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doi: 10.1098/rspa.2007.0106, 3037-30534632007Proc. R. Soc. A

Ciprian D Coman and Andrew P Bassomcircular shearing of annular thin filmsBoundary layers and stress concentration in the

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Boundary layers and stress concentration inthe circular shearing of annular thin films

BY CIPRIAN D. COMAN1,* AN D ANDREW P. BASSOM2

1Department of Mathematics, University of Glasgow, University Gardens,Glasgow G12 8QW, UK

2School of Mathematics and Statistics, University of Western Australia,Crawley 6009, Australia

This work addresses a generalization of Deans classical problem, which sought to explain

how an annular thin elastic plate buckles under uniform shearing forces applied around itsedges. We adapt the original setting by assuming that the outer edge is radially stretchedwhile the inner rim undergoes in-plane rotation through some small angle. Boundary-layer methods are used to investigate analytically the deformation pattern which is set upand localized around the inner hole when this angle reaches a well-defined criticalwrinkling value. Linear stability theory enables us to identify both the critical load andthe preferred number of wrinkles appearing in the deformed configuration. Ourasymptotic results are compared with a number of direct numerical simulations.

Keywords: boundary layers; thin films; circular shearing; wrinkling

1. Introduction

Shear-induced buckling is a form of structural instability whose relevance to thefailure of rectangular plates was first explained by Timoshenko (1921) andSouthwell & Skan (1924). Shortly afterwards, Dean (1924) showed how shear-induced buckling can occur in an annular thin plate whose boundaries aresubjected to uniform but opposite shearing forces. The key to Deans classicalwork was a simplification of the pre-buckling stress distribution that led to anordinary differential buckling equation of Eulerian type, and hence solvable inclosed form. The conclusion of his paper makes it clear that Dean realized thathis analysis could not be extended for when tension is applied on one (or both) ofthe circular boundaries. A review of the vast literature on elastic instabilitiesappearing since suggests that the first attempt to revisit the problem left open byDean was some preliminary work described in Coman & Bassom (2007b). Therewe supposed that the annulus was stretched by applying a uniform displacementfield to the outer boundary while the inner rim was subjected to a small torque.This situation falls within a broader context related to partial wrinkling/bucklinginstabilities due to non-compressive edge loads, a situation which has attractedconsiderable interest over the past few decades.

Proc. R. Soc. A (2007) 463, 30373053

doi:10.1098/rspa.2007.0106

Published online 4 September 2007

* Author for correspondence ([email protected]).

Received 25 June 2007Accepted 13 August 2007 3037 This journal is q 2007 The Royal Society

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There are a number of practical instances where non-compressive edgeloading heralds the onset of localized and regular buckling. Examples includethe azimuthal shearing of stretched annular thin plates ( Li & Steigmann 1993;Miyamura 2000), the in-plane bending of a uniformly stretched rectangularthin plate (Stein & Hedgepeth 1961) and annular-shaped plates, which are

uniformly stretched by imposed displacements on the two boundaries(Haughton & McKay 1995). Traditionally, such problems have been analysedusing various tension field theories, which share the property that they alldisregard the obvious bending stiffness involved. The seminal work ofSteigmann (1990) has proved to be particularly versatile for handling alarge number of interesting situations (see, for example, Haseganu &Steigmann (1994); Li & Steigmann (1995) or Roxburgh et al. (1995)). Tensionfield theory was formulated with the objective of determining the extent of thewrinkled region and the direction of the wrinkles themselves; the price oftackling these relatively large-scale questions is that the details of the fine

structure of the wrinkling pattern are lost. In many cases, the theory has beenshown to be a very good approximation to reality in the advanced post-buckling regime, although recently Iwasa et al. (2004) have questionedwhether this is always so. Most of the problems for which the tension fieldapproach is relevant have the particular feature that there is no branching inthe wrinkling process as the load increases; although wrinkles do grow inamplitude and lengthen, their number does not change.

The genesis of the tension field theory can be traced to the attempts of theGerman engineer Wagner in the 1920s to account for the strength of thin metalwebs and spars, which can carry a shear load well in excess of the initialbuckling threshold. Since it is an approximate theory valid for severe loadingconditions, it is likely to be poor for describing the weakly nonlinear regime thatsets in immediately after wrinkling is first triggered. This is not a majorhandicap, though, as the incipient behaviour is very well suited for classicalbuckling/post-buckling analyses in which the bending rigidity is fully accountedfor. It is such a linearized, buckling regime which will be the main focus of ourstudy in what follows.

Recent developments in cell biomechanics (Burton & Taylor 1997; Boal 2002;Bernal et al. 2007) suggest that the quantitative analysis of wrinkled patternsproduced by living cells crawling on polymer nanomembranes can be applied togive an estimate of the force applied by the cell cytoskeleton. This requires a

delicate modelling that must clearly take into account the bending rigidity ofmaterial surfaces. Using classical plate theory, Geminard et al. (2004) addressedseveral numerical and experimental aspects of this problem for a simplified modelconsisting of an annular thin film with prescribed displacements on bothboundaries. Coman & Haughton (2006) and Coman & Bassom (2007a) extendedthe Geminard et al. (2004) study by investigating the possible relevance ofsingular perturbation methods and showed how systematic analytical progresscan be made. The developments pursued in these two works have proved to havea broader scope and subsequently motivated further studies (Coman 2007;Coman & Bassom in press).

