-
Development of a non-linear triangular prismsolid-shell element
using ANS and EAS techniques
Fernando G. FloresDepartment of Structures, Universidad Nacional
de Crdoba, Casilla de Correo 916,
5000 Crdoba-Argentina, and CONICET, [email protected]
Abstract
This paper extends a previous triangular prism solid element
adequate tomodel shells under large strains to become a solid-shell
element, i.e. thatdiscretizations may include just one element
across the thickness. A totalLagrangian formulation is used based
on a modified right Cauchy-Green de-formation tensor (C). Three are
the introduced modifications: a) an assumedmixed strain approach
for transverse shear strains b) an assumed strain ap-proach for the
in-plane components using a four-element patch that includesthe
adjacent elements and c) an enhanced assumed strain approach for
thethrough the thickness normal strain (with just one additional
degree of free-dom). One integration point is used in the triangle
plane and as many asnecessary across the thickness. The intention
is to use this element for thesimulation of shells avoiding
transverse shear locking, improving the mem-brane behavior of the
in-plane triangle, alleviate the Poisson effect lockingand to
handle quasi-incompressible materials or materials with isochoric
plas-tic flow. Several examples are presented that show the
transverse shear andPoisson effect locking free behavior, how the
improvement in the membraneapproach alleviates the volumetric
locking and the very good performance ofthe introduced element for
the analysis of shell structures for both geometricand material
non-linear behavior.Keywords: Solid-Shell, Prism, Large strains
1. Introduction
Promoted by the continuous improvement in computer facilities
and alsoby the need for improving different aspects of the models
to obtain morereliable simulations, the use of solid elements for
the simulation of shellshas grown a lot in the last fifteen years.
The main advantages when usingsolid-shell elements are: a) general
tridimensional constitutive relations can
Preprint submitted to Elsevier June 11, 2013
-
be used as plane stress state assumption is no needed; b)
contact forces arerealistically applied on the outer surfaces which
is particularly important forfriction; c) large transverse shear
deformations can be considered speciallyif more than one element is
used across the thickness; d) special transitionelements between
shell elements and solid elements are avoided; e)
boundariesnon-parallel to the shell normal or director can be
correctly modeled; f)local triads and rotation vectors, that are
costly in general and difficult toparameterize and update, are not
needed.
For the simulation of strongly non-linear problems due to, for
instance,complex constitutive models or contact, the simplex
(linear interpolation)elements are preferred and if possible with
displacement degrees of freedom(DOFs) only as they are more
reliable and robust. Solid-shell elements, i.e.when only one
element is used across the thickness, developed until noware
hexahedral and particularly the largest developments are devoted to
thetri-linear 8-node brick. It is well known that simplex solid
elements basedon the standard displacement formulation when used to
simulate slenderstructures lock severely. This numerical locking
indicates the inability ofthe interpolation functions (and their
gradients) to fit the solid behaviorthat many times makes the
obtained solutions useless. In bending problemstransverse shear
locking appears that increases with the shell slenderness.Besides
that a linear interpolation (constant gradient in that direction)
doesnot allow to fit a linear through the thickness strain due to
the Poisson effect.If the shell is initially curved artificial
transverse strains and stresses appearunder pure bending due to
curvature thickness locking. Membrane lockingspecially appears on
initially curved shells when bending is preponderantwithout middle
surface stretching. Finally when dealing with incompressibleor
nearly incompressible materials or elastic-plastic materials with
isochoricplastic flow (metals typically) volumetric locking
appears.
