Nonlinear Shell Modeling of Thin Membranes with Emphasis on Structural Wrinkling

11 Pages·2003·0.99 MB·English

by  Tessler, Alexander

NONLINEAR SHELL MODELING OF THIN MEMBRANES
WITH EMPHASIS ON STRUCTURAL WRINKLING

Alexander Tessler*, David W. Sleight*, and John T. Wang*
Analytical and Computational Methods Branch, Structures and Materials Competency
NASA Langley Research Center, Hampton, VA 23681-2199, U.S.A.

Abstract accompanied by greater amplitude and longer structural
Thin solar sail membranes of very large span are wrinkles.
being envisioned for near-term space missions. One A thin-film solar sail is a classical two-dimensional
major design issue that is inherent to these very flexible structure, with a thickness that is much smaller than its
structures is the formation of wrinkling patterns. lateral dimensions. Since the bending stiffness is
Structural wrinkles may deteriorate a solar sail’s negligibly small compared to its membrane stiffness,
performance and, in certain cases, structural integrity. the load-carrying capability using thin, low-modulus
In this paper, a geometrically nonlinear, updated films is predominantly due to tensile membrane
Lagrangian shell formulation is employed using the stresses. One key response phenomenon intrinsic to this
ABAQUS finite element code to simulate the formation class of structures is structural wrinkling. Structural
of wrinkled deformations in thin-film membranes. The wrinkles are local post-buckling patterns that are
restrictive assumptions of true membranes, i.e. Tension manifested by geometrically large transverse
Field theory (TF), are not invoked. Two effective deformations whose magnitudes are much larger than
modeling strategies are introduced to facilitate the membrane thickness. Their formation is generally
convergent solutions of wrinkled equilibrium states. attributed to extremely low compressive stresses
Several numerical studies are carried out, and the supported by extremely low bending stiffness. To
results are compared with recent experimental data. simulate such effects, the analytical model must
Good agreement is observed between the numerical necessarily account for both membrane and bending
simulations and experimental data. deformations undergoing geometrically nonlinear
kinematics with large displacements and rotations, i.e.,
Introduction
a geometrically nonlinear shell model. However,
The exploitation of solar energy for the purpose of obtaining stable equilibrium states using shell-based
near-term space exploration presents a viable and modeling turns out to be a challenging computational
attractive possibility in the minds of NASA scientists problem. Here, the elastic deformations, possessing a
and engineers. The specific propulsion technology is very small amount of strain energy, are accompanied by
called solar sails. Very large, ultra-low-mass, thin- large rigid-body motions, rendering these structures
polymer film (gossamer) structures are now being under constrained. Moreover, the shell models for these
designed and tested for a wide variety of space problems are highly ill-conditioned. The membrane
exploration missions. The difficulties associated with rigidity, being much greater than the bending rigidity,
conducting tests in a weightless environment place may dominate excessively, thus suppressing the
greater emphasis on the reliance on computational formation of wrinkling deformations.
methods. Other gossamer structures that possess similar Because of the aforementioned analytical and
structural characteristics include inflatable space computational challenges, most investigations of the
antennas, sun shields, and radars. A concise overview analysis of thin membranes excluded the bending effect
of gossamer structures and related technologies can be altogether, resulting in Tension Field (TF) theory
found, for example, in [1]. applicable to the so-called true membranes [2-16]. By
A solar sail can gain momentum from incidence of eliminating compressive stresses through a modification
sunlight photons at its surface. Since the momentum of the constitutive relations, TF theory enables the
carried by an individual photon is very small, the solar prediction of the basic load transfer and wrinkle
sail must have a large surface area and a low mass, so orientations in membranes; however, it cannot predict
that sufficient acceleration can be generated. Also, a the out-of-plane wrinkled shapes, wavelengths, and
solar sail requires a highly reflective surface that has amplitudes. Stein and Hedgepeth [6] explored a
minimal wrinkling and billowing under operational modified version of TF theory by identifying partially
conditions. In the presence of significant wrinkling and wrinkled domains. Following their methodology, Miller
billowing, the solar sail may lose efficiency, as and Hedgepeth [10] performed a finite element analysis
compared to a flat sail, due to the oblique incidence of using a recursive stiffness-modification procedure
photons. The billowing problem may be overcome by termed the Iterative Membrane Properties (IMP)
applying sufficient tension to the sail membrane. method. Several computational efforts [11-16]
Higher tensile stresses, however, are commonly employed the IMP and penalty-based formulations of
* Aerospace Engineer, Member AIAA.
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American Institute of Aeronautics and Astronautics
TF theory by implementing appropriate routines into square membrane loaded in tension by corner forces.
