Comput MechDOI 10.1007/s00466-013-0945-5

ORIGINAL PAPER

Static response of elastic inflated wrinkled membranes

Riccardo Barsotti · Salvatore S. Ligarò

Received: 10 April 2013 / Accepted: 12 November 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper we present an effective numericalalgorithm for determining the equilibrium shapes of inflatedelastic membranes susceptible to wrinkling. The use of a two-state constitutive law and the introduction of a suitable cri-terion allow for accounting for wrinkling of the membrane,although in an approximated way. In the active state, thematerial is able to transmit only tensile stresses; vice versa,in the passive state it is stress-free and can contract freely.Equilibrium of the membrane in the current inflated configu-ration is enforced by recourse to the minimum total potentialenergy principle, whereas the Lagrange multipliers methodis used to solve the minimum problem by accounting forthe aforesaid nonlinear constitutive law. We use an expresslydeveloped iterative-incremental numerical algorithm, con-sistent with the established governing set of equations, foraccurately monitoring the evolution of the stress field in themembrane during the inflation process. Specifically, we sup-pose that the membrane reaches its final shape at the endof a four-stage loading process corresponding to the tempo-rary enforcement and the subsequent removal of a fictitiousantagonist plane traction acting uniformly along its entireboundary. By this way it is possible to solve with great accu-racy the set of governing equilibrium equations by means of anumerical procedure in which the membrane’s tangent stiff-ness is always kept different from zero. The soundness of theproposed algorithm is verified by comparing the results withwell-known solutions available in the literature. In particular,for each specific value of pressure, the current configuration

R. Barsotti (B) · S. S. LigaròDepartment of Civil and Industrial Engineering, University of Pisa,Largo L. Lazzarino, 56122 Pisa,Italye-mail: r.barsotti@ing.unipi.it

S.S. Ligaròe-mail: s.ligaro@ing.unipi.it

of the inflated membrane found by assuming that compres-sions are allowed is compared in details to the correspondingpseudo-deformed surface, obtained by assuming a tension-only response.

Keywords Inflatable structures · Large displacementanalysis · Wrinkling · Pseudo-deformed surface

1 Introduction

Inflated textile membranes (usually indicated as “struc-tural membranes” [1]) accomplish nowadays functions inan increasing number of engineering applications. The mostcommon fields are typically aerospace and civil engineering,where they are used, for example, as air ships, weather bal-loons, inflatable domes, temporary shelters, inflatable boatsand energy absorbers, automotive air bags, and landing cush-ions for space vehicles. They are also used in biomedicalengineering as pneumatic muscles, angioplasty balloons, andin ocular biomechanics applications. More over, they areone of the promising technical solutions for many futuregossamer spacecraft systems, such as sunshields for large-aperture telescopes, solar sails, and antennas [2–4]. Finally,it is worth a mention the unusual application of the inflatablemembranes as open ocean aquaculture cages [5].

The reasons for the growing interest in membranesare manifolds. These innovative elements allow for quickdeployment and removal, show large deployed-to-packagedvolume ratio, are vey lightweight, and are able to recovertheir inflated shape after the removal of the load, even forload values beyond their load-bearing capacity. On the otherhand, the growing diffusion of inflated membranes makesthe accurate description of the stress field arising in themembrane during the inflation phase and the subsequent

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service life a present day problem of great relevance. How-ever, regardless the number of studies devoted to suchtheme, even in recent times, the analysis of inflatable mem-branes can still be considered one of the problems in themechanics of structures still not solved in a satisfactory way,because of the many material and geometrical nonlinearitiesinvolved.

As regards the material nonlinearities, even if we limitour attention to the more common case of structural textilemembranes, it can be easily seen that the particular kind, ori-entation and arrangement chosen for the bundle of the resist-ing fibers—the warp and the fill—strongly affect the macro-scopic behaviour of the resulting composite material. Theseeffects may be more pronunced in the case of moderate valuesof the stress level, because in this range the actual arrange-ment and properties of the material at the microscale mayinduce unexpected changes in the stress distribution withinthe membrane [6]. On the contrary, the mechanical responseof highly pressurized membranes, where higher stress levelsare observed, shows a much less marked dependence on thematerial properties at the microscale. Hence, for what con-cerns the constitutive law able to describe the response ofhighly inflated structural membranes, a simple linear elasticlaw may in many cases suffice, assuming that the stressesremain within the elastic range of the material. The specificlinear constitutive law may be determined experimentally bymeans of simple tests aimed to assess some mean values forthe elastic moduli [7,8].

