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STUDY OF RIPPLE FORMATION IN UNIDIRECTIONALLY-TENSIONED MEMBRANES

Bernardo C. Lopez* and Shyh-Shiuh Lih†

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

Jack Leifer‡

University of Kentucky, Paducah, KY, 42002-7380, USA

and

Gladys Guzman§

California State University, Fresno, CA , 93740-8027, USA

The study of membrane behavior is one of the areas of interest in the development of ultra-lightweight and lightweight structures for space applications. Utilization of membranes as load-carrying components or support structure for antenna patch-arrays, collectors, sun-shades and solar-sail reflective surfaces brings about a variety of challenges that require understanding of the ripple-formation phenomenology, development of reliable test and analysis techniques, and solution methods for challenges related to the intended applications. This paper presents interim results from a study on the behavior of unidirectionally tensioned flat and singly-curved membranes. It focuses on preliminary experimental work to explore formation of ripples and on helping establishing a robust finite element analysis (FEA) methodology to correlate and predict their formation on thin membrane models. The term “ripple” is used here to refer to elastic out-of-plane deformation of a membrane (typically called “wrinkles” by the community). Thus, reserving the term “wrinkle” to permanent, plastic deformations.

Nomenclature

x, y = coordinate variables p = focal length dx = small displacement in the direction of coordinate x dy = small displacement in the direction of coordinate y F/D = Focal distance to Diameter ratio in a reflector aperture

* Staff Engineer, Science and Technology Development Section, 4800 Oak Grove Drive, MS 299-100, AIAA Member † Staff Engineer, Thermal and Propulsion Engineering Section, 4800 Oak Grove Drive, AIAA Member ‡ Asst. Professor, Extended Campus Programs, Department of Mech. Engineering, PO Box 7380, AIAA Member § Sophomore Student, Civil Engineering Department *

45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference19 - 22 April 2004, Palm Springs, California

AIAA 2004-1737

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc.The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.All other rights are reserved by the copyright owner.

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Fig. 2 - Test Apparatus

A

B

B

C

D

E

E

Fig. 1 – Antenna Reflector Definition

I. Introduction

The work presented is motivated by the mechanical specifications derived from a project for the National Aeronautics and Space Administration (NASA), in which a precipitation radar antenna prototype is manufactured and tested1. Two main characteristics of the antenna design (high stowage and mass efficiency) lead to the choice of a membrane-based reflector and a mechanism that provides stowage and deployment capabilities2.

At half the scale of the aperture desired for a space article, this prototype has dimensions of 2.65m x 2.65m of projected aperture and an F/D ratio of 0.35. It is a dual-frequency antenna operating at Ku and Ka bands (approximately 13.5 and 35 GHz, respectively), with scanning capabilities. The allowable tolerances on the antenna reflector are expressed as a maximum allowable shape error of 0.250mm RMS.

In order to achieve scanning capabilities, the antenna membrane reflector has the shape of a shell formed by sweeping a parabola along a straight, perpendicular line (or, otherwise defined as the section of a parabolic cylinder, as shown in Figure 1).

Mathematically, the parabolic profile has the standard form given by,

Where p has value of 0.9275m.

Shaping of the membrane is achieved through a rigid definition on the two parabolic edges and application of unidirectional in-plane stress between the edges (in the straight z-direction), with negligible stress in the orthogonal direction. The straight edges are rigid and fully constrained in the prototype model.

This loading condition leads to the spontaneous formation of ripples, as the unidirectional tension causes an effective compressive stress in the orthogonal in-plane direction, which is balanced only as the membrane buckles to reduce the minor stresses to zero.

Ripples are major contributors to reflector shape area in our antenna prototype design, therefore, prediction of membrane ripple amplitude and spatial frequency via FEA is a very desirable capability. To that end, a search of work done by the community using existing computer codes focused attention on work conducted by Leifer, Belvin3, Wong and Pellegrino4-5, whose methodology is followed and improved upon by this work. At the same time, a test apparatus was built for testing of membranes under various loading and boundary conditions (Fig. 2).

It is comprised of a rigid frame (A), two clamping devices (B), a set of flexures (C), tensioning screws (D) and a set of depth gauges (E). One of the clamping devices is fixed to the frame, while the other one, on the side of the tensioning screws, can be rigidly pulled, released or rotated because of its flexure interfaces. Because of this feature, tension loads can be imparted onto the membrane via the screws (D), to explore ripple patterns at a variety of stress conditions. Membrane deflections are read via the depth gauges (E).