the nonlinear finite element codes TENSION6 [17,18] Relevant parametric studies of geometric imperfections
and ABAQUS [19]. The main utility of these TF are performed and provide an improved insight on their
approaches is to enable adequate load-carrying use in the geometrically nonlinear analysis of thin
predictions to be made and to enable the general regions membranes. Comparisons with experimental results are
where structural wrinkles develop to be identified. The also provided and discussed.
major shortcoming of the TF schemes, however, is their
Analysis Framework and Modeling Strategies
inability to predict the wavelengths and amplitudes of
Elastic, quasi-static shell analyses and parametric
wrinkles. This shortcoming may be overcome by
studies of thin-film membranes loaded in plane are
modeling thin-film structures using shell-based finite
carried out using the Geometrically Non-Linear (GNL),
element analysis that includes both membrane and
updated Lagrangian description of equilibrium
bending deformations.
formulation implemented in ABAQUS [19].
Recent advances in nonlinear computational
The selection of a four-node, shear-deformable shell
methods and shell-element technology offered viable
element, S4R5, incorporating large-displacement and
possibilities of simulating highly nonlinear wrinkled
small-strain assumptions, is made because of the
deformations in thin membranes using shell-based
following considerations. The element is based upon
analysis. Lee and Lee [20] developed a nine-node,
Mindlin theory and uses C0-continuous bilinear
quadratic shell element that used artificially modified
kinematics. To allow adequate modeling of thin-shell
shear and elastic moduli to enable locking-free shell
bending, the element employs reduced integration of
analysis of very thin shells. They also employed small
the transverse shear energy and an ad hoc correction
out-of-plane geometric imperfections using
factor that multiplies the transverse shear stiffness. The
trigonometric functions. In addition, a fictitious
latter device imposes the Kirchhoff constraints (i.e.,
damping term was added to the nonlinear equilibrium
zero transverse shear strains) numerically. Both of these
equations to circumvent numerical ill-conditioning due
“computational remedies” are intended to facilitate
to stability issues. They computed a wrinkled
locking-free bending deformations in thin shells. To
deformation state for a square membrane subjected two
improve the element’s reliability, an hourglass control
tensile forces; however, the numerical results were not
method is used to suppress spurious zero-energy
validated with experiment. A similar computational
(hourglass) modes that result from under-integrating the
effort using ABAQUS [19], by Wong and Pellegrino
shear strain energy. Such low-order, C0-continuous
[21], involved the use of superposition of buckling
shell elements are commonly preferred for nonlinear
eigenvectors to describe small out-of-plane geometric
analysis because of their computational efficiency,
imperfections over the entire spatial domain of a
robustness, and superior convergence characteristics.
membrane. They also provided some comparisons with
For a localized structural instability such as
experimental results. Neither of these efforts, however,
wrinkling, the ABAQUS code provides a volume-
examined how the application of various types of
proportional numerical damping scheme invoked by the
geometric imperfections and their spatial distributions
STABILIZE parameter. The stabilization feature adds
may affect the outcome of a nonlinear analysis.
fictitious viscous forces to the global equilibrium
In this paper, several modeling ideas are explored
equations. This enables the computation of finite
for the purpose of aiding the geometrically nonlinear,
displacement increments in the vicinity of unstable
updated Lagrangian shell analysis of thin-film
equilibrium and thus circumvents numerical ill-
membranes with emphasis on the wrinkled
conditioning due to stability issues. The default value of
equilibrium/deformation state. The modeling ideas
the stabilization parameter (2.0×10-4) is used in the
include (1) a simplified and computationally efficient
numerical examples that follow.
way of introducing out-of-plane geometric
Next, quasi-static shell solutions for two thin-film
imperfections, thus ensuring the necessary coupling
membranes are discussed. The deformations in these
between the membrane and bending deformations, and
membranes are associated with the highly nonlinear,
(2) identifying the tension-loaded corner regions as the
low-strain-energy equilibrium states that possess
critical modeling areas. Unless adequate meshing and
structural wrinkles. Enabling modeling strategies for the
load introduction are used, these regions can effectively
solution of these computationally challenging problems
“lock” the wrinkling formation due to an overly stiff
are discussed.
behavior. The undesirable stiffening may also be
directly linked to the boundary restraints in the corner Application of Pseudorandom Geometric
regions, thus preventing the formation of wrinkles. Imperfections
Several numerical studies are carried out using the
When planar membranes are subjected to purely in-
ABAQUS code. These studies include analyses of (1) a
plane loading, no mechanism exists, even in the
flat rectangular membrane loaded in shear, and (2) a flat
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presence of compressive stresses, to initiate the out-of- data, geometry, and loading are shown in Figure 1,