It is helpful to review briefly the rationale behind some of these papers. Aswe have mentioned already, when an elastic film is acted upon by suitableedge loads, it may become susceptible to a wrinkle instability. If, for the

C. D. Coman and A. P. Bassom3038

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moment, we allow ourselves to lapse into relative vagueness, and we shalltighten up our meanings in due course, the articles mentioned considersituations in which a loading parameter, say l, is related to the number ofwrinkles generated, n2N. The form of lZl(n) can be determined via thenumerical solution of a suitable ordinary differential eigenproblem, and, for

any particular film, it would be expected that there is a preferred wrinkledconfiguration. In other words, of all the possible n, there is one for which thecorresponding l is the smallest; this mode number then requires least loadingto excite it and is likely to be the one provoked first in practice. It can beshown that, in the limit where the thickness of the elastic film tends to zero,the underlying ordinary differential problem becomes increasingly singular andnumerical experiments predict that the preferred wrinkling number grows.Moreover, asymptotic analysis then becomes viable and the first theories inthis direction appealed to standard WKB arguments. Coman & Haughton(2006) demonstrated that WKB methods can provide accurate approxi-

mations to the form of l(n) when n[1, but it is not easy to use their resultsto infer the identity of the most-favoured wrinkling mode. In contrast, theunderlying differential system can also be studied using boundary-layer theoryand matched asymptotic expansions. While such techniques are commonplacein many fields of fluid mechanics, their power and versatility seem to havebeen rather less recognized for problems involving the stability of solids. InComan & Bassom (2007a), we demonstrated how these methods can be usedto determine the preferred wrinkle mode for a prestressed annular thin film intension. Although the analysis necessary to isolate this instability pattern canbe carried out relatively quickly, it was found that the resulting predictionsare in excellent agreement with numerical simulations. The picture that hasemerged from these studies looks seductively general: in each of the examplesstudied, wrinkling has been shown to be controlled by a hierarchy ofboundary-layer problems each solvable in terms of Airy functions. In passing,we remark that it should not be concluded that the boundary-layer approachis superior to the WKB results for in many ways the two techniques yieldcomplementary, but distinct, information. The relative merits of the twotechniques have been discussed at length in Coman & Bassom (2007a, inpress), hence will not be repeated here.

Here, our specific concern is with a description of the wrinkling mode that canbe initiated by the circular shearing of an annular thin film. The relevant

eigenproblem, which is presented in 2, has already been the subject of anumerical study by the present authors. The simulations show that a torqueapplied to the inner rim of the annulus triggers a localized wrinkle modeconcentrated in the vicinity of this inner edge, and this problem was analysedusing WKB methods in Coman & Bassom (2007b). For small wrinklewavelengths, it was found that the WKB predictions correlate well with thenumerical results, but it was far from clear how the preferred wrinkle patterncould be isolated; this is tackled here via the boundary-layer approach.Moreover, the analysis described in 2 is rather more than a trivial modificationof studies like Coman & Bassom (2007a) for reasons we describe later.

We proceed as follows. Section 2 provides the background to the partialwrinkling problem that forms the main part of our study. Based on the linearizedDonnellvon Karman bifurcation equation for thin plates, an eigenproblem

3039Boundary layers and azimuthal shearing



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governed by a partial differential equation is derived. With the help of a normal-mode solution, this is then recast as a singularly perturbed SturmLiouvilleproblem for an ordinary differential equation with complex-valued coefficients.The main part of our analysis is detailed in 3, where a boundary-layer techniqueis used to obtain the relationship between the number of wrinkles and a

dimensionless form of the bending stiffness of the plate. The attractive feature ofthe boundary-layer method adopted is that approximations for the criticalwrinkling load and the asymptotic structure of the corresponding localizedeigenmodes are obtained naturally along the way. Comparisons with directnumerical simulations are included in 4, and we conclude in 5 with a summaryand brief discussion.

2. The annular model

The model adopted in this paper was introduced and discussed in Coman &Bassom (2007c), so here we restrict ourselves to highlighting only the mainfeatures. The general setting is depicted in figure 1a: a clamped annular film ofinner radius R1, outer radius R2 and thickness h (h/R2/1) is stretched byimposing the uniform displacement field U0O0 around the outer edge while theinner boundary is rotated through some (small) angle by the application of atorque M. Classical plate theory is used to describe the statics of the thin filmand the notation used is standard.

Assuming an axisymmetric deformation prior to the onset of instability, thepre-bifurcation state of stress is easily deduced by solving the system of equations

for plane stress elasticity. This information is then coupled to the linearizedDonnellvon Karman buckling equation, and so leads to the governingeigenproblem in the usual (r, q) polar coordinates for the (infinitesimal) out-

+

(a) (b)

+

srr

s1

s2

sqq

M

U0

U0

R1

R2

h

++

M

Figure 1. The pre-bifurcation state in a prestressed annular thin film subjected to a uniform

displacement field on the outer boundary (rZR2) and azimuthal shearing on the inner rim(rZR1). The radial and orthoradial stress distributions shown in (a) are both tensile but, asindicated in (b), one of the two principal stresses s1 and s2 can become compressive in a certainpart of the annular domain.




