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Research Papers

On Boundary Condition Implementation Via Variational Principles in Elasticity-Based Homogenization

[+] 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 June 12, 2016; final manuscript received July 16, 2016; published online August 10, 2016. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 83(10), 101008 (Aug 10, 2016) (15 pages) Paper No: JAM-16-1292; doi: 10.1115/1.4034227 History: Received June 12, 2016; Revised July 16, 2016

Convergence characteristics of the locally exact homogenization theory for periodic materials, first proposed by Drago and Pindera (2008, “A Locally-Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases,” ASME J. Appl. Mech., 75(5), p. 051010) and recently generalized by Wang and Pindera (“Locally-Exact Homogenization Theory for Transversely Isotropic Unidirectional Composites,” Mech. Res. Commun. (in press); 2016, “Locally-Exact Homogenization of Unidirectional Composites With Coated or Hollow Reinforcement,” Mater. Des., 93, pp. 514–528; and 2016, “Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers,” ASME J. Appl. Mech., 83(7), p. 071010), are examined vis-a-vis the manner of implementing periodic boundary conditions. The locally exact theory separates the unit cell problem into interior and exterior problems, with the separable interior problem solved exactly in cylindrical coordinates and the inseparable exterior problem tackled using a balanced variational principle. This variational principle leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients in the displacement field representation of the unit cell's different phases. Herein, we compare the solution's convergence behavior based on the balanced variational principle with that based on the constrained energy-based principle originally proposed by Jirousek (1978, “Basis for Development of Large Finite Elements Locally Satisfying All Fields Equations,” Comput. Methods Appl. Mech. Eng., 14, pp. 65–92) in the context of locally exact finite-element analysis. The relevance of this comparison lies in the recently rediscovered implementation of Jirousek's constrained variational principle in the homogenization of periodic materials.

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References

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Figures

Grahic Jump Location
Fig. 1

Repeating unit cells of hexagonal and rectangular arrays of fibers

Grahic Jump Location
Fig. 2

Convergence of homogenized moduli E22∗, G23∗, and G12∗ for unidirectional glass/epoxy, containing 0.20 fiber volume fraction, with the number of harmonics used in the displacement field representation: (a) hexagonal array and (b) square array. Comparison between predictions based on the balanced (left column) and constrained (right column) variational principles.

Grahic Jump Location
Fig. 3

Comparison of σ23(y2,y3) stress fields in unidirectional glass/epoxy containing 0.20 fiber volume fraction for loading by ε¯23=0.01 generated using 2, 8, and 12 harmonics by balanced and constrained variational approaches: (a) hexagonal array and (b) square array. Vertical scale in MPa.

Grahic Jump Location
Fig. 4

Comparison of σ23(y2,y3) stress fields in unidirectional glass/epoxy containing 0.20 fiber volume fraction for loading by ε¯23=0.01 generated using nine harmonics by balanced (left column) and constrained (right column) variational approaches: (a) hexagonal array and (b) square array. Vertical scale in MPa.

Grahic Jump Location
Fig. 5

Comparison of σ22(y2,y3) stress fields in unidirectional glass/epoxy containing 0.20 fiber volume fraction for uniaxial loading by σ¯22 at ε¯22=0.01 generated using nine harmonics by balanced (left column) and constrained (right column) variational approaches: (a) hexagonal array and (b) square array. Vertical scale in MPa.

Grahic Jump Location
Fig. 6

Comparison of σ23(y2,y3) stress fields in unidirectional glass/epoxy containing 0.60 fiber volume fraction for loading by ε¯23=0.01 generated using nine harmonics by balanced (left column) and constrained (right column) variational approaches: (a) hexagonal array and (b) square array. Vertical scale in MPa.

Grahic Jump Location
Fig. 7

Convergence of homogenized moduli E22∗, G23∗, and G12∗ for aluminum with cylindrical holes containing 0.20 porosity volume fraction with the number of harmonics used in the displacement field representation: (a) hexagonal array and (b) square array. Comparison between predictions based on the balanced (left column) and constrained (right column) variational principles.

Grahic Jump Location
Fig. 8

Comparison of σ22(y2,y3) stress fields in aluminum with cylindrical holes containing 0.20 porosity volume fraction for uniaxial loading by σ¯22 at ε¯22=0.01 generated using nine harmonics by balanced (left column) and constrained (right column) variational approaches: (a) hexagonal array and (b) square array. Vertical scale in MPa.

Grahic Jump Location
Fig. 9

Convergence of homogenized transverse Young's modulus E22∗ for unidirectional graphite/epoxy with coated fibers, containing 0.50 fiber volume fraction, with the number of harmonics used in the displacement field representation: (a) hexagonal array and (b) square array. Comparison between predictions based on the balanced (left column) and constrained (right column) variational principles.

Grahic Jump Location
Fig. 10

Comparison of (a) σ22(y2,y3) and (b) σ23(y2,y3) stress fields in the square unit cell representative of unidirectional graphite/epoxy with coated fibers containing 0.50 fiber volume fraction for uniaxial loading by σ¯22 at ε¯22=0.01 generated using 14 harmonics by balanced (left column) and constrained (right column) variational approaches. Vertical scale in MPa.

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