In order to be computationally efficient, it is necessary that
solid-shellelements have a different integration scheme on the
shell tangent plane (as lowas possible) than across the thickness,
where the number of integration pointsmust be arbitrary in order to
capture the non-linearities of the constitutivemodel when bending
is present. This is particularly important for elastic-plastic
models for example springback in sheet metal forming
simulations.The use of a single integration point in the plane of
the shell requires, in orderto maintain the efficiency, the series
expansion of different variables (e.g.the inverse of the Jacobian)
and some control of the spurious deformationmodes due to
under-integration. The first one brings some limitations onthe
allowable distortion of the elements and the latter usually leads
to theintroduction of user-defined factors that must be properly
tuned. Howeverthe advantages are very important and markedly
decrease the storage space
2
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for internal variables and the CPU time, particularly in codes
with explicitintegration of the momentum equations.
The advances in solid and solid-shell elements aimed to cure the
differentlocking problems are numerous. In Reference [21] a
detailed state of the artfor this type of elements can be found. To
avoid transverse shear locking,the classical mixed assumed strain
approximation by Dvorkin and Bathe [6]is the most used when full
integration is considered (see for example [14, 29])and a variation
of it, modifying and increasing the sampling points to avoidthe
occurrence of spurious modes, when reduced integration is preferred
(seefor example [4]). In solid elements under large strains the
volumetric lock-ing has been mostly resolved using mixed elements
with constant pressureas originally proposed by [17], where the
pressure degree of freedom can beremoved locally averaging over the
element the volumetric strain, leading toan element with constant
volumetric strain. This technique is not accept-able for
solid-shell elements because it leads to excessively flexible
elementsand does not allow an adequate normal strain gradient
across the thick-ness. More recently enhanced assumed strain
techniques (EAS) [23] havebeen developed consisting of improving
the deformation gradient with theinclusion of internal degrees of
freedom which are condensed at the elemen-tary level, maintaining
displacements as the only global degrees of freedom.EAS techniques
not only allows to eliminate the volumetric locking, but alsoto
eliminate the problem due to the Poisson effect and can even be
usedto improve the performance of the membrane part. It is not free
of disad-vantages: instabilities may occur under large compressive
strain requiring asignificantly increase of elemental database or
the use of an iterative loop ineach element for the determination
of the internal degrees of freedom at eachglobal Newton-Raphson
iteration. Apparently the solid-shell element withthe best
performance developed until now is that presented in Refs. [20,
21]where it is shown that an additional internal degree of freedom
together witha reduced integration scheme is sufficient to solve
the volumetric locking andthe problem arising from Poisson effect.
The element satisfies the membraneand bending patch-tests, shows
very good convergence properties and worksproperly under large
elastic-plastics strains as shown in [22]. Alternatively[3] have
developed an element with an additional local displacement degreeof
freedom at the center of the element that improves the
interpolation ofthe normal displacement.
In the authors knowledge, the only triangular prism solid
element ade-quate to simulate shell is the one proposed in [7]. In
this Reference transverseshear locking is cured using an assumed
natural strain (ANS) to modify themetric tensor components
associated with shell normal direction. Volumetriclocking is
alleviated on the one hand by averaging volumetric deformation
3
-
over the element (restricted to use at least two elements in the
thickness)and on the other hand by an assumed strain technique for
the in-plane com-ponents that uses pieces of information from
adjacent elements ([9]). Thusa simple element is obtained, which
does not require stabilization due tounder-integration, formulated
in large strains and suitable for contact prob-lems. One obvious
and important advantage of a triangular prism elementis that the
triangular mesh generators are quite more efficient, and
provideelements with a better aspect ratio. Curiously there are not
developmentsfor triangular prism solid-shell elements. Probably the
reason for this is thefew possibilities given by the interpolation
functions of the standard prismelement.
The behavior of the standard (displacement base) 6-node prism
and 8-node brick are quite different, thats why the strategies to
cure the differentlocking problems may be different. The transverse
shear locking of the firstone is quite lower while the latter has a
better in-plane behavior. In a planestrain condition a one point
quadrature eliminates volumetric locking for aquadrilateral but not
for a triangle. Note also that for the same mesh density(measured
in the number of nodes) a reduced integration strategy implies
thedouble number of integration points for the prism than for the
brick.