Thus, formation of ripples and their orientation in response to tension magnitude and distribution, and variations due to membrane thickness and composition can be studied by this method.

y x( )x2

4 p.(1)

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Fig. 3 - Steps in the FEA Approach

The main objective is to study a variety of membranes under various load conditions and correlate the observed behavior using the FEA methods, to establish a viable, practical and efficient methodology with existing code.

II. Finite Element Analysis Approach

The modeling of ripple formation is essentially complicated due to sensibility and low stability of the nonlinear post-buckling or post-rippling phenomena. Recent developments in these efforts3-5 suggest that application of Finite Element Analysis (FEA) to predict membrane ripple or buckling behavior is a viable approach. The techniques discussed in Reference 3 are used and improved in the context of the Advanced Precipitation Radar Antenna.

As commonly known and presented in References 3-5, shell elements which are typically intended to model either plate or membrane behavior in FEA models, do not readily produce out-of-plane deformations such as ripples or wrinkles, unless they are initiated by “seeding” out-of-plane forces or imperfections, in conjunction with uniaxial tension and small shearing in the orthogonal direction in the plane of the membrane. The out-of plane forces are removed later in the process, after the ripples are formed.

The methodology is detailed as follows: (1) A tension field is generated by introducing a very small in-plane enforced edge displacement (2) The ripple behavior is initiated by the application of small out-of-plane seed forces (3) Shear loads are applied gradually by imposing horizontal displacements to the top edge of the membrane in steps. (4) The seed forces are then gradually removed while the ripples remain on the membrane.

COSMOS/M nonlinear finite element package was used in the modeling work. The code allows for nonlinear representation of the large deformation of an elastic membrane. The adaptive steps option in the COSMOS/M is selected to accelerate the computation. This option will adjust the loading steps according to divergence and bifurcation conditions in the computational process.

To verify the validity of the model, the example of shearing a polyimide (popularly referred to as Kapton™ -a brand of Dupont) membrane as presented by Reference 3 is duplicated. The relevant model parameters are shown in Table 1, and loading conditions are similar but not exactly the same due to some parameter variations from the cited paper. But, as shown in Fig. 4 and as much as they could be compared to Ref. 3, the results were duplicated.

Table 1: Properties of Kapton membrane

Length 380 mm Width 128 mm Thickness 25µm Young’s Modulus 3.53 GPa Poisson’s Ratio 0.3

Fig. 4 - Ripple Pattern

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(2)

(1)

Fig. 7 – Model Boundaries and Loading

The above exercise showed ripples initiating at the free vertical edges and propagating to the center. This observation motivated placing only a pair of out-of-plane “seeding” forces in step 2, at these membrane edges. As shown in Figure 5.

Fig. 5 – Seeding Forces Applied at Both Unconstrained Edges

As expected, it was found that this simplified approach also initiated ripple formation at the edges (See Figure 6) and produced the same ripple pattern propagation to the center of the membrane, just as with the baseline approach that uses a larger population of seeding forces. Convergence problems were thus reduced and the analysis process shortened.

Fig. 6 - Ripples initiated from both side of the unconstrained edges

III. Modeling Results

Having duplicated results of Ref. 3 and simplified the approach, the next step was to conduct analysis on a model that could offer opportunity for correlation. Case 1

The first model is provided by fitting a 25.4 µm thick, square polyimide membrane in the test frame shown in Fig. 2, except that instead of curved, the clamps are straight. The model’s relevant information is provided on Table 2

Table 2: Properties of square membrane

Width 952 mm Thickness 25.4 µmYoung’s Modulus 2.549 GPa Poisson’s Ratio 0.3

One end of the membrane was fixed, while the other one was attached to a rigid, unconstrained bar, to which enforced displacements could be imparted to simulate tension loads applied by the tensioning screws, used in the test apparatus.

In this case, loading of the membrane by tightening one of the tensioning screws was simulated. This loading would

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Table 3: Loading Conditions of Case Study 1 Steps Loading Conditions 1 Rigid Bar Displacement dy =1 mm 2 Seed force Fy=1.e-6 N 3 Rigid Bar Tip Displacement dx =1.467mm

Rigid Bar Tip Displacement dy =6.3 mm

cause combined membrane stretching in both the x and y-directions, and rotation of the flexed rigid boundary, as shown in Fig. 7.

The loading conditions follow the methodology described earlier in this paper and is summarized in Table 3. The loads are applied gradually via an adaptive scheme in the finite element code, in adherence to convergence criteria. As in the real model, FEA model results (Fig. 8) showed membrane buckling manifested by the formation of ripples. Which provides acceptable qualitative correlation. However, the amplitude of the analysis ripples (0.014mm) is approximately two orders of magnitude smaller than those observed in the physical model (1.5mm). This difference can be improved by modeling more realistic boundary conditions, as discussed under a different section of this paper.

