In-plane surface modes of an elastic toroidal membrane

International Journal of Engineering Science 60 (2012) 25–36

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In-plane surface modes of an elastic toroidal membrane Ganesh Tamadapu, Anirvan DasGupta ⇑ Department of Mechanical Engineering and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

a r t i c l e

i n f o

Article history: Received 23 August 2011 Received in revised form 17 May 2012 Accepted 30 May 2012 Available online 27 June 2012 Keywords: Inflated structures Toroidal membrane Curvature elasticity Anisotropic membrane Surface modes

a b s t r a c t In this work, the dynamics of in-plane surface deformation modes of an inflated toroidal membrane has been studied. We have considered both isotropic and anisotropic but homogeneous material properties. The covariant form of the equation of motion assuming a general in-plane small amplitude displacement field has been derived from the variational formulation which clearly shows the effect of curvature on the dynamics. The curvature term in the equation of motion may be interpreted as an effective quadratic potential in the Lagrangian with a coupling proportional to the Ricci curvature scalar of the membrane. The variational problem is discretized, and is subsequently analyzed to obtain the eigenfrequencies and modes of vibration. The effect of geometric and material properties on the modal dynamics has been studied. The effect of anisotropy on the modal dynamics of the torus has also been studied. Certain invariant deformation measures have been defined which are found to characterize the modes in terms of presence or absence of nodal curves/ points. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Inflatable structures are of two types. The first type of structures, known as air supported structures, consists of membranes restrained by cables on the ground and pressurized by air from below. The second type of membrane structures are closed and inflated by pressurized air. These inflated structures are used in a large number of the modern applications such as balloons, self deploying structures, and terrestrial and space structures (Jenkins, 2001). Dynamics of such structures can present interesting non-trivial effects due to the presence of curvature, and anisotropy in material properties. The mechanics of inflated structures has been a topic of considerable interest in the recent past. Experimental and analytical studies on the behavior of inflatable fabric panels and tubes at high pressure and subjected to bending loads have been reported in Thomas and Wielgosz (2002, 2004). Studies on in-plane bending and stretching of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure are also available (Le van & Wielgosz, 2005). The analysis of stresses in internally pressurized membranes has been presented by Jordan (1962), Sanders and Liepins (1963), Kydoniefs and Spencer (1965), Kydoniefs (1969) and Hill (1980). The natural frequencies and modes of vibrations of a pressure prestressed toroidal membrane has been investigated numerically by Liepins (1965). Under prestress, low frequency flexural modes, and high frequency meridional and circumferential (collectively known as extensional), or purely circumferential modes have been observed. While the flexural family of modes depend predominantly on the prestress, the later two families of modes are found to be insensitive to prestress. The numerical analysis of vibrations of an inflated toroidal membrane in the context of tire dynamics has been reported by Saigal, Yang, Kim, and Soedel (1986). Approximate

⇑ Corresponding author. E-mail addresses: [email protected] (G. Tamadapu), [email protected] (A. DasGupta). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.05.005