The geometrical nonlinearities, however, cannot be disre-garded even when accounting for a linear constitutive law. Afirst nonlinearity, which may be classified as “global”, i.e. anonlinearity affecting the membrane as a whole, is caused bythe large distance separating the initial configuration, deflatedand stress-free, from the final one reached by the membrane atthe end of the inflation process, where the enclosed volume isa maximum and the resisting fibers are almost straightenedout. Consequently, in the analysis it becomes necessary toaccount for large displacements, while strains may be reason-ably considered small or moderate, especially for structuralmembranes [9]. A second, but by no means negligible, geo-metrical nonlinearity is due to the formation of wrinkles. Thismay be considered a “local” geometrical nonlinearity, sinceit usually affects only small portions of the deformed meansurface of the membrane, and it is a direct consequence of theextremely low values of the bending and torsional stiffnessof the membrane’s wall. In the equilibrium configuration, themechanical response of the membrane is ruled by the partic-ular composition of taut and wrinkled disjoint regions thatcan be observed on the inflated membrane surface. As it wasfirst pointed out by [10], wrinkling itself can be considered asa source of anisotropy, even in the case the material compos-ing the membrane is isotropic. Moreover, from an analyticalstandpoint, wrinkling-induced anisotropy is hard to handle

because the orientation of the axes of elastic symmetry aswell as the elastic moduli are all unknown in this case.

In the present paper, the use of a two-state constitutivelaw and the introduction of a suitable criterion allow foraccounting for wrinkling of the membrane, although in anapproximated way. In the active state, the material is able totransmit only tensile stresses; vice versa, in the passive stateit is stress-free and can contract freely. This simple but effec-tive mechanical model has recently been implemented bythe authors for determining the equilibrium shapes of inex-tensible membranes; in particular, cylindrical and axially-simmetric membranes have been considered [11].

Here, elastic membranes of generic shape are considered.Equilibrium of the membrane in the current inflated config-uration is enforced by recourse to the minimum total poten-tial energy principle, while the Lagrange multipliers methodis used to solve the minimum problem by accounting forthe aforesaid nonlinear constitutive law (Sect. 2). Detailsabout the solving procedure by finite elements are given inSect. 3.

The main novel point is illustrated in Sect. 4. We proposean expressly developed iterative-incremental numerical algo-rithm for accurately monitoring the evolution of the stressfield in the elastic membrane during the inflation process.For each given load increment, equilibrium of the membraneis imposed via the virtual work principle. We suppose that themembrane reaches its final shape at the end of a four-stageloading process corresponding to the temporary enforcementand the subsequent removal of a fictitious antagonist planetraction, acting uniformly along its entire boundary. By thisway it is possible to solve with great accuracy the set ofgoverning equilibrium equations by means of an expresslydeveloped numerical procedure, in which the membrane’stangent stiffness is always kept different from zero.

A comparison with the few solutions available in the lit-erature is made in Sect. 5. In particular, the results for thesquare airbag [12], the rectangular airbag [13] and the cross-shaped [14] inflated membrane are illustrated in detail. Foreach specific value of pressure, the current configurationof the inflated membrane found by assuming that compres-sions are allowed is compared in details to the correspondingpseudo-deformed surface, obtained by assuming a tension-only response.

2 The mechanical model for an inflated wrinkledmembrane

Let us consider the membrane showed in Fig. 1. The mem-brane is an ideal, perfectly flexible, two-dimensional bodymade of an elastic material. The case we want to consider hereis that of a membrane loaded by a uniform internal pressurep, slowly increasing. The inflation process is considered as

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Fig. 1 Symbols and notations

a sequence of quasi-static configurations, thus disregardingany dynamical effect.

2.1 The kinematic relations

In the initial configuration C0, assumed as reference, themembrane is unloaded and stress-free, and coincides withthe boundary Ω0 of the closed region V 0 of the three-dimensional space. Once inflated (p > 0), the membranereaches the deformed configuration C , corresponding to theunknown boundary Ω of the closed region V (Fig. 1).

Let us fix a rectangular system of coordinates in the three-dimensional Euclidean space, both in the initial and in thecurrent configuration of the membrane, and let x0 and x bethe position vectors of any point P in the initial and currentconfiguration, respectively. The position on the surface of thesame point P is also individuated by means of a pair of val-ues (s1, s2) belonging to a suitable two-dimensional auxiliardomain S, i.e. its coordinates with respect to a given intrin-sic curvilinear reference system introduced on the membranesurface.

