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Arch Appl Mech (2006) 76: 549–566DOI 10.1007/s00419-006-0061-5

ORIGINAL

H. Luo · C. Pozrikidis

Buckling of a flush-mounted plate in simple shear flow

Received: 15 February 2006 / Accepted: 28 June 2006 / Published online: 12 August 2006© Springer-Verlag 2006

Abstract The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deformingunder the action of a uniform tangential load due to an overpassing simple shear flow is considered. Workingunder the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, aneigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equationwith position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describesthe deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its ownweight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derivean 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 ofthe resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physicallyrepresenting critical thresholds of the uniform tangential load above which the plate buckles and wrinkles dueto the partially compressive developing stresses. The associated eigenfunctions representing possible modesof deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.

Keywords Plate buckling · Membrane wrinkling · Elastic instability · Linear von Kármán equation ·Finite-element method · Generalized eigenvalue problem

1 Introduction

Buckling and wrinkling of biological and fabricated membranes under the action of a compressive load arisesin several physical contexts including shear flow past liquid capsules enclosed by polymerized membranes [1,2], instability of compressed hard films on soft substrates [3], and undulation instability of lamellar phasesconsisting of multiple molecular sheets in channel flow [4]. Although scaling laws describing the topologyof the wrinkled shapes have been derived and classical results from the theory of plates and shells have beeninvoked to estimate critical buckling thresholds, a detailed theoretical analysis of the conditions under whichbuckling and wrinkling occur under specific circumstances is not available.

In this paper, we consider the buckling of a membrane patch modeled as an elastic plate deforming under theaction of a simple shear flow applied on the upper side. The term membrane refers to a biological object ratherthan to a thin shell with infinitesimal bending stiffness, as is traditionally defined in mechanics. Accordingly,we shall refer to the membrane patch interchangeably as a plate. The shear flow imparts to the membrane auniform shear stress that can be distributed over the cross section to yield a uniform body force tangential tothe undeformed shape. Physiological situations where this occurs include instances where a biological cell orvesicle is captured on a surface [5] or an endothelial cell is subjected to capillary blood flow [6]. Fung and Liu

H. Luo · C. Pozrikidis (B)Department of Mechanical and Aerospace Engineering University of California, San Diego, La Jolla, CA 92093-0411, USAE-mail: [email protected]

550 H. Luo, C. Pozrikidis

[7] discuss the mechanics of the endothelium and suggest that the main effect of an overpassing shear flow isto generate tensions over the exposed part of the cell membrane, while the cell interior is virtually unstressed.In an idealized depiction, the exposed membrane is a thin elastic patch anchored around its edges on theendothelium wall and connected to the basal lamina by side walls. In an alternative physical interpretation,the uniform body force may be attributed to the weight of a uniform vertical plate clamped on a rigid wall.The in-plane deformation of the membrane generates in-plane tensions whose precise distribution dependson the material properties, shape of the anchoring rim, and assumed boundary conditions around the rim.Elementary mechanics indicates that the upstream part of the plate is stretched, while the downstream portionis compressed. Compression raises the possibility of buckling and wrinkling when the uniform load exceedscritical thresholds representing bifurcation points in the solution space. The computation of these bifurcationpoints and associated eigenfunctions is the main objective of our work.

In Sect. 2, the relevant eigenvalue problem is formulated based on the linear von Kármán equation for elas-tic plates. Though solutions of similar eigenvalue problems are available for plates and shells with rectangularand circular shapes, with a few exceptions, previous studies have addressed situations where the second deriv-atives on the right-hand side of the von Kármán equation are multiplied by constant coefficients representingspatially uniform in-plane stresses, and this considerably simplifies the analysis. In Sects. 3 and 4, analyticaland numerical solutions are presented for plates with circular, elliptical, and rectangular shapes, and the effectof the plate shape on the threshold levels for instability is discussed.

2 Theoretical model

We consider an elastic membrane flush-mounted on a rigid wall with the edge clamped around the rim, asillustrated in Fig. 1. The upper surface of the membrane is exposed to an overpassing shear flow along thex-axis with velocity ux = Gz, where G is the shear rate, and the z-axis is normal to the wall, as shown in Fig. 1.The lower surface of the membrane is in contact with a stationary fluid medium that is unable to withstandshear stress. The shear flow imparts to the upper surface of the membrane a uniform hydrodynamic shear stress,τ = μG, where μ is the fluid viscosity. In the context of thin-shell theory for a zero-thickness membrane,the shear stress can be smeared from the upper surface into the whole cross section of the membrane. Whenthis is done, the shear stress effectively amounts to an in-plane body force uniformly distributed over the crosssection with components

bx = τ

h= μG

h, by = 0, (1)

where h is the membrane thickness. Justification for smearing the shear stress is provided in the Appendixwhere the numerical solution of an analogous two-dimensional problem obtained by the finite-element methodis discussed. The emerging problem also describes the buckling of a homogeneous clamped plate flush mountedon a vertical wall and deforming under the action of its own weight.

