FINITE PLANE TWIST OF AN ANNULAR MEMBRANE

FINITE PLANE TWIST OF ANANNULAR MEMBRANE

By X. LI and D. J. STEIGMANN

(Department of Mechanical Engineering, University of Alberta, Edmonton,Alberta, Canada T6G 2G8)

[Received 17 August 1992]

SUMMARYWe present an analysis of the finite deformation of an annular membrane induced by

the rotation of a rigid hub. The membrane is partly wrinkled for certain combinationsof hub radius and rotation angle. For a particular strain-energy function of harmonictype, we obtain solutions that are analytical in the sense that the problem is reducedto an algebraic system in three parameters. Solutions of this system are used tocharacterize various properties of the deformation, including the equilibrium torque-twist relation.

1. Introduction

THE problem of plane axisymmetric twist of an annular membrane was usedby Reissner (1) to illustrate the mathematical development of tension-fieldtheory for infinitesimal deformations. This theory furnishes an idealized modelof the stress and deformation associated with fine-scale wrinkling or post-buckling of a thin plate with vanishingly small bending stiffness. Furtherprogress on the general theory was achieved by Kondo and his coworkers.Their work is summarized in (2), which includes a reconsideration of Reissner'sproblem. A slightly modified version of Reissner's theory was used by Steinand Hedgepeth (3) to study the partly-wrinkled annulus, in addition to otherexamples. In all of these investigations the basic assumptions are that thewrinkles are continuously distributed over a smooth surface and that theycoincide with trajectories of the active principal stress. The second principalstress is taken to be identically zero.

For isotropic elastic membranes with no bending stiffness, Pipkin (4) showedthat all of the basic assumptions of tension-field theory follow as consequencesof the minimum-energy principle. In particular, in certain problems the infimumof the energy is achieved in the smooth limit of a sequence of deformationsinvolving closely-spaced wrinkles. Subsequently, Steigmann (5) used this con-cept to develop a general tension-field theory valid for finite deformations ofarbitrarily curved membranes composed of isotropic materials. The lattertheory is used in the present paper to extend Reissner's analysis to finitedeformations. Specifically, we study the problem of an annular membrane fixedat its outer radius and prestretched by attaching it to a rigid hub at its innerradius. The hub is then rotated through a specified angle. Mikulas (6) used theStein-Hedgepeth theory for infinitesimal deformations to study the relatedproblem of partial wrinkling of a uniformly prestretched solid disc by the

[Q. Jl Meet »ppL M»tii, Vol. 46, PL 4, 1993] © Oxford Unimsity Press 1993

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602 X. LI AND D. J. STEIGMANN

rotation of an attached hub. Our analysis is not directly comparable to his,however, as the annular membrane does not admit a uniform state of prestretchin equilibrium.

We begin in section 2 with a brief account of the equilibrium theory ofisotropic elastic membranes. This is followed in section 3 by a general derivationof the strain-energy function to be used in wrinkled parts of the membrane. Weapply this derivation to strain-energy functions of harmonic type. This class ofstrain energies is used to facilitate analytical solution in the tense part of themembrane. Its application to the analytical treatment of a number of boundary-value problems involving plane deformations has been developed in (7,8), forexample.

A concise summary of the tension-field theory of wrinkled membranes isgiven in section 4. Following some kinematical preliminaries in section 5, thesolution of the axisymmetric twist problem for a particular harmonic materialis developed in sections 6 and 7. The solutions in the tense and wrinkled partsof the membrane are obtained separately in terms of various constants ofintegration. The constants are determined in section 8 by imposing matchingconditions at the boundary between these parts. The location of this boundaryis determined in the course of the analysis. This procedure leads to a couplednonlinear algebraic system of three equations in three parameters, solutions ofwhich are obtained by an iterative method.

In section 9 we present the torque-twist response in graphical form forvarious values of the hub radius. We also show how the wrinkled part of themembrane grows with increasing twist. Finally, we display the principalstretches as functions of initial radius for a particular value of the hub rotationangle.

2. Equilibrium of isotropic elastic membranes

Our analysis is based on the so-called direct theory of elastic membranes, inwhich the membrane is regarded as a two-dimensional continuum endowedwith a strain energy measured per unit area of a reference surface. There is noconsideration of three-dimensional effects and we do not invoke any of theapproximations used to derive membrane theory by descent from three-dimensional elasticity (9). Detailed discussions of the direct theory can be foundin (5, 10 to 12).

For our present purposes it is sufficient to take the reference surface to beflat. Thus consider a membrane that occupies a region Q of a fixed plane withunit normal k. A deformation is a mapping of the material points x e Cl ontoa surface y(x) in three-dimensional space. The deformation gradient F(x) is thelinear transformation defined by

dy(\) = Fd\. (2.1)It can be represented in the form (4)

F = AI<g>L + / * m ® M , (2.2)

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FINITE PLANE TWIST OF AN ANNULAR MEMBRANE 603

where X and fi are the non-negative principal stretches, (1, m) are the ortho-normal principal vectors of strain spanning the tangent plane of the deformedsurface at the particle x, and (L, M) are the corresponding vectors in the initialplane, oriented so that L A M = k. From (2.1) and (2.2) it follows that ?A and/im are the derivatives of y(x) in the L- and M-directions, respectively:

AI = (L.V)y, /an = (M.V)y. (2.3)

Here V() is the two-dimensional gradient operator with respect to positionxef i .

As a constitutive hypothesis, we take the membrane to be perfectly flexible,with a strain energy W(F) per unit area of fi. We further assume that W{ •) isinsensitive to superposed rigid rotations. Then W = W(C), where C = F r F isthe strain:

C = A2L ® L + n2M ® M. (2.4)

For isotropic materials, W(-) depends on C through the determinant det Cand the trace tr C(12). These are in one-to-one correspondence with theinvariants

J = (det C)*, / = (tr C + 2J)*, (2.5)

and thus W = w(I, J). From (2.4) we have

J = X\i and / = X + fi. (2.6)

Therefore, the strain energy of an isotropic elastic membrane is expressible asa symmetric function of the principal stretches: W = w(X, fi).

The force transmitted across an arbitrary material arc dx, with unit normalv and length ds defined by v ds = dx A k, is t ds, where

t = Tv (2.7)

is the traction and T is the Piola stress-resultant. For isotropic membranes, thestress-resultant has the representation (5)

T = wj ® L + wyn ® M, (2.8)

where subscripts X and /* are used to denote partial derivatives.If an arbitrary part P a fi of the membrane is in equilibrium with no

distributed forces, then the integral of t over the closed curve dP vanishes. Thisleads by standard arguments, together with (2.7), (2.8), to the pointwiseequilibrium equation

[V-(wiL)]l + [V.(wl,M)]m + wi(L.V)l + w|1(M.V)in = O, xe f i . (2.9)

3. Energy minimizers and the relaxed-energy density. Harmonic materials

According to the energy criterion of elastic stability, stable deformationsfurnish local minima of the potential energy in some suitable class of kinematic-ally admissible competitors. For the pure displacement problem considered in

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the present paper, this energy is simply the integral of ff(F(x)) over ft, and allcompeting deformations are required to be equal to a prescribed positionfunction on the boundary dQ. It is well known that if a deformation y(x) isstable, then it is equilibrated. It is also necessary that the gradient of thisdeformation furnish a non-negative second variation of the energy. Thiscondition is referred to as infinitesimal stability by Truesdell and Noll (13).