In this paper we propose to modify the assumed strain 6-node
elementprism [7] in order to make it suitable as a solid-shell
element, i.e., that justone element can be used in the thickness of
the sheet and get correct resultsin thin shells with non-zero
Poisson ratios and quasi-incompressible materialsor with isochoric
plastic flow.
The next section summarizes the basic formulation of the solid
element.Then the improvements in the standard element are
presented, starting withthe improvement in the tangent plane of the
sheet, followed by the transverseshear formulation and finally the
EAS technique used to prevent the Poissoneffect locking. Section 5
presents several examples showing the very goodbehavior of the
element and finally some conclusions are summarized.
2. Basic kinematics of the solid finite element
Next the kinematic formulation of the standard 6-node triangular
prismelement is presented. The element configurations are described
by the stan-dard isoparametric interpolations [31]
X () =6I=1
N I () XI (1)
x () =6I=1
N I () xI =6I=1
N I ()(XI + uI
)(2)
4
-
where XI , xI , are uI denote the original coordinates, the
current coordinatesand the displacements of node I respectively.
The shape functions N I ()combine linear polynomials in terms of
the area coordinates (, ) on thetriangular base with a linear
interpolation () along the prism axis:
N1 = zL1 N4 = zL2
N2 = L1 N5 = L2
N3 = L1 N6 = L2(3)
where the third triangular area coordinate and the linear
Lagrangian poly-nomials have been included:
z = 1 L1 =
1
2(1 ) (4)
L2 =1
2(1 + )
In a standard way, defining the Jacobian matrix at each
integration point
J =X
(5)
the computation of the Cartesian derivatives of the shape
functions can beperformed
N IX = J1 N I (6)
At each element center, coincident with the principal orthotropy
direc-tions of the constitutive material, a local Cartesian triad
is defined
R = [t1, t2, t3] (7)
that allows to compute the Cartesian derivatives with respect to
this localsystem (Y)
N IY = RT N IX (8)
and the deformation gradient F as a function of the present
nodal coordinates
Fij =NNI=1
N Iyj xIi (9)
Finally the components of right Cauchy-Green deformation tensor
C areobtained
Cij = FkiFkj (10)
5
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As most of the constitutive equations are easily dealt with
using volu-metric and deviatoric components, at each integration
point the deformationtensor is decomposed in a multiplicative
form
C = [det (C)]13 CD = J
23CD (11)
defining the volumetric and deviatoric strain components as
= ln J
(12)
eD = ln(C
12D
)the logarithmic strain tensor results
e =
31 + eD (13)
The computation of the strains (12) requires the spectral
decompositionof C
C = LT 2 L (14)
where 2 is a diagonal matrix collecting the eigenvalues 2i of C
and Lincludes the associated (unit) eigenvectors.
Adopting the hypothesis of additivity of elastic and plastic
strain compo-nents, the strain tensor is written as
e = ep + ee (15)
For materials with a yield surface independent of the mean press
(Misesor Hills yield functions for instance) it is possible to work
exclusively withthe deviatoric strain an stress tensors, making
easier and computationallycheaper the integration of the
constitutive equations.
As it will be shown in the next section the tensor C is modified
usingassumed strain techniques that in one case includes an
additional internaldegree of freedom , leading to an improved
tensor C. The balance equationsto be solved (variational
formulation) are for the large strain case
g1 (u, ) =
Vo
1
2S(C)
: uC dV0 + gext = 0 (16)
g2 (u, ) =
Vo
1
2S(C)
: C dV0 = 0 (17)
6
-
where S is the second Piola-Kirchhoff stress tensor that can be
related toKirchhoff stress tensor T using the following
expressions:a) defining the rotated tensor
TL = LT T L (18)
b) the relationship between the rotated tensors is (See for
example Reference[5])
[SL] =1
2[TL]
(19)
[SL] =ln (/)12
(2 2
) [TL]c) finally the 2nd Piola-Kirchhoff stress tensor is
S = L SL LT (20)
As an alternative to the logarithmic strain, the spectral
decomposition(14) allows to easily deal with large strain
hyperelastic materials (elastomers),using models such as Ogden,
Mooney-Rivlin, neo-Hookean, etc., that areusually defined in terms
of a strain energy written in terms of the principalstretches.