plane, buckled deformations. One way to overcome this where a square membrane (Mylar® polyester film, edge
difficulty is to perturb slightly the planar geometry out length, a=229 mm) is clamped along the bottom edge
of plane. In this manner, the geometric perturbations and subjected to a uniform horizontal displacement of 1
(imperfections) will engender the necessary coupling mm along the top edge. The span-to-thickness ratio
between the bending and membrane deformations. (a/h) of the membrane is approximately 3×103 and,
In this effort, to initiate the requisite membrane-to- from the perspective of shell theory, is regarded as a
bending coupling, pseudorandom geometric thin shell. Based upon results of a preliminary
imperfections are imposed at the nodes of the originally convergence study (not discussed herein), a highly
planar membrane mesh. The computing effort involved refined mesh is constructed of 104 square-shaped S4R5
in generating such a set of pseudorandom numbers is shell elements. The numerical model that is originally
trivial. In the simplest setting, the imperfections can be planar is augmented by the pseudorandom, nodal
applied at every interior node of the mesh, and the imperfections distributed over the interior nodes of the
imperfection magnitudes may be computed as a model, using the amplitude parameter α=0.1. In
function of the membrane thickness as Figure 2, the out-of-plane deformation contours are
z =αδ h (i=1, N) (1) depicted showing a relatively close agreement between
i i
the experiment and analysis in terms of the number of
where α is a dimensionless amplitude parameter,
wrinkles, their orientation and amplitudes. The
[ ]
δi∈ −1,1 is a pseudorandom number, h is the amplitude of the left-edge wrinkle deflecting
membrane thickness, and N is the number of nodes with downward, which represents the maximum wrinkle
the imposed imperfections. As will be subsequently amplitude, compares within 5% between the experiment
demonstrated, there are many alternative ways of and analysis. It should be noted, however, that in the
spatially distributing these imperfections and choosing experiment, the Mylar film was slightly curved out-of-
the value of α. plane before the application of the horizontal
The imperfection amplitudes, z , that are regulated displacement. This actual initial imperfection was not
i
by α, need to be sufficiently small in relation to the included in the computational model given that the
element size to avoid excessive out-of-plane element main focus of this analysis was to validate the efficacy
distortions; this is particularly significant for planar of the pseudorandom imperfection approach. Naturally,
elements with four or more nodes. To preserve a nearly provided that the actual initial imperfections are
flat membrane, z may need to be small in comparison adequately measured prior to loading, they should be
i
included in a computational model.
to the membrane thickness. However, z should be
i
large enough to enable adequate coupling to take place Square Thin-Film Membrane Subjected to
between membrane and bending deformations. This Symmetric Corner Tensile Loads
aspect is quantified in a parametric study that follows.
When a tensile load is applied at a corner of a thin-
The randomness aspect of Eq. (1) ensures an
film membrane, wrinkles tend to radiate from the
unbiased imperfection pattern. Thus the imperfections
corner; subsequently, the corner wrinkling affects the
neither predefine nor dominate the resulting deformed
wrinkled equilibrium state over the entire membrane
equilibrium state. Geometric imperfections are
domain.
commonly specified over the whole spatial domain of a
Recently, Blandino et al. [23] performed a
membrane, e.g., as in [20,21]. As demonstrated
laboratory test on a 500 mm square, flat membrane
subsequently, strategic application of geometric
made of a KAPTON® Type HN film. The material
imperfections over partial mesh regions may also be
properties, membrane dimensions, and loading are
just as effective. This further points out that conducting
shown in Figure 3. The membrane is subjected to
a computationally intensive analysis to generate
tensile corner loads (F=2.45 N) applied in the diagonal
geometric imperfections based on a related structural
directions via Kevlar threads at the left and bottom
problem, such as a buckling eigenvalue problem in
corners of the membrane. The top and right corners of
[21], may be entirely unnecessary.
the membrane are fixed to the test frame with Kevlar
Square Thin-Film Membrane Subjected to In-plane
threads. The corners are also reinforced on both sides
Shear Loading
with small patches of a transparency film
The choice of the first numerical example is (approximately 10 mm in diameter).
primarily motivated by the availability of recent A suitable analytical model, that is statically
experimental data obtained by Prof. Jack Leifer and his equivalent to the experimental one, would result in the
colleagues at NASA Langley using the photogrammetry loading by four tension corner forces acting in the
technique (refer to Leifer et al. [22]). The problem is opposite directions along the two diagonals of the
closely analogous to that reported in [21]. The material square membrane. In a computational shell model,
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specifying the applied concentrated forces at the corner membrane are not modeled. (It is expected, however,
nodes does not lead to a wrinkled equilibrium state, that including the corner reinforcements in a
