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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 

Thin solar sail membranes of very large span are 
being envisioned for near-term space missions. One 
major design issue that is inherent to these very flexible 
structures is the formation of wrinkling patterns. 
Structural wrinkles may deteriorate a solar sail's 
performance and, in certain cases, structural integrity. 
In this paper, a geometrically nonlinear, updated 
Lagrangian shell formulation is employed using the 
ABAQUS finite element code to simulate the formation 
of wrinkled deformations in thin- film membranes. The 
restrictive assumptions of true membranes, i.e. Tension 
Field theory (TF), are not invoked. Two effective 
modeling strategies are introduced to facilitate 
convergent solutions of wrinkled equilibrium states. 
Several numerical studies are carried out, and the 
results are compared with recent experimental data. 
Good agreement is observed between the numerical 
simulations and experimental data. 

Introduction 

The exploitation of solar energy for the purpose of 
near-term space exploration presents a viable and 
attractive possibility in the minds of NASA scientists 
and engineers. The specific propulsion technology is 
called solar sails. Very large, ultra-low-mass, thin- 
polymer film (gossamer) structures are now being 
designed and tested for a wide variety of space 
exploration missions. The difficulties associated with 
conducting tests in a weightless environment place 
greater emphasis on the reliance on computational 
methods. Other gossamer structures that possess similar 
structural characteristics include inflatable space 
antennas, sun shields, and radars. A concise overview 
of gossamer structures and related technologies can be 
found, for example, in [1]. 

A solar sail can gain momentum from incidence of 
sunlight photons at its surface. Since the momentum 
carried by an individual photon is very small, the solar 
sail must have a large surface area and a low mass, so 
that sufficient acceleration can be generated. Also, a 
solar sail requires a highly reflective surface that has 
minimal wrinkling and billowing under operational 
conditions. In the presence of significant wrinkling and 
billowing, the solar sail may lose efficiency, as 
compared to a flat sail, due to the oblique incidence of 
photons. The billowing problem may be overcome by 
applying sufficient tension to the sail membrane. 
Higher tensile stresses, however, are commonly 



accompanied by greater amplitude and longer structural 
wrinkles. 

A thin-film solar sail is a classical two-dimensional 
structure, with a thickness that is much smaller than its 
lateral dimensions. Since the bending stiffness is 
negligibly small compared to its membrane stiffness, 
the load-carrying capability using thin, low-modulus 
films is predominantly due to tensile membrane 
stresses. One key response phenomenon intrinsic to this 
class of structures is structural wrinkling. Structural 
wrinkles are local post-buckling patterns that are 
manifested by geometrically large transverse 
deformations whose magnitudes are much larger than 
the membrane thickness. Their formation is generally 
attributed to extremely low compressive stresses 
supported by extremely low bending stiffness. To 
simulate such effects, the analytical model must 
necessarily account for both membrane and bending 
deformations undergoing geometrically nonlinear 
kinematics with large displacements and rotations, i.e., 
a geometrically nonlinear shell model. However, 
obtaining stable equilibrium states using shell-based 
modeling turns out to be a challenging computational 
problem. Here, the elastic deformations, possessing a 
very small amount of strain energy, are accompanied by 
large rigid-body motions, rendering these structures 
under constrained. Moreover, the shell models for these 
problems are highly ill-conditioned. The membrane 
rigidity, being much greater than the bending rigidity, 
may dominate excessively, thus suppressing the 
formation of wrinkling deformations. 

Because of the aforementioned analytical and 
computational challenges, most investigations of the 
analysis of thin membranes excluded the bending effect 
altogether, resulting in Tension Field (TF) theory 
applicable to the so-called true membranes [2-16]. By 
eliminating compressive stresses through a modification 
of the constitutive relations, TF theory enables the 
prediction of the basic load transfer and wrinkle 
orientations in membranes; however, it cannot predict 
the out-of-plane wrinkled shapes, wavelengths, and 
amplitudes. Stein and Hedgepeth [6] explored a 
modified version of TF theory by identifying partially 
wrinkled domains. Following their methodology. Miller 
and Hedgepeth [10] performed a finite element analysis 
using a recursive stiffness-modification procedure 
termed the Iterative Membrane Properties (IMP) 
method. Several computational efforts [11-16] 
employed the IMP and penalty-based formulations of 



Aerospace Engineer, Member AIAA. 