3. Modifications of the standard element
The triangular prism element described above must be
substantially im-proved to be used for large strain elastic-plastic
analysis of shells includingcontact constraints. Different
modifications are introduced over the metrictensor C with that
aim.
The discretization of a shell type solid using solid-shell
elements impliestwo steps a) a discretization of the shell middle
surface with three-node tri-angles in the present case and b) a
discretization in the thickness directionusing one (or more) prism
elements based on the triangles defined before.Here it will be
assumed that the 6-node connectivity associates nodes 1-3and nodes
4-6 with planes nearly parallel to the shell middle surface andthat
the latter ones are above the first three nodes along the shell
normalat a distance equal to the thickness. Thus middle surface
normal direction(local y3) is almost coincident with local natural
coordinate .
As shown in previous section the strain tensor is computed from
thespectral decomposition of the right Cauchy-Green deformation
tensor, thus
7
-
an interesting possibility is to directly modify the components
of C associatedto the behavior intended to improve
C =
Cm11 Cm12 Cs13Cm21 Cm22 Cs23Cs31 C
s32 C33
(21)where the components with the upper index m are those that
have the maininfluence on the in-plane (membrane and bending)
behavior of the shell andthose denoted with an s are those
associated with the transverse shear. Thenthe deformation tensor
may be divided into three parts
C = C1 + C2 + C3 (22)
where
C1 = C11t1 t1 + C22t2 t2 + C12
(t1 t2 + t2 t1) (23)
corresponds with the components on the tangent plane,
C2 = C13(t1 t3 + t3 t1)+ C23 (t2 t3 + t3 t2) (24)
are those components mainly associated with the transverse shear
strainsand
C3 = C33t3 t3 (25)
is used to compute the through the thickness strain.The changes
in each of the parts in which tensor C has been divided
are described next. The proposed approximation to C1 is
identical to thatdescribed in [7]. The modification in C2 is
similar to that described in [7] butmodifying the position of the
sampling points for the assumed mixed strainapproach. As for C3 now
an EAS approximation is considered with theinclusion of a single
internal degree of freedom. Moreover unlike the originalreference
that uses 2 fixed integration points in normal direction, here
thenumber of integration points is arbitrary, as usual in
solid-shell elements.For optimization reasons the integration
scheme leads to a different wayof evaluating the equivalent nodal
force vector and the geometric stiffnessmatrix. With regard to the
volume integral (performed with respect to thereference
configuration) the determinant of the Jacobian of the
isoparametricapproach is evaluated at three points, namely the
centers of the faces and theelement center, and is quadratically
interpolated to the integration points.
8
-
1 2
3 78
9
. .
.
P1P2
P3
(a) (b)
Figure 1: Patch of elements. (a) spatial view. (b) parametric
space view.
3.1. Improvements on the in-plane behavior using the adjacent
elementsTo improve the in-plane interpolation the same technique
used for rotation-
free shell elements ([9]) is applied here. A four-element patch,
involving 12nodes, is defined by the element and its three adjacent
ones (see Figure 1.a).This allows to define an in-plane quadratic
interpolation at both the upperand lower surfaces, that is used to
compute the in-plane deformation gradi-ent and the associated
components of the metric tensor. Here we follow theprocedure used
in ([9]) that averages at the element center the metric
tensorcomponents computed at each mid-side points. For the
solid-shell elementexactly the same computations can be performed
at both upper and lowerfaces (see Figure 1.b with the notation of
the lower face), and a subsequentinterpolation to the integration
points. For the lower face the associatedquadratic shape functions
are simply:
N1 = (z + ) N7 = z2
(z 1)
N2 = ( + z) N8 = 2
( 1)
N3 = ( + z) N9 = 2
( 1)
(26)
Thus, at each face defined by three nodes of the element and
anotherthree from the adjacent elements:
1. A local system (t1, t2) on the shell tangent plane is
computed (t1 andt2 are chosen according to the local system R
defined at the elementcenter), with t3 normal to the face .
