Wrinkling Simulation of Membrane Structures under Tensile and Shear Loading

Columbia International Publishing Journal of Vibration Analysis, Measurement, and Control (2015) Vol. 3 No. 1 pp. 17-33 doi:10.7726/jvamc.2015.1002

Research Article

______________________________________________________________________________________________________________________________ *Corresponding e-mail: [email protected] Mechanical & Industrial Engineering Department, Indian Institute of Technology Roorkee, India 2 Group Head, Antenna Systems Mechanical Group, Mechanical Engineering Systems Area, Space

Applications Centre, ISRO, Ahmedabad , India

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Wrinkling Simulation of Membrane Structures under Tensile and Shear Loading

Satish Kumar1*, S H Upadhyay1, and Anil C Mathur2

Received 28 November 2014; Published online 30 May 2015

© The author(s) 2015. Published with open access at www.uscip.us

Abstract

This paper presents a detailed Finite Element Analysis study of the formation and evolution of wrinkle pattern that observed in stretched thin membranes. The model problem is set up under the various load conditions. As a precondition for wrinkling, development of compressive stresses in the transverse direction is found to depend on both the length-to-width aspect ratio & thickness of the rectangular membrane. Shape and size of wrinkle also depends on applied tensile strain and shear strain. The analysis has been done in two parts; in first part we see the effect of thickness of membrane and number of element variation on number of wrinkles and eigenvalue frequency. In second part, two-dimensional stress analysis is performed under the plane-stress condition to finding out stretch-induced stress distribution patterns in the elastic membrane. The analysis has been carried out on the rectangular membrane with tensile loading with assuming imperfections in the structure with the ABAQUS a commercially available finite element package. Wrinkling patterns are presented to show how wrinkle formation with increasing shear loads, tensile loads, membrane thickness and number of elements.

Keywords: Wrinkle; Rectangular membrane; Eigenvalue; Eigen frequency; Tensile load; Shear load; Finite Element Analysis

Nomenclature

FEA : Finite Element Analysis t : Time in sec ρ : Mass Density of the membrane in kg/m3 L : Length of the membrane W : Width of the membrane E : Young’s modulus of the membrane in N/m2

ν : Poisson ratio

Satish Kumar, S H Upadhyay, and Anil C Mathur / Journal of Vibration Analysis, Measurement, and Control (2015) Vol. 3 No. 1 pp. 17-33

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α : Length-to-width aspect ratio RTM : Rectangular Thin Membrane BC : Boundary Condition IMP : Iterative Materials Properties

1. Introduction Over the past decades, new structural concepts for space missions involving thin film surface and high-accuracy membrane structures have been designed. Compared to traditional space structure, a gossamer structure (membrane structure) has lesser mass and package volume. But the materials used in gossamer structure cannot support compressive stress because of their small bending stress. The result of compressive stress is that buckling occurs and wrinkles are formed. The existence of wrinkled regions may affect the performance and reliability of the flexible gossamer structures (as in an antenna or a reflector). These prestressed membrane structures will have to remain partially wrinkled in their outfitted configuration. Tensile forces are applied to remove irregular surface distortions during operation. Thus, the prediction and analysis of wrinkle patterns in a membrane surface is one of the resent interests for material & space researchers. In this paper, the effects of membrane wrinkling are dealt with implicitly as it may be logically understood that wrinkles have a small effect on force magnitudes and environmental conditions (gravity, air, vacuum and temperature). Finite element models of membrane structures, based on thin-shell elements and membrane element, are used to simulate the formation and growth of wrinkles. Removing the wrinkles would require a biaxially tensile stress state, thus significantly increasing the loads transmitted to the edge deployable structure that supports the membrane. Wrinkles can reduce the performance of reflectors and sunshields, or cause difficulties in maneuvering solar sails. Hence, it is now important to details calculation of the wrinkles, such as amplitude and wavelength, in order to determine if the membrane structures meet the requirements of each particular application. Previous numerical studies of wrinkled membranes have mostly focused on determining the region(s) affected by wrinkles and the direction of the wrinkles. It is now possible to compute the actual shape and size of the wrinkles in structures of different shape and size. Here we present a FEA procedure for carrying out such simulations using the commercially available finite element package ABAQUS. It is shown that the accuracy of the wrinkles computed in this way is such that the numerical simulation can replace physical experimentation, although the computer run times are currently still impractically long for the present procedure to be adopted as a design tool. A significant benefit of the present work is that one can probe the simulation results in order to gain additional insights into the characteristics of wrinkles and their evolution under varying, thickness, number of elements, loads or boundary conditions on different shapes. Some of modeling techniques have been recently proposed to study the wrinkling behavior of membrane. In this paper we present a finite element (FE) simulation technique to study the membrane wrinkling, which prognosticates with good accuracy the natural frequencies and mode shapes of wrinkled membrane structures. The simulation has been done both in vacuum and in air.