To derive the governing set of equations, we adopt a totalLagrangian formulation. We admit the deformation to be rep-resented by the strain gradient:

F = ∇x = ∂x∂x0 , (1)

which is a continous and sufficiently regular function. Fur-thermore, we indicate with E the corresponding Green straintensor,

E = 1

2(FT F − I). (2)

Here, for the sake of simplicity, we consider the case wherethe membrane is formed by superposing two congruentplane figures, glued along their common boundaries, andwe assume that the initial configuration of the membraneis flat. Without prejudicing generality, we let the coordinatesof points belonging both to the initial and the current config-uration be expressed in the same reference system. Finally,we assume that the (x0

1 , x02 ) plane coincides with the (s1, s2)

plane. Under such hypotheses, the strain gradient F and theGreen strain E may be expressed as:

F = Fjα(e j ⊗ iα), Fjα = ∂x j

∂sα

= x j,α, (3)

and

E = 1

2(x j,αx j,β − δαβ)iα ⊗ iβ , (4)

respectively, where e j ( j = 1, 2, 3) are the unit vectorsdirected along the axes of the reference system in the cur-rent configuration and iα (α = 1, 2) are those directed alongthe axes of the (s1, s2) plane.

2.2 The constitutive law and the proposed wrinklingcriterion

In order to account for any possible form of local instabil-ity present in the current equilibrium shapes, in place of theactual material, we substitute a new, two-state fictive one.When this material is in the active state, a tensile-only stateof stress arises and the material behaves as linear elastic. Con-trariwise, when the fictive material is in the passive state, anycontraction may freely take place, unaccompanied by com-pressive stresses. By this way, the geometrical nonlinearitydue to wrinkling is artificially replaced with a suitable equiv-alent material nonlinearity [15,16].

This model, which is a direct extension of a previous onepresented by the authors in [11] and regarding inextensiblematerials, sufficiently well describes the effective mechani-cal behaviour of modern fabric textiles, especially in case ofmembranes made of structural materials showing high stiff-ness under tensile stresses. In the following, the material willbe assumed as homogeneous, isotropic, linear elastic. Theextension to the anisotropic case is straightforward.

We assume, according to the ideas formerly introduced byWu [17], that the material overall strain tensor E is the sumof an elastic, E(e), and an inelastic E(w) quote. The inelasticstrain tensor is what was named after by Wu as the wrin-kle strain. It is an additional negative semi-definite inelastic

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strain that may be though as related to the out-of-plane com-ponent of the displacement of points belonging to wrinkledregions of the surface. The original idea by Wu has beenresumed and extended to finite elasticity by many researchesduring the years that followed its first introduction (here,without attempting by no way to give an exaustive list, welimit ourselves to cite [18], among the first works, and [19],among the more recent ones).

The introduction of the wrinkle strain corresponds to somekind of “smearing” of the wrinkles on the average mid-surface of the membrane (the so called “pseudo-deformed”surface). Such a smearing operation can be considered havingan acceptable approximation in the limit case where the mem-brane’s thickness goes to zero, or alternatively the tensilestress goes to infinity. An experimental confirmation of thisitem may be found in [20,21], where it is observed that thenumber of wrinkles increases and their amplitude decreasesfor increasing tensile stresses.

The chosen constitutive law for the Piola-Kirchhoff stresstensor S thus reads⎧⎨

⎩

S = CE(e) = C(E − E(w)) ,

S ≥ 0 , E(w) ≤ 0 ,

S : E(w) = 0 .

(5)

The local mechanical response at any given point on the sur-face Ω depends on the value of the principal stresses. From(5) it can be easily seen that only one of the following threestress regimes may take place:

1. taut, both principal stresses are positive (S > 0, E(w) =0);

2. wrinkled, one principal stress is positive, the other one iszero (S ≥ 0, E(w) ≤ 0);

3. inactive, both principal stresses are zero (S = 0, E(w) =E < 0).

These three stress regimes are characterized by completelydifferent elastic responses. The first one coincides with the“standard” elastic behavior of the material. In the second one(wrinkled regime), instead, the membrane behaves as a sortof orthotropic material having one elasticity modulus equal tozero and whose directions of elastic symmetry change frompoint to point according to that of the principal stress [10].Finally, in the inactive regime the membrane is stress-free.

The overall strain E uniquely determines the stress regimethat will take place. Inactive and taut stress regimes respec-tively correspond to the inequalities:

tr(E) < 0 , det (E) > 0 , (inactive state),

tr(CE) > 0 , det (CE) > 0 , (taut state). (6)

If neither of the two sets of inequalities is verified, the mem-brane is wrinkled.

Contrariwise to the stress regimes that may take place inmembranes subjected to in-plane loads, in inflated structuralmembranes, slack states cannot take place anyhow, becauseequilibrium could not be possible.

Determining the state of stress for assigned strains isstraightforward for both taut and inactive regimes. In caseof wrinkled regime, the non-zero principal value for stressand wrinkle strain and the inclination angle of the principalstress direction have to be determined. In the general caseof an anisotropic material, the set of constitutive relations(5) allows for determining the three unknowns quantities (asillustrated in [22]). For isotropic materials, as are those con-sidered here, the principal directions for stress, strain andwrinkle strain coincide and the non-zero principal values forwrinkle strain and stress, ε(w)

2 and σ1, are given by the simplerelations:

ε(w)2 = ε2 + λ

2μ + λε1 ,

σ1 = 2με1 + λ(ε1 + ε2 − ε(w)2 ) , (7)

where μ and λ are the Lame’s constants of the material, whileε1 and ε2 are the principal values of the Green strain.