We shall assume that the in-plane stresses developing due to the in-plane deformation in the absence ofbuckling, σi j , are related to the in-plane strains, εi j , by the linear constitutive equation

⎡⎣σxxσyyσxy

⎤⎦ = E

1 − ν2

⎡⎣

1 ν 0ν 1 00 0 1 − ν

⎤⎦ ·

⎡⎣εxxεyyεxy

⎤⎦ , (2)

x

yz

θ

u = Gzx

Membrane

Fig. 1 Shear flow past a membrane patch modeled as an elastic plate flush-mounted on a plane wall

Buckling of a flush-mounted plate in simple shear flow 551

where

εkl = 1

2

(∂vk

∂xl+ ∂vl

∂xk

), (3)

(vx , vy) is the tangential displacement of membrane point particles in the x–y plane, E is the membranemodulus of elasticity, and ν is the Poisson ratio. The upper limit, ν = 0.5, corresponds to an incompressiblematerial. Although the Poisson ratio is positive for the vast majority of materials encountered in practice,zero or slightly negative values have been reported for wrinkled membranes [8]. Physically, smoothing of thewrinkles under uniaxial extension leads to expansion in the lateral direction associated with a negative Poissonratio [9] (see also [10].)

Force equilibrium requires the differential balances

∂σxx

∂x+ ∂σyx

∂y+ bx = 0,

∂σxy

∂x+ ∂σyy

∂y+ by = 0, (4)

subject to the boundary conditions vx = 0 and vy = 0 around the clamped rim of the plate.As a specific application, we consider an elliptical plate whose axes are aligned in the x and y directions.

The rim is described by the equation (x/ax )2 + (y/ay)

2 = 1, where ax and ay are the two semi-axes. Thesolution of the plane-stress problem can be found by inspection and is given by

vx = V

(1 − x2

a2x

− y2

a2y

), vy = 0, (5)

where

V = τ

Eh

(1 − ν2) a2x a2

y

(1 − ν) a2x + 2 a2

y(6)

is a constant with dimensions of length. The associated in-plane stresses are given by

σxx = E

1 − ν2

∂vx

∂x= −2

EV

(1 − ν2) a2x

x = −τh

2 a2y

(1 − ν) a2x + 2 a2

yx,

σxy = E

2(1 + ν)

∂vx

∂y= − EV

(1 + ν) a2y

y = −τh

(1 − ν) a2x

(1 − ν) a2x + 2 a2

yy, (7)

σyy = ν σxx ,

independent of the modulus of elasticity, E . For a circular plate of radius a, ax = ay = a, we obtain thesimplified expressions

vx = τ

Eh

1 − ν2

3 − ν(a2 − x2 − y2), vy = 0, (8)

and associated stresses

σxx = − 2

3 − ν

τ

hx, σxy = −1 − ν

3 − ν

τ

hy, σyy = ν σxx . (9)

These expressions confirm that the stream-wise component of the in-plane normal stress, σxx , is positive (ten-sile) on the upstream half, and negative (compressive) on the downstream half of the plate. The transversecomponent of the normal stress, σyy , is also positive or negative depending on the sign of the Poisson ratio.

Compression raises the possibility of buckling and wrinkling when the shear stress, τ , exceeds a criticalthreshold. To compute the transverse deflection along the z axis upon inception of buckling, z = f (x, y), we


work under the auspices of linear elastic stability of thin plates and shells and derive the linear von Kármánequation,

∇4 f ≡ ∇2∇2 f = ∂4 f

∂x4 + 2∂4 f

∂x2 ∂y2 + ∂4 f

∂y4

= h

EB

(σxx

∂2 f

∂x2 + 2 σxy∂2 f

∂x∂y+ σyy

∂2 f

∂y2 − bx∂ f

∂x− by

∂ f

∂y

), (10)

where EB is the bending modulus (e.g., [11] p. 18; [12], p. 305).This is a fourth-order differential equation with position-dependent coefficients multiplying the second

derivatives on the right-hand side. Since the membrane is clamped around the rim, the deflection satisfies thehomogeneous Dirichlet and Neumann boundary conditions f = 0 and ∂ f/∂n = 0, where ∂/∂n denotes thenormal derivative around the rim in the x–y plane.

Substituting the expressions for the in-plane shear stresses in (10), and nondimensionalizing lengths by acharacteristic membrane surface length, a, we derive the dimensionless parameter τ̂ = (τa3)/EB, expressingthe strength of the shear flow relative to the developing bending moments. Equation (11) admits the trivialsolution, f = 0, for any value of τ̂ , and nontrivial eigensolutions at a sequence of discrete eigenvalues. Thecomputation of these eigenvalues and corresponding eigenfunctions for a specified membrane shape is themain objective of our analysis.

3 Fourier series solution for a circular plate

We begin by considering the buckling of a circular plate of radius a. Substituting expressions (9) for thein-plane stresses and expression (1) for the body force in (10), we obtain

∇4 f = − α

a3

[x∂2 f

∂x2 + (1 − ν) y∂2 f

∂x∂y+ ν x

∂2 f

∂y2 + 3 − ν

2

∂ f

∂x

], (11)

where

α ≡ 2τa3

EB(3 − ν)(12)

is a dimensionless parameter. The solution is to be found subject to the homogeneous Dirichlet and Neumannboundary conditions, f = 0 and ∂ f/∂r = 0 at r = a.