9
-
2. At each mid-side point (PK) the in-plane Jacobian (X,X) is
evaluatedand projected over the Cartesian directions (t1, t2)
J =
[X t1 X t1X t2 X t2
](27)
3. Note that at each mid-side point PK only four of the six
derivatives ofthe shape functions defined in (26)
1 2 3 7 8 9N I 1 + 1 z 12 z 12 0N I 1 + z 1 12 z 0 12
are non zero. For instance for mid-side point P1((, ) =
(12, 12
))the
shape function derivatives of nodes 8 and 9 that do not belong
to any ofthe two adjacent triangles are null. The Cartesian
derivatives of thesefour functions are computed as[
N I1N I2
]K= J1K
[N IN I
]K(28)
4. That allows to compute the in-plane deformation gradient (fK1
, fK2 ) andwith it CKij (i, j = 1, 2). These components are
averaged over each faceCf
ij (f = 1, 2 for lower and upper face respectively).5. When and
adjacent element is missing (boundary), as originally pro-
posed for rotation-free shells, the values of the components of
Cij com-puted from the 3-node central triangle are included for the
averaging.
For the prism element nG integration points are used along the
normal direc-tion (). At these points the in-plane components of
the Cauchy-Green tensorare interpolated using (remind that the
modified components are identifiedby an over bar)
Cij () = L1C1ij + L
2C2ij (29)
while their variations (for future use) are
12C1112C22C12
= 12C1111
2C122C112
L1 + 12C2111
2C222C212
L2 = E11E22
2E12
(30)At each face a modified tangent matrix Bf relating the
incremental tensor
10
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components with the incremental displacements u can be written
as
12Cf1112Cf22Cf12
= 13
3K=1
12CK1112CK22CK12
=1
3
3K=1
4J=1
fK1 N
J(K)1
fK2 NJ(K)2(
fK1 NJ(K)2 + f
K2 N
J(K)1
) uJ(K)
=(Bfm)318 u
f (31)
where array uf include only the nodes on each face f (lower or
upper).Then it is possible to write[
Bm]336 =
[L1B1m, L
2B2m]
(32)
Note that each matrix is associated with a different set of
nodes, becausematrix B1m is associated with the nodes on the lower
face and B2m with thenodes on the upper face.
The equivalent nodal force vector stems from the integral
rT1 u =
11
S11S22S12
T [ L1B1m, L2B2m ] Jd u=
11
S11S22S12
T L1JdB1m, 11
S11S22S12
T L2JdB2m u
=
S111S122S112
T B1m, S211S222S212
T B2m u (33)
where
uT =[uT1 u
T2 u
T3 u
T7 u
T8 u
T9 u
T4 u
T5 u
T6 u
T10u
T11u
T12
](34)
3.1.1. Geometric stiffness matrixThe geometric stiffness matrix
results from
uTKgu =
V
u
12C111
2C22C12
T S11S22S12
udV (35)=
nGG=1
V G3
2f=1
Lf3
K=1
4I,J=1
{uI
[N I1N
I2
] [ S11 S12S21 S22
] [NJ1NJ2
]}K11
-
where the sum on G is the numerical integration with nG points
along direc-tion .
It is possible to take advantage of the values S computed above
(33)leading to
uTKgu =
2f=1
3K=1
4I,J=1
{uI
[N I1N
I2
]K [ nGG=1
V G3Lf[S11 S12S21 S22
]] [NJ1NJ2
]KuJ
}
=
2f=1
3
K=1
4I,J=1
{uI
[N I1N
I2
]K [ Sf11 Sf12Sf21 S
f22
] [NJ1NJ2
]KuJ
}f
(36)
where the contributions from each face may be seen independently
one fromthe other. In fact the integrated values Sfij depends on
the strains at bothfaces.
