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Wrinkles in Square Membranes 117
following a similar approach [13], to obtain the following expressions for halfwavelength
and amplitude
and
λ =
4 2π2R1(R1 − r1) 2Et3 sin θ1
3(1 − ν2 )T1 T
(23)
A = 2λ(δ1 + δ2)
π
(24)
Here δ1 and δ2 are the radial displacements of the corners loaded by T1 T and
T2 T , respectively; these corner displacements can be estimated from
δi ≈
Ti T
θi ln Ri
Ai
+(1− ν) + θi − 1
tan θi
2
i = 1, 2 (25)
2Etsin 2 θi
r1
Here, Ai is theareaofapartofthe central, biaxially stressed region that is
associated with the loads Ti T . More refined estimates can be obtained from a
two-dimensional finite-element analysis, see Section 5.
4 Finite-Element Simulations
We have recently shown [14] that wrinkling of a thin membrane can be accurately
modelled using the thin shell elements available in the commercial
finite-element package ABAQUS [15]. The analysis is carried out by introducing
initial geometrical imperfections, obtained from an initial eigenvalue
analysis, followed by a geometrically non-linear post-buckling analysis using
the pseudo-dynamic *STABILIZE solution scheme. This approach, although
expensive in computational terms, is so far the only method that can reveal
full wrinkle details and can be relied upon as an almost exact replication of
physical experimentation. An alternative approach is the Iterative Modified
Properties (IMP) method [9] which uses a combined stress-strain wrinkling
criterion inatwo-dimensional membrane model. The IMP method has been
recently implemented as an ABAQUS user subroutine and has been shown to
accurately predict the extent of the wrinkled regions and the two-dimensional
stress distribution —but of course not the details ofthe wrinkles.
Both of these modelling techniques were used to simulate a 0.025 mm thick,
500×500 mm 2 square Kapton membrane (E=3530 N/mm 2 and ν =0.3). The
membrane was loaded at each corner through a spreader beam by a 0.1 mm
thick, 25 mm × 20 mmKapton tabs—as in the experiment of Section 2.
A uniform mesh of 200 by 200 square elements was used to model the
whole structure, in order to capture the fine wrinkle details in the corners.
In the shell model, the Kapton membrane and the corner tabs were modelled
using S4R5 thin shell elements of different thickness. At the corners, the
shell elements were connected to “Circ” beam elements through the *MPC,
R 2 i