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.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Gibson, L. J. , and Ashby, M. F. , 1999, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK.
Lin, T.-C. , Chen, T.-J. , and Huang, J.-S. , 2012, “ In-Plane Elastic Constants and Strengths of Circular Cell Honeycombs,” Compos. Sci. Technol., 72(12), pp. 1380–1386. [CrossRef]
Chung, J. , and Waas, A. M. , 2001, “ In-Plane Biaxial Crush Response of Polycarbonate Honeycombs,” J. Eng. Mech., 127(2), pp. 180–193. [CrossRef]
D'Mello, R. J. , and Waas, A. M. , 2013, “ Inplane Crush Response and Energy Absorption of Circular Cell Honeycomb Filled With Elastomer,” Compos. Struct., 106, pp. 491–501. [CrossRef]
Papka, S. D. , and Kyriakides, S. , 1998, “ In-Plane Crushing of a Polycarbonate Honeycomb,” Int. J. Solids Struct., 35(3–4), pp. 239–267. [CrossRef]
Poritski, H. , and Horvay, G. , 1951, “ Stresses in Pipe Bundles,” ASME J. Appl. Mech., 18, pp. 241–250.
Chung, J. , and Waas, A. M. , 2000, “ The Inplane Elastic Properties of Circular Cell and Elliptical Cell Honeycombs,” Acta Mech., 144(1), pp. 29–42. [CrossRef]
Lin, T.-C. , and Huang, J.-S. , 2013, “ In-Plane Mechanical Properties of Elliptical Cell Honeycombs,” Compos. Struct., 104, pp. 14–20. [CrossRef]
Eringen, A. C. , 1967, “ Theory of Micropolar Elasticity,” Princeton University, Office of Naval Research Department of the Navy, Technical Report No. 1.
Bazant, Z. P. , and Christensen, M. , 1972, “ Analogy Between Micropolar Continuum and Grid Frames Under Initial Stress,” Int. J. Solids Struct., 8(3), pp. 327–346. [CrossRef]
Kumar, R. S. , and McDowell, D. L. , 2004, “ Generalized Continuum Modeling of 2-D Periodic Cellular Solids,” Int. J. Solids Struct., 41(26), pp. 7399–7422. [CrossRef]
Chen., J. Y. , Huang, Y. , and Ortiz, M. , 1998, “ Fracture Analysis of Cellular Materials: A Strain Gradient Model,” J. Mech. Phys. Solids, 46(5), pp. 789–828. [CrossRef]
Wang, X. L. , and Stronge, W. J. , 1999, “ Micropolar Theory for Two-Dimensional Stresses in Elastic Honeycomb,” R. Soc., 445, pp. 2091–2116. [CrossRef]
Mora, R. J. , and Waas, A. M. , 2007, “ Evaluation of the Micropolar Elasticity Constants for Honeycombs,” Acta Mech., 192(1), pp. 1–16. [CrossRef]
Chung, J. , and Waas, A. M. , 2009, “ The Micropolar Elasticity Constants of Circular Cell Honeycombs,” Proc. R. Soc. A, 465(2101), pp. 25–39. [CrossRef]
Onck, P. R. , 2002, “ Cosserat Modeling of Cellular Solids,” C. R. Méc., 330(11), pp. 717–722. [CrossRef]
Fatemi, J. , Onck, P. R. , Poort, G. , and Keulen, F. V. , 2003, “ Cosserat Moduli of Anisotropic Cancellous Bone: A Micromechanical Analysis,” J. Phys., 105, pp. 273–280.
Bellis, M. L. D. , and Addessi, D. , 2011, “ A Cosserat Based Multi-Scale Model for Masonry Structures,” J. Multiscale Comput. Eng., 9(5), pp. 543–563. [CrossRef]
Trovalusci, P. , Ostoja-Starzewski, M. , Bellis, M. L. D. , and Murrali, A. , 2015, “ Scale-Dependent Homogenization of Random Composites as Micropolar Continua,” Eur. J. Mech. A, 49, pp. 396–407. [CrossRef]
Starzewski, M. O. , 2008, Microstructural Randomness and Scaling in Mechanics of Materials, Chapman and Hall, London.
Jasiuk, I. , and Ostoja-Starzewski, M. , 2003, “ From Lattices and Composites to Micropolar Continua,” Micromech. Nanoscale Effects, 10, pp. 175–212.
Li, X. , and Liu, Q. , 2004, “ A Version of Hill's Lemma for Cosserat Continuum,” Acta Mech. Sin., 25(4), pp. 499–506. [CrossRef]
Hohe, J. , and Becher, W. , 2001, “ Effective Stress–Strain Relations for Two-Dimensional Cellular Sandwich Cores: Homogenization, Material Models, and Properties,” ASME Appl. Mech. Rev., 55(1), pp. 61–87. [CrossRef]


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



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In