If a deformation is infinitesimally stable, then its gradient F(x) is necessarilysuch that the function of e defined by iV(F(x) + ea <g> b) has a non-negativesecond derivative at e = 0, for each x e ft. This is the well known Legendre-Hadamard inequality (13, 14). Here a is an arbitrary three-dimensional vectorand b is an arbitrary vector in the plane of ft. For isotropic membranes, Pipkin(4) has derived restrictions on the derivatives of the strain energy w(A, //) thatare necessary and sufficient for the Legendre-Hadamard inequality. These are

w^SsO, w,,SsO, wu^0, w^SsO, a 5=0 (3.1)and

(w^w^)* - wX/1 ̂ b - a, (wuw^)* + wXll^ -b - a, (3.2)where

a = (Aw, - /mv)/(A2 - n2) and b = <JMX - Aw^)/(A2 - /z2). (3.3)

We are particularly interested in the inequalities (3.1)1>2, which require thatthe principal stresses furnished by an energy minimizer be non-negative at everypoint in ft. These inequalities have no counterparts in the Legendre-Hadamardinequality for three-dimensional elasticity (15). Typically, a given strain-energyfunction, adapted for use in membrane theory, will violate these restrictions incertain regions of the (A, ̂ )-plane. Then it is possible to formulate boundary-value problems in membrane theory that have no stable solution.

To accommodate such problems, Pipkin (4) introduced a relaxed strainenergy, defined in such a way that (3.1), (3.2) are automatically satisfied for allA, \t > 0. The construction of this energy can be understood by considering thedeformation of a strip under uniaxial tension. Thus suppose that a unit squareof the membrane is deformed into a rectangle of dimensions A > 1 and ft = g(X),where g(h) is the solution of w^k, •) = 0. This is the natural width in simpletension (4), and we assume that it is uniquely determined. We also assume that0(1) = 1. One would expect that, for fixed A, a compressive force wM < 0 wouldbe required to make the strip narrower than #(A). The resulting deformationwould then be unstable according to (3.1)2.

Pipkin has shown that smooth deformations with 0 < \i < g(X) can beconstructed as limits of sequences of finely-wrinkled configurations involvingclosely-spaced folds parallel to the tensile axis. The value of the strain energyin each member of the sequence, and therefore in the limit, is equal to the strainenergy at the natural width. Thus the relaxed energy is equal to w(^ g{).)) inthat part of the (A, /i)-plane where A > 1 and 0 < /i < #(A). Since it is independ-ent of/i in this region, it satisfies (3.1)2 as an equality. Additional properties ofthe relaxed energy follow from the remaining inequalities in (3.1), (3.2). In

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particular, if A > 1 and \i ^ #(A), then it is necessary that f'{X) ^ 0 and /(A) ^ 0,where / = dw(X, g(X))/dX is the uniaxial force-stretch relation. Then all of theinequalities in (3.1), (3.2) are automatically satisfied if g(X) < A for A > 1. Thelatter restriction is consistent with experimental data on rubber (16).

A typical strain-energy function delivers positive principal stresses in theregion of the (X, /i)-plane defined by X > g(ji) and // > g(X), and is equal to therelaxed energy in this region if it satisfies the remaining Legendre-Hadamardinequalities there. The membrane is tense for deformations with principalstretches in this region. For stretches X > 1 and 0 < fi ^ g(X), the membranemay be regarded as finely wrinkled, and the relaxed energy is w(X, g(A)). Fromthe symmetry of the function w(X, /z), it follows that the relaxed energy isw{g(ji), n) if fi > 1 and 0 < X < g(ji). Finally, deformations with stretches0 ^ (A, fi) ^ 1 can be constructed as limits of sequences with folds along bothprincipal axes. This construction involves no stress. Accordingly, we take therelaxed energy to be identically zero if both stretches are less than unity (4).

In the present paper, we analyse the plane-twist problem for harmonicmaterials (7, 8, 17) with strain energies of the form

w(/,J) = 2 G [ F ( / ) - J ] , (3.4)

where G is a positive constant with dimensions of force/length and F(-) is atwice-differentiable function. It is easily verified with the aid of (2.6) that thefunction of X and /i defined by (3.4) satisfies (3.1)3j4i5 and (3.2) if and only if

F'(I)>0 and F"(I) ^ 0. (3.5)

Varley and Cumberbatch (8) have used particular harmonic strain energiesto study deformations of thin sheets containing elliptical holes. In theirformulation it is only necessary to specify the uniaxial force-stretch relation/(A) and the natural width g(X). In the family of harmonic materials that theyconsidered, the particular material defined by

= 2G(A-A- 1 ) , 0(A) = A-' (3.6)

was found to yield solutions that gave particularly good agreement withexperiments on rubber (8).

To derive the form of F(I) associated with (3.6), we recall that /(A) is thevalue of wA corresponding to wM = 0. Then from (2.6), (3.4) and (3.6) it followsthat F'(I) = A, where A is a root of / = A + A"1. The latter equation has realsolutions if and only if 1/2 ^ 1, and only one of them furnishes a function F " ( )that satisfies (3.5)2 when 1/2 > 1. The associated function F ' ( ) is

F'(I) = 1/2 + [(//2)2 - 1]*, (3.7)

and the principal stresses are

wx = 2G[_F\I) - /i], wt, = 2G[F'(I)-X2. (3-8)

From (3.6)2 it is evident that J > 1 in the region of the (A, /i)-plane defined

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by X > g(ji) and y > g(X). According to (2.6), I2 - 4J = (A - y)2, so 1/2 > 1 inthis region. Then (3.7) makes sense and the strict inequalities in (3.5) aresatisfied. The second of these implies that w^ > 0, and therefore w/1> 0 sincey > g(X) and w^ vanishes at y = g(X). It follows from the symmetry of w(X, y)that wx > 0 also. Thus all of the strict inequalities in (3.1), (3.2) are satisfied inthe region {(A, y): X > g(ji), y > #(A)}. In the regions {(A, y): X > 1,0 < y ^ g(X)}and {(A, y): y > 1,0 < X < g(ji)}, the relaxed energy furnishes principal stresses{wi, wj = {/(A), 0} and {0,/Qi)}, respectively, where / ( • ) is given by (3.6),.With the energy identically zero for 0 ^ (X, y) ^ 1, it can now be verified that theLegendre-Hadamard inequality is satisfied for all X, y ^ 0.