even with the inclusion of the geometric imperfections computational model would result in improved
discussed in the previous example. Here, two model- correlation with the experiment.) The originally planar
related pitfalls that prevent the development of finite element mesh is augmented by the
wrinkling can be inferred. First, a concentrated corner pseudorandom, nodal imperfections distributed over the
force, causing a near-singular membrane stress field, interior nodes of the model, using the amplitude
may bring about pathological performance in the parameter α=0.1 (A parametric study on the selection
neighboring elements, since conventional elements of α is discussed subsequently.)
cannot model singular stress fields. Thus the dominant The contour plots depicting the deflection
membrane strain energy may suppress an already very distributions in the experiment and the present
small bending energy, causing severe ill-conditioning geometrically nonlinear shell analysis are shown in
and eliminating the influence of the local bending Figure 5. The computer simulation is able to effectively
energy altogether. The second modeling concern is of predict four wrinkles radiating from the truncated
kinematical nature. The corner elements are often corner regions just as those from the measured
distorted, meeting at a single corner node (i.e., experimental results. The analysis also shows that
quadrilaterals collapsed into triangles) at which curling occurs at the free edges as observed in the
kinematic boundary conditions may be imposed. experimental results; however, the amplitudes of the
Tessler [24] demonstrated, in closed form and experimental deflections are somewhat greater.
numerically, that acute bending stiffening (and even Although intended to be symmetric, the experimental
“shear locking”) may result in otherwise perfectly well results are somewhat asymmetric. As in the previous
performing Mindlin elements strictly due to over example, the actual initial geometric imperfections, not
constraining of the element kinematics because of the incorporated in the analysis, may have contributed to a
nodal restraints. This severe bending stiffening results significant asymmetry in the experiment and the
from the Kirchhoff constraints (zero transverse shear differences with the analysis. This again is evidence to
strain conditions) that engender spurious kinematic the fact that such ultra-flexible and lightly stressed
relations in the elements situated along the boundaries spatial structures are not only difficult to model
(and lines of symmetry) where the kinematic restraints analytically but also to test experimentally, requiring
are imposed. Consequently, with the over constrained, further refinements in the experimental methods for
stiff corners, no wrinkling can be initiated. these thin-film membranes.
The above arguments lead to a basic conclusion that
1. A study of imperfection amplitudes
eliminating sharp-corner meshes, in the regions where
The practical question of how to decide on the
concentrated loads are applied, may be beneficial for
appropriate value of the amplitude parameter, α, in Eq.
the modeling of wrinkled equilibrium states. Truncating
a corner a short distance inward may result in an (1) may be satisfactorily answered through a parametric
improved load transfer and mesh quality in that local study in which the distribution of the pseudorandom
region. This truncation will necessitate replacing the imperfections is kept constant, and the value of α is
concentrated force with a statically equivalent varied. To minimize the computational effort, it suffices
distributed traction. The benefits of this strategy are to model a symmetric quadrant of the membrane as
twofold: (a) removal of a severe stress concentration shown in Figure 6, where the imperfections are defined
(note that in practice, the load introduction into a over all interior nodes. The appropriate truncations of
structure is closer to a distributed load than a the corners, the application of statically equivalent
concentrated force), and (b) improvements in the distributed tractions, and the symmetry boundary
kinematics in the critical corner regions from which conditions are all implemented as previously described.
wrinkles radiate. The amplitude parameter, α, is varied in the range
Consistent with the present modeling philosophy, 0.001≤α≤5.0, and for each fixed value of this
the corners of the square membrane are “cut-off” such parameter, a geometrically nonlinear analysis is
that the length of each corner edge (which is normal to performed. The results from this study are presented in
the direction of the applied load) is set to be small, as Figure 7, where a normalized deflection, W /h, at the
mid
shown in Figure 4. The prescribed kinematic boundary center of the membrane free edge (and which represents
restraints are also depicted in the figure. The domain of the maximum value of the deflection across the entire
the entire membrane is discretized with a relatively domain), is plotted versus α. There are three basic
refined mesh (4,720 elements) for the purpose of a ranges in the graph. For very small amplitudes
suitable comparison with the experiment. For simplicity (α<0.005), the imperfections are not large enough to
and for the purpose of validating the efficacy the corner provide sufficient bending-to-membrane coupling. No
cut-off modeling, the reinforced corners of the wrinkling deformations can be simulated. In the range
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0.01≤α≤1.0, all analyses predict practically the same of deformation coupling between bending and
