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Research Papers

Micropolar Constitutive Relations for Cellular Solids

[+] Author and Article Information
Armanj D. Hasanyan

Department of Aerospace Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: armanj@umich.edu

Anthony M. Waas

Fellow ASME
Department of Aeronautics and Astronautics,
University of Washington,
Seattle, WA 98195
e-mail: awaas@aa.washington.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 4, 2015; final manuscript received November 25, 2015; published online January 4, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(4), 041001 (Jan 04, 2016) (10 pages) Paper No: JAM-15-1534; doi: 10.1115/1.4032115 History: Received October 04, 2015; Revised November 25, 2015

With the recent development of micromechanics in micropolar solids, it is now possible to characterize the macroscopic mechanical behavior of cellular solids as a micropolar continuum. The aim of the present article is to apply these methods to determine the micropolar constitutive relation of various cellular solids. The main focus will be on the hexagonal packed circular honeycomb to demonstrate how its constitutive relationship is obtained. In addition, the same method will be applied to determine the material properties of a grid structure and a regular hexagon honeycomb. Since we model the cellular solid as an assembly of Euler–Bernoulli beams, we know that the macroscopic material properties will depend on the cell wall thickness, length, and Young's modulus. From this, and in conjunction with nondimensional analysis, we can provide a closed form solution, up to a multiplicative constant, without resorting to analyzing the governing equations. The multiplicative constant is evaluated through a single numerical simulation. The predicted values are then compared against assemblies with different local properties, using the numerical result as a benchmark since it takes into account higher order thickness effects. It is concluded that our closed form expressions vary from the numerical predictions only when the thickness of the beams increase, as expected since shear effects must be taken into account. However, for most engineering applications, these expressions are practical since our closed form solution with the Euler–Bernoulli assumption only produces about 10% error for most extreme cases. Our results are also verified by comparing them against those reported in the literature.

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Figures

Grahic Jump Location
Fig. 1

Configuration of a honeycomb at the global and local scales. (a) In-plane view of a hexagonally packed circular cell honeycomb and (b) diamond shape RVE of the honeycomb.

Grahic Jump Location
Fig. 2

Effect of thickness t on material constants: (a) μ¯, (b) λ¯, (c) μ¯c, and (d) γ¯c

Grahic Jump Location
Fig. 3

(a) Grid and (b) regular hexagon structures along with their RVE under consideration

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