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free vibration solution neglecting the curvature of the membrane for tires of small cross-section compared to the radius has also been obtained. The static and dynamic analysis of inflated geomembrane tubes (used for flood protection) on rigid and flexible foundations have been reported by Plaut and Cotton (2005). However, only a simplified two dimensional analysis of the cross-section has been performed, which is applicable for straight tubes only. In the existing literature, the effect of geometry and material properties on the dynamics of membrane structures with curvature has not been explored. A covariant formulation of the dynamics of such membranes is expected to clarify the role of curvature. A knowledge of the nodal curves/points of the structural modes is important in deciding the support/attachment points for minimization of support stress and disturbance isolation. This aspect is crucial in, for example, an inflatable antenna, and similar precision applications. These issues motivate the present study, which is directed at establishing a fundamental understanding of the effects of geometric and material properties on the dynamics of membrane structures. In this work, we study the in-plane surface dynamics of an inflated toroidal membrane with isotropic and anisotropic but homogeneous material properties. This includes the high frequency extensional dynamics reported by Liepins (1965). Our focus on this dynamics is motivated by the following two requirements. Firstly, the shape control and disturbance rejection in such structures will be realized by using piezo-actuator patches which will force the in-plane modes. Further, actuator placement and control design will be guided by the modal frequencies and mode shapes. A covariant form of the equation of motion for a general in-plane small amplitude motion of a torus is first derived from the variational formulation. The effect of the curvature and anisotropy has been clearly brought out. The curvature term in the equation of motion may be interpreted as a quadratic potential in the Lagrangian with a coupling (distributed stiffness coefficient) proportional to the Ricci curvature scalar of the membrane. The variational problem has been discretized and subsequently analyzed to obtain the eigenfrequencies and modes of vibration of the membrane. The variation of the eigenfrequencies of the system with geometric and material properties (radius and Lamé parameters, respectively) are studied and discussed. A new feature of the eigenspectrum has been pointed out and explained, which is not found in Liepins (1965) paper. Materials typically used in the construction of inflatable structures are anisotropic due to fiber reinforcements. Using an anisotropy (orthotropy) model proposed in Ignatovich and Phan (2009), we have studied the effects of the magnitude and direction of anisotropy on the inplane dynamics of the membrane. It is observed that certain modes remain insensitive to the variation of the anisotropy parameter. On the other hand, beyond a certain range of the anisotropy parameter, we observe divergence instability in certain modes. In order to understand the contribution of dilatoric and deviatoric components in a mode, we propose certain invariant deformation measures based on the strain tensor. These measures are found to characterize the modes in terms of the presence/absence of nodal curves/points. It is observed that modes which are axisymmetric only can have real nodal curves. The approach of this work and the results obtained can be used to understand certain theoretical aspects of the dynamics and wave propagation in engineering applications involving inflated structures. The paper is organized as follows. The mathematical formulation of the problem is presented in Section 2. In Section 3, the discretization and modal analysis are presented. The numerical results are presented and discussed in Section 4. We conclude the paper with Section 5 by highlighting the main results obtained, and also indicate the scope of further work in this direction. 2. Problem formulation 2.1. Kinematics of deformation Consider a circular inflated toroidal membrane with a ring radius R and a circular section radius r, as shown in Fig. 1. The density of the material is q and the Lamé parameters of the material are k and l. Let x1 (meridional) and x2 (equatorial) represent the co-ordinates on the surface of the torus, as shown in the figure. The deformation kinematics of surfaces with curvature have been discussed previously (Green & Zerna, 1992; Naghdi, 1972) which we follow here. The infinitesimal distance ds between two neighboring points on the surface of torus can be found from geometry as

ds2 ¼ r 2 ðdx1 Þ2 þ ðR þ r cos x1 Þ2 ðdx2 Þ2 ¼ g ij dxi dxj

ðsummation conventionÞ;

where i, j = 1, 2, and the metric tensor gij on the torus surface is given by

g ij ¼

r2

0

0

ðR þ r cos x1 Þ2

!

:

ð1Þ

Consider a general in-plane small amplitude displacement field u1(x1, x2, t) and u2(x1, x2, t) on the torus in the x1 and x2 directions, respectively. The (linear) strain tensor can be written as

ij ¼

1 1 Lu g ij ¼ ðg ik rj uk þ g jk ri uk Þ; 2 2

ð2Þ

where Lu X ij represents the Lie derivative of a tensor Xij with respect to the displacement vector field (u1, u2). Here, rj is the covariant derivative with respect to the jth co-ordinate in the undeformed co-ordinate basis, and gij = (gij)1. In the co-ordinate basis, the four strain components can be written as

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27

Y

R x2

Z

X

r x1 X

Fig. 1. Circular torus with meridional (x1) and equatorial (x2) co-ordinate system.

11 ¼ r2 u1;1 ;

ð3Þ

22

ð4Þ

12

  ru1 sin x1 ¼ ðR þ r cos x1 Þ2 u2;2  ; ðR þ r cos x1 Þ   1 2 1 r u;2 þ ðR þ r cos x1 Þ2 u2;1 : ¼ 21 ¼ 2

ð5Þ

It may be noted that these strain components ij have dimensions of (length)2 in the co-ordinate basis. The strain tensor in the frame basis can be easily derived which will be non-dimensional. 2.2. Deformation measures The strain tensor can be decomposed into two irreducible parts, namely the dilatoric (trace part) and the deviatoric (trace free part) strain tensors as

ij ¼

1 #g þ cij ; 2 ij

where the dilatation scalar # = gklkl, and cij is the trace-free deviatoric strain tensor. These tensors allow us to define two invariant deformation measures over the membrane as follows. The dilatoric measure may be defined as

J# ¼

Z 2p Z 2p 0

pffiffiffi ðg kl kl Þ2 g dx1 dx2 ;

0

which reflects the stretching of the membrane. The deviatoric measure may be defined as

Jr ¼

Z 2p Z 2p 0

0

pffiffiffi

cij cij g dx1 dx2 ;

which quantifies the amount of shear suffered by the membrane. These invariant deformation measures can be used to quantify a mode in terms of the dilatoric and deviatoric components. Consider the axisymmetric displacement field u1 = 0 and u2 = u2(x1, t) for which 11 = 0 and 22 = 0, as can be easily checked. In this case, the dilatation scalar # = gijij = 0, implying J# = 0. Such a displacement field, therefore, represents a pure shear deformation of the torus. In general, J# = 0 whenever the displacement field is divergence-free, i.e., riui = 0. 2.3. Constitutive relations The strain energy density of deformation of a homogeneous isotropic elastic material is usually expressed as   F = F(I1, I2, I3) where I1 = gijij, I2 ¼ I21  ij ij =2 and I3 = Det (ij) are the strain invariants and ij is the contravariant strain