wrinkled deformation equilibrium state as evidenced by membrane deformation was addressed. By applying
the constant value of W /h across this range of α. The pseudorandom out-of-plane imperfections to the
mid
solutions corresponding to the values α>1 can be initially planar membrane surface, the requisite
viewed as erroneous because this range of α values membrane-to-bending coupling is invoked at the
commencing stage of the nonlinear solution process.
implies that the imperfection amplitudes are too large;
Using relatively small and unbiased imperfections,
in fact, they are on the order of the membrane
converged wrinkled equilibrium states can be obtained
thickness. One way to interpret this is by examining the
which are independent of the initial imperfections.
transverse element distortions that may have been
For the class of thin-film problems in which corner
caused by the large imperfections: in general, one of the
regions are subjected to tension loads, as in the second
four element nodes will be out of plane, thus violating
example problem, the need for improved modeling of
the basic element formulation.
such corner regions was identified. The introduction of
2. A study of spatial distribution of imperfections
truncated corners and the replacement of concentrated
In this study, several alternative schemes of spatial loads with statically equivalent distributed tractions
distributions of the imperfections are examined in enabled successful geometrically nonlinear wrinkled
relation to their effect on the wrinkled equilibrium stress states to be determined.
states. Consider three distinct imperfection distribution The present modeling schemes were used in several
models corresponding to α = 0.10 (from the acceptable numerical studies involving thin-film membranes
range of values), as shown in Figure 8. In Model 1, the subjected to mechanical loads. First, converged
imperfections are imposed across all of the interior wrinkled deformation modes were predicted for a
nodes (in the shaded region). In Models 2 and 3, the square membrane loaded in shear. These results
imperfections are only focused on the corner regions. compared favorably with an experiment recently
The results from this study are presented in Figure 9, conducted at NASA Langley. In the second example, a
which shows a wrinkled wave along the A-B line (see square membrane subjected to four-corner tensile loads
Figure 6). Noticeably, the three imperfection schemes was analyzed and exhibited major wrinkling emanating
produce the same wrinkled deformations, and this is from the corners. The corner regions were truncated in
consistent across the whole membrane. Thus the the model in order to “design out” the near singular
imperfection schemes may be deemed equivalent from stress fields associated with concentrated loads and to
the perspective of determining the appropriate wrinkled improve element topology to avoid overly stiff corners.
deformations for this problem. This is not surprising A noticeably close correlation with the experiment was
since the dominant wrinkling modes emanate from the also observed for this very challenging computational
corner regions. For this reason, the imperfections need problem. It is expected that even closer correlation may
only be imposed over these key regions. be possible once the corner reinforcements are
Concluding Remarks represented in a computational model in sufficient
detail. Also, a greater insight was developed through a
In this paper, careful modeling considerations were
set of parametric studies of the amplitudes and spatial
explored to simulate the formation of highly nonlinear
distributions of the pseudorandom imperfections.
wrinkling deformations in thin-film membranes. The
Our experience with the geometrically nonlinear,
analyses were carried out within the framework of the
updated Lagrangian shell analysis of thin-film
geometrically nonlinear, updated Lagrangian shell
membranes suggests that the current state-of-the-art
formulation using a commercial finite element code
computational methods have the potential for
ABAQUS. The underlying shell relations employ the
adequately simulating the structural response of such
assumptions of small strains, large displacements, and
highly flexible and under constrained wrinkled
do not rely on the classical membrane assumptions of
structures. There exist, however, numerous challenging
Tension Field theory. The finite element models utilize
issues requiring in-depth analytic, computational, and
a Mindlin-type quadrilateral shell element, S4R5. The
experimental pursuits. Issues related to robustness,
element employs reduced integration of the transverse
nonlinear solution convergence, sensitivity to boundary
shear energy, and has an ad hoc correction factor
restraints and applied loading, mesh dependence, and
multiplying the shear stiffness to ensure locking-free
the shell-element technology, specifically addressing
bending behavior even for very thin shells. Moreover,
ultra-thin behavior, need to be closely examined and
the hourglass control scheme is used to suppress
addressed. Computational opportunities also exist in
spurious zero-energy (hourglass) modes that result as a
exploring the explicit dynamics nonlinear analyses that
consequence of the reduced integration.
may have the advantage of obviating the stability issues
To achieve convergent geometrically nonlinear
in modeling the wrinkling phenomenon.
solutions that correlate well with experiment, the issue