An eigensolution of (11) can be expressed as a Fourier series with respect to the plane polar angle θ , definedsuch that x = r cos θ and y = r sin θ , as shown in Fig. 1, in the form

f (r, θ) = 1

2p0(r)+

∞∑n=1

(pn(r) cos nθ + qn(r) sin nθ

)=

∞∑n=−∞

Fn(r) exp(−inθ), (13)

where i is the imaginary unit, pn(r), qn(r) are real functions,

Fn(r) ≡ 1

2

(pn(r)+ i qn(r)

)(14)

for n ≥ 0, Fn(r) = F∗−n(r), and an asterisk denotes the complex conjugate. A straightforward computationyields

∇4 f =∞∑

n=−∞n(r) exp(−inθ), (15)

where

n(r) ≡ F ′′′′n + 2

rF ′′′

n − 1 + 2n2

r2 F ′′n + 1 + 2n2

r3 F ′n + n2 n2 − 4

r4 Fn, (16)


and a prime denotes a derivative with respect to r . To express the right-hand side of (11) in a Fourier series aswell, we use the Cartesian-to-plane-polar transformation rules

∂ f

∂x= cos θ

∂ f

∂r− sin θ

r

∂ f

∂θ,

∂2 f

∂x2 = cos2 θ∂2 f

∂r2 − sin 2θ

r

∂2 f

∂r∂θ+ sin2 θ

r

∂ f

∂r+ sin 2θ

r2

∂ f

∂θ+ sin2 θ

r2

∂2 f

∂θ2 ,

(17)∂2 f

∂y2 = sin2 θ∂2 f

∂r2 + sin 2θ

r

∂2 f

∂r∂θ+ cos2 θ

r

∂ f

∂r− sin 2θ

r2

∂ f

∂θ+ cos2 θ

r2

∂2 f

∂θ2 ,

∂2 f

∂x∂y= 1

2

(sin 2θ

∂2 f

∂r2 + 2cos 2θ

r

∂2 f

∂r∂θ− sin 2θ

r

∂ f

∂r− 2

cos 2θ

r2

∂ f

∂θ− sin 2θ

r2

∂2 f

∂θ2

).

Using Euler’s formula to express the cosines and sines in terms of the complex exponential exp(−iθ), andsubstituting the Fourier expansion, we find

∂ f

∂x= 1

2

∞∑n=−∞

[ (F ′

n + nF

r

)eiθ +

(F ′

n − nF

r

)e−iθ

]exp(−inθ),

∂2 f

∂x2 = 1

2

∞∑n=−∞

[Hn + Rn e2iθ + Tn e−2iθ

]exp(−inθ),

(18)∂2 f

∂y2 = 1

2

∞∑n=−∞

[Hn − Rn e2iθ − Tn e−2iθ

]exp(−inθ),

∂2 f

∂x∂y= − i

2

∞∑n=−∞

[Rn e2iθ − Tn e−2iθ

]exp(−inθ),

where

Hn ≡ F ′′n + F ′

n

r− n2 Fn

r2 , Rn ≡ 1

2

[F ′′

n + (2n − 1)F ′

n

r+ n(n − 2)

Fn

r2

],

(19)

Tn ≡ 1

2

[F ′′

n − (2n + 1)F ′

n

r+ n(n + 2)

Fn

r2

].

If the function Fn is real, in which case Fn = F−n , the functions Rn and Tn are related by Rn = T−n .Using the preceding expressions, we find that the term enclosed by the square brackets on the right-hand

side of (11) takes the simple form

1

2

∞∑n=−∞

[n eiθ +�n e−iθ

]exp(−inθ) = 1

2

∞∑n=−∞

[n+1 +�n−1

]exp(−inθ), (20)

where

n = r F ′′n +

[3 + ν

2+ (1 − ν)n

]F ′

n + n

(1 + ν

2− νn

)Fn

r,

(21)

�n = r F ′′n +

[3 + ν

2− (1 − ν)n

]F ′

n − n

(1 + ν

2+ νn

)Fn

r.

Substituting (20) and (15) in (11), and equating corresponding Fourier coefficients, we derive an infinitetridiagonal system of ordinary differential equations,

n = − α

2a3 (n+1 +�n−1), (22)

for n = 0,±1,±2, . . . . Approximate eigenvalues can be computed by truncating the system at a finite level,n = ±N , and solving the eigenvalue problem defined by the retained ordinary differential equations. In the


case of eigensolutions with left-to-right symmetry with respect to the z–x plane, the Fourier series only involvescosine terms, the component functions Fn are real, Fn = F−n , and −n = �n . The general system (22) thenreduces to

0 = − α

a3 1, n = − α

2a3 (n+1 +�n−1), (23)

for n = 1, 2, . . . , N . On the other hand, in the case of eigensolutions that are antisymmetric with respect to thez–x plane, the Fourier series only involves sine terms, the component functions Fn are imaginary, Fn = −F−n ,−n = −�n , and 0 = 0.

To satisfy the boundary conditions F = F ′ = 0 around the rim, we set

Fi = (a − r)2 Hi (r), (24)

and approximate the modulating functions Hi (r) with M th-degree polynomials in r ,

Hi (r) = ci0 + ci1r + · · · + ci Mr M , (25)

where M is a specified polynomial order. Substituting (24) and (25) into the governing differential equations,computing the derivatives analytically by differentiating the polynomials, and enforcing the resulting equationsat the Chebyshev collocation points

r j = cos

[(j − 1

2

)π

M + 1

], (26)

for j = 1, 2, . . . ,M + 1, we obtain a generalized eigenvalue problem of the form

A · c = α B · c, (27)

where the vector c contains the polynomial coefficients ci j . The banded matrices A and B arise from the orthogo-nal collocation. Specifically, the matrix A consists of 2N+1 diagonal blocks with dimensions (M+1)×(M+1),and the matrix B consists of 2N super-diagonal and sub-diagonal blocks with same dimensions.

