Research Papers

On the Buckling of a Two-Dimensional Micropolar Strip

[+] Author and Article Information
Armanj D. Hasanyan

Department of Aerospace Engineering,
University of Michigan,
Ann Arbor, MI 48109-2102
e-mail: armanj@umich.edu

Anthony M. Waas

Felix Pawlowski Collegiate Professor
Department of Aerospace Engineering and
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109-2102

*Present address: Boeing-Egtvedt Chair and Chairman, Department of Aeronautics and Astronautics, University of Washington, Seattle WA 98195-2400. E-mail: awaas@aa.washington.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 27, 2014; final manuscript received January 26, 2015; published online February 26, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(4), 041006 (Apr 01, 2015) (10 pages) Paper No: JAM-14-1612; doi: 10.1115/1.4029680 History: Received December 27, 2014; Revised January 26, 2015; Online February 26, 2015

This study examines the buckling of a single strip of material, modeled as a two-dimensional (2D) micropolar solid. The effects of material microstructure are incorporated by modeling the material using micropolar theory. By setting the micropolar constants to zero, the equations of classical elasticity are obtained and these results are compared to the buckling analysis performed by previous authors on elastic materials. In the limiting case, when the thickness of the strip becomes small in comparison to the overall length, the micropolar beam equations are developed. Because buckling analysis requires the consideration of geometric nonlinearity, nonlinear micropolar equations are derived using a variational procedure, which also results in variationally consistent boundary conditions. Due to the complexity of micropolar theory, its application has been limited to linear analysis with a few exceptions.

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Voigt, W., 1892, “Theoretische Studien über die Elasticitätsverhältnisse der Krystalle I,” Abh. d Kön. Ges. d Wiss. Göttingen, pp. 3–52. http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/voigt_-_elasticity_of_crystals_-_i.pdf
Cosserat, E., and Cosserat, F., 1909, Theory of Deformable Bodies, Scientific Library, A. Hermann and Sons, Paris.
Dyszlewicz, J., 2004, Micropolar Theory of Elasticity, Springer-Verlag, Berlin.
Eringen, A. C., and Suhubi, E. S., 1964, “Nonlinear Theory of Simple Micro-Elastic Solids—I,” Int. J. Eng. Sci., 2(2), pp. 189–203. [CrossRef]
Suhubi, E. S., and Eringen, A. C., 1964, “Nonlinear Theory of Simple Micro-Elastic Solids—II,” Int. J. Eng. Sci., 2(4), pp. 389–404. [CrossRef]
Kafadar, C. B., and Eringen, A. C., 1971, “Micropolar Media—I the Classical Theory,” Int. J. Eng. Sci., 9(3), pp. 271–305. [CrossRef]
Erbay, S., Erbay, H. A., and Dost, S., 1991, “Nonlinear Wave Modulation in Micropolar Elastic Media—I. Longitudinal Waves,” Int. J. Eng. Sci., 29(7), pp. 845–858. [CrossRef]
Gauthier, R. D., 1974, “Analytical and Experimental Investigations in Linear Isotropic Micropolar Elasticity,” Ph.D. thesis, University of Colorado, Boulder, CO.
Gauthier, R. D., and Jahsman, W. E., 1975, “A Quest for Micropolar Elastic Constants,” ASME J. Appl. Mech., 42(2), pp. 369–374. [CrossRef]
Mora, R. J., and Waas, A. M., 2007, “Evaluation of the Micropolar Elasticity Constants for Honeycombs,” Acta Mech., 192(1–4), pp. 1–16. [CrossRef]
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., and Huang, Y., 1998, “Fracture Analysis of Cellular Materials: A Strain Gradient Model,” J. Mech. Phys. Solids, 46(5), pp. 789–828. [CrossRef]
Trovalusci, P., Starzewski, M. O., Bellis, M. L. D., and Murrali, A., 2014, “Scale-Dependent Homogenization of Random Composites as Micropolar Continua,” Eur. J. Mech.—A. Solids, 49, pp. 396–407. [CrossRef]
Trovalusci, P., Bellis, M. L. D., Starzewski, M. O., and Murrali, A., 2014, “Particulate Random Composites Homogenized as Micropolar Materials,” Meccanica, 49(11), pp. 2719–2727. [CrossRef]
Yang, J. F. C., and Lakes, R. S., 1980, “Effect of Couple Stresses in Compact Bone: Transient Experiments,” Adv. Bioeng., 2, pp. 65–67. [CrossRef]
Yang, J. F. C., and Lakes, R. S., 1982, “Experimental Study of Micropolar and Couple Stress Elasticity in Compact Bone in Bending,” J. Biomech., 15(2), pp. 91–98. [CrossRef] [PubMed]
Starzewski, M. O., 2008, Microstructural Randomness and Scaling in Mechanics of Materials, Chapman and Hall, London.
Ji, W., and Waas, A. M., 2010, “2D Elastic Analysis of the Sandwich Panel Buckling Problem: Benchmark Solution and Accurate Finite Element Formulations,” Z. Angew. Math. Phys., 61(5), pp. 897–917. [CrossRef]
Starzewski, M. O., and Jasiuk, I., 1995, “Stress Invariance in Planar Cosserat Elasticity,” Proc. R. Soc. A, 451(1942), pp. 453–470. [CrossRef]
Ramezani, S., Naghdabadi, R., and Sohrabpour, S., 2009, “Analysis of Micropolar Elastic Beams,” Eur. J. Mech. A. Solids, 28(2), pp. 202–208. [CrossRef]
Ambartzumian, S. A., 2001, “The Theory of the Transversal Isotropic Plate With the Account of the Couple Stresses,” Proc. Natl. Acad. Sci. Armenia Mech., 54(1), pp. 3–16.


Grahic Jump Location
Fig. 1

Configuration of the problem studied

Grahic Jump Location
Fig. 2

Comparison between Timoshenko theory (red), elasticity solution according to Ref. [19] (blue), micropolar theory (green), Euler–Bernoulli beam theory (black) when k˜ = γ˜ = 0 and λ˜ = 2.0

Grahic Jump Location
Fig. 3

The effect of the micropolar constant (a) k˜, (b) γ˜, and (c) their coupling effect on the buckling load s˜ = σ/2μ: (a) k˜≠0,γ˜ = 0, and λ˜ = 2.0; (b) k˜ = 0,γ˜≠0, and λ˜ = 2.0; (c) k˜≠0,γ˜≠0, and λ˜ = 2.0

Grahic Jump Location
Fig. 4

Comparison of 1DMB theory (red) with the micropolar solution (blue): (a) γ˜ = 0,k˜ = 0; (b) γ˜ = 0,k˜ = 0.4; (c) γ˜ = 0.2,k˜ = 0; and (d) γ˜ = 0.2,k˜ = 0.4

Grahic Jump Location
Fig. 5

Deformation modes for an elastic solid (k˜ = γ˜ = 0) for low and high values of L/2πh: (a) L/2πh = 0.6; (b) L/2πh = 6.0; (c) L/2πh = 0.6; (d) L/2πh = 6.0; (e) L/2πh = 0.6; (f) L/2πh = 6.0

Grahic Jump Location
Fig. 6

Deformation modes for a micropolar solid (k˜ = 0.4,γ˜ = 0.2) for low and high values of L/2πh: (a) L/2πh = 0.6; (b) L/2πh = 6.0; (c) L/2πh = 0.6; (d) L/2πh = 6.0; (e) L/2πh = 0.6; and (f) L/2πh = 6.0



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