1 



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TF theory by implementing appropriate routines into 
the nonlinear finite element codes TENSI0N6 [17,18] 
and ABAQUS [19]. The main utility of these TF 
approaches is to enable adequate load-carrying 
predictions to be made and to enable the general regions 
where structural wrinkles develop to be identified. The 
major shortcoming of the TF schemes, however, is their 
inability to predict the wavelengths and amplitudes of 
wrinkles. This shortcoming may be overcome by 
modeling thin-film structures using shell-based finite 
element analysis that includes both membrane and 
bending deformations. 

Recent advances in nonlinear computational 
methods and shell-element technology offered viable 
possibilities of simulating highly nonlinear wrinkled 
deformations in thin membranes using shell-based 
analysis. Lee and Lee [20] developed a nine-node, 
quadratic shell element that used artificially modified 
shear and elastic moduli to enable locking-free shell 
analysis of very thin shells. They also employed small 
out-of-plane geometric imperfections using 
trigonometric functions. In addition, a fictitious 
damping term was added to the nonlinear equilibrium 
equations to circumvent numerical ill-conditioning due 
to stability issues. They computed a wrinkled 
deformation state for a square membrane subjected two 
tensile forces; however, the numerical results were not 
validated with experiment. A similar computational 
effort using ABAQUS [19], by Wong and Pellegrino 
[21], involved the use of superposition of buckling 
eigenvectors to describe small out-of-plane geometric 
imperfections over the entire spatial domain of a 
membrane. They also provided some comparisons with 
experimental results. Neither of these efforts, however, 
examined how the application of various types of 
geometric imperfections and their spatial distributions 
may affect the outcome of a nonlinear analysis. 

In this paper, several modeling ideas are explored 
for the purpose of aiding the geometrically nonlinear, 
updated Lagrangian shell analysis of thin-film 
membranes with emphasis on the wrinkled 
equilibrium/deformation state. The modeling ideas 
include (1) a simplified and computationally efficient 
way of introducing out-of-plane geometric 
imperfections, thus ensuring the necessary coupling 
between the membrane and bending deformations, and 
(2) identifying the tension-loaded corner regions as the 
critical modeling areas. Unless adequate meshing and 
load introduction are used, these regions can effectively 
"lock" the wrinkling formation due to an overly stiff 
behavior. The undesirable stiffening may also be 
directly linked to the boundary restraints in the corner 
regions, thus preventing the formation of wrinkles. 

Several numerical studies are carried out using the 
ABAQUS code. These studies include analyses of (1) a 
flat rectangular membrane loaded in shear, and (2) a flat 



square membrane loaded in tension by comer forces. 
Relevant parametric studies of geometric imperfections 
are performed and provide an improved insight on their 
use in the geometrically nonlinear analysis of thin 
membranes. Comparisons with experimental results are 
also provided and discussed. 

Analysis Framework and Modeling Strategies 

Elastic, quasi-static shell analyses and parametric 
studies of thin-film membranes loaded in plane are 
carried out using the Geometrically Non-Linear (GNL), 
updated Lagrangian description of equilibrium 
formulation implemented in ABAQUS [19]. 

The selection of a four-node, shear-deformable shell 
element, S4R5, incorporating large-displacement and 
small-strain assumptions, is made because of the 
following considerations. The element is based upon 
Mindlin theory and uses C^-continuous bilinear 
kinematics. To allow adequate modeling of thin-shell 
bending, the element employs reduced integration of 
the transverse shear energy and an ad hoc correction 
factor that multiplies the transverse shear stiffness. The 
latter device imposes the Kirchhoff constraints (i.e., 
zero transverse shear strains) numerically. Both of these 
"computational remedies" are intended to facilitate 
locking-free bending deformations in thin shells. To 
improve the element's reliability, an hourglass control 
method is used to suppress spurious zero-energy 
(hourglass) modes that result from under-integrating the 
shear strain energy. Such low-order, C^-continuous 
shell elements are commonly preferred for nonlinear 
analysis because of their computational efficiency, 
robustness, and superior convergence characteristics. 

For a localized structural instability such as 
wrinkling, the ABAQUS code provides a volume- 
proportional numerical damping scheme invoked by the 
STABILIZE parameter. The stabilization feature adds 
fictitious viscous forces to the global equilibrium 
equations. This enables the computation of finite 
displacement increments in the vicinity of unstable 
equilibrium and thus circumvents numerical ill- 
conditioning due to stability issues. The default value of 
the stabilization parameter (2.0x10""^) is used in the 
numerical examples that follow. 