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tensor obtained as ij = gikgjl kl. For the anisotropic (orthotropic) material case, we define a direction through a unit vector ai which satisfies gijaiaj = 1. We then propose a fourth invariant J = aiajij which represents the projection of the strain tensor in the direction of ai. The strain energy density, considered only up to second order in the strain tensor elements, is then expressed as

F ¼ FðI1 ; I2 ; JÞ ¼

k þ 2l 2 I1  2lI2  fJ 2 ; 2

ð6Þ

where k and l are the Lamé parameters, and f is an anisotropy parameter having the same dimension as k and l. The elasticity tensor may be obtained from (6) as

C ijkl ¼

@2F ¼ kg ij g kl þ lðg ik g jl þ g il g jk Þ  fai aj ak al : @ ij @ kl

ð7Þ

It is interesting to note that the elasticity tensor for the linearized equivalent (planar) Mooney–Rivlin and neo-Hookean hyperelastic materials can also be represented by (7). 2.4. Governing equations The strain energy density of the membrane may be expressed using (7) as

h i   b ¼ 1 C ijkl ij kl ¼ 1 kðrl ul Þ2 þ l rl ui ri ul þ g jl g rl ui rj us  fðak al g rl us Þ2 : V is ks 2 2

ð8Þ

The kinetic energy density of the membrane can be written as

1 Tb ¼ qg ij ui;t uj;t 2   2  2  1 ¼ q r 2 u1;t þ ðR þ r cos x1 Þ2 u2;t : 2

ð9Þ ð10Þ

where q is the density of the material. Hence, the Lagrangian for the system is given by

L¼

Z 2p Z 2p 0

b pffiffiffi Lh g dx1 dx2 ;

ð11Þ

0

bV b is the Lagrangian density. The equation of motion and b ¼T where h is the thickness of the membrane, g = Det(gij) and L the boundary conditions are conveniently obtained from the Hamilton’s principle dS = 0, where

S¼

Z

t2

Ldt

t1

is the action integral. Variation of S gives

dS ¼

1 2

Z

Z 2p Z 2p h

t2

t1

0

0

2qg ij ui;t duj;t  2krl ul rk duk  lfrl ui ri dul þ ri ul rl dui g þ g jl g is frj us rl dui þ rl ui rj dus g

pffiffiffi  2fak al g ks rl us ai aj g im rj dum h g dx1 dx2 dt ¼ 0: Simplifying the above expression with integration by parts and using periodicity conditions on the torus given by

ui ðx1 þ 2mp; x2 þ 2np; tÞ ¼ ui ðx1 ; x2 ; tÞ;

m; n 2 Z;

we have

Z

t2

t1

Z 2p Z 2p h 0

0

i pffiffiffi qg ik ui;tt þ krk rl ul þ lrl rk ul þ lg ki g jl rj rl ui  fg rk g il rj ðar aj am al rm ui Þ duk g dx1 dx2 dt ¼ 0:

ð12Þ

For arbitrary independent variations duk, (12) is satisfied if and only if

qg ik ui;tt þ krk rl ul þ lrl rk ul þ lg ki g jl rj rl ui  fg rk g il rj ðar aj am al rm ui Þ ¼ 0:

ð13Þ

Using the properties of the Riemann curvature tensor, one can write (Spain, 1960)

rl rk ul ¼ rk rl ul þ Rpk up ;

ð14Þ

where Rij is the Ricci tensor. Substituting (14) in (13) and simplifying yields the equation of motion in covariant form as

l qui;tt  ðk þ lÞg ik rk rl ul  lg jk rj rk ui  Rui þ fg sl rj ðai aj ak al rk us Þ ¼ 0; 2