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Development of a discretely pre-tensioned, pressurised membrane requires more complexity and will be the subject of further study. We predict that the out-of-plane pressure can usefully remove some wrinkles by overwhelming compression in the bulk of membrane.

2. Review of Previous Works A few years past, a much interest on the wrinkling of membranes has magnetized, starting from the study that the thin-walled shell can carry loads well above the initial buckling value; which incited the progress of tension field theory by Wagner (1929). In 1938 (Reissner), 1961(Stein and Hedgepeth), Simplified the formulations of tension field theory and Mansfield (1969, 1989) made it possible to find solutions to non parallel tension-lines quandaries. In 1986 Pipkin proposed generalize theory of tension field and, more recently, Epstein and Forcinito (2001). A rudimentary prevalent to all of these formulations, and withal to the accompanying numerical solutions that have been pursued alongside, is that the membrane is modelled as a no-compression, two-dimensional continuum with negligible bending stiffness. Visually perceive Jenkins and Leonard (1991) for an extensive list of references and Adler (2000) for a more recent perspective; visually perceive additionally Liu et al. (2000) for a numerical study in which wrinkle amplitudes were prognosticated. Hence, it is surmised in effect that an illimitable number of wrinkles of infinitesimally diminutive amplitude will compose. This is not, of course, what authentically transpires and it is prominent that, albeit the stress fields engendered by these theories are essentially veridical, the out-of-plane displacements are significantly different from those observed in practice. Wrinkling theory was the Iterative Materials Properties model (IMP) firstly developed (Miller and Hedgepeth 1982; Miller et al. 1985). It is predicated on the observation that if during a simulation a membrane element is deemed to be wrinkled, the geometric strain in the direction perpendicular to the direction of the wrinkles, due to out-of-plane deformation of the material, can be modelled by introducing a variable efficacious Poisson’s ratio for the element. Hence, in lieu of utilizing the standard “taut” modulus matrix, predicated on Hooke’s law for plane stress and given by

2

1 0

1 01

10 0 (1 )

2

t

vE

M vv

v

Miller et al. used the “wrinkled” modulus matrix

2(1 ) 0

0 2(1 ) ,4

1

w

A BE

M A B

B B

(1)

(2)


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Where A = (εx – εy) / (ε1- ε2 ) and B = γxy /(ε1 – ε2); εx , εy, γxy are the engineering components of plane strain; ε1, ε2 are the major and minor principal strains ε1 ε2; and the directions 1 and 2 are parallel and perpendicular to the wrinkles, respectively. For later on, note that σ1, σ2 are the major and minor principal stresses (σ1 σ2). This model is implemented as a user-defined material (UMAT) subroutine in the finite element package ABAQUS (Adler 2000). A standard ABAQUS iteration Adler’s IMP subroutine calculated at any stage the principal strain and stresses at any point using Mt , initially assuming the element to be taut, and then checks:

If σ2 0, the element is taut and so no change is needed; If σ2< 0 and ε1 0, the element is slack and so all stress components are zero; If σ2< 0 and ε1> 0, the element is wrinkled, so the stress components are recomputed using

Mw.