2.3 The equilibrium problem

Let the current inflated configuration C be a stable equilib-rium configuration. Assuming the existence of a strain energyfor the membrane, W , in the form:

W =∫

S

1

2CE(e) : E(e)da , (8)

that is, by assuming a relaxed energy exists [15,23], the totalpotential energy of the system

(x, E(w)) =∫

S

1

2C(E − E(w)) : (E − E(w))da

− p

3

∫

S

(x · (x,1 × x,2) − x0 · (x0,1 × x0

,2))da , (9)

i.e. the difference between the stored internal strain energyand the work performed by the pressure p is a minimum inthe current inflated configuration [11,24]. By assuming, asalready stated, that the initial configuration is flat and belongsto the (x0

1 , x02 ) plane, symmetry considerations allow to limit

the analysis to only one half of the membrane. Therefore, in(9) the region S will indicate the map of the points belongingto the upper half of the membrane.

Eventual equilibrium solutions have to satisfy the condi-tion:

min(x,E(w))∈A

Π , (10)

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where the minimum must be sought within a suitable spaceA of admissible configurations. Here, we assume that theposition vector x and the wrinkle strain tensor E(w) are suf-ficiently regular, continuous functions of (s1, s2) coordinatesand that the wrinkle strain tensor is negative semi-definite.

The constraint over the wrinkle strain can be incorporatedwithin the minimum problem (10) via the Lagrange multipli-ers method. To this aim, we search for the stationary valuesof the Lagrangian functional:

L(x, E(w),�, M) =∫

S

1

2C(E − E(w)) : (E − E(w))da

− p

3

∫

S

(x · (x,1 × x,2) − x0 · (x0,1 × x0

,2))da

+∫

S

� : (E(w) + M2)da , (11)

where � and M are two tensors whose components are mul-tipliers and slack variables, respectively.

The Kuhn-Tucker conditions [25] for the functional (11)are composed by the field equations:

F(∇ · S) + (∇F)S = p(x,1 × x,2) ,

� = S = C(E − E(w)) ,

E(w) = −M2 ≤ 0 , S ≥ 0 ,

S : E(w) = 0 , (12)

where ∇ · S = Sαβ,β iα and ∇F = xk,αβ(ek ⊗ iα ⊗ iβ), repre-sent equilibrium, constitutive and compatibility conditions,respectively. On the border of S, proper boundary terms haveto be added.

3 The solution procedure

Due to the above-mentioned geometrical and material non-linearities, the solution must be obtained numerically, byresorting to a recursive scheme. Here, we suggest a simpleincremental-iterative procedure based on a Newton-Raphsonscheme, as explained below.

For each given load increment, equilibrium of the mem-brane is imposed via the virtual work principle. At any givenpressure level p, the equilibrium equations have been lin-earized in the load increment.

Let x be the position vector of any generic point of themembrane in the last known configuration and let Δx bethe increment in the position vector produced by the givenincrement in the load, Δf . Disregarding terms of second orhigher order, the linearised equilibrium equations may beformally written as:

KG(Δx) + KE (Δx) + Kp(Δx) = Δf . (13)

The linear differential operators KG and KE represent thetangential geometrical and elastic stiffness, respectively,while Kp is the stiffness term accounting for the followerload p. They can be formally expressed as:

KG(x) = ∇ · [(∇x)S] , (14)

KE (x) = ∇ ·(

FC1

2[(∇x)T F + FT ∇x]

)

, (15)

Kp(x) = p(x,2 × x,1 + x,2 × x,1) , (16)

where F = ∇x, E = 1/2(FT F − I) and S = C(E − E(w))

are known quantities calculated in the last known equilibriumconfiguration of the membrane, x, before the load incrementΔf is applied.

In general, a non-symmetric stiffness matrix may corre-spond to expression (16). However, a symmetric stiffnessmatrix accounting for the pressure load is obtained for all theexamples we deal with in the present paper [26].

The cartesian components of KG , KE and Kp are:

KG(x) · ei = Sαβ xi,αβ + Sαβ,β xi,α , (17)

KE (x) · ei =[

FiαCαβγ δ

1

2

(Fjδx j,γ + Fjγ x j,δ

)]

,β

, (18)

Kp(x) · ei = pεi jk(x j,2xk,1 + x j,2 xk,1) , (19)

where εi jk is the alternating symbol.At every internal point of the membrane, the load term on

the right side of Eq. (13) is expressed as:

Δf = (Δp + p)(x,1 × x,2) − FS , (20)

where Δp is the increment in the pressure and p is the pres-sure level applied at the previous load step.

