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Simulation of light-weight membrane structures by wrinkling model

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng
2005;
62
:2127–2153Published online 14 February 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1266
Simulation of light-weight membrane structuresby wrinkling model
Riccardo Rossi
1
, Massimiliano Lazzari
1
, Renato Vitaliani
1
,
∗
,
†
and Eugenio Oñate
2
1
Department of Structural and Transportation Engineering
,
University of Padua
,
35131 Padua
,
Italy
2
International Center for Numerical Methods in Engineering (CIMNE)
,
Universidad Politécnica de Cataluña
,
Campus Norte UPC
,
08034 Barcelona
,
Spain
SUMMARYThe computational challenge in dealing with membrane systems is closely connected to the lack of bending stiffness that constitutes the main feature of this category of structures. This manifestsnumerically in badly conditioned or singular systems requiring the use of stabilized solution procedures,in our case of a ‘pseudo-dynamic’ approach. The absence of the ﬂexural stiffness makes the membranevery prone to local instabilities which manifest physically in the formation of little ‘waves’ in‘compressed’ areas. Current work presents an efﬁcient, sub-iteration free ‘explicit’, penalty materialbased, wrinkling simulation procedure suitable for the solution of ‘static’ problems. The procedure isstabilized by taking full advantage of the pseudo-dynamic solution strategy, which allows to retainthe elemental quadratic convergence properties inside the single solution step. Results are validatedby comparison with published results and by setting up ‘numerical experiments’ based on the solutionof test cases using dense meshes. Copyright
2005 John Wiley & Sons, Ltd.
KEY WORDS
: geometrical non-linear behaviour; membrane structures; wrinkling
1. INTRODUCTIONWhen the ratio between the thickness and the other dimensions of a structural shell systemsgets very low, the ﬂexural contribution to the response to the external loads loses importance,making appealing the use of a mathematical model which neglects the bending contribution,namely a purely membrane model. The immediate consequence of such an idealization is thatthe motion of the structure outside its own plane is not restrained making possible both ‘global’rigid body motions and ‘local’ instabilities (e.g. References
[
1–3
]
).This behaviour is physical, particularly it can be readily veriﬁed that real membranes showlocal instability phenomena, made evident by the formation of little
waves
in zones in which
∗
Correspondence to: R. Vitaliani, Department of Structural and Transportation Engineering, University of Padua,35131 Padua, Italy.
†
E-mail: rvit@caronte.dic.unipd.it
Received 9 June 2003 Revised 13 October 2004
Copyright
2005 John Wiley & Sons, Ltd.
Accepted 13 October 2004
2128
R. ROSSI
ET AL.
compression tends to appear. The phenomena manifests as a consequence of the formationof compressive stresses as in this conditions any initial imperfection immediately leads to anout of plane displacement which reduces the resistance of the membrane to the compres-sion (making the compression vanish). The shape of those ‘waves’ or ‘wrinkles’ is governedby the state of stress and by the local bending stiffness of the system, high bending resis-tance resulting in few large ‘waves’, low resistance in numerous (at the limit inﬁnite) littlewrinkles.Generally speaking the kinematic model for a FE membrane formulation does not allowthe description of wrinkles of size smaller then the element’s dimension. This results in anon-physical compressive resistance shown by ‘coarse’ FE models.Correct description of the wrinkles is possible by increasing the mesh density up to alevel at which the single wave is subdivided between a number of elements. This approach ispossible and needed when accurate prediction of the wrinkle’s size and distribution is needed;different authors References
[
4–6
]
addressed the problem by using rotation free thin shellsmodels. Numerical prediction of the wrinkle’s size and distribution together with experimentalvalidation can be found in References
[
7–10
]
with reference to space structures, for whichthe formation of the wrinkles becomes a crucial aspect of the structural behaviour. This worksaddress the simulation of the so-called ‘Kapton’s membrane’ by the use of ABAQUS S4R5 four-nodes shell elements, describing in detail the effect of initial imperfections and the loading andunloading behaviour of the different systems. Such approaches allow an excellent descriptionof the system’s deformation however involve the use of a mesh density inversely proportionalto the expected size of the wrinkles which makes the technique computationally unaffordablefor large systems.In many cases on the other hand a ‘deterministic’ prediction of the wrinkling ﬁeld is notneeded or the system’s unknowns (construction method, imperfections, etc.) make it unreliable.In those cases it is of interest to obtain a sort of ‘averaged membrane response’ in the formof an individuation of the wrinkled areas and the elimination of the non-physical compressivestresses from the model’s response.It is worth noting that this approach is not necessarily less precise than the previous one:indeed no information on the wrinkling size is provided, but the global stress ﬁeld is properlydescribed. As well it is important to highlight how in civil engineering structures, the positionof wrinkles is never known because of its strong dependence on the initial imperfections; inthis case the only reliable result is the identiﬁcation of the wrinkled areas.Over the years this problem was addressed using basically two different strategies: a ﬁrst one
[
11,27
]
based on the modiﬁcation of the deformation gradient to take in account the formationof the wrinkles, a second one taking in account the formation of the wrinkles by modifyingthe constitutive law.It is commonly recognized that the introduction of the wrinkling leads often to numericalinstabilities (‘
...