The relaxed strain energy is also locally convex in the sense that the secondderivative of W(F + eA), evaluated at e = 0, is non-negative for every A. To seethis we use Pipkin's necessary and sufficient conditions for local convexity (4).These consist of inequalities (3.1) and

wjuw*, - wl > 0, \a\ > b, (3.9)where a and b are defined in (3.3). For general strain energies of harmonic type,(3.9), 2 are equivalent to

F"(/)Ssl/2 and F'(I)^I/2 (3.10)

respectively. Thus the relaxed energy defined by (3.6), (3.7) is convex for allX, \i ^ 0, and strictly convex if X > g(ji) and y. > g(X). Since local convexity issufficient for non-negativity of the second variation (18), we conclude that anequilibrium deformation obtained by using the relaxed energy automaticallysatisfies the infinitesimal stability criterion.

This framework can be used to establish partial uniqueness of solutions inthe following sense (18). Let yj(x) and y2(x) = y, + u(x) be two equilibriumdeformations, and suppose that yi(x) is such that the membrane is tense insome compact subset O, of Q. If u is sufficiently small, then it satisfies theequation obtained by linearizing (2.9) about the deformation yu together withu = 0 on XI The second variation of the potential energy vanishes for all suchu (18). But the second variation is strictly positive if u is non-zero on Cln sincethe relaxed energy is strictly convex there. Thus it follows that u(x) = 0 for allx g n,, that is, the tense part of the deformation is uniquely determined withininfinitesimal perturbations.

4. Tension-field theory

When the relaxed energy is used, deformations associated with stretches/i < g(A) and X > 1 furnish a state of stress that is locally uniaxial: T = /(A)l ® L.We refer to such states as tension fields. For these it follows from (2.7) that thetraction on an element of arc dx with unit normal v is

t = /(A)l(L.v). (4.1)

For tension fields the equilibrium equation (2.9) reduces to

(V. [/(A)L])I + /(A)(L. V)l = 0. (4.2)

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Because 1 is a unit vector, it is orthogonal to its derivative. Thus for / # 0 wehave

V.[/(A)L] = 0 and (L.V)l = 0. (4.3)

The first of these can be used with Green's theorem to show that the forcechannelled between two L-curves is constant along their length (5).

To advance the analysis of (4.3)2, we introduce scalar-valued functions <f>(x)and iZ'(x) that are defined to be solutions of the differential equations

L.V«£ = A, L.V</> = 0. (4.4)

Since {L, M} is an orthonormal basis at every point in ft, we can write

V</> = AL + (M. \4>)M, \\fi = (M. Vi/OM, (4.5)

and thereby derive

x. (4.6)

Thus \p takes constant values on the trajectories M . dx = 0. On these samecurves we use (2.1), (2.2) and (4.6), to obtain

dy = Fdx = \d<f>. (4.7)

It follows that <f> measures arc length along the images of these curves on thedeformed surface.

We assume that <p(x) and \p(x) are functionally independent, that is,

k.V</> A \\j> = XM.\\p # 0 . (4.8)

Then the parameters <fi and ip define a curvilinear coordinate system on ft, andthe gradient operator can be written as

V( •) = V0d( • )/d<p + Vt/^( • )/d\p. (4.9)

This implies that L.V(-) = Ad(-)/d<f>, which in turn can be used to write (2.3)l

and (4.3)2 as the linear first-order system

y, = i, i, = o. (4.10)

The general solution furnishes the family of ruled surfaces defined by

y(x) = 0(x)l(<Kx)) + uGKx)), (4.H)

where u(-) is an arbitrary vector-valued function. Thus a particular curve\p = const, is mapped onto a straight line on the deformed surface, and thestress transmitted along this line is equal to /(A).

To proceed further we need expressions for the functions <£(x) and iKx), whichare described by (4.4). To this end we derive a system of two equations for thefunctions L(x) and A(x). Solutions of this system furnish the coefficients of thelinear, uncoupled partial differential equations (4.4)lj2, which can then be solvedfor the functions <p a n d 41 appearing in (4.11). One equation for L and A is

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given by (4.3)!. To obtain the second equation, we use (4.9) and (4.11) to write(2.3)2 as

urn = (M. Vfll + (M.V«/O((£1V + uv). (4.12)

Then the orthogonality condition l.m = 0 can be put into the form M.V<f> =—h(ij))M.Vip, where h(ip) = l.uv . From this result and (4.5)1>2 it follows that

XL = VU+ h(x)dx . (4.13)

Thus the basic system for L and k consists of (4.3)t and

k.VA(AL) = 0. (4.14)

We note that the tension-field problem is statically determinate in the sensethat L and k are described by equations that do not involve the deformationexplicitly. After the deformation (4.11) has been completely specified, (2.3) or(4.12) can be used to verify that p. ^ g(k) and to locate curves on which p = g(?j)in the reference plane. These curves are the boundaries between the tension fieldand the tense parts of the membrane.

The detailed solution procedure for plane axisymmetric deformations ispresented in section 7, where we also verify the functional-independencecondition (4.8). The basic theory outlined in the present section has been usedelsewhere (19) to solve problems involving cylindrical reference surfaces. Acomplete discussion of the general theory, accounting for lateral pressureloading and arbitrarily curved surfaces, can be found in (5).

5. Kinematics of plane axisymmetric deformations with twist

Consider a flat sheet that occupies the annular region rt < r ^ r0, 0 ^ 9 < 2Kin its reference configuration, where r and 9 are polar coordinates. Position inthis configuration is described by x = ri(0), where i is a unit vector in the radialdirection. In a plane axisymmetric twist of the sheet, the point with coordinates(r, 9) is displaced to the position y(x) = p(r)$9 + F(r)), where p is the deformedradius and F is the twist angle. We find it convenient to write the deformationin the form

y(x) = u(r)i(0) + i;(r)j(0), (5.1)

where u = p cos F, v = p sin F and j = i'(0) = k A i(0).To obtain an expression for the deformation gradient F(x), we write dy($

as a linear combination of dr = \(9).d\ and d9 = r~1\(8).dx. Comparison with(2.1) yields

F = u'(r)i ® i + i/(r)j ® i + (u/r)j ® j - (o/r)i ® j , (5.2)

and the associated strain is

C = [(«O2 + (f')2]i ® i + [(«/r)2 + (<Vr)2]j ® j

+ [iu/r)vT - (p/r)a'](i ® j + j ® i). (5.3)

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If we stipulate that the determinant det F be non-negative, then it follows from(2.5), and (5.2) that

J = det F = (u/r)u' + (v/r)v'. (5.4)

Then (2.5)2, (5.3) and (5.4) result in

/ = [(u' + u/r)2 + (i/ + v/r)2]*. (5.5)

Let y and a be the angles formed by i(0) and the principal vectors L and Iappearing in (2.2):

L = cos yi(0) + sin yj(0), 1 = coscri(0) + sin ocj(0), ]> (5.6)

M = —sin yi(6) + cos yj(0), m = —sin ai(6) + cos aj(0)J

Using these we can write (2.2) in the form

F = (A cos a cos y + \i sin a sin y)i ® i + (k cos a sin y — fi sin a cos y)l ® j

+ (?. sin a sin y + fi cos a cos y)j ® j + (X sin a cos y — n cos a sin y)j <g> i.