To solve the nonlinear system of Eq. (12), we make use ofa specifically developed FE code using plane linear isopara-metric elements. By writing in weak form the linearized equi-librium Eq. (13), the stiffness matrices as well as the externalload vector can be easily obtained [27].

4 A technique for improving the algorihm’s accuracy

In general, solving the equilibrium problem for a wrinkledelastic membrane, presents some not negligible difficulties,even in the simpler case where only in-plane traction forcesare applied without any pressure (see, for example, [28,29]).A more recent attempt for solving the inflation problem, how-ever without accounting for wrinkling, may be found in [24].Here we introduce an expressly developed numerical proce-dure that seems able to effectively handle with the equilib-rium problem of inflated wrinkled elastic membranes.

The aim of the numerical procedure of solution is to deter-mine the equilibrium shape of the initially flat membraneinflated by an internal pressure up to the value p f in . It is aneasy matter to verify that for p = 0 all the stiffness terms

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Fig. 2 The proposed four-stages loading process

given by relations (14)–(16) become equal to zero for a flatmembrane. Therefore, in order to assure a sufficient degreeof stability of the numerical algorithm, during some stagesof the inflation process, a uniform traction is applied alongthe border of the membrane. More precisely, we will adopt aloading process composed by the four different stages listedbelow (Fig. 2).

1. Tensioning stage (OA): an increasing uniform tractionTmax is applied all along the border at the onset of theloading process, so as to make the stiffness matrix non-singular also in the first stages of the inflation process.To this aim, small values of Tmax proved to be sufficientwhen dealing with tension-only membranes. Conversely,in the compression-allowed case the values adopted forTmax were not small compared to the tensile stresses thatappear in the membrane once it is inflated. Although wedid not provide a general criterion for establishing Tmax ,we suggest a semi-empirical way for determing it in orderto increase the stability of the numerical procedure. In par-ticular, we found that Tmax should be of the same order ofthe mean traction that would be present in an inextensiblesphere of the same area as the membrane, inflated by thesame pressure p f in . A detailed analysis of this point willbe illustrated in a forthcoming paper by the authors.

2. Inflating stage (AB): the internal pressure is increasedfrom zero up to pmax , while mantaining the tractions con-stant to Tmax . This stage is subdivided into several incre-mental steps, each one corresponding to a given incrementin the pressure. By this way a suitable iterative procedureperformed at each incremental step allows for a progres-sive updating of the wrinkle strains within each element.

3. Detensioning stage (BC): the border tractions are progres-sively reduced until they go back to zero, while mantainingthe internal pressure equal to pmax . This stage, as the pre-

vious one, is subdivided in several incremental steps, eachone corresponding to a given decrement of the traction forthe same reason explained above.

4. Deflating stage (CD): the internal pressure is progres-sively reduced from the initial value pmax down to itsfinal value p f in .

During the first and the third stage (i.e., tensioning anddetensioning), according to the presence of the traction forceson the edge, T = T1e1 + T2e2, the boundary conditions onthe unconstrained part of the border may be expressed as:

FSn · eα = Tα , (α = 1, 2) , (21)

where n is the unit vector orthogonal to the border of thereference domain S, pointing outward. In case no tractionis applied, the inflated membrane is orthogonal to the planecontaining its initial configuration.

The first tensioning stage is performed in one single loadincrement and no wrinkle strain appears. The other threestages, instead, are performed by progressively increasingthe loads. At each given load step, i.e. for a given incrementin the pressure Δp or in the traction forces ΔT , an iterativeprocedure is used to determine the corresponding incrementsin the displacement and wrinkle strain components.

At the beginning of each iteration we update the nodalcoordinates. By means of the shape functions, in each ele-ment we update the coordinates of the membrane surface:

x = x + Δx , (22)

then we check whether each element is taut, wrinkled or slackby using the wrinkle criterion (6) and finally we update thewrinkle strain components:

E(w) = ∇(x) (slack elements) ,

E(w) = ε(w)2 (n2 ⊗ n2) (wrinkled elements) , (23)

where the non-zero principal wrinkle strain ε(w)2 is obtained

by using the first of (7) and the unit vector n2 of (s1, s2) planeis orthogonal to the tensile principal stress direction.

Once all quantities are updated, the new value for the loadterm Δf is calculated according to (20), the load vector andthe stiffness matrices are assessed and a new iterative step isperformed. The iterative cicle is stopped and the next loadstep is applied once the conventional value for the error, e,becomes lower than a fixed threshold value. For any givenmesh, the error is defined as:

e = ΔX · ΔX(X − X0) · (X − X0)

, (24)

where ΔX is the vector collecting the increment in the nodalcoordinates, while X and X0 are the current and initial coordi-nates vector, respectively. A plot of the error recorded duringa single load step is shown in Fig. 3.