tension ﬁeld behaviour may exhibit erratic behaviour during the solutionprocess and possibly prevent convergence..’
[
12
]
). In the authors experience the inclusion of the wrinkling phenomena as an additional source of non-linearity leads to a slower convergenceof the Newton–Raphson algorithm or eventually to a ‘high’ number of subiterations. An increasein the number of iterations was reported in the literature
[
11
]
even for the case in which theconsistent tangent operator was used.In current work we address an efﬁcient ‘static’ solution procedure for membrane systemsincluding the wrinkling correction. Our procedure is based on a penalty material model which
Copyright
2005 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2005;
62
:2127–2153
SIMULATION OF LIGHT-WEIGHT MEMBRANE STRUCTURES
2129is similar to the one proposed in Reference
[
12
]
the main innovation being connected to thesolution strategy used.It is well known that straight-forward ‘static’ solution of membrane systems is not possiblebecause of impending local rigid body motions which make the stiffness matrix badly con-ditioned or singular. The use of a ‘pseudo-static’ solution procedure based on an equivalentzero-mass damped dynamical system allows smooth convergence to the static solution. Thisprocedure allows the determination of the ﬁnal ‘static’ conﬁguration by performing a numberof ‘dynamic steps’ which should be intended as ‘artiﬁcial states’ without real physical meaning.Our technique is based on an ‘explicit’ correction of the material which is kept constantinside each solution step. No guarantee is given that the compressions are correctly removedat the end of each solution step, however when the dynamic procedure converged to a staticconﬁguration no further change for the strain ﬁeld manifests which ensures convergence of thewrinkling procedure.It should be noted that keeping the material constant inside each solution step ensures thatthe convergence properties of the membrane ﬁnite element are kept, follows immediately thatthe only additional cost is connected to the modiﬁcation of the constitutive law.Stability of the procedure is greatly increased by the presence of the dynamic terms, a stabi-lization for the material modiﬁcation phase is however proposed which allows the oscillationsof the stress ﬁeld to be damped out in a shorter time.A number of ‘numerical experiments’ were ﬁnally set up, using dense unstructured meshesto allow the formation of wrinkles. The same examples were run using much coarser meshesincluding the wrinkling correction. The results were compared both in terms of second Piola–Kirchoff (PK2) stresses and displacements.2. ELEMENT TECHNOLOGYThe problem discussed here is connected with the description of the motion of thin structures inthe space. In the analysis large displacements have to be taken in account because the problemis geometrically non-linear. The non-linear ﬁnite element procedure is developed according tothe total Lagrangian formulation using appropriate stress and strain measures as the Green–Lagrange strain and the PK2 stress (e.g. References
[
13–15
]
). Forms of material non-linearitycan be introduced at a later stage.Focus is here given on the formulation of the elements to be used in the analysis. Thecrucial point is that the stress ﬁeld is contained in the plane (or axis) of the element, but theelement itself can rotate in the 3D space. As a result, the contribution to the equilibrium isconnected with its actual orientation.From a practical point of view this means that the constitutive law can be formulated interms of in-plane (or axial) strains and of in-plane stresses. This is greatly simpliﬁed by theintroduction of a co-rotational system of co-ordinates.Although it is possible to formulate higher order elements, it is well known that in presenceof severe distortions low-order elements are much more reliable (e.g. Reference
[
16
]
). Thereforestress is given to the formulation of an efﬁcient triangular 3D membrane element.The reference paper for the formulation used is Reference
[
17
]
, while details on the derivationof the various terms involved can be found in References
[
18,19
]
. Next lines give a brieﬂyoutline of the formulation used, together with the basic steps needed for the implementation.
Copyright
2005 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2005;
62
:2127–2153
2130
R. ROSSI
ET AL.