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Acknowledgements Finite Elements in Analysis and Design, Vol. 37,
233-251, 2001.
The authors would like to thank Profs. Jack Leifer,
University of Kentucky, and Joe Blandino, James 14. Liu, X., Jenkins, C. H., and Schur, W. W., “Fine
Madison University, for providing the experimental Scale Analysis of Wrinkled Membranes,” Int. J.
results. Computational Eng. Sci. Vol. 1, 281-298, 2000.
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Dharamsi, U. K., “The Effect of Asymmetric
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Mechanical and Thermal Loading on Membrane
Structures,” Chapter 1 in Gossamer Spacecraft:
Wrinkling,” 43rd AIAA/ASME/ASCE/AHS
Membrane and Inflatable Structures Technology for
Structures, Structural Dynamics, and Materials
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Conference, Denver, CO, AIAA-2002-1371, April
191, Progress in Astronautics and Aeronautics,
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AIAA, 1-33, 2001.
16. Johnston, J. D., “Finite Element Analysis of
2. Wagner, H., “Flat Sheet Metal Girders with Very
Wrinkled Membrane Structures for Sunshield
Thin Metal Web,” Z. Flugtechnik
Applications,” 43rd AIAA/ASME/ASCE/AHS
Motorluftschiffahrt, Vol. 20, 200-314, 1929.
Structures, Structural Dynamics, and Materials
3. Reissner, E., “On Tension Field Theory.”
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2002.
92, 1938.
17. Lo, A., “Nonlinear Dynamic Analysis of Cable and
4. Kondo, K., Iai, T., Moriguti, S., and Murasaki, T.,
Membrane Structures.” Ph.D. Dissertation, Oregon
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State University, Corvalis, 1981.
Study of Basic Problems In Engineering Science by
18. Jenkins, C. H. and Leonard, J. W., “Dynamic
Means of Geometry, Vol. 1, Gakujutsu, Bunken
Wrinkling Of Viscoelastic Membranes," J Appl
Fukyo-Kai, Tokyo, 61-85, 1955.
Mech, 60, 575-582, 1993.
5. Mansfield, E. H., “Load Transfer via a Wrinkled
19. ABAQUS/Standard User’s Manual, Version 6.3.1,
Membrane,” Proceedings of the Royal Society of
Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket,
London, Vol. 316, 269-289, 1970.
RI, 2002.
6. Stein, M., and Hedgepeth, J. M., “Analysis of Partly
20. Lee, K., and Lee, S. W., “Analysis of Gossamer
Wrinkled Membranes,” NASA TN D-813, 1961.
Space Structures Using Assumed Strain
7. Wu, C. H, and Canfield, T. R., “Wrinkling in Finite
Formulation Solid Shell Elements,” 43rd
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AIAA/ASME/ASCE/AHS/ASC Structures,
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8. Pipkin, A. C., “The Relaxed Energy Density for Denver, CO, AIAA-2002-1559, April 2002.
Isotropic Elastic Membranes,” IMA J. Applied
21. Wong, Y. W., and Pellegrino, S., “Computation of
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Pressurized Membranes,” ASME Journal of Applied Structures Dynamics, and Material Conference,
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10. Miller, R. K. and Hedgepeth, J. M., “An Algorithm 22. Leifer, J., Black, J. T., Belvin, W. K., and Behun,
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Membranes,” AIAA, 20,1761–1763, 1982. Wrinkle Reduction in Thin Film Membrane
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“Static and Dynamic Analysis of Partially Wrinkled Structures, Structural Dynamics and Materials
Membrane Structures,” 41st Conference, Norfolk, VA, April 2003.
AIAA/ASME/ASCE/AHS/ASC Structures, 23. Blandino, J. R., Johnston, J. D., and Dharamsi, U.
Structures Dynamics, and Material Conference, K., “Corner Wrinkling of a Square Membrane due
Atlanta, GA, AIAA-2000-1810, April 2000. to Symmetric Mechanical Loads,” Journal of
12. Schur, W. W., “Tension-field Modeling by Penalty Spacecraft and Rockets, Vol. 35 No. 9, Sept/Oct
Parameter Modified Constitutive Law,” Proceedings 2002.
of the 24th Midwestern Mechanics Conf., Ames, IA, 24. Tessler, A., “A Priori Identification of Shear
1995. Locking and Stiffening in Triangular Mindlin
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Deflection Analysis of Pneumatic Envelopes Using 53, 183-200, 1985.
a Penalty Parameter Modified Material Model,”
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American Institute of Aeronautics and Astronautics
∆ = 1 mm