This is kenned as the coalesced wrinkling criterion, as a coalesced stress/strain condition has to be satiated for a wrinkle to subsist. Wrinkling criteria predicated pristinely on stress or strain have probable shortcomings and are less precise (Liu et al. 2001). Prosperous prognostications of the shape and pattern of the wrinkled regions in a square membrane subjected to point loads, and withal in inflated balloons of different shapes were obtained by Adler (2000). The main quandary was that the solution inclined to diverge in the presence of many slack regions. Johnston (2002) utilized the same approach to analyze the static and dynamic demeanor of the sunshield for a space telescope. This sunshield consists of several reflective foils which wrinkle extensively. An alternate tension field model was developed by Liu et al. (1998). The main distinguishment between this method and IMP is that, in lieu of modifying the material properties iteratively, the utilizer preselects a soi-disant penalty tension field parameter to provide a modicum of stiffness in the direction transverse to the wrinkles. This avails to surmount the numerical singularities associated with vanishingly minute diagonal terms in the tangent stiffness matrix. Liu et al. (1998) carried out a simulation of the deployment of a parachute. Modelling issues, including the cull of the penalty term, influence of the order of integration and local remeshing in the wrinkled regions are all discussed in this paper. Liu et al. (2000) amalgamated the approach of their earlier paper with the semi analytical resoluteness of the impending buckling mode by Lin and Mote (1996), The wrinkle wavelength and amplitude, by applying Lin and Mote’s eigenvalue analysis to determine the number of wrinkles. The wrinkle amplitude is then determined through an argument essentially identically equivalent to that put forward by Wong and Pellegrino (2006b). It is implicitly surmised that the number of wrinkles will not vary once the wrinkles have commenced to compose (which is not correct), and that the wrinkled region can be surmised to deport as a simply fortified rectangular plate. Liu et al. (2000) have shown this approach to provide plausibly precise results for a square membrane subjected to a concrete coalescence of tension and shear. Several iterative schemes that use no-compression material models have been proposed. In their simplest form, these schemes begin by assuming that the behavior of the membrane is linear elastic.


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Then, any compressive principal stresses are set to zero and the associated stiffness matrix coefficients are also set to zero. The principal stresses are recalculated at every iteration, to eschew history dependency in the results. An early study of airbag inflation predicated on this approach (Contri and Schrefler 1988) set a sample quandary that many others have subsequently tackled. An analogous approach was endeavored by the present authors, utilizing the *NO COMPRESSION option in ABAQUS, but poor convergence was observed. Determinately, a number of membrane finite elements that incorporate wrinkling within their formulation have been derived from a continuum mechanics approach. The methods proposed range from utilizing a modified deformation tensor (Roddeman et al. 1987), to a geometrically modified (Nakashino and Natori 2005) or energetically modified (Haseganu and Steigmann 1994; Barsotti and Ligaro 2000) stress strain tensor.

3. Membrane Materials Many materials with improved combinations of properties are used by space scientist / engineer to select their specific mission requirement. The smart materials are listed in the Table 1 (Gajbhiye et al 2012). Membrane structures consist of two dimensional pre-stress elastic thin membrane as a major structural element. A membrane has no compression or bending stiffness, therefore it has to be prestressed to act as a structural element. In practice, any two-dimensional elastic continuum resists bending moment. However, if the tension is large and the curvatures are small, the effect of bending moment can be neglected. Thus, the membrane can be imagined as an extension of the string to two dimensions. For analysis of membrane structure following assumptions are made as;

i. Effect of gravity on the membrane is negligible. ii. Displacement is only in vertical direction (Gajbhiye et. al 2012). iii. Membrane is thin enough to neglect its volume and only consider its area. iv. Magnitude of pre-stress remains constant and mass density assumed uniform throughout

the membrane. Table 1 Material properties of membrane material

S.No. Name of Materials Mass Density

[kg/m3] Young's Modulus

[N/m2] Poisson's Ratio(ν)