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Fig. 3 FE iterative solution procedure, error versus iteration number(compression-free membrane)

A numerical procedure of solution which is somewhatsimilar to our is illustrated in [9]. We observe however thathere the required stiffness of the membrane in the very begin-ning of the loading process is guaranteed by the tractionforces applied along the border, while in the cited work apseudo-dynamic method is used.

The incremental-iterative method makes use of a globalstiffness matrix which is never singular, thus avoiding thedrawbacks that often appears when dealing with wrinkledmembranes, such as spurious oscillations in the stress values(to this regard, see [30] where penalty factors are used tolimit such undesiderated effects).

5 Applications

In the following examples it is assumed that the membranesurface is obtained by the superposition of two congruentplane figures, sealed along their common boundaries. Byassuming, as already stated, that the initial configuration ofthe membrane is flat and belongs to the (x0

1 , x02 ) plane, sym-

metry considerations allow to limit the analysis to only onehalf of the membrane. Hence, the points initially lying in theplane region S will be mapped to the points belonging to theupper half of the membrane (Fig. 4).

In the following, three application examples are illus-trated, namely: the square, the rectangular and the cross-shaped airbag. In each example, two different constitutiveassumptions, identified as “tension-only” and “compression-allowed” membrane, are accounted for. The use of thesetwo different constitutive relations allows for highlightingthe controversial aspects of the solutions obtained by meansof the standard linear elastic constitutive law, as well as theirmuch lower rate of convergence.

When tension-only state of stress are considered, ournumerical procedure proved to be able to converge even ifvery small values of the external traction T were appliedalong the border. To this regard, Fig. 5 clearly illustrates thatfor tension-only state of stress the solution is practically unaf-fected by the value of Tmax .

Fig. 4 Axonometric view of initial and current inflated configuration(upper half of the membrane)

Fig. 5 Vertical displacement of the central point of the square airbag,vC , versus external traction T (tension-only state of stress)

On the contrary, when compression-allowed membranesare considered, i.e. a standard linear elastic law is assumed,the numerical solution procedure showed to be more unstableand both the external traction Tmax and the overpressure pmax

have to be suitably chosen in order to assure convergenceof the procedure. It is worth noting, in addition, that smallvariations in the solution were observed after a change in thevalues assigned to Tmax or pmax .

Finally, we found that the deflating stage was not neces-sary at all for tension-only membranes, while it was indis-pensable for compression-allowed membranes. Thus, thenumber of stages of the loading process was three for tension-only membranes, and four for compression-allowed mem-branes.

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Fig. 6 Rate of convergence for tension-only (top) and compression-allowed (bottom) elements

5.1 The square airbag

This first example is a classic benchmark problem [12]. Inthe initial configuration the square airbag is flat and has adiagonal length of 1,200 mm with a thickness of 0.6 mm. Thematerial is linear elastic and isotropic with Young’s modulusE = 588 MPa and Poisson’s coeffcient ν = 0.4. The internalpressure is set equal to 5 kPa. Because of symmetry, only one-quarter of the upper part of the airbag has been modeled.

In order to give an estimation of the rate of convergence ofthe numerical procedure, this example has been solved withdifferent meshes, each characterized by a different number ofd.o.f. In Fig. 6, the vertical displacement of the central pointof the square, vc, is plotted versus the d.o.f. number. Thedisplacements are made dimensionless by dividing them bythe displacement v0

c calculated with the more refined mesh(having 663 d.o.f.). The two plots in the figure highlightthe strong differencies existing in the rate of convergenceof the numerical method corresponding to tension-only andcompression-allowed elements. A much smoother and fasterconvergence is observed for tension-only elements. On thecontrary, the presence of the compressive stresses is respon-sible for the slower and slightly more erratic convergence forcompression-allowed elements, as it was expected to someextent.

The main results relative to the more refined mesh (663d.o.f.) are showed in Figs. 7 and 8. Let us first consider thecase in which a pure tension-only state of stress is allowed.The vertical displacement of the central point is 21.5 cm; the

(a)

(b)

Fig. 7 Top view of the whole airbag: tension-only state of stressallowed, pseudo-deformed inflated shape (a), compressive stressesallowed, inflated shape (b)

horizontal displacement of point A and the displacement ofpoint B along the diagonal are 11.9 and 4.9 cm, respectively(Fig. 7a). The mean values of the normal stresses in the 16elements surrounding the origin are equal to σ11 = σ22 =3.92 MPa (Fig. 8, top). These results match very closelywith the corresponding ones recently obtained for the sameproblem by other authors [9,30]. When compressive stressesare allowed, the vertical displacement of the central pointdecreases to 20.0 cm (Fig. 7b), while the mean values of thenormal stresses in the 16 elements surrounding the originraise to σ11 = 4.82 MPa and σ22 = 4.76 MPa (Figure 8,bottom).