Mylar® Polyester Film Properties
Edge length, a (mm) 229
Thickness, h (mm) 0.0762
Elastic modulus, E (N/mm2) 3790 a
Poisson’s ratio, ν 0.38

a
Figure 1. Square thin-film membrane (Mylar® film) clamped along bottom edge and subjected to prescribed
displacement along top edge.

w (mm) w (mm)
+0.67 +1.18
+0.36 +0.82
+0.04 +0.45
-0.27 +0.01
-0.59 -0.27
-0.90 -0.63
-1.22 -0.99
-1.53 -1.35
-1.85 -1.71
-2.16 -2.07
-2.48 -2.43
-2.79 -2.79
-3.11 -3.15
(a) Experiment (Photogrammetry) (b) GNL Shell FEM (ABAQUS)

Figure 2. Wrinkling deformations of clamped square thin-film membrane (Mylar® film) subjected to prescribed
displacement along top edge: (a) Experiment (Photogrammetry) [22], and (b) GNL/FEM shell analysis (ABAQUS-
S4R5).
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American Institute of Aeronautics and Astronautics
KKeevvllaarr tthhrreeaaddss
KAPTON® Type HN Film Properties
Edge length, a (mm) 500
aa Thickness, h (mm) 0.0254
Elastic modulus, E (N/mm2) 2590
Poisson’s ratio, ν 0.34