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of-plane film displacement whw(r, q). This system can be written as

Pm :mK2L0wKALC1 wK

B

r2LK1 wC

l

r3L2wZ0; for r;q2h;1!0;2p;

wr;qZvw

vr r;qZ0; for r;q2fh;1g!0;2p;

8>>>>>>>:

where the dimensionless radial co-ordinate rhr/R2 and hZR1=R2 denotes theaspect ratio of the annulus. Here we have also introduced the differentialoperators

L2hv2

vqvrK

1

r

v

vq; LG1 h

v2

vr2G

1

r

v

vrG

1

r2v2

vq2and L0h LC1

2;

together with the two auxiliary constants

Ah1Cn

1Kh2and Bh

h21Kn1Kh2

; 2:1

where n is the Poisson ratio of the material. The analysis that leads to (Pm), andwhich is detailed in Coman & Bassom (2007b), also identifies two significantcombinations of physical parameters, which play key roles in the description ofthe wrinkling instability. In particular,

lhM1Kn2pEhU0R2

and m2h12U0R2

h2

; 2:2

where E is the appropriate Youngs modulus. We see that l is a dimensionlessquantity proportional to the ratio between the shear stress and the initial tensionin the annulus, while m measures the dominance of membrane action overbending effects. The assumptions used tacitly in the asymptotic analysis includedin 3 are: (i) hZOS(1) and (ii) m[1. The second assumption is quite natural asmfR2/h and reflects our interest in the thin-film limit of the configurationdescribed in figure 1a. We mention in passing that Deans original work and arelated numerical contribution by Bucciarelli (1969) dealt with the comp-lementary case m/0.

If shs(r, l, h, n) denotes the second-order plane stress tensor that

characterizes the pre-bifurcation state due to the applied loads, it is knownthat this tensor has exactly one negative eigenvalue within the annular region

h!r!rh4h41Kn2Cl2 1Kh2 2

41Cn2( )1=4

; 2:3

as long as

lO4h2

ffiffiffin

p1Kh2

hl0: 2:4

The expression that appears on the right-hand side of the inequality (2.4)represents the loading parameter threshold that marks the onset of compressivestresses in the film; in the case of a true membrane (i.e. mZN), this corresponds




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to the critical wrinkling load. For the sake of completeness, a sketch of the pre-buckling principal stress distribution in the annular domain is included infigure 1b.

Solutions of the eigenproblem (Pm) are sought in the formwr; qZWrexpinq WrhW1rC iW2r 2:5

with the understanding that the real part of (2.5) represents the physicalquantity, and where n2N is the mode number (the number of identical half-waves of the wrinkling pattern). Using the normal-mode ansatz (2.5), it is foundthat the complex amplitude WhW(r) satisfies the differential equation

W000 0CArW000CBrW00CCrW0CDrWZ0; 2:6where

Arh2r

; BrhK 2n2C1

r2Cm

2 ACB

r2

!;

Crh1r

2n2C1

r2Km

2 AKB

r2C

iln

r2

!and

Drh 1r2

n2n2K4r2

Cm2 n2 AK

B

r2

Ciln

r2 !& '

:

Equation (2.6) is supplemented with the boundary conditions

WhZW1Z 0 and W0hZW01Z 0; 2:7which follow directly from the original form of (Pm).

It is the system (2.6) and (2.7) that was tackled numerically in Coman &Bassom (2007b). A sample of those results is illustrated in figure 2, which showsthe form of the eigenvalue l as a function of mode number n and aspect ratio h.

The graphs displayed are quite representative of the general behaviour of l asn2N varies. If we think of some chosen h, then for smallish nthe corresponding lis relatively large. As ngrows, l falls before reaching a minimum value, and then

0.1 0.3 0.5 0.7

1

2

3

4

l

h h h

(a) (b) (c)

0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7

Figure 2. Representative solutions of the system (2.6) and (2.7), with mZ40 and Poisson rationZ0.5. Shown is the eigenvalue l as a function of the aspect ratio and the mode number(a) n2[2,6] and (b) n2[7,19], with the arrows indicating the behaviour of the solution withincreasing n. The combined results of (a) and (b) are shown in (c).




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it rises again with further increase ofn. In a physical situation, it is probable thatthe mode number with corresponding least l is the most susceptible to excitation,on the obvious grounds that it requires the smallest applied torque for wrinklingto be induced.

