Buckling of a flush-mounted plate in simple shear flow

Abstract

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.

Buckling of a pre-compressed or pre-stretched membrane in shear flow

“…The computation of these bifurcation points and associated eigenfunctions is the main objective of our work. The analysis and results extend the recent work of Luo and Pozrikidis (2006) on the buckling of elastic plates with arbitrary shapes flush mounted on a rigid wall in the absence of pre-stress.…”

“…The trend continues as b becomes zero and then positive, where pre-compression transforms into pre-stretching. The critical value for an unstressed membrane, a = 72.71 for b = 0, was identified previously by Luo and Pozrikidis (2006) by analytical methods and confirmed by finite element calculations. Buckled shapes for pre-compressed and a pre-stretched membrane, b = À10 and 10, are shown in Fig.…”

“…The first solution branch in this case originates from the critical point b = À26.3746, a = 0, where the membrane buckles under compression in the absence of hydrodynamic load. The critical value, a=136.33 for b = 0, is identical to that computed by Luo and Pozrikidis (2006) for an unstressed membrane. Buckled shapes for pre-compressed and pre-stretched membranes exhibiting notable oscillations are shown in Fig.…”

Surface wrinkling of a channelized flow

“…Perhaps the closest earlier work to our current study is by Luo & Pozrikidis (2006, 2007. They considered the buckling of an arbitrarily shaped elastic section in an otherwise rigid plate suspended in flowing fluid.…”

Flow-induced deformation of an elastic membrane adhering to a wall

“…The flow-induced deformation of vesicles and red blood cells has been studied in great detail using elementary and sophisticated models (e.g., Keller and Skalak, 1982;Pozrikidis, 2003;Skotheim and Secomb, 2007;Zhang et al, 2008). Luo and Pozrikidis (2006 recently discussed the buckling of a flat membrane patch with arbitrary shape clamped on a plane wall representing the exposed surface of an endothelial cell.…”