(5.7)

For the present class of deformations, comparison with (5.2) furnishes fourequations that can be summarized concisely as

(A + n) exp(/a;) = u' + u/r + i(v' + v/r); a> = a — y,~)( (5-8)

{X — n) exp(ifi) = u' — u/r + i(v' — v/r); Q = a + y.)

These can be used to calculate the individual principal stretches and the anglesa, y. Alternatively, the stretches may be obtained directly as the roots of

x2 - Ix + J = 0, (5.9)

which follows from (2.6). The angle a> delivered by (5.8), determines the localrotation: I = cos GJL + sin coM.

6. Tense solution and incipient wrinkling. Harmonic materials

To analyse axisymmetric equilibrium deformations with twist in fully tenseparts of the membrane, one may combine (2.9) and (5.8) to derive a system foru(r) and v(r) for the particular strain-energy function at hand. It is much moreconvenient, however, to proceed directly with the Euler-Lagrange equationsfor the potential energy

£[y] = 2K I ° H(u, v; u', v')r dr, (6.1)

whereH = vv[/(u, v; u', v'), J(u, v; u', •>')]. (6.2)

These arerHu = (rHu.)' and rHB = (rHv.)', (6.3)

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where subscripts are used to denote partial derivatives and the primes arederivatives with respect to r.

From (5.4) and (5.5) we obtain

J» = u'/r, JH. = u/r, ly = (riyl(u' + u/r), /„. = / " V + u/r), (6.4)

together with similar expressions for Jv, Jv., etc. These can be used in relationslike

Hu = w,Iu + WjJu, Hu. = w,Iy, + WjJu., (6.5)

etc. to derive the system

(rHu.y = rHu + r [ / " ' *,(«' + «/r)]' + (vv^' j(rHv.)' = rHv + r [ /"»tf / i / + o/r)]' ( t f ) ' J

Comparison with (6.3) furnishes equilibrium equations valid for arbitraryisotropic materials:[/- lw,(u' + «/r)]' + 0W«/ r = 0, [/" • w,(v' + v/r)J + (w^'v/r = 0. (6.7)

For homogeneous harmonic materials defined by (3.4), Wj = — 2G(const.),and (6.7) can be integrated once immediately:

/ " ' dF/dI(u' + u/r) = const., / " ' dF/dI(v' + v/r) = const. (6.8)

On squaring and adding these results, we find with the aid of (5.5) thatdF/dl = const, in equilibrium. Since we are considering materials for whichd2F/dI2 > 0, it follows that / = const., and (6.8) can be integrated once againto obtain

u(r) = axr + bjr, v(r) = a2r + bjr, (6.9)

where aubu etc. are constants. Ogden and Isherwood (7) used a complex-variable formulation to demonstrate that a wide variety of boundary-valueproblems for harmonic materials can be solved by taking the invariant / to beconstant.

The principal stretches are determined by substituting the general solution(6.9) into equations (5.8). Taking A ^ \i for definiteness, we obtain

k = {a\ + a\)± + r-2(b\+b\)\ / / = (fl? + flf)* - r"2(6? + fef)*, (6.10)

and it follows immediately from (2.6), or directly from (5.4), (5.5) and (6.9), that

J = a\ + a\- r~\b\ + b\), (I/I)2 =J + r~\b\ + b\). (6.11)

For the particular harmonic material defined by (3.4), (3.6), (3.7), themembrane is tense if and only if J > 1. Since J is a monotone increasing functionof r, the present solution is valid in the annulus (r, r0], where r is the value ofr obtained by setting J = 1 in (6.1 l)l. The circle r = r is the boundary betweenthe tense and wrinkled parts of the membrane.

In the present paper we take the membrane to be fixed at the outer boundaryr = ro, and attached to a rigid hub of radius p\ < r, at the inner boundary r = r,.

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This hub is rotated through the prescribed angle T. Then the data are

"o = ro. <>o = 0; ui = picosx, w, = pi sin T. (6.12)

We use the notation uo = u(ro), u, = ufa), etc.Suppose that the values of p{ and T are such that the membrane is partly

wrinkled (rt < f < r0), and define

u = u(f), v = v(f). (6.13)

Then from (6.9), (6.12), 2 and (6.13), we easily obtain the integration constants

«i = O-o - uf)/tf - r2), b, = rr20(u - r)/(r2

o - r 2 ) ,(6.14)

~r2), b2 = rr2v/(r2 - f2).

The parameters r, u and v are determined by imposing matching conditionsbetween the tense solution in the annulus (r, r0] and the tension field in theannulus [rh f) (section 8).

If the data are such that the entire membrane is tense, then (6.9) is valid inrx ^ r < ro, and equations (6.12) give

«i = ('o2 - «,r,)/(ro2 - rf), 6, = rxr

2o{ux - O/(ro

2 - r,2),}r (6-15)

a2 = -rlVi/(r2

0 - r2), b2 = r^v-M - r2). J

A state of incipient wrinkling exists if the data are adjusted so that J = 1 atr = r,. Using (6.12) and (6.15) in (6.1 l)i, we find after lengthy calculation thatthis condition can be put into the form

2r02 cos T = (r2 - r2 )r-JPi + (r2 + rf )Pi/r,. (6.16)

For fixed rjro, cos T has a single minimum with respect to the ratio pjrx at

. Pi/r, = \_{T\ - r2)/(r2o + r 2 ) ]* . (6.17)

This is the value that maximizes the hub-twist angle |T| required to achieveincipient wrinkling at the inner boundary. From (6.16) it follows that T = 0 at

PiAs = 1 and pjrx = (r20 - rf)/(r2

o + rf). (6.18)

The first value corresponds to the undeformed configuration, in which wrinklingof the entire membrane occurs at the slightest twist (1, 6). The second valuegives the deformation required to initiate wrinkling without twist. For smallervalues of pi, the membrane is partly wrinkled due to a purely radial displacement.

The non-monotone behaviour of cos z in the interval with limits (6.18)! ? iseasily understood. For p,/Vj = 1, wrinkling is instantaneous at |T| = 0+. Likewise,as the limit (6.18)2 it approached, wrinkling is imminent and a small rotationof the hub suffices to produce it. Between these limits, substantial rotation isrequired to cause wrinkling.

To compute the torque required to twist the hub, we first consider the tractiontransmitted across a circle r = const, by the material in the annulus (r, ro].