Next, quasi-static shell solutions for two thin-film 
membranes are discussed. The deformations in these 
membranes are associated with the highly nonlinear, 
low-strain-energy equilibrium states that possess 
structural wrinkles. Enabling modeling strategies for the 
solution of these computationally challenging problems 
are discussed. 

Application of Pseudorandom Geometric 
Imperfections 

When planar membranes are subjected to purely in- 
plane loading, no mechanism exists, even in the 



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presence of compressive stresses, to initiate the out-of- 
plane, buckled deformations. One way to overcome this 
difficulty is to perturb slightly the planar geometry out 
of plane. In this manner, the geometric perturbations 
(imperfections) will engender the necessary coupling 
between the bending and membrane deformations. 

In this effort, to initiate the requisite membrane-to- 
bending coupling, pseudorandom geometric 
imperfections are imposed at the nodes of the originally 
planar membrane mesh. The computing effort involved 
in generating such a set of pseudorandom numbers is 
trivial. In the simplest setting, the imperfections can be 
applied at every interior node of the mesh, and the 
imperfection magnitudes may be computed as a 
function of the membrane thickness as 



z^ =ab^h (i = l,N) 



(1) 



where a is a dimensionless amplitude parameter. 
Si g [- 1,1] is a pseudorandom number, h is the 
membrane thickness, and A^ is the number of nodes with 
the imposed imperfections. As will be subsequently 
demonstrated, there are many alternative ways of 
spatially distributing these imperfections and choosing 
the value of a . 

The imperfection amplitudes, z^ , that are regulated 
by a , need to be sufficiently small in relation to the 
element size to avoid excessive out-of-plane element 
distortions; this is particularly significant for planar 
elements with four or more nodes. To preserve a nearly 
flat membrane, z- may need to be small in comparison 
to the membrane thickness. However, z- should be 
large enough to enable adequate coupling to take place 
between membrane and bending deformations. This 
aspect is quantified in a parametric study that follows. 

The randomness aspect of Eq. (1) ensures an 
unbiased imperfection pattern. Thus the imperfections 
neither predefine nor dominate the resulting deformed 
equilibrium state. Geometric imperfections are 
commonly specified over the whole spatial domain of a 
membrane, e.g., as in [20,21]. As demonstrated 
subsequently, strategic application of geometric 
imperfections over partial mesh regions may also be 
just as effective. This further points out that conducting 
a computationally intensive analysis to generate 
geometric imperfections based on a related structural 
problem, such as a buckling eigenvalue problem in 
[21], may be entirely unnecessary. 

Square Thin-Film Membrane Subjected to In-plane 
Shear Loading 

The choice of the first numerical example is 
primarily motivated by the availability of recent 
experimental data obtained by Prof Jack Leifer and his 
colleagues at NASA Langley using the photogrammetry 
technique (refer to Leifer et al. [22]). The problem is 
closely analogous to that reported in [21]. The material 



data, geometry, and loading are shown in Figure 1, 
where a square membrane (Mylar® polyester film, edge 
length, a=229 mm) is clamped along the bottom edge 
and subjected to a uniform horizontal displacement of 1 
mm along the top edge. The span-to-thickness ratio 
(a//?) of the membrane is approximately 3x10^ and, 
from the perspective of shell theory, is regarded as a 
thin shell. Based upon results of a preliminary 
convergence study (not discussed herein), a highly 
refined mesh is constructed of 10"^ square-shaped S4R5 
shell elements. The numerical model that is originally 
planar is augmented by the pseudorandom, nodal 
imperfections distributed over the interior nodes of the 
model, using the amplitude parameter a = 0.1. In 
Figure 2, the out-of-plane deformation contours are 
depicted showing a relatively close agreement between 
the experiment and analysis in terms of the number of 
wrinkles, their orientation and amplitudes. The 
amplitude of the left-edge wrinkle deflecting 
downward, which represents the maximum wrinkle 
amplitude, compares within 5% between the experiment 
and analysis. It should be noted, however, that in the 
experiment, the Mylar film was slightly curved out-of- 
plane before the application of the horizontal 
displacement. This actual initial imperfection was not 
included in the computational model given that the 
main focus of this analysis was to validate the efficacy 
of the pseudorandom imperfection approach. Naturally, 
provided that the actual initial imperfections are 
adequately measured prior to loading, they should be 
included in a computational model. 