1 Kevlar Reinforced

Film 790 11.9 x 10ˆ9 0.3

2 Kapton 1420 2.50x10ˆ9 0.34

3 Mylor 1390 8.81x10ˆ9 0.38

4. Results and Discussions 4.1 Rectangular Membrane with shear loading Finite element modelling of the formation of wrinkles in at membranes is challenging if the


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membrane is assumed to be perfectly elastic, at which is most convenient for setting up the initial mesh. The elastic analysis cannot proceed without some out-of-plane distortion, in order to seed the potential for bending and, hence, wrinkling. Therefore, the mesh must be imperfect initially, and one successful approach is to superpose artificial displacements from a stability analysis onto the initially at membrane before proceeding with main analysis. The buckled shapes of the first eigenmodes are then extracted via a linear stability analysis. Here we have taken Kapton, Kevlar & Mylar for analysis and the size of membrane element 1.5 m x 0.5 m and thickness of material will varies from 0.01x10-3 m to 0.1x10-3 m. The properties of the membrane are given in Table1. Each simulation consists of steps, as shown in Fig. 1. The first step consists of pretensioning the membrane by moving the upper edge by 0.0005 m, in the y-direction. Then, a geometrically nonlinear equilibrium check was performed. The geometric stiffness provided by the prestress has the effect of increasing the out-of-plane stiffness of the thin membrane. Only translation in the y-direction was allowed for the two side edges. And all six degrees of freedom of the bottom edge were completely constrained using *ENCASTRE. In the second step, an eigenvalue buckling analysis was carried out with a prescribed horizontal displacement of 0.003 m and 0.03 mm at the upper edge as shown in Fig. 2a. The model boundary conditions were modified by using the *BOUNDARY, OP=MOD option. This has the effect of moving the upper edge nodes in the horizontal x-direction by the prescribed displacement. The chosen geometrical imperfections were then seeded onto the pristine mesh using the *IMPERFECTION command. The out-of plane displacements following loading are obtained, showing the number of wrinkles, their amplitudes and their locations on the membrane. The first principal stresses are also distilled, given that wrinkles tend to form in their direction. But a more useful measure of wrinkle location stems from examining the second principal stresses. When they are negative, they indicate compression and hence regions where wrinkles can be expected. Note that, under increasing pre-tension, the wrinkle amplitudes do increase generally but their distribution tends not to once a dominant pattern has formed. Consequently, the compressive is largely invariant in plan form after wrinkles have formed, thereby providing useful information without having to specify load magnitudes. The analysis has been done for two types of element first one is rectangular shell element (S4R) and second one is triangular shell element (S3R). In both the cases uniform meshing has been used. In Fig. 2b total number of uniform rectangular shell element is 4800 and in Fig.2c total number of uniform triangular shell element is 9600. Kevlar membrane with rectangular element has thickness of 0.01 x10-3 m and 27 number of wrinkle has been formed as shown in Fig. 3. Similarly in Fig. 4 kevlar membrane with triangular element has same thickness but has formed only 13 wrinkles. This means the number of wrinkles varies with the type of elements. Also when thickness of membrane increases the number of wrinkles decreases as shown in Tables 2 and 3. As Figs. 5 and 6 show that the effect of number of uniform element (S4R and S3R) on wrinkling behaviour of membrane is more when thickness of membrane is less and vice versa. If the thickness of membrane increased the convergence on number of wrinkles is more.


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Fig.1. Flow chart for wrinkle analysis in ABAQUS

Fig. 2. a. A rectangular Membrane with bottom side clamped and upper Side is subject to Shear Load


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Fig. 2.b. Membrane mesh (4800 Rectangular Element)

Fig. 2.c. Membrane mesh (9600 Triangular Element)


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Fig. 3. Kevlar (t=0.01x10-3 m) eigenvalue (0.26213)

Fig. 4. Kevlar (t=0.01x10-3m) eigenvalue (0.26941)

Table 2 Comparison of Eigenvalue and number of wrinkles when the pre-stressed of 0.0005 m is

applied to three different smart materials of different thickness taking rectangular elements (S4R).