3.2. Transverse shear formulationTo cure transverse shear
locking, following most of the literature, an in-
terpolation in natural coordinates of mixed tensorial components
is used.In Reference [19] a general methodology for
Reissner-Mindlin shell elementsis presented, that is particularized
for the quadratic 6-node triangular ele-ment, leading to a linear
variation of the transverse shear strain tangent tothe side.
Assuming a constant value of the shear strain at each side,
thetechnique was also used for a linear 3-node element ([30]). For
the presentelement the latter case is helpful to write the relevant
(mixed) componentsof the right Cauchy-Green tensor as
[C3C3
]=
[ 1 1
] 2C1t3C23C33
= P (, ) c (37)with
c =
2C1t3C23C33
= 2f1t f13f2 f23
f3 f33
(38)where the most relevant components to transverse shear (C3 ,
C3) has beenwritten with respect to a mixed coordinate system that
includes the in-planenatural coordinates (, ) and the spatial local
coordinate in the transversedirection (y3). These components are
written in terms of the transverse shearstrain tangent to the side
computed at each mid-side point (1
( = = 1
2
),
2( = 0, = 1
2
)and 3
( = 1
2, = 0
), see Figure 2.a). Similarly to hexahe-
dral solid-shell elements with reduced integration (one point in
the shellplane)[4] the sampling points chosen are those located at
lower and upper
12
-
faces ( = 1). As a result six sampling points located at
mid-side of eachtriangular face are necessary as shown in Figure
2.b. Besides that, the nu-merical integration is performed along
the prism axis ( = = 1
3), whereby[
C3C3
]= P
( =
1
3, =
1
3
)c = Pac =
2C1t3 C23 + C33
3
[ 1+1
]+
[C33C23
](39)
1 2
3
ft2
ft3
ft1
P2
P3
P1
f31
ft3
ft1
ft2 f3
1
f32
312
(a) (b)
Figure 2: Points for the computation of the transverse shear
strains.
Replacing (38) into (39) allows to compose
C2 = C3(t t3 + t3 t)+ C3 (t t3 + t3 t) (40)
where the dual base vectors[
t t t3]computed from the local base[
t t t3]
=[X
X
Xy3
]have been used. The modified Cartesian com-
ponents (again denoted by an over bar) are computed projecting
over thelocal Cartesian base
C13 = t1 [C3
(t t3 + t3 t)+ C3 (t t3 + t3 t)] t3
= C3
(a1a
33 + a
31a3
)+ C3
(a1a
33 + a
31a3
)(41)
where the symbol aji = ti tj (with i = 1, 2, 3 and j = , , 3)
has beenintroduced. Noting that aji =
ji it simplifies to
C13 = C3a1 + C3a
1 (42)
Proceeding similarly for component C23 we can write in a single
expression[C13C23
]=
[a1 a
1
a2 a2
] [C3C3
]= J1p
[C3C3
](43)
13
-
where J1p is the inverse of the in-plane Jacobian of the
isoparametric map-ping. Note that due to the way in which the local
system has been definedthe components in (43) are null in the
reference configuration.