The stability characteristics of elastic plates are usually deduced from the

plots of the neutral stability curves akin to those shown in figure 2. (Theterminology is borrowed from the literature on hydrodynamics, but has a slightlydifferent connotation in the present context.) Although the numerical solution ofthe normal-mode system (2.6) and (2.7) is not difficult (but the problem doesbecome increasingly tricky once mT500), there remains the task of identifyingthat value of n2N for which l is least; in other words, we need to find theenvelope of the neutral stability curves. While WKB methods can be used toapproximate this envelope when m[1, and can lead to really quite accurateresults for minimal effort (see Coman & Bassom 2007b), it is not at all obvioushow one can adapt such analysis to infer the identity of the most dangerous

azimuthal mode. The results discussed in Coman & Bassom (2007b) suggestthat as m/N, the mode number of the most important wrinkling mode grows,the corresponding eigenvalue l appears to tend to a constant, and theeigenfunction of the system (2.6) and (2.7) seems to be concentrated near theinner rim rZh. The objective of the rest of this paper is to put these statementson a firmer footing, and we do so using boundary-layer analysis combined withasymptotic matching.

Before we launch into a detailed discussion of the properties of eigensolutionsof (2.6) and (2.7), it is worth reconsidering some of the findings described inComan & Bassom (2007a). The eigenproblem under discussion there, thatdescribing the wrinkle instability of a purely tensioned annulus, has, at firstglance, a very close parallel to (2.6). In fact, the only clear distinction is that theimaginary parts of the coefficients in C(r) and D(r) of (2.6) are absent. (Weremark that the suppression of these terms would seem to remove the eigenvaluel completely from the eigensystem! However, in the pure tension problem, thecorresponding forms of the auxiliary constants A and B defined in (2.1) are infact functions of the eigenvalue.) In Coman & Bassom (2007a) it was shown that,when the geometrical parameter m[1, the most important modes reside in thescaling regime nZO(m3/4). For such values n2N, the corresponding eigenfunc-tions are compressed into a thin region of depth O(mK1/2) attached to rZh.Here, the leading-order solutions are governed by a scaled form of an Airy-like

equation, which can be solved so that the eigenfunction vanishes away from theinner rim of the annulus and satisfies the first equation of the boundaryconditions (2.7). However, the derivative parts of (2.7) cannot be immediatelyenforced on rZh, and this points to the presence of a second layer embeddedwithin the main O(mK1/2)-depth layer. With this solution structure, it is quitestraightforward to deduce the form of the eigenvalue lwOS(1) as a function of nand see that the nwm3/4 regime contains the most dangerous mode sought.

Given the apparent close similarity between the problem in Coman & Bassom(2007a) and that discussed here, it would seem reasonable to assume that theunderlying solution structure detailed earlier should transfer across with minimal

difficulty; at worst, only a slight reworking of that theory, as suggested by thevarious minor modifications, is needed. Unfortunately, it soon becomes clear thatthe inclusion of the imaginary terms in the coefficients C(r) and D(r) of (2.6) is




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rather more than small perturbations and, indeed, if we naively adopt thescalings suggested by the paper mentioned, then these terms are actually thedominant ones in the underlying equation. The presence of these relatively largeextra terms suggests that the eigenfunctions of the problem oscillate quickly on ashort scale: this short scale is much less than the principal

O(mK1/2) Airy-

equation layer identified in Coman & Bassom (2007a, in press)although it islong compared with the extent of the embedded layer. Rapidly oscillatingeigenfunctions are consistent with the numerical simulations discussed inComan & Bassom (2007b; see, in particular, fig. 2) and means that theasymptotic description of wrinkles when m[1 is somewhat more than a trivialextension of calculations performed before. With these various comments inmind, we now embark on a detailed analysis of (2.6) and (2.7).

3. The boundary-layer analysis

The rapid spatial oscillations in eigensolutions of our problem leave us with thepotential difficulty of matching together solutions over three distinct lengthscales: the principal region attached to the inner rim; its small embedded sub-zone; and the intermediate scale of the oscillations. Rather than try to accountfor all of these simultaneously, it appears easiest to strip out the spatialoscillations right from the outset and search for solutions of the particular type

WrZVrexpfikm3=4rKhg; 3:1with k2R, a constant to be tied down in due course. The cost of writing thesolution in this way is that the new equation for the (slowly varying) amplitude

V(r) is rather more involved than that for W(r). Nevertheless, it can be arrangedconveniently in a form similar to (2.6), so that

V000 0CbArV000CbBrV00CbCrV0CbDrVZ0; 3:2where the more complicated coefficients bA, bB,bCand bD are listed in appendix A.We note that the required boundary conditions for (3.2) follow immediately from(2.7), so that V(h)ZV(1)Z0 and V0(h)ZV0(1)Z0.

Guided by the analysis discussed at the end of 2, we seek the form of thesolution in an O(mK1/2) neighbourhood of rZh, where the coordinate X isdefined by

rZ

hCm

K1=2

X with XZOS1:

The size of this region, together with the O(mK3/4) length scale on which theeigenfunction oscillates, suggests that we seek a solution in which V(r), theeigenvalue l and the mode number n all expand in series of the types

VXZV0XCV1XmK1=4CV2XmK1=2CV3XmK3=4C/; 3:3alZ l0Cl1m

K1=4Cl2m

K1=2Cl3m

K3=4C/ 3:3b

andnZN0m

3=4CN1m

1=2CN2m

1=4CN3C/ ; 3:3c

here, the expectation that lZOS(1) and nZO(m3/4

) at leading orders issuggested in the work of Coman & Bassom (2007a). On substituting theseexpressions into (3.2), we find at the leading (zeroth) order that non-trivial