22..4455 NN

KKeevvllaarr tthhrreeaaddss

22..4455 NN

Figure 3. Square thin-film membrane (KAPTON® Type HN film) loaded in tension by corner forces as tested by
Blandino et al. [23].

FFF
UUnniiffoorrmm
TTrraaccttiioonn
BB..CC..’’ss
FFF FFF
{{vv,,ww,,θθ ,,θθ}}== 00
xx zz
θθθ
yyy
yyy,,,vvv
θθθ
xxx,,,uuu xxx
zzz,,, www,,,θθθ
zzz BB..CC..’’ss
FFF
{{uu,,ww,,θθ ,,θθ}}== 00
yy zz
777 mmmmmm

Figure 4. Square thin-film membrane (KAPTON® Type HN film) loaded in tension by corner tractions: Full FEM
model with truncated corners using GNL S4R5 shell elements in ABAQUS code.

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American Institute of Aeronautics and Astronautics
w (mm) w (mm)
(a) Experiment (Capacitance sensor measurement) (b) GNL/FEM Shell Results (ABAQUS)

Figure 5. Wrinkling deformations of square thin-film membrane (KAPTON® Type HN film) loaded in tension by
corner tractions (a) Experiment (Capacitance sensor measurement) [23], and (b) GNL/FEM shell analysis
(ABAQUS-S4R5).

FFF///222
θθθ
yyy,,,vvv yyy
θθθ
xxx,,,uuu xxx
’s’s’s zzz,,, www,,,θθθ
C.C.C. WWW zzz
B.B.B. mmmiiiddd
y y y
rrr
etetet
mmm
mmm
yyy
SSS
UUUnnniiifffooorrrmmm
BBB
TTTrrraaaccctttiiiooonnn
xxx
FFF///222
SSSyyymmmmmmeeetttrrryyy BBB...CCC...’’’sss AAA BBB...CCC...’’’sss
{{{vvv,,,www,,,θθθ ,,,θθθ}}}=== 000
xxx zzz
Figure 6. Square thin-film membrane (KAPTON® Type HN film) loaded in tension by corner tractions: Symmetric-
quadrant model with truncated corners used in parametric studies.

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American Institute of Aeronautics and Astronautics
18.0

16.0

14.0
12.0
Wmid/h
10.0

8.0

6.0

4.0

2.0

0.0
0.001 0.01 0.1 1.0 10.0

Imperfection amplitude parameter, α

Figure 7. Square thin-film membrane (KAPTON® Type HN film) loaded in tension by corner tractions: Effect of
imperfection amplitude on the development of wrinkling deformations in GNL/FEM shell models.

Region with imposed
imperfections

(a) Imperfection Model 1 (b) Imperfection Model 2 (c) Imperfection Model 3

Figure 8. Square thin-film membrane (KAPTON® Type HN film) loaded in tension by corner tractions: Symmetric-
quadrant GNL/FEM shell models showing regions of imposed random imperfections.

10
American Institute of Aeronautics and Astronautics