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According to (2.7), (2.8) and (3.8), this is Ti(0), where

(2G)~ 'T = dF/dI(l ® L + m ® M) - Oil ® L + Am ® M). (6.19)

For the present class of purely planar deformations, the deformation gradientis invertible if J > 0 and (2.2), (2.6),>2 yield

J F ~ r = / i l® L + A m ® M , F + J F - T = /(l<g> L + m ® M), (6.20)

where the superscript — T is used to denote the transpose of the inverse. Thesepermit (6.19) to be written in the form

(2G)~1T = r'dF/dliF + JF~T) - JF~T. (6.21)

Using (5.2) in this expression, we eventually obtain the traction

(2C)- ! t = / " l dF/dliiu' + u/r)i(0) + (i/ + i>/r)j(0)] - r" 'y, (6.22)

where y is given by (5.1).The resultant torque is

JoM = r\ y A t dd. (6.23)

Substituting the general solution (6.9) into (5.1) and (6.22), we find thatM = Mk, where

M = SnGI'1 dF/dI(a2bl - atb2). (6.24)

This is independent of r, as required by equilibrium. The torque transmitted tothe membrane by the hub is equal in magnitude to M, but has the opposite sign.

7. Solution in the wrinkled region

If the hub radius and rotation angle are such that partial wrinkling of themembrane is indicated, then the analysis of the previous section applies onlyin the annulus (r, ro] in which the membrane is tense. In the remaining part[rh r), we assume that A > 1 and /i < g{?>) = I/A. This assumption is verified insection 9. Then the tension-field theory of section 4 is applicable for re[r{, f).

In view of (5.6), equations (4.3), and (4.14) can be written as

( r / cos y\ + ( / sin y)e = 0 and (rl sin y)r - (A cos y)g = 0, (7.1)

respectively. Equations (5.8) imply that A and y are independent of 9 for thepresent class of axisymmetric deformations. Then for homogeneous materials,(7.1) can be integrated to yield

/ cos y = a/r and A sin y = b/r, (7.2)

where a and b are constants to be determined in section 8. (These constantsshould not be confused with the functions of A and n defined in (3.3).) Thesecombine to give the results

r2 = [a//(A)]2 + W)2, tan y = X' '/Wb/a, (7.3)

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which were first derived by Wu and Canfield (20) without the benefit ofthe general theory of section 4.

It is easily verified using (7.3)t that if /(A) and /'(A) are positive, then r is astrictly decreasing function of A for any fixed values of a and b. Then for thematerial defined by (3.6)x, A is uniquely determined as a monotone decreasingfunction of r if A > 1. Because it is given only implicitly, it is convenient to useA in place of r as the independent variable. The orientation of the principal axesin the initial plane is then determined from (7.3)2.

To construct the deformation (4.11), we need expressions for the functions<p and <lr. These are nearly arbitrary solutions of (4.4), subject only to thefunctional independence condition (4.8). In the present problem, (4.4) are

cos y(f>r + r " 1 sin y<pB = A, cosyip, + r " 1 sin y^je = 0. (7.4)

The second of these determines the ratio of the derivatives of \ji. If we takeipB = 1, then \jj = 6 — <5(r), where <5'(r) = r" 1 tan y. We can also satisfy (7.4)x bysetting <j>g = 0. Taking <p = 0 and if/ = 0 at r = r,, we obtain

= A sec y dx and i/f = 0 - d(r); 5{r) =J n * r,

4>{r) = A sec y dx and \j/ = 9 - d(r); 3(r) = x~l tan y dx. (7.5)

We can convert <f> and <5 to functions of A by using (3.6)j, (7.2) and (7.3)(Appendix A).

The function <p(r) measures the deformed length of the tension trajectoriesif/ = const, in the annulus [r,, r). It is evident from (7.2)i that the integrandA sec y = rlf(X)/a, and this is positive or negative in the annulus if and only ifa is positive or negative, respectively. Thus we assume that a > 0, so that thelength is a positive, increasing function of radius. If a were zero, then it wouldfollow from (7.2) and (5.6)t that L is purely azimuthal: L = ±j(0). Thus thetension trajectories would be concentric circles, and because no traction istransmitted across them (see (4.1)), the torque required to twist the membraneand the force required to deform it radially would vanish identically.

Using (7.2) and (7.5) we calculate

V</> = </,'(r)Vr = a " ' rA/(A)i(0), V^ = V0 - 5'(r)Vr = r " ' [j(0) - tan yi(

(7.6)Then

k.V</> A \if> = a-lXf(k). (7.7)

This is strictly positive for r e [>,, r), so the assumption of section 4 that <f> and\p are functionally independent is verified.

From (4.11) and (7.5), the derivatives of the deformation y(x) with respectto the polar coordinates are

yr = A sec y\ - r~l tan y(<f>lv + u v ) , yg = 4>\v + u v . (7.8)

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Then from (2.3)2 and (5.6) we have

/im = - s inyy , + r" 1 cosyy9 = - A tan yl + r" 1 secy(^lv + uv), (7.9)

and the orthogonality condition 1. m = 0 furnishes the restriction

I.uv, = rAsiny = fe, (7.10)

where b is the constant in (7.2).At the inner boundary r = rt, <f> vanishes, $ = 0, and (4.11) reduces to

y = u(0). Then the data (6.12)34 yield

u(0) = «i,i(0) + »J(0) = p,i(0 + t), (7.11)

where

i(0 + T) = cos ri(0) + sin ij(0), j(0 + x) = - s i n ri(0) + cos rj(0). (7.12)

From this result we infer that

uOA) = uiiOA) + vjty) = pjty + T), (7.13)where i(ip + T) is obtained by substituting \ji in place of 0 in (7.12)!.

Let p be the angle between Ity) and i(i/f):

1 = cos Pity) + sin /?j(t/O, m = - s i n 0i(i/O + cos ^j(i^). (7.14)

This is not the same as the angle a defined in (5.6). For deformations involvingtwist, these angles are equal only at r = r,. Using (7.13) in (7.10), we obtainfc/p, = l.j(«A + T). Substitution of (7.14) and (7.12)2, with 0 replaced by ip,leads to

b/Pi = sin(£ - t ) . (7.15)

Thus the tension trajectories in the deformed configuration form a constantangle with i(0 — <5(r)). A fixed ray i(c), c = const., in the deformed configurationcorresponds to the curve 0 + F(r) = c in the reference plane. On this curve0 — S(r) is a particular function of r, equal to some other function of thedeformed radius p. Proceeding along the ray, different tension lines areencountered, and each makes an angle with the ray that depends on radius(Fig. 1).