ð15Þ

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29

where we have used the (Einstein-space) property (Spain, 1960) Rij ¼ Rg ij =2, and R ¼ g ij Rij is the Ricci curvature scalar of the membrane. In the case of flat spaces R ¼ 0, while for a torus R ¼ 2 cos x1 =ðRr þ r 2 cos x1 Þ. It may be noted that the Ricci scalar for a torus can have positive, negative and zero values depending on the meridional co-ordinate x1. The governing equation (15) represents an extension of the Navier’s equation for an infinite elastic continuum with curvature and anisotropy. The conventional Navier’s equation for isotropic elastic bodies in flat space consists of only the first three terms of (15). The fourth term with Ricci scalar incorporates the effect of the curvature of the membrane on the dynamics. This term may be interpreted as a flexible bedding of stiffness lR/2, which can be represented by a quadratic potential in the Lagrangian. The presence of this term can have interesting consequences on the wave propagation characteristics in a continuum. For example, the dilatoric and deviatoric wave components may get coupled through this term. An imprint of this effect is observed in the eigenmodes of a toroidal membrane, as discussed later. The last term in (15) is due to the anisotropy in the material. This term can also couple the dilatoric and deviatoric modes of the membrane. The equation of motion (15) may be written in component form. For example, for an isotropic (f = 0) torus, the equations of motion are given by

u1;tt 

ðk þ 2Þ 1 1 ðk þ 1Þ 2 ðk þ 2Þ 2 sin x1 u;11  u1;22  u;12 þ sin x1 u1;1  u2 2 2 2 1 r r rð1 þ r cos x Þ rð1 þ r cos x1 Þ ;2 ð1 þ r cos x1 Þ k cos x1 þ rðk þ 1  cos 2x1 Þ

u1 ¼ 0; rð1 þ r cos x1 Þ2 1 ðk þ 2Þ ðk þ 1Þ rðk þ 2Þ 3 sin x1 u2;tt  2 u2;11  u2;22  u1;12 þ sin x1 u1;2 þ u2 ¼ 0; 2 2 3 1 1 1 r rð1 þ r cos x1 Þ ;1 ð1 þ r cos x Þ ð1 þ r cos x Þ ð1 þ r cos x Þ þ

ð16Þ ð17Þ

where we have non-dimensionalized using the substitutions

k!

k

l

;

f!

f

l

;

r!

r ; R

t!

t R

rffiffiffiffi

l q

and the scaling l = 1, q = 1 and R = 1. The non-dimensional k is actually related to the Poisson ratio as k = 2m/(1  2m). It may be noted that, as m ? 0.5 (signifying a planar incompressible material), k ? 1. On the other hand, for the linearized equivalent (planar) Mooney–Rivlin and neo-Hookean hyperelastic materials, k = 2. In the above set of non-dimensional parameters, which has been used throughout the rest of this paper, we refer to r as the geometric parameter, k as the material parameter, and f as the anisotropy parameter. While r varies the curvature of the membrane, k modifies the compressibility of the material. These are, therefore, the important parameters of the problem. It may be noted that in (17), the Ricci term containing u2 is canceled by a term containing u2 contributed by gjkrjrkui. This cancellation of the u2 term is to be expected since the angular momentum of the membrane in x2 direction is conserved. However, a similar statement cannot be made for the u1 terms since the strain energy expression (8) depends on u1 displacement as a result of curvature. 3. Discretization and modal analysis Admissible functions satisfying the periodicity conditions of the problem may be formed from the product of two functions, one from the set (cosmx1, sinpx1), and the other from the set (cosnx2, sinqx2), where m, n = 0, 1, . . . , 1 and p, q = 1, 2, . . . , 1. Thus, the admissible functions Uk(x1, x2) for different values of m, n, p and q are of the form

½cos mx1 cos nx2 ; sin px1 cos nx2 ; cos mx1 sin qx2 ; sin px1 sin qx2 : The solutions are then constructed using these admissible functions as

u1 ðx1 ; x2 ; tÞ ¼

N X U k ðx1 ; x2 Þck ðtÞ;

u2 ðx1 ; x2 ; tÞ ¼

k¼1

N X

k

U k ðx1 ; x2 Þd ðtÞ;

ð18Þ

k¼1

where N represents number of admissible functions, and ck(t) and dk(t) are the coefficients of kth admissible function for u1 k and u2, respectively. Substituting (18) into (11) and integrating over space yields the discrete Lagrangian L ¼ Lðck ; d ; c_ k ; d_ k Þ. The discretized equations of motion are obtained from

  d @L @L  k ¼ 0; dt @ z_ k @z

k

where zk ¼ ck ; d

ð19Þ

consists of 2N second order ordinary differential equations in time, which may be represented as

€ þ KX ¼ 0: MX

ð20Þ k

k

Here X is the coefficient vector formed with all the coefficients c (t) and d (t), M is the mass matrix and K is the stiffness matrix which is a function of r, k and f. We assume a complex modal solution of (20) of the form X = X0eixt where X0 is a constant amplitude vector, and x is the non-dimensional frequency. The dimensional frequency is obtained from the

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relation Xn ¼ xn

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi l=qR2 . The eigenvectors X0 are used in (18) to obtain the eigenfunctions of the membrane. For example

the Pth eigenfunction vector corresponding to xp can be constructed as

(

W iP ðx1 ; x2 Þ

¼

)

