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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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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