Square Thin-Film Membrane Subjected to 
Symmetric Corner Tensile Loads 

When a tensile load is applied at a corner of a thin- 
film membrane, wrinkles tend to radiate from the 
corner; subsequently, the comer wrinkling affects the 
wrinkled equilibrium state over the entire membrane 
domain. 

Recently, Blandino et al. [23] performed a 
laboratory test on a 500 mm square, flat membrane 
made of a KAPTON® Type HN film. The material 
properties, membrane dimensions, and loading are 
shown in Figure 3. The membrane is subjected to 
tensile corner loads (F=2.45 N) applied in the diagonal 
directions via Kevlar threads at the left and bottom 
comers of the membrane. The top and right corners of 
the membrane are fixed to the test frame with Kevlar 
threads. The comers are also reinforced on both sides 
with small patches of a transparency film 
(approximately 10 mm in diameter). 

A suitable analytical model, that is statically 
equivalent to the experimental one, would result in the 
loading by four tension corner forces acting in the 
opposite directions along the two diagonals of the 
square membrane. In a computational shell model. 



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specifying the applied concentrated forces at the corner 
nodes does not lead to a wrinkled equilibrium state, 
even with the inclusion of the geometric imperfections 
discussed in the previous example. Here, two model- 
related pitfalls that prevent the development of 
wrinkling can be inferred. First, a concentrated corner 
force, causing a near-singular membrane stress field, 
may bring about pathological performance in the 
neighboring elements, since conventional elements 
cannot model singular stress fields. Thus the dominant 
membrane strain energy may suppress an already very 
small bending energy, causing severe ill-conditioning 
and eliminating the influence of the local bending 
energy altogether. The second modeling concern is of 
kinematical nature. The corner elements are often 
distorted, meeting at a single corner node (i.e., 
quadrilaterals collapsed into triangles) at which 
kinematic boundary conditions may be imposed. 
Tessler [24] demonstrated, in closed form and 
numerically, that acute bending stiffening (and even 
"shear locking") may result in otherwise perfectly well 
performing Mindlin elements strictly due to over 
constraining of the element kinematics because of the 
nodal restraints. This severe bending stiffening results 
from the Kirchhoff constraints (zero transverse shear 
strain conditions) that engender spurious kinematic 
relations in the elements situated along the boundaries 
(and lines of symmetry) where the kinematic restraints 
are imposed. Consequently, with the over constrained, 
stiff corners, no wrinkling can be initiated. 

The above arguments lead to a basic conclusion that 
eliminating sharp-corner meshes, in the regions where 
concentrated loads are applied, may be beneficial for 
the modeling of wrinkled equilibrium states. Truncating 
a corner a short distance inward may result in an 
improved load transfer and mesh quality in that local 
region. This truncation will necessitate replacing the 
concentrated force with a statically equivalent 
distributed traction. The benefits of this strategy are 
twofold: (a) removal of a severe stress concentration 
(note that in practice, the load introduction into a 
structure is closer to a distributed load than a 
concentrated force), and (b) improvements in the 
kinematics in the critical corner regions from which 
wrinkles radiate. 

Consistent with the present modeling philosophy, 
the corners of the square membrane are "cut-off such 
that the length of each corner edge (which is normal to 
the direction of the applied load) is set to be small, as 
shown in Figure 4. The prescribed kinematic boundary 
restraints are also depicted in the figure. The domain of 
the entire membrane is discretized with a relatively 
refined mesh (4,720 elements) for the purpose of a 
suitable comparison with the experiment. For simplicity 
and for the purpose of validating the efficacy the corner 
cut-off modeling, the reinforced comers of the 



membrane are not modeled. (It is expected, however, 
that including the corner reinforcements in a 
computational model would result in improved 
correlation with the experiment.) The originally planar 
finite element mesh is augmented by the 
pseudorandom, nodal imperfections distributed over the 
interior nodes of the model, using the amplitude 
parameter a = 0.1 (A parametric study on the selection 
of a is discussed subsequently.) 

