orientation (bottom) shortest-distance distribution functions for the RSA and equilibrium configurations

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Fig. 10

Definition of angles λ and η. The spherical shell of the reference fiber j is divided into concentric bins of angular size Δλ = 10 deg.

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Fig. 11

The function H(ρ,λ,η) is plotted in terms of λ and η showing (a) RSA, and (b) equilibrium orientational structures at the most probable separation ρ = 8.25d

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Fig. 12

The function H(ρ,λ,η) plotted (a) versus η at fixed λ = 0 and π/2, and (b) versus λ at fixed η = 0 and η = π/2. In all cases, ρ is fixed to ρmax = 8.25d.

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