The numerical task is to compute as many eigenvalues as possible, beginning with the smallest eigenvalueand moving upward. Physically, the smallest eigenvalue represents the minimum shear stress for buckling.Numerical experimentation showed that the best results are obtained by first inverting the matrix A, and thensolving the standard eigenvalue problem, D · c = λ c, for λ ≡ 1/α, concentrating on the largest eigenvalue, λ,and moving downward, where D ≡ A−1 · B. Since the matrix A arises from the discretization of the ellipticbiharmonic operator, it is nonsingular and well conditioned. By contrast, the matrix B is poorly conditioned.The computation of the eigenvalues was carried out using the Matlab eig function.

Eigenvalues come in degenerate positive–negative pairs representing identical buckling states that arisewhen the shear flow is oriented toward the positive or negative direction of the x-axis. All the correspondingeigenvectors, c, turn out to be real, which means the eigenfunctions are either symmetric or antisymmetricwith respect to the z–x plane, designated by S or A, respectively. Symmetry or antisymmetry can be detectedby comparing the signs of the coefficients of the polynomials H−n and Hn , and was verified by solving (23).Multiple eigenvalues with distinct eigensolutions arise only at specific values of the Poisson ratio, as will bediscussed in the next section.

By way of illustrating the performance of the numerical method and demonstrating the convergence ofthe numerical results, in Table 1a we list numerical values for the smallest eigenvalue for ν = 0.5, against theFourier truncation level, N , and polynomial degree, M . The corresponding mode of deformation is designatedas symmetric mode, S1, as shown in Fig. 2. The results in Table 1a suggest that accuracy up to the fifth decimalpoint for this mode can be achieved with a moderate Fourier truncation level, N = 6, and polynomial degreetruncation level, M = 25. Table 1b illustrates the convergence of the sixth smallest eigenvalue correspondingto the symmetric mode, S4, and Table 1c illustrates the convergence of the twelfth smallest eigenvalue corre-sponding to the symmetric mode, S7. The associated eigenfunctions are shown in Fig. 2. Larger eigenvaluesrequire higher truncation levels, resulting in matrices of large size whose eigenvalues can be identified withsufficient accuracy only up to a point. In the next section, results will be presented for those eigenvalues thatcould be extracted with confidence to shown accuracy.


Table 1 a Convergence of the smallest eigenvalue, α = 71.5795 (S1 mode) for ν = 0.5, listed with respect to the Fouriertruncation level, N , and expansion order, M . b Convergence of the eigenvalue α = 275.18 (S4 mode) for ν = 0.5. c Convergenceof the eigenvalue α = 452 (S7 mode) for ν = 0.5

M (a) N = 1 N = 2 N = 3 N = 4 N = 5 N = 6

14 78.77035 95.12050 58.18914 80.58670 79.79018 79.7730817 78.77057 73.47277 70.94721 70.82917 70.81285 70.8121818 " 72.05807 71.61643 71.60297 71.60484 71.6051021 " 72.07656 71.60049 71.57975 71.57926 71.5792625 " 72.07594 71.60076 ” 71.57951 ”

M (b) N = 5 N = 6 N = 7 N = 8 N = 9 N = 10

22 286.02569 282.62062 280.65468 280.08687 280.00046 280.0294425 282.33248 277.94532 275.83596 275.29922 275.22545 275.2187930 " 277.91176 275.80062 275.26349 275.18968 275.18355

M (c) N = 5 N = 6 N = 7 N = 8 N = 9 N = 10

22 405.02099 517.40551 483.33009 476.57590 461.19203 449.0649925 446.92262 436.43061 469.63320 457.21055 453.31981 451.8913230 450.28448 442.29069 464.69797 454.91600 452.68277 451.94615

3.1 Results and discussion

Table 2 displays the computed eigenvalues for three Poisson ratios, listed in order of increasing magnitude.Concentrating on the first column for ν = 0.5 corresponding to an incompressible membrane, we note theinterlaced appearance of symmetric (S) and antisymmetric (A) modes. A left-to-right break of symmetry inthe solution space occurs for the antisymmetric modes. The component functions, Fn , and correspondingeigenfunctions are plotted in Fig. 2. All functions Fn except for F0, tend to zero at the origin, r = 0, whichis necessary for the deflected shape to be single-valued. A truly incompressible membrane will not be ableto exhibit transverse deflection without reducing the area of the base enclosed by the clamped circular edge.However, the change in area due to the deflection is of second order with respect to the maximum deflectedheight, and the results for ν = 0.5 are relevant to a nearly incompressible material.

To interpret the computed eigenvalues from a physiological standpoint, we reverse the definition of αand find that the critical shear rates where buckling occurs are given by G = α EB (3 − ν)/(2μa3). TakingEB � 1 × 10−12 dyn cm, which is typical of a biological membrane, α = 5μm, and μ = 1.2 cp = 1.2 mPa s, wefind G = α (1 − ν/3) s−1. For ν = 0.5, buckling in the S1 mode will occur when G = 71.6 (1 − 1/6) s−1 �60.0s−1. In human circulation, the shear stress varies in the range 1–2 Pa through all branches, corresponding toG ∼ 100 s−1. This estimate exceeds the S1 buckling threshold but not subsequent thresholds. In flow througha cylindrical capillary of radius b, the wall shear rate is related to the mean velocity, Um, by G = 4Um/b.Taking b = 10 μm, we find that, as Um is raised from 0.001 to 1 cm/s, the wall shear rate increase from 4 to4,000 s−1, which includes several buckling modes.