HP ðx1 ; x2 Þ ; UP ðx1 ; x2 Þ

ð21Þ

where

HP ðx1 ; x2 Þ ¼

N X U k ðx1 ; x2 ÞX k0P ;

UP ðx1 ; x2 Þ ¼

k¼1

2N X

U kN ðx1 ; x2 ÞX k0P

k¼Nþ1

are the eigenfunctions in x1 and x2 directions, respectively, and X0P is the Pth eigenvector. The eigenfunction vectors may be normalized such that

Z 2p Z 2p 0

0

pffiffiffi g ij W iP W jQ g dx1 dx2 ¼ dPQ :

ð22Þ

4. Numerical results and discussion In order to understand the effect of geometry on the in-plane dynamics of the toroidal membrane, we first present and briefly discuss the results of modal analysis of a flat torus. Subsequently, we take-up the curved toroidal membrane and study the effect of variations of the non-dimensional radius r, the material parameter k, and the anisotropy parameter f and anisotropy direction ai on the modes of the membrane. The effect of anisotropy on the modal dynamics is studied subsequently. 4.1. Surface modes of a flat torus The flat torus is a cylindrical surface with periodic closure at the ends. Thus, it is bi-periodic and doubly symmetric, which is distinct from a curved torus which has only one symmetry (though bi-periodic). For an isotropic flat toroidal membrane, the equation of motion comprises only the first three terms in (15). The curvature term is absent since the Ricci curvature scalar R ¼ 0. The solutions are straightforward and can be obtained analytically. The expressions for the discrete non-dimensional eigenfrequencies of the flat torus in terms of non-negative integer indices m and n corresponding to the coordinates x1 and x2, respectively, are given by



xPm;n

2

 ¼

   2 m2 m2 2 S 2 ðk þ 2Þ and ; þ n x ¼ þ n m;n r2 r2

ð23Þ

where m, n = 0, 1, 2, . . . , 1, and the superscript P or S indicates dilatoric or deviatoric frequency, respectively. The two rigid body motions of the flat torus corresponding to m = n = 0 are the roll mode and the translation mode. There are four types of modal groups for which one of the displacement field variables is zero and the other is independent of either x1 or x2. These modes occur for the frequencies xS0;n , xP0;n , xSm;0 and xPm;0 , and are the torsional (u1 = u1(x2, t), u2 = 0), axial (u1 = 0, u2 = u2(x2, t)), shear (u1 = 0, u2 = u2(x1, t)) and meridional (u1 = u1(x1, t), u2 = 0) modes, respectively. When k ? 1, we have only the shear modes. It is interesting to note the functional dependence of the natural frequencies on the geometric parameter r of the flat torus. Some of these generic features are also retained in a curved torus, as discussed in the following sections. The first few modes of the flat torus are presented in Fig. 2. with r = 0.5 and k = 60. The corresponding eigenfrequencies, number of nodal points Np, number of nodal curves Nc and the dilatoric and deviatoric measures J# and Jr, respectively, are also indicated in the figure. For the modes with m = 0 or n = 0, there is one degenerate modal companion. A twofold degeneracy is observed for the modes with m – 0 and n – 0, as shown in Fig. 2(d) and (e), which have three degenerate modal   companion. Identical eigenfrequencies xS1;0 ¼ xS0;2 are observed for two different mode shapes shown in Fig. 2(b) and (c), due to the intersection of two distinct modal branches at the chosen value of r. Two types of nodal curves corresponding to u1 = 0 (thin curve) and u2 = 0 (thick curve) are shown in figure. Exceptions are observed in Fig. 2(a)–(c) and (f) where a thick curve represents a real nodal curve on which u1 = u2 = 0 simultaneously. The S;P real nodal curves are observed only for the modes with the non-dimensional eigenfrequencies xS;P 0;n and xm;0 . Hence there are four types of modes which have real nodal curves due to the double symmetry. Intersection of the two types of nodal curves yields a nodal point for all the other modes with m – 0 and n – 0. Depending upon the values of the deformation measures, the modes may be classified as pure shear modes when J# = 0, or mixed modes J# – 0 and Jr – 0. All the modes shown in Fig. 2 are pure shear modes. In mixed modes, the membrane also undergoes stretching, and consequently are associated with high frequencies. It may be noted that all dilatoric modes are mixed modes. The variation of the eigenspectrum of the flat torus with r is presented in Fig. 3(a). A part of this figure is magnified in Fig. 3(b) for the frequency range x = 0 to 7. Two sets of bunching of frequency curves are observed in Fig. 3(a) depending on the values of the deformation measures. The first bunching set of modes are all pure shear modes with J# = 0 (including the low frequency horizontal curves), which can be seen clearly in Fig. 3(b). On the other hand, second bunching set of modes

G. Tamadapu, A. DasGupta / International Journal of Engineering Science 60 (2012) 25–36

31

Fig. 2. First few modes of the flat toroidal membrane for r = 0.5 and k = 60. In (a)–(c) and (f), the thick curve represents a real nodal curve u1 = u2 = 0. For the remaining modes, a thin curve represents u1 = 0 and a thick curve represents u2 = 0.