We remark that, for the “tension-only” membrane theinflated shape that has been obtained has to be considereda pseudo-deformed configuration, i.e. a surface somewhataveraging the actual one, as illustrated by Wu [17]. Forthe “compression-allowed” membrane, instead, the inflatedshape would theoretically coincide with the actual one, if thebending stiffness of the membrane were correctly accountedfor and if the numerical procedure were able to exactly solvethe problem, without making any error.

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(a)

(b)

Fig. 8 Plot of σ11 (blue circles) and σ22 (red squares) values alongthe positive side of the x1 axis: tension-only state of stress allowed (a),compressive stresses allowed (b). (Color figure online)

In the case considered here, however, the inflated shapeobtained for the compression-allowed membrane (Fig. 7b)is the result of a spurious mesh-dependent effect, althoughit might seem more realistic than the pseudo-deformed one(showed on top of the same figure). In fact, the distribution,number and magnitude of the clearly visible foldings alongthe contour, produced by the compressive stresses, stronglydepend on the characteristic length of the elements. Thefolding changes if the characteristic length of the elementis changed or even if small changes are made on the Tmax

or pmax values used in the numerical procedure of solution.Therefore, no reliable conclusion can be drawn whenevera “compression-allowed” membrane approach is used. Wenote, anyhow, that the numerical procedure that we devel-oped shows a quite good stability, since the resulting shapeof the airbag is regular and symmetric, and the spurious fold-ing is almost uniformly distributed, also in the case wherecompressive stresses are allowed.

Figure 8 (top diagram) clearly shows that when no com-pressive stress is allowed a wrinkled region is to be expectedin the middle of each side of the square airbag. When, on the

Fig. 9 Top view of a PVC coated polyester inflated square airbag (top);wrinkled regions in the upper-right quarter of the numerical model(bottom)

contrary, the membrane is assumed able to transmit compres-sive stresses (Fig. 8, bottom diagram), spurious compressivestresses appear. It is worth observing that the wave-length ofthe oscillations in the compressive stress is of the same orderof the length of the element.

A qualitative comparison between the numerical resultsand those obtained from experiments has also been made. Theinflated shape of the airbag obtained by means of the numer-ical model turned out to be almost close to the experimentalone (Fig. 9). In particular, the position and the extension ofthe wrinkled regions observed experimentally are in goodagreement with those obtained from the numerical model.To this regard, in the same figure a plot of the distributionof the wrinkle strain in the upper-right quarter of the numer-ical model is shown, by drawing a small arrow in each ele-ment of the mesh in the direction of the principal wrinklestrain, having a length proportional to the same principalvalue.

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(a)

(b)

(c)

Fig. 10 Experimental test on a PVC coated polyester rectangularairbag (a), top view of the pseudo-deformed inflated configuration fortension-only material (b), top view of the deformed inflated configura-tion for compression-allowed material (c)

5.2 The rectangular airbag

The second example is a simple but interesting variation ofthe preceding problem [13]. We consider an initially plane,rectangular airbag whose sides are long 1,800 and 566 mmrespectively. The membrane’s thickness, t , and elastic prop-erties, E and ν, are the same as the preceding case. A differ-ential pressure between the inside and the outside of 5 kPa isassumed.

The top view of the pseudo-deformed inflated configura-tion of the numerical model is shown in Fig. 10. The com-parison between the numerical results and those obtainedfrom an experimental test performed on a PVC coated poly-ester membrane shows a good agreement. As in the previousexample, when a compression-allowed material is used, amesh-dependent folding is observed.

As it was expected, the central part of the inflated airbagis almost of cylindrical shape, showing a diameter-to-lenghtratio of about 1:5. A clearly detectable wrinkled zone extend-ing from the ends towards the middle of the airbag for a lengthof the same order of the width of the rectangle may be seenat both ends of the airbag (Fig. 11b).

Both the hoop and the axial stresses are far from beinguniform up to a distance from the end which can be muchlarger than the width of the rectangle (Fig. 11a). Moreover, in

(a)

(b)

(c)

Fig. 11 Upper-right quarter of the numerical model: mean nodal val-ues of the principal stresses (a) and wrinkle strain (b), level curves of therelaxed strain energy density (×10−8 J/mm2) per unit square millimeterof surface (c)

the lateral part of the cylinder a relevant stress concentrationis observed (Fig. 11c).

