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Research Papers

Numerical Evaluation of Single Fiber Motion for Short-Fiber-Reinforced Composite Materials Processing

[+] Author and Article Information
Dongdong Zhang

Douglas E. Smith1

 Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211SmithDoug@missouri.edu

David A. Jack

 Department of Mechanical Engineering, Baylor University, Waco, TX 76798David_Jack@baylor.edu

Stephen Montgomery-Smith

 Department of Mathematics, University of Missouri, Columbia, MO 65211stephen@missouri.edu

1

Corresponding author.

J. Manuf. Sci. Eng 133(5), 051002 (Sep 13, 2011) (9 pages) doi:10.1115/1.4004831 History: Received March 10, 2011; Revised July 25, 2011; Published September 13, 2011; Online September 13, 2011

This paper presents a computational approach for simulating the motion of a single fiber suspended within a viscous fluid. We develop a finite element method (FEM) for modeling the dynamics of a single rigid fiber suspended in a moving fluid. Our approach seeks solutions using the Newton–Raphson method for the fiber’s linear and angular velocities such that the net hydrodynamic forces and torques acting on the fiber are zero. Fiber motion is then computed with a Runge-Kutta method to update the fiber position and orientation as a function of time. Low-Reynolds-number viscous flows are considered since these best represent the flow conditions for a polymer melt within a mold cavity. This approach is first used to verify Jeffery’s orbit (1922) and addresses such issues as the role of a fiber’s geometry on the dynamics of a single fiber, which were not addressed in Jeffery’s original work. The method is quite general and allows for fiber shapes that include, but are not limited to, ellipsoidal fibers (such as that studied in Jeffery’s original work), cylindrical fibers, and bead-chain fibers. The relationships between equivalent aspect ratio and geometric aspect ratio of cylindrical and other axisymmetric fibers are derived in this paper.

FIGURES IN THIS ARTICLE
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Copyright © 2011 by American Society of Mechanical Engineers
Topics: Fibers , Motion
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Figures

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Figure 1

Coordinate systems of a single ellipsoidal fiber in Jeffery’s theory: x′ axis is along the fiber’s semi-long axis and y′ axis is defined by rotating vector θ∧ with respect to x′ axis by ψ

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Figure 2

Example calculations of Jeffery’s orbit with orbit constants C = 0.3, 1 and +∞

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Figure 3

(a) Three-dimensional finite element model; (b) mesh model in the yz plane with two velocity boundaries; (c) velocity distribution of fluid domain in the yz plane

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Figure 4

Fiber motion in the yz plane: (a) in the vertical direction (φ = 0); (b) in the horizontal direction (φ = π/2)

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Figure 5

Initial set-up of the motion of a single ellipsoidal fiber in a simple shear flow (Uz=γ·y)

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Figure 6

Numerical results of three-dimensional single fiber motion in half period: (a) evolution of φ, θ, ψ; (b) evolution of φ·, θ·, ψ·; (c) evolution of xc , yc , zc ; (d) evolution of x·c, y·c, z·c

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Figure 7

(a) Cylindrical fiber fixed in the vertical direction; (b) cylindrical fiber fixed in the horizontal direction

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Figure 8

Equivalent aspect ratios of cylinders: the comparison of our numerical data (diamonds) and fitted curve (solid curve) with Cox’s theoretical curve (dashed curve) for slender fibers and experimental data (crosses)

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Figure 9

(a) Bead-chain fiber fixed in the vertical direction; (b) bead-chain fiber fixed in the horizontal direction

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Figure 10

Equivalent aspect ratios of bead-chain fibers: the diamonds represent our numerical data and the solid curve is our fitted curve of the data

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Figure 11

Evolution of a cylindrical fiber with geometric aspect ratio re  = 6 and an ellipsoidal fiber with re  = 4.8: (a) evolution of φ, θ, ψ; (b) evolution of φ·, θ·, ψ·

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Figure 12

Evolution of a bead-chain fiber with the geometric aspect ratio re  = 6 and an ellipsoidal fiber with re  = 5.4: (a) evolution of φ, θ, ψ; (b) evolution of φ·, θ·, ψ·

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Figure 13

Comparison of the motions of ellipsoid, cylinder, and bead-chain fiber with the same geometric aspect ratio re  = 6 (period T given as shown)

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