Fig. 3. Variation of the non-dimensional eigenfrequency with the geometric parameter r for a flat toroidal membrane with material parameter k = 60.

are all mixed modes with dominant J# value, which also includes next level of horizontal curves, as shown in Fig. 3(a). Since high energy is associated with stretching of the membrane, the second set of curves corresponds to high frequency modes. A clear decoupling of the pure shear (J# = 0) and mixed mode curves is observed in the frequency spectrum due to the absence of the Ricci curvature coupling in a flat torus. It may be observed from (23) that eigenfrequencies corresponding to m = 0 are independent of r in both the sets. The eigenfrequencies of all the other modes decrease monotonically. From (23), it can be observed pffiffiffiffiffiffiffiffiffiffiffiffi that the first bunching set of modes is independent of the material parameter (k), while the other is proportional to k þ 2. 4.2. Surface modes of an isotropic torus For a curved toroidal membrane with r = 0.5 and k = 60, the first few modes of vibration are presented in Fig. 4. The corresponding eigenfrequencies and other details are also indicated in the figure. It is interesting to observe that, for all the non-degenerate modes, one of the displacement field variables is zero and the other is independent of the co-ordinate x2 (axisymmetric modes). Depending upon the values of the deformation measures,

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we observe that the sixth mode (Fig. 4(d)) with u1 = u1(x1, t) and u2 = 0 is a mixed mode, since both J# and Jr are non-zero. On the other hand, the third and ninth modes (respectively, Fig. 4(b) and (f)) with u1 = 0, u2 = u2(x1, t) and J# = 0 are pure shear modes. Therefore, except for the axisymmetric modes with u1 = 0, all other modes are mixed modes. Two types of nodal curves corresponding to u1 = 0 (thin curve) and u2 = 0 (thick curve) are shown in Fig. 4. Exceptions are observed in Fig. 4(b) and (f) where the thick curve represents a real nodal curve on which u1 = u2 = 0 simultaneously. Each mode may have nodal points, or nodal curves, or there may be none. The non-axisymmetric fundamental mode in Fig. 4(a) has four nodal points at (x1, x2) = {(0, p/2), (p, p/2), (0, 3p/2), (p, 3p/2)}. On the other hand, an axisymmetric mode may have either nodal equators, or no nodal curve or point. For example, the motion of all points on the torus in the sixth mode in Fig. 4(d) is along the meridional direction. This mode has no nodal curves or points because u1 – 0 for any x1. On the other hand, the third and ninth modes have nodal equators at x1 = 1.87819, 5.01978 and x1 = 0, p, respectively. An interesting effect of curvature is observed in the mode shown in Fig. 4(d) which has only meridional motion. In the flat torus case, this mode corresponds to a zero frequency roll mode. This meridional symmetry of the flat torus is spontaneously lost in the curved torus, and a curvature induced stiffening results in the non-trivial purely meridional mode shown in Fig. 4(d). Since the meridional symmetry is lost due to curvature in the case of curved torus, all the torsional and axial modes corresponding to m = 0 in the case of flat torus are transformed into mixed modes (J# – 0) with only nodal points in the curved torus. Since circumferential symmetry remains in the curved torus, all the modes corresponding to n = 0 in the case of flat torus retain the property of the real nodal curves and J# value in the curved torus. Thus, in the case of a curved toroidal membrane, only axisymmetric modes can have real nodal curves (equators), except for the mode shown in Fig. 4(d). 4.2.1. Effect of r The variation of the eigenfrequencies with the geometric parameter r is presented in Fig. 5(a). A part of this figure is magnified in Fig. 5(b) for the frequency range x = 0 to 7. It is observed that the fundamental mode shapes for different values of r are different due crossings in the frequency loci of certain modes. For example, for r = 0.2, 0.5 and 0.9, Fig. 4(d), (a) and (b) are the fundamental mode shapes, respectively. The first three modes for r = 0.2 are Fig. 4(d), (a) and (c), and this order changes at r = 0.5 due to crossing of the frequency curves. It is noted from Fig. 5(b) that the eigenfrequency of the mode shape shown in Fig. 4(d) diverges with increase in r. This behaviour may be understood from denominator term (R + r cos x1) in the potential energy expression (when written in the expanded form). As r/R ? 1, for x1 = p, the potential energy diverges for any finite displacement u1. This results in stiffening of the membrane mode, an effect induced due to the increase in the curvature at the inner equator of the torus. Two sets of bunching of frequency curves are observed in Fig. 5. The first bunching set of modes (including the low frequency curves, which are horizontal at low r value) largely has dominant u2 displacement, and in some cases, u1 = 0 exactly. This latter set of modes are the pure deviatoric modes, i.e., J# = 0. On the other hand, the second bunching set of modes (including the high frequency curves, which are horizontal at low r value) largely has dominant u1 displacement, and in some cases, u2 = 0 exactly and J# – 0 for this set. Since high strain energy is associated with dilatoric stretching, the second set of curves largely correspond to high frequency modes.