To establish a further point of reference for the numerical results, we consider a circular membrane thatis compressed around the edges with a uniform force per unit length, Pr. Classical analysis shows that thesymmetric S1 buckling mode occurs at the critical eigenvalue βS1 ≡ Pra2/EB � 14.68 ([12], p. 368) whichis nearly five times smaller than the present eigenvalue for the corresponding mode, αS1 = 71.58. This largedifference is expected in light of the partial compression of the membrane patch in shear flow presently con-sidered. The uniformly compressed plate also exhibits antisymmetric buckling modes associated with a breakof radial symmetry. The first antisymmetric mode, A1, occurs at the critical eigenvalue βA1 � 28.87 ([13], seee.g., [11], p. 230) which is also nearly five times less than the present eigenvalue for the corresponding mode.

The results in Table 2 show that the Poisson ratio can have an important effect on the spectrum of eigen-values. The most striking effect is that, as ν is reduced from 0.5 to 0.25, the relative position of symmetric andantisymmetric eigenmodes is altered. For example, when ν = 0.25, the eigenvalue of the S2 mode is lowerthan that of the A1 mode, and is thus expected to appear first in an experiment where the shear rate is graduallyincreased from zero. Because of mode crossing, there is a critical Poisson ratio where the eigenvalues of the A1and S2 modes are identical. Our computations show that this occurs when ν = 0.42703 and α = 138.308. Theeigenfunctions of this double eigenvalue are arbitrary superpositions of symmetric and antisymmetric modes,and may thus have an arbitrary orientation is space. As the Poisson ratio is further decreased, the ordering of


Fig. 2 Fourier modes and buckled shapes for ν = 0.5, corresponding to the eigenvalue: a α = 71.5795 (S1 mode), b α = 132.128(A1 mode), c α = 137.29 (S2 mode), d α = 208.409 (S3 mode). e α=212.107 (A2 mode) , f α = 275.18 (S4 mode), g α = 300.13(A3 mode), h α = 318.1 (S5 mode), i α=364.5 (A4 mode), j α = 403 (S6 mode), k α = 429 (A5 mode), l α = 452 (S7 mode)

the eigenvalues changes even further. Families of eigenfunctions for Poisson ratios ν = 0.25 and 0 are shownin Figs. 3 and 4. Comparing these shapes with those shown in Fig. 2 for ν = 0.5, we find that the low modes,such as S1, S2, A1, and A2, are virtually identical. The Poisson ratio affects the shape of higher modes bychanging elastic properties of the membrane.

Because in the mathematical model the upper and lower surface of the membrane are indistinguishable,the eigenfunctions can be flipped with respect to the x–y plane in order to facilitate the comparison. In thecontext of the zero-thickness membrane model, the transverse deformation is indeterminate. In reality, theinner surface of a cell membrane is supported by the cytoskeleton and the outer surface of a membrane in


Fig. 2 (contd.)

the endothelium is coated with the glycocalyx and exposed to the shear flow. Numerical solutions of a modelproblem discussed in the Appendix suggest that the indeterminacy is removed for a finite-thickness membrane.

4 Finite-element solution for arbitrary membrane shapes

A dual finite-element method was implemented for describing the buckling of a non-circular plate. The numeri-cal procedure involves two steps: determination of the in-plane membrane tensions in the absence of transversedeflection, and solution of the eigenvalue problem based on the linear von Kármán equation. For ellipticalmembranes aligned with the flow, the analytical solution presented in Sect. 2 was used to generated the in-plane


Fig. 2 (contd.)

tensions. For more general shapes, the tensions were found by performing a plane-stress analysis using theGalerkin finite-element method with six-node quadratic descendant elements (e.g., [14]).

Grid generation was carried out by successively subdividing a parental element structure into smaller ele-ments. The results of the plane-stress code were confirmed to reproduce available analytical solutions. Analternative of this dual implementation would be a unified approach wherein the eigenvalue problem is solvedat one stage in the context of shell buckling. The present formulation reduces the number of unknowns at theexpense of solving a fourth-order differential equation.

Since the von Kármán equation involves the fourth-order biharmonic operator on the left-hand side,Hermitian conforming elements must be used to ensure continuity of the solution and its gradient across


Table 2 Eigenvalues for a circular membrane at three Poisson ratios; S denotes a symmetric mode, and A denotes an antisymmetricmode

Mode ν = 0.5 ν = 0.25 ν = 0

1 71.5795000 (S1) 77.4095411 (S1) 83.4655475 (S1)2 132.128774 (A1) 140.1698392 (S2) 141.6740559 (S2)3 137.294138 (S2) 154.5573454 (A1) 178.89886 (A1)4 208.409 (S3) 225.54982 (A2) 233.834 (A2)5 212.107 (A2) 242.6051 (S3) 263.6254 (S3)6 275.18 (S4) 295.922 (S4) 320.74 (S4)7 300.13 (A3) 340.92 (A3) 365.4 (A3)8 318.1 (S5) 346.67 (S5) 372.55 (S5)9 364.5 (A4) 413.8 (S4) 442 (A4)10 403 (S6) 442 (S6)11 429 (A5) 472 (S7)12 452 (S7) 487 (A5)

Fig. 3 Eigenfunctions for a circular membrane with Poisson ratio ν = 0.25

the element edges, adding to the complexity of the formulation. We have adopted the Hrieh-Clough-Tocher(HCT) triangular element defined by three vertex nodes, three edge nodes, and one hidden interior node (e.g.,[14]). In the formulation, each element is subdivided into three subelements, and the solution over each subel-ement is approximated with a complete cubic polynomial in x and y involving 10 coefficients, for a total of 30coefficients per element. After imposing constraints the C1 continuity condition, the HCT element is endowedwith 12 degrees of freedom consisting of the values of the solution and its Cartesian or directional derivatives.The finite-element grid is identical to that employed for solving the plane-stress problem. The input to thebuckling code includes the components of the in-plane stress tensor at the element vertex nodes.