The contour plots depicting the deflection 
distributions in the experiment and the present 
geometrically nonlinear shell analysis are shown in 
Figure 5. The computer simulation is able to effectively 
predict four wrinkles radiating from the truncated 
corner regions just as those from the measured 
experimental results. The analysis also shows that 
curling occurs at the free edges as observed in the 
experimental results; however, the amplitudes of the 
experimental deflections are somewhat greater. 
Although intended to be symmetric, the experimental 
results are somewhat asymmetric. As in the previous 
example, the actual initial geometric imperfections, not 
incorporated in the analysis, may have contributed to a 
significant asymmetry in the experiment and the 
differences with the analysis. This again is evidence to 
the fact that such ultra-flexible and lightly stressed 
spatial structures are not only difficult to model 
analytically but also to test experimentally, requiring 
ftirther refinements in the experimental methods for 
these thin-film membranes. 

1. A study of imperfection amplitudes 

The practical question of how to decide on the 
appropriate value of the amplitude parameter, a , in Eq. 
(1) may be satisfactorily answered through a parametric 
study in which the distribution of the pseudorandom 
imperfections is kept constant, and the value of a is 
varied. To minimize the computational effort, it suffices 
to model a symmetric quadrant of the membrane as 
shown in Figure 6, where the imperfections are defined 
over all interior nodes. The appropriate truncations of 
the comers, the application of statically equivalent 
distributed tractions, and the symmetry boundary 
conditions are all implemented as previously described. 

The amplitude parameter, a , is varied in the range 
0.001<a<5.0, and for each fixed value of this 
parameter, a geometrically nonlinear analysis is 
performed. The results from this study are presented in 
Figure 7, where a normalized deflection, Wmid/h, at the 
center of the membrane free edge (and which represents 
the maximum value of the deflection across the entire 
domain), is plotted versus a . There are three basic 
ranges in the graph. For very small amplitudes 
(a < 0.005), the imperfections are not large enough to 
provide sufficient bending-to-membrane coupling. No 
wrinkling deformations can be simulated. In the range 



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0.01 < a < 1.0 , all analyses predict practically the same 
wrinkled deformation equilibrium state as evidenced by 
the constant value of Wmid/h across this range of a. The 
solutions corresponding to the values a > 1 can be 
viewed as erroneous because this range of a values 
implies that the imperfection amplitudes are too large; 
in fact, they are on the order of the membrane 
thickness. One way to interpret this is by examining the 
transverse element distortions that may have been 
caused by the large imperfections: in general, one of the 
four element nodes will be out of plane, thus violating 
the basic element formulation. 

2. A study of spatial distribution of imperfections 

In this study, several alternative schemes of spatial 
distributions of the imperfections are examined in 
relation to their effect on the wrinkled equilibrium 
states. Consider three distinct imperfection distribution 
models corresponding to a = 0.10 (from the acceptable 
range of values), as shown in Figure 8. In Model 1, the 
imperfections are imposed across all of the interior 
nodes (in the shaded region). In Models 2 and 3, the 
imperfections are only focused on the comer regions. 
The results from this study are presented in Figure 9, 
which shows a wrinkled wave along the A-B line (see 
Figure 6). Noticeably, the three imperfection schemes 
produce the same wrinkled deformations, and this is 
consistent across the whole membrane. Thus the 
imperfection schemes may be deemed equivalent from 
the perspective of determining the appropriate wrinkled 
deformations for this problem. This is not surprising 
since the dominant wrinkling modes emanate from the 
comer regions. For this reason, the imperfections need 
only be imposed over these key regions. 

Concluding Remarks 

In this paper, careful modeling considerations were 
explored to simulate the formation of highly nonlinear 
wrinkling deformations in thin-film membranes. The 
analyses were carried out within the framework of the 
geometrically nonlinear, updated Lagrangian shell 
formulation using a commercial finite element code 
ABAQUS. The underlying shell relations employ the 
assumptions of small strains, large displacements, and 
do not rely on the classical membrane assumptions of 
Tension Field theory. The finite element models utilize 
a Mindlin-type quadrilateral shell element, S4R5. The 
element employs reduced integration of the transverse 
shear energy, and has an ad hoc correction factor 
multiplying the shear stiffness to ensure locking-free 
bending behavior even for very thin shells. Moreover, 
the hourglass control scheme is used to suppress 
spurious zero-energy (hourglass) modes that result as a 
consequence of the reduced integration. 

To achieve convergent geometrically nonlinear 
solutions that correlate well with experiment, the issue 



of deformation coupling between bending and 
membrane deformation was addressed. By applying 
pseudorandom out-of-plane imperfections to the 
initially planar membrane surface, the requisite 
membrane-to-bending coupling is invoked at the 
commencing stage of the nonlinear solution process. 
Using relatively small and unbiased imperfections, 
converged wrinkled equilibrium states can be obtained 
which are independent of the initial imperfections. 