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solutions for V0(X) can exist only if a certain algebraic constraint between N0, l0and k holds. First-order terms throw up a simple first-order linear differentialequation for V0(X); it is possible to obtain bounded non-trivial solutions only ifthe coefficients of the V0 and dV0/dXterms individually vanish. Taken together,these three requirements lead to

k2 AC

B

h2

CkN0l0

h3

CN20h2

AKB

h2

Z 0; 3:4

2k ACB

h2

C

N0l0

h3Z0 3:5

and an expression for l1. The second of these relations can be used to express k interms of N0 and l0, whence it follows that

l20Z 4A2h4KB2: 3:6

It is remarked that this value is precisely the one appearing in the inequality(2.4), denoting the loading parameter threshold for an idealized membrane. Itthen follows that

k2ZN20h2

Ah2KB

Ah2CB

; 3:7

with k!0 (from (3.5)) and, in turn, l1Z0.It is at the following order that the equation which fixes the structure of the

eigensolution appears. In particular, it is found that

D2d2V0dX2

CD1dV0dXCD0V0Z 0;

where Dj (jZ0,1,2) are

D2hK ACB

h2

; D1hK

iN1l0h3

;

D0h4A2N20

hAh2CB !

XC4A2N40

Ah2CB2Cl2kN0

h3

C

N21h2

AKB

h2

:

This is just a rescaled Airy equation, hence, in order to ease its solution, thedependent variable is changed according to

V0XZQ0Xexp KiN1l0

2hAh2CB X& '

: 3:8

Elementary manipulations show that the amplitude Q0(X) in (3.8) must satisfythe equation

d2Q0dX2

KaXCbQ0Z 0;where

ah4A2N20 h

Ah2C

B2 and bh

4A2N40 h2

Ah2C

B3Cl2

kN0

h

Ah2CB !: 3:9In order to obtain solutions confined to the XZOS(1) region, it is necessary that

Q0(X)/0 as X/N. Thus, by choosing the proportionality constants




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conveniently, Q0 can be given in terms of the Airy function of the first kind,

Q0ZAiZCaK2=3b; ZZa1=3X: 3:10Furthermore, since the inner rim of the annulus is restrained against out-of-planedisplacements, we also need aK2=3bZKz0, where Kz0 is the first zero of Ai

(z0z

2.3381). This relation can then be rewritten with the aid of (3.9), thusl2

l0Z

h2

2

Ah2CB

Ah2KB

N20

4A2h2

Ah2CB3 !

Cz0NK2=30

4A2h

Ah2CB2 !2=3( )

: 3:11

Now, we see the way in which the minimum possible loading parameter can bedetermined. Whereas the leading order l0 satisfies (3.6) and is independent ofN0,and l1h0, the correction term l2 does vary with N0 and grows without bound asboth N0/0 and N0/N. It is a simple matter to check that l2 is minimized when

N0ZN0 h

z0

3 3=8 Ah2CB5=8

h1=2

2A1=4

;

3:12

and then

l2Z l2h

27=3A4=3h8=3l0

3Ah2KBAh2CB1=3N2=30z0: 3:13

(a) Higher-order terms

Given the initial scalings for our various quantities, the analysis summarizedthus far is neither particularly long winded nor complicated. Our predictions forthe critical mode number N0 and corresponding eigenvalue should be formallyvalid as m/N, but we would be fortunate indeed if these effectively leading-order expressions proved to be accurate approximations for all but very large m.In order to extend the probable usefulness of our results, it would be convenientto have a little more information at hand. Although it can be anticipated that theassociated algebraic manipulations will become increasingly tedious as we moveto further orders, modern symbolic algebra tools make the task quite tractable.

(i) Third order

If we seek a solution with

V1XZQ1Xexp K iN1l02hAh2CB X& '; 3:14then expressed in terms of ZZa1/3X, the amplitude Q1 is found to satisfy

d2Q1dZ2KZK z0Q1Z q11Cq12ZQ0Cq13Cq14Z

dQ0dZ

; 3:15where the constants are defined according to

q11hh2

a2=3

Ah2CB

8AN30 N1h2

Ah2CB

C2N1N2

h2AK

B

h2

C

k

h3N1l2CN0l3

K

N1l0N0l2CN2l02h4Ah2CB K

2kN1l0hAh2CB k

2C

N20h2

C

iN0l02h4

3Ah2CB

Ah2CB

!;




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q12h8A2hN0N1

aAh2CB2 ;

q13hKih2

a1=3Ah2CB4k k2C

N20

h2 C 1h3 N0l2CN2l0 !;q14h

il0N03Ah2CBa2=3h2Ah2CB2 :

Properties of the Airy operator can be used to find the particular integral of(3.15) analytically. Routine calculations show that

Q1ZZ1

4q14

d4Q0dZ4

C1

3q12

d3Q0dZ3

C1

2q13Cz0q14

d2Q0dZ2

Cq11Cz0q12Kq14dQ0dZ

;

and the boundary conditions for V require that Q1(Z)/0 as Z/0. Hence, wededuce that

q11C2

3z0q12K

1

2q14Z0: 3:16

This is a complex-valued constraint, which, when simplified, shows that atcritical conditions the correction term

l3h0:

Note also that when N0ZN0 , the coefficient of the terms proportional to N1 in

(3.16) automatically vanishes so that we can derive no information at this stageabout the optimum value N1 . At first sight, since the previous order problemfixed N0 , we might have expected that knowledge of N1 would appear here butthis is not the case, solely because our interest is in the neighbourhood of theminimum of the hypersurface lZl(Nj) (jZ0, 1, 2,.).