The foregoing results can be used to relate the integration constants a, b in(7.2) to the torque required to twist the membrane. From (4.1), (5.6), and (7.2) u

the traction exerted by the material in the annulus (r, f) across the circler = const, is

t = / c o s y l = (a/r)l. (7.16)

Then (4.11), (7.13) and (7.14) yield

y A t = (a/r)u A 1 = (a/r)Pi sin(£ - r)k. (7.17)

Using this and (7.15) in (6.23), we find the resultant torque M = Mk, where

M = 2nab. (7.18)

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Tense region—v ^ ~^s_,— Wrinkled region

s—Tension line

FIG. 1. Deformed configuration of the partly-wrinkled membrane

As in section 6, the torque exerted by the hub on the membrane is — M. Wetake the hub rotation to be counterclockwise (Fig. 1). To obtain a torque ofthe same sense, we assume that b ^ 0, since we have already assumed that a ispositive.

To facilitate matching of the present solution with the solution of the previoussection at r = r, we need to write (4.11) in the form (5.1). From (4.11), (7.13)and (7.14), it follows that

y = (<f> cos 0 + p, cos t)i(t/O + (4> sin fi + p, sin T)j(i/0. (7.19)

This can be resolved in the basis (i(0),j(0)} by substituting (7.12)12, with Treplaced by — <5(r). Comparison with (5.1) then gives

u(r) = 4> cos(/? - 5) + Pi COS(T - 5),)\ (7.20)

v(r) = 4> sin(0 - S) + pt sin(t - S). J

We also require that the two expressions (2.2) and (5.2) for the deformationgradient be equal. In particular, equality of Fi(0) implies that

u'(r)i(0) + t/(r)j(0) = AI cos y - urn sin y, (7.21)

where we have used (5.6). Substituting (7.14) and eliminating i(i/f) and j(i/>) asbefore, we derive

u'(r) = A cos y cos(/? — 5) + \i sin y sin(/? — <5),*j

v\r) = A cos y sin(/? — d) — n sin y cos(/? — 5). j

Finally, we impose equality of Fj(0) to obtain

- (»/r)i(0) + (u/r)j(0) = AI sin y + /an cos y. (7.23)

Scalar-multiplication of this equation with itself, and use of (7.2) l i2 in the

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Tension line

FIG. 2. Tension lines are tangential to the hub in the limiting state (T = TmlI)

resulting equality, furnishes the result

u2 + v2 = + b2.

It is easily verified using (5.4) that

d(u2 + v2)/dr = 2rJ > 0, re[rhr).

(7.24)

(7.25)

Then (7.24) implies that n/f is a monotone increasing function of r. The presentsolution is therefore limited by the condition n(r{) = 0, corresponding to a localcollapse of the materal at the inner boundary. This collapse entails no energeticpenalty, since the relaxed strain energy is independent of fj. in the wrinkledregion. According to (7.24) and (7.15), the limiting condition occurs at b = — pt

and T = xmMX, whereU-^ = k (7-26)

From (7.5) and (7.14) it is evident that ft is the angle formed by i(0) and atension line at the point (/•,, 6) where it intersects the hub. Then (7.26) impliesthat tension lines are tangential to the hub in the limiting deformation (Fig.2). For T > !„,„ we expect that tension lines would wrap around the hub. Oursolution does not account for this, however.

8. Matching conditions

We assume the deformation to be of class C1, so that y(x) and F(x) arecontinuous at r = r. This assumption entails no essential loss of generality. For,Steigmann (5) has shown that F may be discontinuous at the boundariesbetween tense and wrinkled parts of the membrane only if these boundariesare tension trajectories. In the present problem the boundary is circular, andsince tension trajectories are not azimuthal (section 7), F is necessarilycontinuous. Thus we require that u(r), v{r) and their derivatives be continuousat r = f. This is the circle on which the condition /i = g(X) is satisfied.

We use overbars to denote the values of functions at r = r. Then from (6.9)

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and (7.22), continuity of u' and v' implies that

A cos y cos(/? — 5) + g sin y sin(fi - S) = a, — b,/r2,~), \ (8-1)

A cos y sin(/? ~ 8) - g sin y cos(/? — <5) = a2 — bjr1,)

where g = g(X) and a , ,b , , etc. are the integration constants in (6.14). Thesedepend on u and iJ, which in turn are determined from (7.20):

u = 4> cos(/J — 5) + Pi COS(T — 5), v = 4> sin(/? — 8) + px sin(t — S). (8.2)

Using (6.14) we calculate

a, - b,/r2 = 2/i2 - / t , u , a2-b2/f2 = -A^v, (8.3)

whereAl=r\\^f2lrl)A2 and A2 = {\ - r2/^)"1. (8.4)

To simplify (8.1)12) we multiply the first by cos(/? — 5), the second bysin(j5 — <5), and then add. Next, we multiply (8.1)2 by cos(/? — 5) and subtractthe result from the product of sin(/J — 5) and (8.1),. Substitution of (8.2) and(8.3) into the resulting equations delivers the system

A cos y + Ailip + p{ cos(/? — T)] = 2A2 cos(/? — 5-

g sin y + i4,Pj sin(/? — T) = 2-42 sin(^ — 5).

Squaring and adding (8.2)12 and substituting into (7.24), we obtain

4>2 + 24>Pi cos(£ - t) + pf = (ag/j)2 + b2, (8.6)

and elimination of b2 from (7.15) results in

~ (8.7)

For deformations with no twist (/? = T = 0), the term in square brackets is thedeformed radius of the circle r = r. In the state of maximum twist defined by(7.26), this term gives the deformed length of a tension line in the wrinkledregion. Both values of this term are associated with the positive square root of(8.7). For intermediate twist angles, 0 < T — /? < \n (Fig. 2), and the positiveroot continues to apply. Thus we have

4> + Pi cos(£ - T) = ag/f. (8.8)

The constant a can be expressed as a function of the stretch A, = A(r,) andthe angle /? by using (7.15) in (7.3)t:

a = M ~ 7 ( W ? ( r . / P i ) 2 - sin2(/? - t)]*. (8.9)

Then (7.2), (7.3), and (7.15) can be used to write r, cosy, sin y, Ax and A2 asfunctions of f$, ).x and A. Moreover, (A.4) and (A.I 1) of Appendix A give 0 and6 as functions of these same three parameters. Collecting these varous functionsinto (8.5), 2 and (8.8), we obtain a system of three highly nonlinear algebraicequations in three variables. This system is presented in terms of dimensionless

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functions in Appendix B. A similar system of three coupled nonlinear equationswas derived by Mikulas (6) in a study of the infinitesimal-deformation problem.

Solutions of this system furnish the constants u and v (see (8.2)), which inturn can be substituted into (6.15) to obtain the deformation in the tense partof the membrane. According to the analysis of section 7, these solutions alsodetermine the deformation in the wrinkled region as a function of the stretchA e [A, AJ. Since the relation between A and r is one-to-one, the entire solutioncan, in principle, be specified as a function of the radius r for fixed values ofr,, ro and the data pi and T.

