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Research Papers

Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers

[+] Author and Article Information
Guannan Wang, Marek-Jerzy Pindera

Civil Engineering Department,
University of Virginia,
Charlottesville, VA 22904-4742

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 21, 2016; final manuscript received April 13, 2016; published online May 11, 2016. Assoc. Editor: Harold S. Park.

J. Appl. Mech 83(7), 071010 (May 11, 2016) (11 pages) Paper No: JAM-16-1096; doi: 10.1115/1.4033430 History: Received February 21, 2016; Revised April 13, 2016

The elasticity-based, locally exact homogenization theory for unidirectional composites with hexagonal and tetragonal symmetries and transversely isotropic phases is further extended to accommodate cylindrically orthotropic reinforcement. The theory employs Fourier series representations of the fiber and matrix displacement fields in cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. Satisfaction of periodicity conditions for the inseparable exterior problem is efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement solution convergence with relatively few harmonic terms. As demonstrated in this contribution, this also applies to cylindrically orthotropic reinforcement for which the eigenvalues depend on both the orthotropic elastic moduli and harmonic number. The solution's demonstrated stability facilitates rapid identification of cylindrical orthotropy's impact on homogenized moduli and local fields in wide ranges of fiber volume fraction and orthotropy ratios. The developed theory provides a unified approach that accounts for cylindrical orthotropy explicitly in both the homogenization process and local stress field calculations previously treated separately through a fiber replacement scheme. Comparison of the locally exact solution with classical solutions based on an idealized microstructural representation and fiber moduli replacement with equivalent transversely isotropic properties delineates their applicability and limitations.

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References

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Figures

Grahic Jump Location
Fig. 1

Repeating unit cell of a hexagonal array of fibers with cylindrically orthotropic microstructures: (a) transversely isotropic, (b) circumferentially orthotropic, and (c) radially orthotropic

Grahic Jump Location
Fig. 2

Eigenvalues for the in-plane displacement field representation in graphite fibers with different cylindrical orthotropy ratios as a function of the harmonic number

Grahic Jump Location
Fig. 3

Comparison of converged stress distributions σ22(y2,y3) in a hexagonal unit cell with a dilute fiber volume fraction subjected to uniaxial loading by σ¯22≠0 only: (a) matrix only, (b) radially orthotropic fiber only with Err/Eθθ≈28, and (c) circumferentially orthotropic fiber only with Err/Eθθ≈1/28. Vertical scale in MPa.

Grahic Jump Location
Fig. 4

Convergence of selected homogenized moduli with the number of harmonics for a unidirectional composite with radially and circumferentially orthotropic (R-O and C-O, respectively) graphite fibers and υf=0.50

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Fig. 5

Comparison of homogenized moduli of a unidirectional composite with radially and circumferentially orthotropic graphite fibers predicted by the locally exact theory and Hashin's CCA model based on equivalent transversely isotropic fiber moduli

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Fig. 6

Comparison of stress fields in the matrix phase of the hexagonal unit cell with a radially orthotropic fiber and the CCA model with an equivalent transversely isotropic fiber of a unidirectional composite with the fiber volume fraction υf=0.65 under pure axial shear loading by σ¯12≠0 at the applied axial shear strain of ε¯12=0.01 : (a) σ12(y2,y3) and (b) σ13(y2,y3). Vertical scale in MPa.

Grahic Jump Location
Fig. 7

Selected homogenized moduli predicted by the locally exact theory based on cylindrically orthotropic graphite fibers and comparison with predictions based on equivalent transversely isotropic graphite fiber moduli (top and bottom left). Verification of transverse isotropy retention (bottom right).

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Fig. 8

Matrix transverse shear stress fields σ23(y2,y3) predicted by the locally exact theory based on cylindrically orthotropic graphite fibers (left) and their transversely isotropic equivalents (right) in a unidirectional composite with the fiber volume fraction υf=0.65 under pure transverse shear loading by σ¯23≠0 at the applied transverse shear strain of ε¯23=0.01 : (a) radially orthotropic fiber and transversely isotropic equivalent and (b) circumferentially orthotropic fiber and transversely isotropic equivalent. Vertical scale in MPa.

Grahic Jump Location
Fig. 9

Fiber transverse shear stress fields σ23(y2,y3) predicted by the locally exact theory based on cylindrically orthotropic graphite fibers (left) and their transversely isotropic equivalents (right) in a unidirectional composite with the fiber volume fraction υf=0.65 under pure transverse shear loading by σ¯23≠0 at the applied transverse shear strain of ε¯23=0.01 : (a) radially orthotropic fiber and transversely isotropic equivalent and (b) circumferentially orthotropic fiber and transversely isotropic equivalent. Vertical scale in MPa.

Grahic Jump Location
Fig. 10

Comparison of homogenized moduli of hexagonal and square arrays predicted by the locally exact theory

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