Fig. 4 Eigenfunctions for a circular membrane with Poisson ratio ν = 0

To develop the Galerkin finite-element method for the buckling problem, we multiply (10) by each one ofthe global interpolation functions associated with the available degrees of freedom, and integrate the productover the patch surface, D, to obtain

∫∫

D

φi (x, y) ∇4 f dx dy =∫∫

D

φi (x, y) w(x, y) dx dy, (28)

where w stands for the right-hand side of (10). Integrating by parts on the left-hand side to reduce the order ofthe biharmonic operator, ∇4 f , we write

∫∫

D

φi ∇4 f dx dy =∫∫

D

φi (x, y) ∇ ·(∇(∇2 f )

)dx dy

=∫∫

D

∇ ·(φi ∇(∇2 f )

)dx dy −

∫∫

D

∇φi · ∇(∇2 f ) dx dy

=∮

C

φi n · ∇(∇2 f ) dl −∫∫

D

∇φi · ∇(∇2 f ) dx dy, (29)

where C is the boundary of D, n is the unit vector in the x–y plane that is normal to C and points outwardfrom D, and l is the arc length along C . Integrating by parts once more the last integral in (29), we find


∫∫

D

φi ∇4 f dx dy =∮

C

φi n · ∇(∇2 f ) dl

−∮

C

∇2 f (n · ∇φi ) dl +∫∫

D

∇2φi ∇2 f dx dy. (30)

Finally, we substitute this expression in (28), and rearrange to obtain∫∫

D

∇2φi ∇2 f dx dy = −∮

C

φi n · ∇(∇2 f ) dl

+∮

C

∇2 f n · ∇φi dl +∫∫

D

φi (x, y) w(x, y) dx dy. (31)

The contour integrals on the right-hand side of (31) are non-zero only if φi is a boundary mode associatedwith a boundary node. If φi is a mode associated with an interior node, these integrals are zero, yielding thesimplified equation

∫∫

D

∇2φi ∇2 f dx dy =∫∫

D

φi (x, y) w(x, y) dx dy. (32)

Since the rim of the membrane is assumed to be clamped, the transverse displacement and its gradient areknown along the boundary, C , the corresponding projections are excluded from the finite-element formulation,and the simplified formulation applies.

Assume that the finite-element expansion involves N mG modes associated with the available degrees of

freedom,

f (x, y) =N m

G∑j=1

hGj φ j (x, y), (33)

where the coefficients hGj represent either the values of the solution or its spatial derivatives at the unique

global nodes. Inserting this expansion in (32), we derive a generalized eigenvalue problem of the form

Ki j hGj = τ̂ Ri j hG

j , (34)

involving N mG unknowns, where τ̂ = (τa3)/EB is the dimensionless membrane tension, and summation over

j is implied on the left-hand side. The global bending stiffness matrix, Ki j , can be compiled in the usual wayfrom the element bending stiffness matrices,

A(l)i j ≡∫∫

El

∇2ψi ∇2ψ j dx dy, (35)

where El denotes the lth element. The matrix Ri j can be compiled from corresponding element matrices. Toeliminate spurious eigenvalues, the final algebraic system is condensed by eliminating the constrained bound-ary degrees of freedom. The best results are obtained by transforming the generalized eigenvalue problem intoa standard eigenvalue problem for the matrix D ≡ K−1 · R, whose eigenvalues are λ = 1/τ̂ . Since we areinterested in the lowest eigenvalues, τ̂ , representing the threshold in the shear flow where buckling occurs, weconcentrate on the highest inverse eigenvalues, λ, which we compute using the Matlab eig function.

The plate buckling code was validated by comparing the results with available solutions. In a first test, thebuckling of an entirely clamped square plate with side length a was considered, subject to compression alongall four edges with a uniform force per unit length, p. The in-plane stress field admits the simple isotropicrepresentation σxx = −p/h, σxy = 0, σyy = −p/h, corresponding to the null body force, bx = 0, by = 0.Dimensionless eigenvalues, p̂ ≡ pa2/EB, computed with two-element discretizations are shown in Table 3a.


Table 3 a Dimensionless eigenvalues, p̂, for a clamped square plate compressed along all four edges; NE is the number ofelements, NF is the number of condensed degrees of freedom, and an asterisk denotes a double eigenvalue. b Dimensionlesseigenvalues, τ̂ , for a clamped circular membrane in shear flow, and Poisson ratio, ν = 0.5; S indicates a symmetric mode, and Aindicates an antisymmetric mode

Finite-element method Finite-element method Separation of variables [15]NE = 128, NF = 323 (a) NE = 512, NF = 1,411

52.824 52.391 52.35893.437* 92.263* 92.157*132.157 128.684 128.302156.788 154.391 154.226171.083 167.469 167.066198.552* 190.702* 189.790*– 247.112* 246.416*– 270.520 269.503

Mode (b) Finite-element method Finite-element method Fourier seriesNE = 64, NF = 163 NE = 256, NF = 707

S1 96.478 90.794 89.474A1 182.621 168.501 165.161S2 185.222 174.222 171.618S3 290.422 267.233 260.511A2 271.613 265.133S4 352.741 343.979A3 387.715 375.162

The third column shows numerical results obtained by the method of separation of variables [15]. The finite-element solution with 512 elements and 1,411 degrees of freedom captures with remarkable accuracy the firsteight eigenvalues up to the second or third significant figure.