(ii) Fourth order

The method of solution at this stage follows the now-established pattern. If werelate V2(Z) to Q2(X) in the obvious way given in (3.8) and (3.14), this functionsatisfies

d2Q2dZ2KZK z0Q2ZX

i;j

cijZi djQ0

dZj;

where the numerous cij2C are complicated constants, which can be expressed interms of quantities introduced at previous orders. Despite the large number ofterms, it is purely a mechanistic task to construct a particular solution for Q2(Z)and it follows that this function tends to Q20hQ20l4;N0; N12R (someconstant) as Z/0. Rather than give the general form of the constant, which iscomplicated and not particularly illuminating, it is worthwhile commenting onsome of its properties. Given the pattern established so far, it can be anticipated

that the expression for Q20 should contain the eigenvalue correction term l4, aquadratic function ofN1 as well as various powers ofN0; this is already indicatedin the above notation for the constant. In order to determine the smallest




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possible value for the loading, it follows that the mode number correction N1needs to be chosen to minimize the quadratic; in fact, the details of thecalculation for Q20 show that the term linear in N1 is absent so that the desiredminimization occurs when we choose N1ZN

1 h0. Then, the substitution of the

remaining critical values yields the result

Q20hh2Ai0Kz0

Ah2CB

kN0 l4h3C

16 N0 2z20A28A2h4C4ABh2K17B245h2a2=3Ah2CB2Ah2KB

& '; 3:17

where Q20ZQ20 l4;N0 ; N

1 .

(b ) The embedded layer

The solutions obtained so far in the XZOS(1) layer have the property thatalthough Q0(X) and Q1(X) vanish as X/0, Q2(X) does not and, furthermore, theconstraints on the derivative of V(X) at the inner rim of the annulus are notfulfilled. All these matters can be rectified by considering the form of the solution

in a thin embedded layer, where we have

rZhCmK1Y with YZOS1;

and where the eigenfunction is expanded as VZmK1=2bV0YC/. To the orderof our working we need to compute only the leading order terms in this innerlayer. Routine manipulations show that bV0 satisfies the equation

d4bV0dY4

K ACB

h2

d2bV0dY2

Z0:

By demanding

bV0Zb

V00Z0 on YZ0, the resulting solution leads to the

asymptotic behaviourbV0YfYK hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAh2CB

p C/ as Y/N;and then matching with the main solution gives the critical value of l4,

l4Z

h10=32A2=3Ah2CB1=3N0 4=3Ah2KB1=2 1C 4z

208A2h4C4ABh2K17B2

45h2Ah2CB1=2Ah2KB

( ): 3:18

4. Numerical results

With the calculations outlined previously, it follows that for large m the criticallZl

islZ l0Cl

2mK1=2Cl

4mK1COmK5=4; 4:1

where lj (jZ0, 2, 4) are given by (3.6), (3.13) and (3.18). The correspondingcritical value of the mode number is available directly from (3.12), so that

ncrZN0 m

3=4COm1=4: 4:2

In order to assess the usefulness of our predictions, it is necessary to compare

these asymptotics with some direct numerical simulations. The neutralstability curves for two typical cases are illustrated in figure 3, in which weshow the results for mZ200 and 500. The behaviour depicted here is identical to




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that indicated in figure 2, but a few more remarks are helpful. The dashed lines

correspond to those pairs (n, m) that satisfy n/O(m1/2) or, following theterminology adopted in Coman & Bassom (2007b), they fall within the so-calledmembrane-like regime in which the response curves depend monotonically onboth h and n. When mZ200, the mode numbers up to 10 lie in the membrane-like regime, but, once mZ500, it is only the first 14 modes that comprise thisregime. The presence of a small bending rigidity in the film does not allow thenumber of wrinkles to grow indefinitely, and it appears that within the rangeO(m1/2)!n/m the film identifies a neutrally stable energy configuration withn wrinkles. Then, the film is in the plate-like regime which is distinguished bythe continuous lines in figure 3.

The stability envelopes are indicated by the white circles, and it is clear that inboth cases the asymptotic formula (4.1) faithfully reproduces the numericalsimulations; moreover, as would be expected, the accuracy improves as m grows.