Fig. 4. First few modes of the toroidal membrane for r = 0.5 and k = 60. In (d), the motion consists of axisymmetric meridional (rolling) oscillations, and has no nodal curve/point. In (b) and (f), the thick curve represents a real nodal curve u1 = u2 = 0. For the remaining modes, a thin curve represents u1 = 0 and a thick curve represents u2 = 0.

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Fig. 5. Variation of the non-dimensional eigenfrequency with the geometric parameter r for a curved toroidal membrane with material parameter k = 60.

To understand the effect of curvature further, we compare the eigenspectra of the flat and curved toroidal membranes in Figs. 3 and 5, respectively. There are two regimes in these figures that are interesting to analyze. When r ? 0, most of the eigenfrequencies are very high in both figures. This feature can be ascribed primarily to the decrease in the inertia of the membrane in this regime, as shown by Liepins (1965). The curvature of the curved torus in this case is very small, and the geometry may be comparable to the flat torus. On the other hand, when r ? 1, some of the eigenfrequencies of the curved toroidal membrane are observed to increase. This is due to the effect of large (negative) curvature at the inner equator of the curved torus. Modes which experience this kind of curvature induced stiffening have predominant u1 displacement. Modes with predominant u2 displacement, however, become compliant as r ? 1. Thus, these two kinds of modes get distinguished as r ? 1, an effect that is absent in the flat toroidal membrane. This feature of the eigenspectrum is not observed in the work reported by Liepins (1965). 4.2.2. Effect of k The effect of variation of the material parameter k on the eigenfrequencies is presented for two values of r in Fig. 6. Based on this figure, we may identify two kinds of modes, namely those which do not vary with k, and those which do. It is clear that, for the former kind, the term rlul = 0 in the strain energy expression (8). These modes are characterized by J# = 0 (deviatoric modes) as discussed previously. For the toroidal geometry, one may also conclude from the potential energy that the sensitivity (i.e., the slope of variation in Fig. 6) will depend on the radius ratio r/R due to the denominator term 1 + rcosx1/R (which goes to zero as r/R ? 1). It is found that the eigenfrequencies of modes belonging to the first bunching set of curves in Fig. 5(a) are almost insensitive to the variation in k, unlike those belonging to the second set.

Fig. 6. Variation of the non-dimensional eigenfrequency with the material parameter k for a curved toroidal membrane.

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4.3. Surface modes of an anisotropic torus We consider an anisotropy direction through the unit vector

ai ¼



   cos a sin a cos a sin a ; ; pffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffi ¼ r ð1 þ r cos x1 Þ g 11 g 22

where a is the angle of the anisotropy direction measured from the x1 direction. For a torus with r = 0.5, k = 60 and anisotropy directions with a = {0, p/6, p/4, p/3, p/2}, the fundamental modes are presented in Fig. 7 for f = 2 along with the corresponding eigenfrequencies and the deformation measures J# and Jr. In this figure, a thin line represents u1 = 0 nodal curve, while a thick line represents u2 = 0 nodal curve. It may be observed that there is no mode with a real nodal curve. This is due to the fact that, in the presence of anisotropy, the deviatoric and dilatoric modes can no longer be decoupled, as mentioned earlier.

Fig. 7. Fundamental mode of a membrane with r = 0.5, k = 60 and f = 2 for five anisotropy directions with a = {0, p/6, p/4, p/3, p/2}. A thin curve represents u1 = 0 and a thick curve represents u2 = 0.

Fig. 8. Variation of non-dimensional eigenfrequencies with anisotropy parameter f for a curved toroidal membrane with k = 60 and r = 0.5 for different values of anisotropy direction a.