For the class of thin-film problems in which corner 
regions are subjected to tension loads, as in the second 
example problem, the need for improved modeling of 
such corner regions was identified. The introduction of 
tmncated corners and the replacement of concentrated 
loads with statically equivalent distributed tractions 
enabled successful geometrically nonlinear wrinkled 
stress states to be determined. 

The present modeling schemes were used in several 
numerical studies involving thin-film membranes 
subjected to mechanical loads. First, converged 
wrinkled deformation modes were predicted for a 
square membrane loaded in shear. These results 
compared favorably with an experiment recently 
conducted at NASA Langley. In the second example, a 
square membrane subjected to four-corner tensile loads 
was analyzed and exhibited major wrinkling emanating 
from the corners. The corner regions were tmncated in 
the model in order to "design out" the near singular 
stress fields associated with concentrated loads and to 
improve element topology to avoid overly stiff corners. 
A noticeably close correlation with the experiment was 
also observed for this very challenging computational 
problem. It is expected that even closer correlation may 
be possible once the comer reinforcements are 
represented in a computational model in sufficient 
detail. Also, a greater insight was developed through a 
set of parametric studies of the amplitudes and spatial 
distributions of the pseudorandom imperfections. 

Our experience with the geometrically nonlinear, 
updated Lagrangian shell analysis of thin-film 
membranes suggests that the current state-of-the-art 
computational methods have the potential for 
adequately simulating the stmctural response of such 
highly flexible and under constrained wrinkled 
stmctures. There exist, however, numerous challenging 
issues requiring in-depth analytic, computational, and 
experimental pursuits. Issues related to robustness, 
nonlinear solution convergence, sensitivity to boundary 
restraints and applied loading, mesh dependence, and 
the shell-element technology, specifically addressing 
ultra-thin behavior, need to be closely examined and 
addressed. Computational opportunities also exist in 
exploring the explicit dynamics nonlinear analyses that 
may have the advantage of obviating the stability issues 
in modeling the wrinkling phenomenon. 



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Acknowledgements 

The authors would like to thank Profs. Jack Leifer, 
University of Kentucky, and Joe Blandino, James 
Madison University, for providing the experimental 
results. 

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1 mm 



Mylar® Polyester Film Properties 


Edge length, a (mm) 


229 


Thickness, h (mm) 


0.0762 


Elastic modulus, E (N/mm^) 


3790 


Poisson's ratio, v 


0.38 




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



w (mm) 



w (mm) 




(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). 



American Institute of Aeronautics and Astronautics 



Kevlar threads 



2.45 N 




KAPTON® Type HN Film Properties 


Edge length, a (mm) 


500 


Thickness, h (mm) 


0.0254 


Elastic modulus, E (N/mm^) 


2590 


Poisson's ratio, v 


0.34 



Kevlar threads 



2.45 N 

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




i 



Uniform 
Traction 



B.C.'s 

{v,w,e,,0j=o 



rTTTI^B.C.'s 

tf^ {u,w,ey,0j= 

7 mm 



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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w (mm) 



w (mm) 




I 




0.34 

0.27 

0.20 

0.13 

0.06 

-0.01 

-0.08 

-0.15 

-0.22 

-0.30 

-0.37 

-0.44 

-0.51 

-0.58 

-0.65 

-0.72 

-0.79 

-0.86 



(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). 




Uniform 
Traction 



B.C.'s 

{v,w,e,,ej=o 

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. 



Symmetry B.C.'s 



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W^M/h 



18.0 
16.0 
14.0 
12.0 
10.0 
8.0 
6.0 
4.0 
2.0 
0.0 



0.001 0.01 0.1 1.0 

Imperfection amplitude parameter, a 



10.0 



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



Region with imposed 
imperfections 






iiiinu\v\\\\\eci^ 
llllllnVwwiGS^Hu 



nmimniim^ 




(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 
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0.60 n 

0.50 
0.40 
0.30 
0.20 



-^ Model 1 
-i- Model 2 
--Models 







-0.30 ^ 



Figure 9. Square thin-film membrane (KAPTON® Type HN film) loaded in tension by corner tractions: Wrinkling 
deflection along section A-B corresponding to three imperfection models for a fixed value of the imperfection 
amplitude parameter, a=0.10. 



11 
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