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Research Papers

Buckling of a Circular Plate Resting Over an Elastic Foundation in Simple Shear Flow

[+] Author and Article Information
Haoxiang Luo1

Department of Mechanical Engineering, Vanderbilt University, VU Station B 351592, 2301 Vanderbilt Pl, Nashville, TN 37235-1592haoxiang.luo@vanderbilt.edu

C. Pozrikidis

Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411

1

Corresponding author.

J. Appl. Mech 75(5), 051007 (Jul 15, 2008) (6 pages) doi:10.1115/1.2937137 History: Received July 03, 2007; Revised March 27, 2008; Published July 15, 2008

The elastic instability of a circular plate adhering to an elastic foundation modeling the exposed surface of a biological cell resting on the cell interior is considered. Plate buckling occurs under the action of a uniform body force due to an overpassing simple shear flow distributed over the plate cross section. The problem is formulated in terms of the linear von Kármán plate bending equation incorporating the body force and the elastic foundation spring constant, subject to clamped boundary conditions around the rim. The coupling of the plate to the substrate delays the onset of the buckling instability and may have a strong effect on the shape of the bending eigenmodes. Contrary to the case of uniform compression, as the shear stress of the overpassing shear flow increases, the plate always first buckles in the left-to-right symmetric mode.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 6

First few eigenvalues, α=τ̂, plotted against ν for a circular membrane with the spring stiffness (a) γ=0, (b) γ=625, and (c) γ=4096. From bottom to top along ν=0, the curves represent modes S1, S2, A1, A2, and S3.

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Figure 7

(a) The lowest eigenvalues of the one-dimensional model, α=τ̂ (solid line), are compared with the S1 eigenvalues of the two-dimensional model (dashed line). (b) and (c) The solid lines illustrate the eigenfunctions of the one-dimensional model for (b) γ=0 and (c) γ=625. The profiles of the two-dimensional eigenfunction S1 at y=0 are shown as dashed lines.

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Figure 1

Shear flow past a membrane patch modeled as an elastic plate flush mounted on a plane wall. The lateral deformation of the membrane is resisted by an elastic material supporting the membrane from underneath.

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Figure 5

Comparison of the buckling mode profiles for ν=0.25 and γ=0 (dash-dotted line), γ=625 (dashed line), and γ=6561 (solid line), and buckling modes (a) S1, (b) S2, and (c) S3

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Figure 4

Buckling eigenmodes for ν=0.25, γ=625, and (a) τ̂=217.24 (S1), (b) τ̂=282.93 (S2), (c) τ̂=291.82 (A1), and (d) τ̂=371.08 (A2)

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Figure 3

Effect of the elastic foundation constant on the square root of the lowest eigenvalue, α=τ̂, for Poisson ratio (a) ν=0 and (b) ν=0.25. From bottom to top, the curves represent modes S1, S2, A1, A2, and S3, where “S” denotes a symmetric mode and “A” denotes an antisymmetric mode.

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Figure 2

(a) Effect of the elastic foundation constant γ on the lowest eigenvalues of a radially compressed circular plate for n=0 (solid line), n=1 (dashed line), and n=2 (dotted line). This figure reproduces Fig. 1 of Wang (3). (b) and (c) Eigenfunctions, pn, for γ1∕4=3, 4, 5, 6, and (b) n=0 and (c) n=1.

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