Another view of the usefulness of the asymptotic predictions is provided infigure 4. Figure 4ashows the form of the critical mode number for three values of

inner rim radius h; the solid line denotes the asymptotic result (4.2) while thesquares show the outcome of direct numerical solutions. Of course, there is aslight inconsistency in our description of ncr; in practice, this quantity must bean integer, but the asymptotic expansions do not impose this restriction. Thenumerical work sought to minimize the predicted loading parameter l over n2Nand so, to facilitate a fair comparison, what is shown by the continuous line isthe nearest integer function applied to the prediction (4.2). This gives rise to thestaircase-like form ofncr, which is observed to be in excellent agreement with thenumerical results. We remark that although the theory can be relied upon onlyfor formally large values of m, it appears to provide surprisingly accurate

predictions for even quite modest-sized m.Figure 4b shows how the critical load prediction compares with its actual valuebased on numerical solutions. While the inclusion of the l2 term gives a

0.1 0.2 0.3 0.4 0.5 0.60

0.5

1.0

1.5

2.0(a) (b)

h

l

h0.1 0.2 0.3 0.4 0.5 0.6

Figure 3. Comparisons between the asymptotic approximations of the neutral stability envelopeand the direct numerical simulations of (2.6) and (2.7): (a) mZ200 and (b) mZ500; Poisson rationZ0.3 in both (a) and (b). The circles show the two-term asymptotic approximation obtained from(4.1); they are joined in order to indicate the overall trend of the stability envelope.




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reasonable approximation to l, the next O(mK1) contribution improves theagreement quite substantially. Now, especially at values of m greater than acouple of hundred, the asymptotic result is really quite excellent and justifies theeffort involved in the higher-order work described in 3. Of course, the accuracydeteriorates for smaller m, but, if it is remembered that the asymptotic analysiswas conducted under the implicit assumption that m1/4[1, it is perhapssurprising that the agreement is as good as it is. The relative accuracy (RA) forthe sets of data shown is best for the larger values of h, although this is notimmediately obvious on inspection offigure 4b. When mZ500 and ifhZ0.3, thenRAz0.6 and 3.2% for the three- and two-term approximations (4.1),respectively; these results change to RAz9.4 and 22%, respectively, whenmZ50. To give an indication of how these results vary with h, it suffices tomention that the counterparts for hZ0.1 of figure 4a,b are RAz2.9 and 8.7%,respectively, when mZ500.

5. Remarks

In this work, we have revisited a problem left open by Dean (1924), which isrelevant to the buckling instability experienced by annular thin elastic platesunder circular shearing. It has long been known that such plates are susceptibleto a wrinkling, which takes the form of a large number of equiangular spiralwrinkles that are concentrated around the inner rim of the annulus. Figure 5shows the type of pattern that is seen in practice; it is observed how a largenumber of wrinkles are focused in a small neighbourhood of the inner boundary

and the individual wrinkles spiral around the centre of the hole. This spirallingeffect is the physical manifestation of the intermediate-scale oscillation in thesolution eigenfunction. The presence of tensile loads at the outer boundary gives

100 200 300 400 5000

10

20

ncr 30

40

50

60(a) (b)

100 200 300 400 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

h = 0.3

h = 0.2

h = 0.1

l*h = 0.3

h = 0.2

h = 0.1

m m

Figure 4. (a) The critical mode number ncr as given by applying the nearest integer function to(4.2) is designated by the staircase-like line. The symbols indicate the result of direct numericalintegration of the eigenproblem (2.6) and (2.7). (b) The form of the critical wrinkling load l forlarge m. Open circles, numerical results; dashed lines, the truncation of (4.1) up to the O(mK1/2)term; solid lines, the full form of (4.1) up to and including the O(mK1) term.




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rise to a situation in which the linearized bifurcation equation is not amenable toclosed-form solution so that Deans approach becomes inoperable. The existenceof the natural large parameter m, which is related to the initial state of prestress,means that boundary-layer methods can be used to determine both the criticalload and the preferred number of wrinkles. It is surprising how the computationof the first few terms in the asymptotic series for the loading l and the number ofwrinkles n give rise to expressions which are in very good accord with directnumerical simulations. We have already mentioned that, in theory at least,symbolic algebra methods combined with matching could be used to calculatefurther terms in (4.1) and (4.2) but, given the impressive accuracy of theexisting terms as evidenced by the results in figure 4, it is by no means certainthat such extra work would be commensurate with the gain.

The situation explored has the tantalizing prospect that it could be related toother physical scenarios. Lindsay (1992) and Haughton & Lindsay (1993)investigated the deformation of a hyperelastic annular slab caused by the

rotation of a rigid shaft passing through the hole and bonded to the annulus.Assuming that the lateral curved surface is kept fixed, a second-order elasticitytheory was used to describe the deformation of the top surface of the slab. For avery broad class of constitutive equations, a pinching deformation in thevicinity of the shaft was noted, and fig. 3 of Haughton & Lindsay (1993) reveals aboundary-layer behaviour akin to that in the present work. Those authorsconducted an axisymmetric analysis and it would be of significant interest toincorporate azimuthal terms. This would both increase substantially the range ofpossible deformations and enable us to determine which is preferred in practice.Moreover, the studies of Lindsay (1992) and Haughton & Lindsay (1993)

demonstrate, to a certain extent, a finite elasticity version of the classicalWeissenberg effect for highly viscous fluids (e.g. Reiner 1958). In its simplestform, this phenomenon appears when a rod, which is rotated in a non-Newtonian

Figure 5. Birds-eye view of the wrinkles around the inner hole of the annulus. Here, the aspectratio hZ0.3, mZ300 and lz0.391.