S.No. Thickness (x 10 -3m)

Kevlar Kapton Mylar

Number of

Wrinkles

Eigenvalue (for first

mode shape)

Number of

Wrinkles


mode shape)

Number of

Wrinkles


mode shape)

1 0.01 27 0.26213 27 0.29591 27 0.33297 2 0.02 25 0.26272 25 0.29652 23 0.33359 3 0.03 21 0.26336 21 0.29716 20 0.33435 4 0.04 20 0.26400 20 0.29782 19 0.33494 5 0.05 18 0.26465 19 0.29894 17 0.33563 6 0.1 13 0.26794 14 0.30187 13 0.33914


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Fig. 5.a. Effect of thickness of Kapton membrane with respect to number of rectangular elements

Fig. 5.b. Effect of thickness of Kevlar membrane with respect to number of rectangular elements


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Fig. 5.c. Effect of thickness of Mylar membrane with respect to number of rectangular elements

Table 3 Comparison of Eigenvalue and no wrinkles when the pre-stressed of 0.0005 m is applied to three different smart materials of different thickness taking triangular elements (S3R).

S.No. Thickness (x 10 -3m)

Kevlar Kapton Mylar

Number of

Wrinkles


mode shape)

Number of

Wrinkles


mode shape)

Number of

Wrinkles


mode shape)

1 0.01 13 0.26941 13 0.30289 13 0.33964 2 0.02 13 0.26967 13 0.30314 13 0.33989 3 0.03 13 0.26995 13 0.30342 13 0.34017 4 0.04 12 0.27024 12 0.30371 12 0.34048 5 0.05 12 0.27054 12 0.30402 12 0.34081 6 0.1 12 0.27233 12 0.30593 11 0.34287

Fig. 6.a. Effect of thickness of Kapton membrane with respect to number of triangular elements


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Fig. 6.b. Effect of thickness of Kevlar membrane with respect to number of Triangular elements

Fig. 6.c. Effect of thickness of Mylar membrane with respect to number of triangular elements

As the pre-tensioning is increased, major wrinkles begin to form by propagating along straight lines between the loaded vertices. These also closely follow the regions of negative principal stresses, which show more clearly thin bands of compression on the free edges where wrinkles may be expected in practice. Number of wrinkle increases when element size decreases as Figs. 5 & 6 also eigenvalues varies with varies with the type of elements, number of elements and thickness of membrane as Table 2 & 3. Eigenvalue decrease by one tenth when the shear forces increase by ten times. That means eigenvalue inversely proportional shear force. At the end of this simulation, the wrinkling pattern bears a remarkable results, two major inclined wrinkles are present along with a third central wrinkle spanning halfway across the membrane. 4.2 Rectangular Membrane with different α (length-to-width aspect ratio) Fig.7.a. Schematic illustration of a rectangular membrane structure with two clamped-ends subjected to Tensile Load in the longitudinal direction. Before applying tensile load, the dimensions of the rectangular membrane are: length L0, width W0, and thickness t0. The nominal strain is


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defined as ε = L/L0-1, where L is the stretch length of Rectangular Membrane. Due to the constraint of the clamped ends, the deformation and stress in the sheet are inhomogeneous. The geometry of the rectangular sheet is characterized by a single dimensionless ratio (α) between the length (L0) and the width (W0), α = L0/W0. There is no effect of thickness on stress analysis but play important role in wrinkling analysis.

Fig. 7.a. Schematic illustration of a rectangular membrane with two clamped-ends, subject to

Tensile Load

Fig. 7.b. Wrinkled rectangular membrane under Tensile Load (ε ~ 10%).