In the second test, the buckling of a circular plate of radius a discussed in Sect. 3 was considered. Dimen-sionless eigenvalues, τ̂ , computed using the finite-element method for two-element discretizations are shownin Table 3b for Poisson ratio, ν = 0.5. The numerical results with the fine discretization comprised of 256elements and 707 degrees of freedom are in good agreement with the highly accurate results based on theFourier series. The difference escalates from 1.5% for the first eigenvalue to 3% for the seventh eigenvalue.

Results for non-circular shapes are expected to carry a comparable amount of error.

4.1 Results and discussion

Table 4 documents the effect of the aspect ratio of an elliptical membrane on the spectrum of eigenvalues forν = 0.5 and 0.0. The membrane with aspect ratio 2:1 is oriented in the direction of the flow, and the membranewith aspect ratio 1:2 is oriented transversely to the flow. In all cases, the reduced critical shear stress is definedas τ̂ ≡ τa3/EB; the equivalent membrane radius, a, is related to the membrane surface area by A = πa2. Themost significant effect of the membrane aspect ratio is an increase in the threshold for the onset of bucklingwith respect to the circular shape. A second effect is a change in the order of appearance of symmetric andantisymmetric modes. In particular, as the membrane becomes more elongated in the direction of the flow, thesymmetric modes aggregate at the lowest thresholds. On the other hand, as the membrane elongates laterally,antisymmetric modes are favored near the lowest thresholds. Figure 5 illustrates the first six buckling modes asthey arise from the finite element solution for ν = 0.5. The shapes for ν = 0 are qualitatively similar. Table 5

Table 4 Dimensionless eigenvalues, τ̂ , for a clamped elliptical membrane with Poisson ratio, ν = 0.5 (first three columns) and0 (last three columns), computed with NE=256 elements and NF=707 degrees of freedom

ax/ay = 2.0 ax/ay = 1.0 ax/ay = 0.5 ax/ay = 2.0 ax/ay = 1 ax/ay = 0.5

122.987 (S1) 89.474 (S1) 158.843 (S1) 235.514 (S1) 125.198 (S1) 180.467 (S1)164.837 (S2) 165.161 (A1) 219.173 (A1) 265.156 (S2) 212.511 (S2) 277.782 (A1)262.202 (S3) 171.618 (S2) 295.629 (S2) 421.099 (S3) 268.348 (A1) 383.081 (S2)277.502 (A1) 260.511 (S3) 360.295 (S3) 514.724 (S4) 350.751 (A2) 413.458 (S3)339.901 (A2) 265.133 (A2) 382.463 (A2) 545.558 (A1) 395.438 (S3) 502.843 (A2)368.626 (A3) 343.975 (A2) 466.346 (A3) 558.298 (A2) 481.110 (S4) 588.446 (A3)448.904 (A4) 300.13 (A3) 489.130 (S4) 714.967 (S5) 548.100 (A3) 650.265 (S4)


S1 S2

–1.5–1

–0.50

0.51

1.5 1

0.5

0

0.5

1

–0.5

0

0.5

1

y

x

z

–1.5–1

–0.50

0.51

1.5 1

0.5

0

0.5

1

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

y

x

z

S3 A1

–1.5–1

–0.50

0.51

1.5 –1

–0.5

0

0.5

10

y

x

z

–1.5–1

–0.50

0.51

1.5 1

0.5

0

0.5

1

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

y

x

z

A2 A3

–1.5–1

–0.50

0.51

1.5 1

0.5

0

0.5

1

–3

–2

–1

0

1

2

3

y

x

z

–1.5–1

–0.50

0.51

1.5 1

0.5

0

0.5

1–0.6

–0.4

–0.2

0

0.2

0.4

0.6

yx

z

Fig. 5 Buckled shapes of an elliptical membrane with aspect ratio 2:1, for Poisson ratio ν = 0.5

summarizes the effect of the membrane aspect ratio on the lowest threshold for instability, corresponding to theS1 mode, for ν = 0.5. Recalling that the numerical error is on the order of 1%, we can state with confidencethat, given the membrane surface area, the circular shape has the lowest buckling threshold.

All computations presented thus far correspond to the aligned elliptical shape, wherein the in-plane stressfield of the undeflected shape is a known linear function of x and y. Next, we consider a case where the in-planestress field is determined from the finite-element solution. Figure 6 shows the distribution of the in-plane stress,σxx , on a square membrane with edge length 2a, together with the first five buckled shapes occurring at thecritical shear stresses, τ̂ = 73.294, 129.577, 141.174, 203.630, and 212.307. These results demonstrate that thepresence of corners does not have a profound effect on the overall features of the buckling modes.