The disuniformity observed in the state of stress stronglydepends on the length-to-width ratio of the rectangle. Figure12 shows the diagrams of the principal stresses in the cross-section of the rectangle taken at x1 = 0 for two rectangleshaving different length (1,800 or 3,600 mm) and the samewidth of 566 mm (because of symmetry only positive valuesfor x2 are considered). The figure clearly shows that in thelonger rectangle (having a diameter-to-length ratio of 1:10)the stresses in the middle cross-section are almost uniform.On the contrary, in the shorter rectangle (diameter-to-lengthratio of 1:5) strong variations are observed in both the hoopand the axial stresses. In particular, the axial stress presentsvery high values, up to ten times the corresponding values

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Fig. 12 Principal tensile stresses in the upper-right quarter of the cross-section of the cylinder at x1 = 0 (top hoop stress; bottom axial stress)for two different lengths of the rectangle: 1,800 mm (red curves) and3,600 mm (blue curves). (Color figure online)

in the longer rectangle, which are concentrated in a smallregion around x2 = 0.

5.3 The cross-shaped inflated membrane

This problem has been proposed in [14]. The particular shapeadopted in this case allows for drawing some qualitative con-siderations on the distribution of stresses and wrinkles at theintersection between beams and columns in structures madeup by suitably assembling inflated membranes.

Because of symmetry, only one-quarter of the upper partof the airbag is modeled. A total number of 843 d.o.f. isused. A mesh-dependent spurious folding is observed alongthe border in the compression-allowed case (Fig. 13b), as itwas already observed for both the square and the rectangularairbag.

The state of stress is all but uniform. An evident stressconcentration is located in the neighbourhood of the inter-secting lines between the central and the lateral parts of theairbag. In the same region larger values of the wrinkle strainare observed (Fig. 14). The aforementioned stress concen-tration can also be inferred by looking at the sharp changein the inclination clearly detectable along the intersectingline between the central and the lateral parts of the air-bag(Fig. 13a). The results obtained in this case match with thecorresponding ones available in the literature [14], thus rep-resenting a validation of the numerical method used.

(a)

(b)

Fig. 13 Top and side view of the whole airbag: tension-only stateof stress allowed, pseudo-deformed inflated shape (a), compressivestresses allowed, inflated shape (b)

6 Conclusions

A robust algorithm for the direct determination of the equi-librium shapes of elastic inflatable structural membranes sus-ceptible to wrinkling has been derived.

Two kinds of geometrical nonlinearities were consid-ered. The first (“global”) one stems from the large dis-tances that separate the initial stress-free, deflated configu-ration from the current inflated configuration. To this regard,large displacements/rotations have necessarily been consid-ered. The second type of nonlinearity (usually called “local”)is instead inevitably due to wrinkling phenomena. The lat-ter geometrical nonlinearity is accounted for in a simpleralthough approximated way by suitably modifying the con-stitutive law for the material.

The proposed method is based on the assumption that themechanical response of the membrane’s constituent mater-ial can be described as that of a two-state material. Whenthe material is in the active state, a linear elastic constitu-tive law is assumed and the membrane is subject to tensilestresses only; vice versa, when it is in the passive state, the

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Fig. 14 Upper-right quarter of the numerical model: mean nodal val-ues of the principal stresses (top) and wrinkle strain (bottom)

membrane is unable to sustain any compressive stress andcontracts freely.

The inflation process has been considered as a sequenceof quasi-static configurations, disregarding any dynamicaleffect. Equilibrium of the membrane in the current configu-ration has been enforced by recourse to the minimum totalpotential energy principle. The Lagrange multipliers methodhad enabled solving the minimum problem by accounting forthe aforesaid nonlinear constitutive law.

We proposed an expressly developed iterative-incrementalnumerical algorithm for accurately monitoring the evolutionof the stress field in the elastic membrane during the inflationprocess. We adopt a loading process composed by four stages,namely: tensioning, inflating, detensioning and deflating. Aboundary fictitious traction is enforced during the tension-ing stage and it is subsequently removed in the detensioningstage; an increasing and a decreasing pressure is applied dur-ing the inflating and the deflating stages, respectively. Foreach given load increment, equilibrium of the membrane hasbeen imposed via the virtual work principle. By this wayit has been possible to solve with great accuracy the set of

governing equilibrium equations by keeping the membrane’stangent stiffness always different from zero.

Three airbags of different shape have been solved. In allthe examined cases, the results obtained match well withthe corresponding ones available in the literature, thus prov-ing the soundness of the proposed algorithm. In each case,a comparison between the results obtained by means of theproposed nonlinear constitutive law and those obtained byassuming a standard linear elastic law (thus allowing for com-pressive stresses) has been made.

When tension-only state of stress were considered, theprocedure proved to be able to converge very quickly towardsthe equilibrium configuration, even if very small values of theexternal traction were applied along the border. More over, nodeflating phase had to be performed. On the contrary, whencompressive stresses were allowed, the numerical solutionprocedure showed to be more unstable and the deflating stagewas indispensable to assure the convergence. In all the threeexamples, a spurious, mesh-dependent folding was observed.

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