The minor principal stress is shown in Fig. 9, where zero values correspond to areas where ripples are formed.

Fig. 8 - Ripple Pattern

Fig. 10 - Minor Principal Stress (GPa)

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Fig. 11 – Loading Conditions of Parabolic Membrane

Case 2

A second simulation focuses this work on the reflector of interest, which is the precipitation radar antenna prototype described in the introduction. This requires a new FEA model of larger dimension, a singly-curved membrane profile, and greater material thickness. See Figures 1 and 11. The parabolic shape is the same one defined by equation (1), with a focal point at p=92.75cm, in the y-direction. Young’s modulus and Poisson’s ratio are the same as those of Table 2, except that material thickness is 177 µm (5mil ).

In the FEA model the curved edge at one end is fixed, while at the other end the membrane edge is connected to a rigid bar of the same shape. A rigid body translation along the secant line as shown in Figure 11, and a single out-of-plane “seeding” force are applied to trigger ripple formation. This force was placed near the center of the membrane, based on a preliminary study that showed ripples initiating near the center. Otherwise, the analysis procedure is the same as previously discussed.

Figure 12 is a contour plot with preliminary results showing 15 fully developed ripples, the peak amplitude is 1.6mm. Qualitatively, these features are comparable to those observed on physical models.

Detailed qualitative and quantitative correlation between these results and physical models will be presented in later publications.

Fig. 13 – The ripple pattern of a single-curved membrane.

IV. Influence of Boundary Conditions

Experimentally, it is observed that the out-of-plane amplitude and the spatial ripple separation can be influenced by boundary conditions such as:

• discontinuities in boundary loading that can lead to localized relaxation in the membrane, • geometric discontinuities at the boundaries which result in localized triggering of ripples, • systemic shrinking of epoxies used to attach the membrane to the supporting structure.

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Factors like these seem to bias the formation of ripples around localized areas of membrane relaxation, and increase their amplitude and number. Ripples are also triggered at locations with load-discontinuities at membrane boundaries.

Prior studies, including those performed by Wong and Pellegrino4,5 and Leifer and Belvin3, demonstrated FEA models that produced ripples qualitatively similar to those observed experimentally for membranes under edge load. Even when using seeding forces, however, these models exhibited certain shortcomings: - The number of ripples predicted in the FEA models under edge shear loading conditions is generally less than

the number of ripples actually observed in the physical membrane loaded under the same conditions (Fig. 14).

Figure 14 - The number of wrinkles predicted in ANSYS3 and COSMOS for a 1 mil Kapton membrane with a top-edge shear displacement of 3 mm is less than that observed experimentally5

- Ripples produced when the membrane is subject to a uniform edge tension (Fig. 15) cannot be reproduced computationally using simple edge boundary conditions.

Figure 15 – Ripples in 1 mil Kapton membrane supported vertically in gravity under horizontal uniaxial edge tension.

These observations indicate that the simple edge loading conditions typically used in FEA models do not adequately represent the true conditions along the membrane supports. The models shown in Fig. 7 and 15, which contains a rigid bar located along one edge of the membrane, are typical of these. Rotation and/or lateral translation of the rigid bar in this FEA model induces the rippling along the surface of the membrane due to the transmission of shear loads into the membrane, but uniform upward motion of the edge in the model does not produce ripples. A more complex FEA model is required to produce the ripple configurations observed under uniaxial tension alone.

A discrepancy between the lower number of ripples predicted by FEA than those observed in physical models is observed, which is believed to be related to FEA models predicting first buckling modes, whereas boundary conditions in physical models may be triggering higher order buckling modes, which would produce a higher number of ripples. In some cases, the rigid bars used for membrane support are clamped to a soft, compliant

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material such as rubber or cork sandwiched between the bar and the membrane; in other cases, an adhesive layer is used to permanently affix the membrane to the rigid bar. It is believed that these intermediate compliant layers can produce lateral compressive strains in the clamped portion of the membrane, as in Fig. 16, which schematically shows how lateral compression in the adhesive layer produces an inward (compressive) lateral strain in the membrane. A similar strain does not occur in the bar-element model definition of these support members due to their high stiffness.

Figure 16 – Cross-section through membrane boundary edge supports. Lateral compression that develops within the compliant layer causes strain in the membrane. Away from the boundary supports, this strain causes local buckling when the membrane is supported in unilateral tension.