Table 5 Effect of the aspect ratio of an elliptical membrane aligned with the flow on the lowest threshold for buckling instabilityfor ν = 0.5

Aspect ratio τ̂ Aspect ratio τ̂

1.0 89.474 1.0 89.4741.2 89.223 0.9 94.5241.4 92.850 0.8 101.2221.6 100.103 0.7 112.2461.8 110.262 0.6 129.9592.0 122.987 0.5 158.843

–1–0.5

00.5

1 10.5

00.5

1–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

yx

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 –1

–0.5

0

0.5

1

0

0.2

0.4

0.6

y

x

z

–1–0.5

00.5

1 –1–0.5

00.5

1–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

yx

z

–1–0.5

00.5

1 –1

–0.5

0

0.5

1

–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

y

x

z

–1–0.5

00.5

1 –1

–0.5

0

0.5

1

–1.5

–1

–0.5

0

0.5

1

1.5

yx

z

–1–0.5

00.5

1 –1

–0.5

0

0.5

1

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

y

x

z

Fig. 6 Distribution of the in-plane stress, σxx , and the first five buckled modes of a square membrane for Poisson ratio ν = 0.5


κ

τ

Fig. 7 Schematic illustration of the solution diagram in the shear stress – center-point curvature solution plane for a flat membrane(solid lines), and for a curved membrane (broken line)

5 Discussion

The computed eigenvalues represent bifurcation points in a certain solution plane, such as the plane of thehydrodynamic shear stress, τ , and curvature at the center of a circular or elliptical membrane, κ . Branches ofpossible shapes originating from these eigenvalues are schematically drawn with solid lines in Fig. 7. In thelinearized analysis presently undertaken, only shapes near the bifurcation points are described to first orderwith respect to the membrane deflection amplitude. To describe substantially deformed shapes we must carryout a nonlinear post-buckling analysis based on the full von Kármán equation (e.g., [16]). The graphs in Fig. 7indicate that, for a given shear stress, there may be one, two, or a higher number of buckled states. Which onewill occur depends on the stability of the eigenfunctions, which is presently unknown.

The simplifying assumption of a perfectly flat undeformed membrane patch is responsible for the suddenappearance of buckled shapes at the lowest critical shear stress. In blood flow through a capillary over anendothelial cell, because the height of the cell can be as high as 10% the lateral diameter, the cell membranepossesses a certain amount of curvature in the undeformed configuration [10].

The solution diagram for a curved membrane originates from a point above the origin of the κ axis, as shownwith the broken line in Fig. 7. As the shear stress is raised, the resting shape is continuously deformed andthen undergoes a rapid transition near the bifurcation points. In this light, the results of the idealized analysisfor a flat patch are useful in that they provide estimates for the conditions under which sudden changes in themembrane shape are most likely to occur.

A second consequence of the flat resting-shape approximation is that the shear stress is constant overthe entire upper surface of the membrane, corresponding to the simple shear flow. A curved shape causes adisturbance flow with an associated perturbation shear stress that is highest near the elevated center of themembrane and lowest around the depressed region around the rim. Moreover, when a membrane patch isnaturally curved or buckles in shear flow, the normal hydrodynamic stress varies over its area as in hydro- andaeroelasticity, and the additional normal load may increase the critical load. However, because the perturbationshear stress and pressure are first-order effects with respect to the cell height, neglecting them is consistentwith the small-deflection buckling analysis based on the linearized von Kármán equation.

Acknowledgments Support for this research has been provided by the National Science Foundation.

Appendix

Deformation of a sheared elastic slab

To support the argument that a shear stress over the upper surface of a thin membrane is tantamount to aparallel body force distributed over the cross section of the membrane, we consider a two-dimensional modelwherein a rectangular slab of an elastic material is subjected to a shear stress on the upper surface, while thetraction is required to be zero on the lower surface. The deformation is restricted by rigid side walls. The slabmaterial is assumed to obey the constitutive equations of linear elasticity involving a modulus of elasticity andthe Poisson ratio.


(a) (b)

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1–0.15

–0.1

–0.05

0

0.05

0.1

0.15

x

y

10.5

0–0.5

–1

–0.15 –0.1 –0.05 0 0.05 0.1 0.15

–0.06

–0.04

–0.02

0

0.02

0.04

0.06

xy

Fig. 8 Deformation of an elastic slab subjected to a shear stress on the upper surface. a Finite-element nodes before (crosses)and after (circles connected by lines) deformation. b Corresponding distribution of the horizontal normal stress, σxx

The equations of plane stress for the model problem were solved using a standard finite-element method withsix-node quadratic elements. Figure 8a shows the finite-element nodes before (crosses) and after deformation(circles connected by lines), for a slender slab with aspect ratio equal to 10 and Poisson ratio ν = 0.25. Toavoid the occurrence of corner singularities, the distribution of shear stress on the upper surface is parabolic,reaching a maximum at the center and dropping to zero at the two ends, σxy ∼ 1 − x2. The results revealthat the shear stress causes a sinusoidal deformation with the crest occurring at the left half and the troughoccurring at the right half of the slab.

Figure 8b shows the corresponding distribution of the horizontal normal stress, σxx . Even though the aspectratio of the slab is only moderately small, σxx exhibits a mild variation across the slab cross section far fromthe side walls. If the shear stress on the upper surface were smeared uniformly from the upper surface into thewhole cross section of the slab, σxx would be constant over each cross section. The variation of σxx with xarising from the finite element solution is consistent with the theoretical prediction based on a zero-thicknessmembrane model, σxx ∼ x − x3/3.

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proposal for the origin of a cortical stress. J. Biomech. Eng. ASME 117, 171–178 (1995)11. Bloom, F., Coffin, D.: Handbook of Thin Plate Buckling and Postbuckling. Boca Raton: Chapman & Hall/CRC (2001)12. Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability, 2nd edn. New York: McGraw Hill (1961)13. Mossakowski, J.: Buckling of circular plates with cylindrical orthotropy. Arch. Mech. Stos. (Poland) 12, 583–596 (1960)14. Pozrikidis, C.: Introduction to Finite and Spectral Element Methods using Matlab. Boca Raton: Chapman & Hall/CRC (2005)15. Muradova, A.D.: Numerical techniques for linear and nonlinear eigenvalue problems in the theory of elasticity. In: Proceed-

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