9. Examples

In all the examples that we consider we set ro/rt = 2 for illustrative purposes.Then the undeformed membrane occupies the interval r/r[ e [1, 2]. For valuesof pjr-, between the limits given in (6.18), the membrane remains fully tense forT between zero and the value obtained from (6.16). The solution is thendetermined explicitly from the analysis of section 6, after a suitable non-dimensionalization. For larger values of T, the membrane is partly wrinkledand the system (B.4) of Appendix B must be solved for fi, X{ and X. This isaccomplished by using a standard Newton-Raphson iterative procedure. Toapply this procedure we rewrite (B.4) in the form Fl(xl, x2, x3) = 0; / = 1, 2, 3,where x, = fi, x2 = X{ and x3 = X, and then calculate the elements dFJdXj ofthe Jacobian by using the chain rule and (A.4), (A. 12), (B.I) to (B.3). Theresulting expressions are too lengthy to be recorded here.

To compute the response of the membrane as a function of t for fixed pjrt,we prescribe a value of t slightly larger than the value (6.16) corresponding toincipient wrinkling. This induces wrinkling in a narrow band adjacent to thehub. Because the band is so narrow, the extreme values X{ and X of the stretchin the band are expected to be nearly equal. Thus we start the iterations bysetting A; and A equal to the value of X(r,) obtained from (6.10), and (6.15) atthe point of incipient wrinkling. At this point we can also use (5.8) and (6.10)to calculate the angle a at r = r, and use this value as the initial guess for fi.After the iterations converge, we prescribe a small additional increment in Tand then repeat the process, using the converged solution for the previousproblem to supply the initial values for the present cycle of iterations. Theprocedure is terminated when T is approximately equal to i H I (see (7.26)).

The Jacobian matrix is singular at X, = X = 1, and therefore the algorithmfails if pjrt = 1, that is, if the membrane is undeformed before a small incrementin twist is prescribed. According to the theory for infinitesimal deformations(1, 6), a small twist superposed on the undeformed configuration results inwrinkling of the entire membrane. Using the finite-deformation theory, we findthat the smallest reduction in pjrt restores the algorithm. Moreover, in suchcases wrinkling spreads rapidly with increasing r when z is small, though itnever covers the entire membrane.

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An alternative procedure is used if the prescribed value of pjri is smallerthan the value (6.18)2 corresponding to incipient wrinkling without twist. Inthis case we divide the interval between (6.18)2 and the prescribed value intosmall increments and compute a sequence of purely radial deformations withT, /?, y and <5 all identically zero. Then (B.4)2 is an identity and the remaining twoequations simplify considerably. In the first member of the sequence, Pi/r, isset equal to a value slightly smaller than (6.18)2. The iterations are then startedby setting A, and A equal to the value of ^.(r,), computed from (6.10)!, associatedwith incipient wrinkling. The iteration cycles for subsequent members of thesequence are initiated by using the converged solution for the immediatelypreceding member, as before. This process is terminated when the prescribedvalue of Pi/f; is reached. Then small increments in T are prescribed and theremainder of the solution proceeds in the manner described previously.

The torque-twist relation can be calculated using (6.24). Alternatively, (7.18)can be used once wrinkling has begun. The foregoing solution procedure can beused with equations (B.I) to obtain a* and b* for fixed values of pjrt and T.This in turn furnishes the non-dimensional torque M* exerted by the hub onthe membrane:

M* = -M/(Grf) = -2M*b*(p iA i )2 , (9.1)

where M is given by (7.18). The computed values of M* are plotted as functionsof T in Figure 3 for various values of pjr{. In each case the curve terminatesat the associated value of zmM1.

Increasing values of pjrx correspond to decreasingly severe states of prestretch.For the larger values a relatively small twist produces tension lines that are

20-00

M*

15-00:

1000:

5 00:

0-00000 2000 4000 6000 8000 10000 12000

Hub rotation (degrees)

FIG. 3. Equilibrium torque-twist response

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inclined at large angles to the radial direction. This is analogous to thebehaviour of a sheared slab. If the slab is prestretched, then an increment inthe amount of shear induces an incremental change in the principal directions.However, the change becomes pronounced as the prestretch decreases inseverity if the amount of shear is also small. Because of the large initial anglesof the tension lines, a relatively small additional hub rotation is sufficient tocause them to become tangential to the hub. Thus xmtx decreases with increasingp,/r,. Moreover, although the membrane is less severely strained, the relativelylarge moment-arm of the tension lines results in a stiff torque-twist response.Conversely, for small pKlr{ the membrane is more severely strained initially, butthis stiffening influence is overshadowed by the smaller moment-arms of thetension lines. The result is a softer torque-twist response.

To avoid crowding, Fig. 3 does not show the torque-twist response for thecase of extremely small prestretch (pjr{ = 0-998). In this case the membranewrinkles almost immediately and the torsional stiffness is initially quite low.This stiffness increases very gradually due to the moment-arm effect. The resultis a torque slightly smaller than that associated with pjrx = 0-9 for smallhub-rotation angles, but slightly larger for larger angles.

The evolution of the wrinkled part of the membrane is shown in Fig. 4 forvarious states of prestretch. Curves that intersect the vertical axis correspondto states of prestretch sufficient to cause partial wrinkling without twist. Forthe remaining examples the membrane remains completely tense until a certainamount of hub rotation is reached. For the case of slight prestretch (pjrt = 0998),

1-50

1-40

fir-.

1:0-2002:0-3003:0-4004:0-5005:0-6006:0-7007:0-8008:0-9009:0-998

1-30:

i-2o-;

110

1 0 0000 2000 4000 6000 80-00 10000 12000

Hub rotation (degrees)

FIG. 4. Evolution of the boundary f/r, separating tense and wrinkled parts ofthe membrane

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2-50-

200 -

1-50-

1-00•00 1-20 1-801-40 1-60

Initial radius, r/r,

FIG. 5. Variation of A with initial radius at x = zmtx

200

1-20

0-80-

0-40-

000

Pi/r,1:0-2002:0-3003:0-4004:0-5005:0-6006:0-7007:0-8008:0-9009:0-998

100 1-20 1-40 1-60 1-80 200

Initial radius, r/rt

FIG. 6. Variation of \i with initial radius at T = xa

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2-00-

J

1 50-

1 0 0 -

0-50-

000

1:0-2002:0-3003:0-4004:0-5005:0-600

6:0-7007:0-8008:0-9009:0-998

•00 I -20 1-40 1-60

Initial radius, r/r.

1-80 2-00

FIG. 7. Variation of J with initial radius at x - xmn. The membrane is wrinkledwhere J < 1 and tense where J > 1

a very small rotation is sufficient to initiate wrinkling. Subsequent growth ofthe wrinkled region is initially quite rapid, but tapers off gradually as zmal isapproached. The rate of growth of the wrinkled region is seen to decreasesubstantially as pjr-, decreases. It is also evident that after wrinkling has begunit continues to cover an ever increasing portion of the membrane. Completewrinkling is not achieved, however.