The uniform tension applied to the membrane edge supports, the compression within the boundary causes rippling in the portion of the membrane that extends beyond its supported edges, and over the entire surface. To test this premise, a finite element model of the 25 micrometer Kapton membrane shown in Fig. 7 was developed using ANSYS 6.1. The 1 m2 model was comprised of approximately 10,000 SHELL 181 elements, and approximately 400 20-node brick elements were used to form the membrane bar supports. Rather than model the complex boundary configuration shown in Fig. 16, a 1 mm axial compression was applied directly to the elements forming the bar supports, prior to applying the unixial tension. This applied compression produced a negative strain of 0.001, distributed uniformly along the length of each bar. The simulation results in Fig. 17 show the ripples produced under this simultaneous boundary compression and uniaxial membrane tension. In order to induce the out-of-plane deflections within the model, the same small “seed forces” described previously were used to induce the out-of-plane deflections within the membrane, and were removed once the initial ripple pattern became established. Maximum out-of-plane peak-to-peak deflection was approximately 1.05 mm for this particular loading configuration.

Figure 17 – Contour plot of ripples developed in a membrane under uniaxial edge tension (horizontal forces T). Compression in the adhesive layers between the bars and the membrane was simulated in the FEA model by applying a uniaxial compression to each of the bars located on the horizontal edges of the membrane elements.

Although the method used to model compression within the membrane edge supports was quite simple, the ripple contours produced qualitative resemblance to physical models, which consistently show a greater number of

Membrane

Bar support

Adhesive or compliant material

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ripples than those produced by previous modeling approaches. Also, the ripple amplitude of 1.05mm from this model, is much closer to the physical model ripple amplitude of 1.5mm. Further assessment of the validity of this approach and other modeling approaches will be conducted, as well as parametric studies to arrive at better modeling of boundary support configurations and their effect on membrane rippling.

V. Conclusions

The preliminary results obtained in this work, provide further indication that finite element modeling using commercially available FEA codes, may offer viable approaches to correlate non-linear behavior of tensioned membrane. Value in these methods are to be compared to alternative methodologies being proposed and under development, inasmuch as they can enable realistic modeling of actual models or articles and provide credible results. While the results presented here show good qualitative correlation to results from physical models, more work is underway to produce well-founded quantitative correlations, while continuing to improve the modeling methodology and solution convergence.

Acknowledgements

The authors wish to thank Josh Ward, student of California State University, and Justin Washington, student of University of Mississippi, for their work in helping design the test apparatus used in the related experimental work. And to Michael Lou of Jet Propulsion Laboratory for feedback on some of the results presented here. The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, with funding from NASA.

References

1. Im, E., Durden, S.L., Kakar, R. K., Kummerow, C.D., and Smith, E.A., “The Next Generation of Spaceborne Rain Radars: Science Rationales and Technology Status”, SPIE’s Third Symposium on Microwave Remote Sensing of the Atmosphere and Environment, Hangzhou, China, 10/02. 2. Lin, J.K., Sapna, G.H. III, Scarborough, S.E., and Lopez, B.C., “Advanced Precipitation Radar Antenna Singly Curved Parabolic Antenna Reflector Development”, AIAA Paper 2003-1651, 4th Gossamer Spacecraft Forum, 44th AIAA SDM Conference, Norfolk, VA, 4/03. 3. Leifer, J. and Belvin, W. K., “Prediction of Wrinkle Amplitudes in Thin Film Membranes via Finite Element Modeling”, AIAA Paper 2003-1983, 4th Gossamer Spacecraft Forum, 44th AIAA SDM Conference, Norfolk, VA, 4/03 4. Wong, Y.W. and Pellegrino, S., “Computation of Wrinkle Amplitudes in Thin Membranes”, 3rd Gossamer Spacecraft Forum, 43rd AIAA SDM Conference, Denver, CO, 4/02. 5. Wong, Y.W., and Pellegrino, S., “Amplitude of Wrinkles in Thin Membranes, in New Approaches to Structural Mechanics, Shells and Biological Structures, H. Drew and S. Pellegrino, ed., Kluwer Academic Publishers, 2002.

6. Jenkins, C.H., Haugen, F. and Spicher, W.H., “Experimental Measurement of Wrinkling in Membranes Undergoing Planar Deformation,” Experimental Mechanics, 38, 2, June 1998, pp. 147 – 152. 7. Stein, M. and Hedgepeth, J.M., Analysis of Partially Wrinkled Membranes, NASA TN D-813, July 1961.