Research Papers

Optimization of Structures Made From Composites With Elliptical Inclusions

[+] Author and Article Information
Christopher D. Kozuch

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
1206 West Green Street,
Urbana, IL 61801
e-mail: ckozuch2@illinois.edu

Iwona M. Jasiuk

Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
1206 West Green Street,
Urbana, IL 61801
e-mail: ijasiuk@illinois.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 29, 2018; final manuscript received August 14, 2018; published online September 14, 2018. Assoc. Editor: Haleh Ardebili.

J. Appl. Mech 85(12), 121006 (Sep 14, 2018) (13 pages) Paper No: JAM-18-1376; doi: 10.1115/1.4041225 History: Received June 29, 2018; Revised August 14, 2018

This paper seeks to determine the relationship between the parameters that define microstructures composed of a matrix with periodic elliptical inclusions and the effectiveness of structural optimization through the application of existing methods. Stiffness properties for a range of microstructures were obtained computationally through homogenization, and these properties were used to conduct separate homogeneous topology optimization and heterogeneous microstructural optimization on two canonical structural problems. Effectiveness was evaluated on the basis of final total strain energy when compared to a reference configuration. Local minima were found for the two structural problems and various microstructure configurations, indicating that the microstructure of composites with elliptical inclusions can be fine-tuned for optimization. For example, when applying topology optimization to a cantilever beam made from a material with soft, horizontal inclusions, ensuring that the aspect ratio of the inclusions is 2.25 will yield the stiffest structure. In the case of heterogeneous microstructural optimization, one of the results obtained from this analysis was that optimizing the aspect ratio of the inclusion is much more impactful in terms of increasing the stiffness than optimizing the inclusion orientation. The existence of these optimal designs have important implications in composite component design.

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Fig. 1

Unit cell of a periodic composite with elliptical inclusions

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Fig. 2

MBB Beam boundary condition definition

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Fig. 3

Half-MBB beam boundary condition definition

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Fig. 4

Cantilever beam boundary condition definition

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Fig. 5

Stiffness tensor results of the homogenization process for stiff inclusions: (a) binary map for ζ = 1, (b) resulting tensor for ζ = 1, (c) binary map for ζ = 3, θ = 0, (d) resulting tensor for ζ = 3, θ = 0, (e) binary map for ζ = 3, θ = π/4, and (f) resulting tensor for ζ = 3, θ = π/4

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Fig. 6

Design results for half-MBB problem with E1 = 100 MPa, E2 = 1 GPa, and ζ = 4.73

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Fig. 7

Design results for cantilever problem with E1 = 100 MPa, E2 = 1 GPa, and ζ = 4.73

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Fig. 8

Normalized strain energy versus aspect ratio and rotation angle (note that the plots have been rotated for the best viewing angle): (a) cantilever problem with soft inclusion, (b) cantilever problem with stiff inclusion, (c) half-MBB problem with soft inclusion, and (d) half-MBB problem with stiff inclusion

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Fig. 9

Normalized strain energy versus rotation angle: (a) stiff inclusion, ζ = 3 and (b) soft inclusion, ζ = 3

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Fig. 10

Normalized strain energy versus aspect ratio: (a) stiff, horizontal inclusion, (b) stiff, vertical inclusion, (c) soft, horizontal inclusion, and (d) soft, vertical inclusion

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Fig. 11

Heterogeneous optimization of rotation angle (θ) for soft inclusions: (a) convergence for MBB beam, (b) convergence for cantilever beam, (c) solution for MBB beam (π radians), and (d) solution for cantilever beam (π radians)

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Fig. 12

Heterogeneous optimization of angle (θ) for stiff inclusions: (a) convergence for MBB beam, (b) convergence for cantilever beam, (c) solution for MBB beam (π Radians), and (d) solution for cantilever beam (π Radians)

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Fig. 13

Heterogeneous optimization of aspect ratio (ζ) for soft inclusions: (a) convergence for MBB beam, (b) convergence for cantilever beam, (c) solution for MBB beam, and (d) solution for cantilever beam

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Fig. 14

Heterogeneous optimization of aspect ratio (ζ) for stiff inclusions: (a) convergence for MBB beam, (b) convergence for cantilever beam, (c) solution for MBB beam, and (d) solution for cantilever beam

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Fig. 15

Color representation of example rotation angles

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Fig. 16

Color representation of example aspect ratios



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