In this case we had design models of different α (length-to-width aspect ratio) in commercially available finite element software ABAQUS. All degree of freedom of one side of rectangular membrane fixed by *ENCASTRE and opposite side had only one degree of freedom (translation in only positive X-direction) and all degrees of freedom of the nodes along the two side edges were completely free, to simulate the actual situation. This compares finite element procedures for the first load-case of a uniformly pre-tensioned membrane. Without seeding an initial imperfection, there is no wrinkling for elastic-only behaviour. This is not unexpected, and when the imperfection is restored, wrinkling is observed. When plasticity is specified in the constitutive material model,


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wrinkling also occurs, and convergence with only elastic behaviour is achieved when the corner displacement (pre-tension) is large enough. However, surprisingly, wrinkling occurs for the plastic case when there is no initial imperfection. Initially, there is no departure from the elastic-perfectly at case but it bifurcates suddenly and moves onto the wrinkled path before achieving the same wrinkled state as the others. In addition to the length-to-width aspect ratio, the thickness of the sheet becomes important in both the eigenvalue and post-buckling analyses. In the present study, we take the width (W) is constant and initial width-to-thickness ratio (W0/t0) is 20000. The eigenvalue increases with decreases in width- to- thickness (W0/t0) ratio increases as shown in Fig. 8. For such a membrane, the rectangular shell elements (S4R) in ABAQUS are used in the finite element. A uniform mesh is used for analysis. At present α increase number of wrinkle decrease but the size of wrinkle increase as shown in Fig. 9. The amplitude of wrinkle increase when load increase. The corresponding out-of-plane displacements are re-scaled to the same amplitude such that, when these sets of displacements are added together and then to there at mesh, the maximum imperfection height is equal to the membrane thickness. Table 4 Comparison of eigenvalue with different thickness (t) and aspect ratio (α = L/W)

S.No. Thickness α=1.5 α=2 α=3

(x 10 -3m) Eigenvalue(for first mode shape)

Eigenvalue(for first mode shape)

Eigenvalue(for first mode shape)

1 0.025 0.02624 0.00025 0.01225 2 0.05 0.14434 0.00312 0.06836 3 0.075 0.34113 0.08225 0.16225 4 0.1 0.61843 0.15126 0.31056

Fig. 8. Effect of thickness (t) and aspect ratio (α = L/W)on eigenvalue


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Fig 9.a. Effect of α (Length-to-width aspect ratio) on wrinkles (α=1.5, eigenvalue=0.026)

Fig. 9.b. Effect of α (Length-to-width aspect ratio) on wrinkles (α=2, eigenvalue=0.00025)

Fig. 9.c. Effect of α (Length-to-width aspect ratio) on wrinkles (α=3, eigenvalue=0.012)

5. Conclusion This paper presents a formulation for the modal analysis for the predicating the behavior of inflatable rectangular membrane structures. The membrane structures have been analyzed with different thickness subjected to different tensile and shear load condition. Pre-stretch condition has


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been also imposed to finding eigenvalue frequency. The FE analysis results obtained using continuum membrane element and the variation in natural frequency is observed using three different smart materials (Kapton, Kevlar and Mylar). Modified Displacement Component method used in this paper is an accurate wrinkling calculating method which has been verified by the experiment (Wong and Pellegrino2006). We successfully apply the modified displacement method in the numerical simulation with a direct perturbed method. In the simulation, we remove the initial imperfection from our model timely to eliminate the influence of the imperfection on the post-wrinkling characteristics. The different modes due to uniform pre-stressed are found. We analyzed the problems of the wrinkle prediction of rectangular membrane under variation of thickness, types of element, number of element, Length-to-width aspect ratio (α), tensile and shear load. This is highly significant because of the interest in inflatable structures for space applications.

Acknowledgements The authors are truly thankful to Mr. Kripa Shanker Singh, Scientist/Engineer-SD, Antenna Systems Mechanical Group (ASMG), Mechanical Engineering Systems Area(MESA), Space Applications Centre (ISRO), Ahmedabad for their valuable and significant suggestions which substantially improved the manuscript. This work is supported by the Indian Space Research Organization, Government of India (grant number ISRO/RES/3/675/14-15).

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Wrinkling Simulation of Membrane Structures under Tensile and Shear Loading

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