As mentioned in the opening, a convenient co-rotational system of axis is introduced. In thissystem of co-ordinates the deformation of the element becomes a two-dimensional plane stressproblem
[
14
]
. The main advantage of the present formulation in comparison with a classicalco-rotational approach is that no explicit change of base is needed, allowing a greater efﬁciencyto be achieved.Given the vectors
x
1
,
x
2
,
x
3
x
I
=
x
I
1
x
I
2
x
I
3
T
is the co-ordinate of the node ‘
I
′
describingthe current position of the nodes in the global co-ordinate system, Figure 1, it is useful todeﬁne the vectors:
x
21
=
x
2
−
x
1
,
x
31
=
x
3
−
x
1
,
x
32
=
x
3
−
x
2
(1)Taking advantage of this notation a local co-ordinate system can be deﬁned by:
v
1
=
x
21
x
21
,
v
3
=
x
21
×
x
31
,
n
=
v
3
v
3
,
v
3
=
2
A
n
,
v
2
=
(
n
)
×
(
v
1
)
(2)Vectors
v
1
,
v
2
,
n
form an orthonormal base with
v
1
and
v
2
describing the plane of the ele-ment.
A
is the current area of the element and the relations between
v
3
and
n
is a consequenceof the properties of the vector produce
(
×
)
[
20
]
.The representation of the position vectors (for a given node ‘
I
’) in the local co-ordinate
(
v
1
,
v
2
,
n
)
system is given by:
y
I
=
x
I
1
•
v
1
x
I
1
•
v
2
x
I
1
•
n
T
,
x
I
1
=
x
I
−
x
1
(3)Since the structure is planar, two co-ordinates only are needed to describe it fully. This isreﬂected by the fact that the last of the scalar products is identically zero
(
n
⊥
x
I
1
by (2)).It should also be noted that, for
I
equal to 1, the
y
I
in (3) is identically zero. The proposedco-ordinate system is deﬁned during all of the deformation process.Here it has been used the standard notation, indicating with the capital letter all the quantitiescalculated in the reference conﬁguration.
X
3
X
1
X
2
V
2
V
1
123n
Figure 1. Co-ordinate systems used with membrane element.
Copyright
2005 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2005;
62
:2127–2153
SIMULATION OF LIGHT-WEIGHT MEMBRANE STRUCTURES
2131
2.1. Kinematic model
In order to develop the formulation of membrane element, two assumptions are made: thatdisplacement ﬁeld inside each element is constrained in the plane of the element itself and thatapproximation for the displacement ﬁeld is linear.The use of area co-ordinates
1
,
2
, and
3
allows to express conveniently the position of any point inside the membrane
(
y
1
=
0
→
1
y
1
=
0
,
Y
1
=
0
→
1
Y
1
=
0
, Figure 1) as:
y
=
2
y
2
+
3
y
3
,
Y
=
2
Y
2
+
3
Y
3
(4)In order to go further in the analysis of the deformation process, it is useful to take inaccount the deformation gradient
F
; this can be opportunely done by the introduction of thetwo tensors
j
(
j
=
y
/
)
and
J
(
J
=
Y
/
)
:
F
=
y
Y
=
y
Y
=
jJ
−
1
(5)Note that
F
is a 2
×
2 tensor even if work is developed in a three-dimensional environment(the three-dimensional problem is here reduced to a two-dimensional one by working in asuitable co-ordinate system). This observation is crucial for the success of the formulation.Trace of the rigid body motion of the element in the 3D space (translation and rotation) iskept in the deﬁnition of the local base. However the deformation gradient (5) is unaffected byrigid body motions. Systematic use of deﬁnitions (2) allows to write the tensors
j
and
J
as:
j
=
y
=
x
21
x
31
•
x
21
x
21
0
v
3
x
21
,
J
=
Y
=
X
21
X
31
•
X
21
X
21
0
V
3
X
21
=
G
−
1
(6)It should be noted that
G
=
J
−
1
is independent from the deformation and can be calculatedonce and stored. In writing (6) it was taken in account that:
y
2
=
x
21
•
v
1
x
21
•
v
2
=
x
21
0
,
y
3
=
x
31
•
v
1
x
31
•
v
2
=
x
31
•
x
21
x
21
v
3
x
21
(7)in which it was used the relation
v
3
=
x
21
x
31
•
v
2
v
1
×
v
2
→
x
31
•
v
2
=
v
3
x
21
.Introducing the tensor
g
as:
g
=
j
T
j
=
x
21
2
x
31
•
x
21
x
31
•
x
21
x
31
•
x
21
x
21
2
+
v
3
•
v
3
x
21
2
=
x
21
•
x
21
x
31
•
x
21
x
31
•
x
21
x
31
•
x
31
(8)
Copyright
2005 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2005;
62
:2127–2153

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