Figures 5 and 6 show the distributions of the principal stretches X and /i inthe membrane at the limiting condition T = rmM, for each value of pjrx. FromFig. 3 or Fig. 4 it is evident that the range of values of Tmni is quite substantial,the difference between the largest and smallest values being about 40°. Themaximum principal stretch in the limiting state occurs at the hub (r/rt = 1).The largest among these maxima is the largest stretch encountered in all of theexamples. Its value lies within the range in which (3.6)i approximates uniaxialdata on rubber with reasonable accuracy. Finally, the distribution of theinvariant J ( = A//) in the limiting states is shown in Fig. 7. This distributionconfirms our assumption that wrinkling ( J < 1) occurs in an annular regionadjacent to the hub.

Acknowledgement

We gratefully acknowledge the support of the Natural Sciences and Engineer-ing Research Council of Canada through grant NSERC OGP 0041743.

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REFERENCES

1. E. REISSNER, Proceedings of the Fifth International Congress on Applied Mechanics(1938) 88-92.

2. K. KONDO, T. IAI, S. MORIGUTI and T. MURASAKI. In Memoirs of the Unifying Studyof the Basic Problems in Engineering Sciences by Means of Geometry, Vol. I, C-V(Gakujutsu Bunken Fukyu-Kai, Tokyo 1955).

3. M. STEIN and J. M. HEDGEPETH, Analysis of partly wrinkled membranes. NASA TND-813 (1961).

4. A. C. PIPKIN. IMA J. appl. Math. 36 (1986) 85-99.5. D. J. STEIGMANN, Proc. R. Soc. A 429 (1990) 141-173.6. M. M. MIKULAS, Behaviour of a flat membrane wrinkled by the rotation of an

attached hub. NASA TN D-2456 (1964).7. R. W. OGDEN and D. A. ISHERWOOD, Q. Jl Mech. appl. Math. 45 (1978) 219-249.8. E. VARLEY and E. CUMBERBATCH, J. Elast. 10 (1980), 341-405.9. A. E. GREEN and J. E. ADKINS, Large Elastic Deformations, 2nd edition (University

Press, Oxford 1970).10. J. J. STOKER, Topics in nonlinear elasticity (notes by R. W. Dickey), Courant Inst.

Math. Sci. (1964).11. P. M. NAGHDI. In Handbuch der Physik, Vol. VI a/2 (ed. S. Flugge; Springer, Berlin

1972).12. and P. Y. TANG, Phil. Trans. R. Soc. A 287 (1977) 145-187.13. C. TRUESDELL and W. NOLL. In Handbuch der Physik, Vol. III/3 (ed. S. Flugge,

Springer, Berlin 1965).14. L. M. GRAVES, Duke Math. J. 5 (1939) 556-560.15. R. W. OGDEN, Nonlinear Elastic Deformations (Ellis Horwood, Chichester 1984).16. M. F. BEATTY and D. O. STALNAKER, ASME Jl appl. Mech. 108 (1986) 807-813.17. F. JOHN, Comm. pure appl. Math. 13 (1960) 239-296.18. R. HILL, J. Mech. Phys. Solids 5 (1957) 229-241.19. D. J. STEIGMANN and A. C. PIPKIN, Q. Jl Mech. appl. Math. 42 (1989) 427-440.20. C. H. Wu and T. R. CANFIELD, Q. appl. Math. 39 (1981) 179-199.

APPENDIX AIn this Appendix we use (7.2) and (7.3) to convert </>(r) and <5(r) to functions of the

stretch ).. For the sake of notational convenience, we use the same symbols to denotefunctions of stretch and radius. The intended functions are indicated by exhibiting thearguments explicitly.

From (7.2), and (7.5), we have

xf{x)r(x)r\x)dx, (A.I)

where r(/>.) is the positive square root of (7.3), and A, = A(r,) is the stretch at the innerboundary. According to (7.3),,

rr\).) = -(a2lp)fV-) ~ b2/P. (A.2)

Substituting into (A.I) and integrating by parts, we obtain

*(/.) = a\xlf(x) \i - T dx/f(x)\ - ^ J \ - 7 « dx. (A.3)

For the function /(/.) defined by (3.6),, the integrals can be evaluated explicitly; we

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calculate

M) = (a/2G)\_x\x2 - I)"1 - |ln(x2 - 1)]|£ - (2Gb2/a)(\n x + K 2 ) | £ . (A.4)

The conversion of the function <5(r) is somewhat more involved. First we use (7.3)2and (7.5)2 to write

5(r)

An integration by parts gives

= - ['k-lf{kyx-*dx. (A.5)

W = b- | [ r '/(A) In xX - Jin x d(f/X)\. (A.6)

Next we use (7.3), to obtain

In r = \n{alf) + \ ln(l + r2f2b2/a2). (A.7)

Then the integral in (A.6) is the sum

fln(fl//)<f(A-'/&/*) +A fln(l + r2f2b2/a2) d(rxfbla), (A.8)

the first term of which can be written as

r 700 ln(fl//W)]li + I x-7'(x)dxj. (A.9)

The second term in (A.8) can be evaluated explicitly. Combining this with (A.7) and(A.9), we reduce (A.6) to

<5(A) = b- \x~ 7(x)|i - T x' 7 'W ix\ ~ {arctan[x- 7(*Wa]}li. (A. 10)

and another integration by parts yields

r x-2f{x)dx + {aKtznix-lf{x)b/a]}\i. (All)

For the function /(A) defined in (3.6), we finally derive

5(1) = (2Gfc/a)(ln x + ix" 2 )^ + {arctan[(2Gb/a)(l - x-2)]}|J. (A.12)

A P P E N D I X B

To reduce the system (8.5)12 a nd (8.8) to a form suitable for computation, we

introduce the following dimensionless quantities:

/ • = fjG = 2(1 - I-'), b* = fc/p, = sin(/? - T),|

a* = p-r'a/G = 2(1 - Af2)[(r,/p,)2tf - (^*)2]*, / (B.I)

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Using these in (7.3), and (7.2), 2, we obtain

\ cosy = aM3 , sin y = l~lb*f*Ai. (B.3)

From (8.4) we have Ax = pfMf, where

A* = (r*)->[l + (r'fWrjftAj, A2 = [\- [f*)2{pjro)2yl. (B.3)

We also define 4>* = </i(A)/pi. This can be obtained from (A.4) by substituting a* in placeof prla/G and (b*)2/a* in place of p^G^/a. Similarly, 5 = d(X) follows from (A.12)after replacement of Gb/a by b*/a*. Then (8.5)^ 2 and (8.8) furnish three equations for/}, Aj a n d A:

+ cos(0 - T)] = 2A2 cos(fi - J),1

sin(^ - r) = 2A2 sin(fi -5), ) (B.4)

in which the condition g = A~' has been used.

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FINITE PLANE TWIST OF AN ANNULAR MEMBRANE