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fluid, leads to the latter climbing up the rod; a comprehensive mathematicalanalysis was given by Joseph & Fosdick (1973) and Joseph et al. (1973). Weintend to return later to probe this apparently unexplored link between our workdescribed here and these other mathematical problems.

C.D.C. acknowledges with gratitude the financial support received from the Royal Society for avisit to the University of Western Australia. We would like to thank the referees for theircomments on this work.

Appendix A

The expressions of the coefficients that appear in equation (3.2) are

bArZ

bA0rC

bA1rm3=4;

bBrZbB0rCbB1rm3=4CbB2rm3=2CbB3rm2;bCrZbC0rCbC1rm3=4CbC2rm3=2CbC3rm2CbC4rm9=4CbC5rm11=4;bDrZbD0rCbD1rm3=4CbD2rm3=2CbD3rm2CbD4rm9=4CbD5rm11=4

CbD6rm3CbD7rm7=2;where

bA0h2rbB0hK 2n2C1

r20@ 1A bC0h2n2C1

r3bD0hn2n2K4

r4

bA1h4ik bB1h6ikr

bC1hK2ik 2n2C1r2

0@ 1A bD1h2n2C1 ikr3

bB2hK6k2 bC2hK6k2r

bD2h2n2C1k2r2

bB3hK ACB

r

20@ 1A bC3hK1

r

AKB

r

20@ 1AKiln

r

3 bD3hn2

r

2AK

B

r

20@ 1ACiln

r4

bC4hK4ik3 bD4hK2ik3r

bC5hK2ik ACBr2

0@ 1A bD5hKik 1r

AKB

r2

0@ 1ACilnr3

24 35bD6hk4bD7hk2 ACBr20@ 1A




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http://dx.doi.org/doi:10.1063/1.2437672http://dx.doi.org/doi:10.1038/385450a0http://dx.doi.org/doi:10.1007/s00033-006-6036-0http://dx.doi.org/doi:10.1016/j.jmps.2007.01.006http://dx.doi.org/doi:10.1177/1081286507077107http://dx.doi.org/doi:10.1007/s00707-005-0307-2http://dx.doi.org/doi:10.1098/rspa.1924.0068http://dx.doi.org/doi:10.1098/rspa.1924.0068http://dx.doi.org/doi:10.1140/epje/i2004-10041-1http://dx.doi.org/doi:10.1016/0020-7683(94)90173-2http://dx.doi.org/doi:10.1098/rspa.1993.0125http://dx.doi.org/doi:10.1098/rspa.1993.0125http://dx.doi.org/doi:10.1016/0020-7225(94)00068-Uhttp://dx.doi.org/doi:10.1115/1.1767171http://dx.doi.org/doi:10.1007/BF00253044http://dx.doi.org/doi:10.1007/BF00253045http://dx.doi.org/doi:10.1093/qjmam/46.4.601http://dx.doi.org/doi:10.1016/0020-7462(95)00019-Khttp://dx.doi.org/doi:10.1093/qjmam/45.4.529http://dx.doi.org/doi:10.1016/S0141-0296(99)00101-7http://dx.doi.org/doi:10.1016/0020-7225(94)E0035-Hhttp://dx.doi.org/doi:10.1098/rspa.1924.0040http://dx.doi.org/doi:10.1098/rspa.1990.0055http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://dx.doi.org/doi:10.1098/rspa.1990.0055http://dx.doi.org/doi:10.1098/rspa.1924.0040http://dx.doi.org/doi:10.1016/0020-7225(94)E0035-Hhttp://dx.doi.org/doi:10.1016/S0141-0296(99)00101-7http://dx.doi.org/doi:10.1093/qjmam/45.4.529http://dx.doi.org/doi:10.1016/0020-7462(95)00019-Khttp://dx.doi.org/doi:10.1093/qjmam/46.4.601http://dx.doi.org/doi:10.1007/BF00253045http://dx.doi.org/doi:10.1007/BF00253044http://dx.doi.org/doi:10.1115/1.1767171http://dx.doi.org/doi:10.1016/0020-7225(94)00068-Uhttp://dx.doi.org/doi:10.1098/rspa.1993.0125http://dx.doi.org/doi:10.1098/rspa.1993.0125http://dx.doi.org/doi:10.1016/0020-7683(94)90173-2http://dx.doi.org/doi:10.1140/epje/i2004-10041-1http://dx.doi.org/doi:10.1098/rspa.1924.0068http://dx.doi.org/doi:10.1098/rspa.1924.0068http://dx.doi.org/doi:10.1007/s00707-005-0307-2http://dx.doi.org/doi:10.1177/1081286507077107http://dx.doi.org/doi:10.1016/j.jmps.2007.01.006http://dx.doi.org/doi:10.1007/s00033-006-6036-0http://dx.doi.org/doi:10.1038/385450a0http://dx.doi.org/doi:10.1063/1.2437672

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