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The nodal points are the intersection of the two types of nodal curves. It is observed from the figures that the nature of fundamental mode is changing with the variation of the anisotropy direction and the sign of the anisotropy parameter. For irrational values of tana, the nodal curves for u2 = 0 (thick lines) are observed to be intertwined. Further, all modes are mixed modes due to the anisotropy. The variations of the eigenfrequencies with the anisotropy parameter f for the anisotropy directions a = {0, p/6, p/4} are presented in Fig. 8 for k = 60 and r = 0.5. Two main features observed in these figures are (i) presence of an instability, and (ii) insensitivity of certain modes to variation of f. These features are discussed below. Anisotropy may be of stiffening (f < 0) or slackening (f > 0) type. Presence of a slackening type anisotropy in a material may make the stiffness matrix non-positive-definite depending upon the value of the anisotropy parameter. The value of f at which the eigenfrequency becomes zero (divergence instability) for a certain mode is the limiting value of f. It is found that different modes (not necessarily the fundamental mode) may become critical. It is observed from the figures corresponding to a = 0 and p/4 (i.e., for self closing anisotropy directions) that certain modes are insensitive to the variation of f. For such modes, aiajij = 0. 5. Conclusions and outlook The dynamics of in-plane deformation of an inflated toroidal membrane has been considered in this paper. The equation of motion has been derived in covariant form. Additional terms due to curvature of the surface and anisotropy of the membrane material have been obtained. Certain invariant deformation measures have been defined to characterize the surface modes in terms of the deviatoric and dilatoric components present. Modal analysis with isotropic and anisotropic but homogeneous material properties has been carried out and compared with the results of a flat toroidal membrane. In a flat torus, the exact solutions of the eigenfrequencies and modes of vibrations have been determined. It is observed that the square of the eigenfrequencies fall-off as square of the radius of the torus. Further, certain modes are independent of the radius of the torus. In the case of a curved torus, we have studied the effects of variation of geometric and material properties of the membrane on the modal dynamics. The main results are summarized as follows:  The variation of the eigenfrequency spectrum of a curved toroidal membrane with (non-dimensional) radius can be divided into two bunching sets of curves depending upon the nature of the modes. The first set has dominant equatorial displacement, while the second set has dominant meridional displacement.  All modes with meridional displacement are mixed modes, i.e., involve both dilatoric and deviatoric strains. This is due to the broken symmetry in the meridional direction (as compared to the flat torus).  The surface modes are observed to have either nodal curves, or nodal points, or none. Real nodal curves (equators) can exist only in axisymmetric modes. Non-axisymmetric modes can have only nodal points. There is only one axisymmetric mixed mode that has no nodal curve or point.  Curvature induced stiffening (slackening) is observed for modes in the second set (first set) with the variation of the nondimensional radius r. Further, some pure shear modes in the case of a flat torus are observed to convert to mixed modes due to curvature in the curved torus.  With the variation of the non-dimensional material constant k, the non-dimensional frequency of the first set of curves remains almost unchanged.  The existence of a limiting value of the anisotropy parameter (depending on the geometric and other material constants) for stability of the membrane was observed. Variation of the anisotropy direction was found to have an effect on the stiffness of the membrane. The analysis presented in this paper can be extended to study the transverse dynamics of a membrane with curvature and anisotropy. Understanding the effect of the curvature term on the wave propagation characteristics is an interesting theoretical problem. Some investigations in this direction related to geodesic flows on toroidal media has already been initiated (DasGupta, Nandan, & Kar, 2009). The issue of existence of nodal surfaces/curves/points/none in three-dimensional elasticity problems is an important outstanding issue which needs to be explored further. Studies on the effect of nonlinearity, internal pressure and coupling with normal displacement have applications in engineering problems involving inflated structures. We intend to address some of these issues in future. References DasGupta, A., Nandan, H., & Kar, S. (2009). Kinematics of flows on curved, deformable media. International Journal of Geometric Methods in Modern Physics, 6, 645–666. Green, A. E., & Zerna, W. (1992). Theoretical elasticity. New York: Dover Publications. Hill, J. M. (1980). The finite inflation of a thick-walled elastic torus. The Quarterly Journal of Mechanics and Applied Mathematics, 33(4), 471–490. Ignatovich, V. K., & Phan, L. T. N. (2009). Those wonderful elastic waves. American Journal of Physics, 77(12), 1162–1172. Available from: arXiv:0906.2392v1 [physics.gen-ph], 1–26. Jenkins, C. H. M. (2001). Gossamer spacecraft: Membrane and inflatable structures technology for space applications. American Institute of Aeronautics and Astronautics, 191. Jordan, P. F. (1962). Stresses and deformations of the thin walled pressurized torus. Journal of Aerospace Science, 29, 213–225. Kydoniefs, A. D., & Spencer, A. J. M. (1965). The finite inflation of an elastic torus. International Journal of Engineering Science, 3, 173–195.

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