At the sampling points the necessary deformation gradient
componentsare ft (natural coordinate derivative) and f3 (local
Cartesian coordinate deriva-tive). At sampling points ft are for
the lower (1) and upper (2) faces respec-tively 2f1tf2
f3
1 = x3 x2x1 x3
x2 x1
2f1tf2f3
2 = x6 x5x4 x6
x5 x4
(44)while f3 can be expressed as
f3 =6I=1
N I3xI =
[f f f
] y3y3y3
= (x) jT3 (45)where jT3 are the components in direction y3 of
the inverse of the Jacobianof the isoparametric mapping (third
column j13 ). Finally the modified trans-verse shear Cartesian
components emerge from replacing equations (38) into(39) and these
into (43)
[C13C23
]= J1p Pa
f1t f13f2 f23f3 f33
(46)while the right Cauchy-Green tensor components at
integration points areobtained interpolating the values at each
face[
C13C23
]() =
[C13C23
]1L1 +
[C13C23
]2L2 (47)
Note that the Green-Lagrange strain components associated to the
trans-verse shear at each face are directly the interpolated
values[
2E132E23
]() =
[C13C23
]()
The tangent matrix Bs, relating displacement increments with
strain in-crements, is also obtained by interpolating from both
faces
Bs () = B1sL
1 + B2sL2 (48)
14
-
that requires first computing at the sampling points (for each
face)
Bsue =
f1t f13 + f1t f13f2 f23 f2 f23f3 f33 + f3 f33
(49)where (with xe = ue)
2f1tf2f3
1 = x3 x2x1 x3x2 x1
2f1tf2
f3
2 = x6 x5x4 x6x5 x4
(50)and
[
f13 f23 f
33
]=
[u1 u2 u3 u4 u5 u6
]
N1(1)3 N
1(2)3 N
1(3)3
N2(1)3 N
2(2)3 N
2(3)3
N3(1)3 N
3(2)3 N
3(3)3
N4(1)3 N
4(2)3 N
4(3)3
N5(1)3 N
5(2)3 N
5(3)3
N6(1)3 N
6(2)3 N
6(3)3
(51)
then interpolate to the element axis as in (39) and finally
convert to theCartesian system
Bfs =(Jfp)1
PaBfs (52)
The associated nodal equivalent forces are computed as:
rT2 =
V
[S13S23
]T [Bs]218 dV
=
11
QT[B1sL
1 + B2sL2]218 Jd
=
11
QTL1JdB1s +
11
QTL2JdB2s
= QT41
[B1sB2s
]418
(53)
where the generalized shear forces Q at each Gauss point are
defined as
Q41 = 11
[S13S23
]L1[
S13S23
]L2
Jd (54)
15
-
3.2.1. Geometric stiffness matrix using QThe above expressions
allow to advance in obtaining the geometric stiff-
ness matrix
uTKsGu =
{[B1sB2s
]u
}TQ (55)
where at each face we have
Bfs u =J1p
3
[2f1t f13
2f1t f13 + f2 f23 + f2 f23 + 2f3 f33 + 2f3 f33
2f1t f13 +
2f1t f13 + 2f2 f23 + 2f2 f23 + f3 f33 + f3 f33
]
= J1p
{1
3
[ B1s B2s + 2B3sB1s 2B2s + B3s
]218
}u (56)
considering for instance increments of strain components in the
lower face[Q1, Q2
](B1su
)results
[Q1, Q2
] J1p3
[ 2f1t f13 2f1t f13 + f2 f23 + f2 f23 + 2f3 f33 + 2f3 f332f1t
f13 +
2f1t f13 + 2f2 f23 + 2f2 f23 + f3 f33 + f3 f33
]denoting by [
Q1, Q2
]=[Q1, Q2
]J1p (57)
we have
[Q1, Q2
](B1su
)=
1
3
(Q1 + Q2) (2f1t f13 +2f1t f13 )+(Q1 2Q2) (f2 f23 f2 f23 )+(2Q1 +
Q
2
) (f3 f33 + f3 f33
)(58)
In the numerical implementation the derivatives NJ(K)3 related
to f3 arekept in an array but not those related to ft because its
values are: 0, 1, and-1 at each side.
3.3. Improvement of the transverse behaviorTo avoid locking due
to the Poisson effect when bending is important and
also to help in alleviating the volumetric locking (in
quasi-incompressibleproblems) an enhanced assumed strain
formulation for the component C3 isused.
3.3.1. Enhanced assumed strain (EAS) techniqueThe standard EAS
method interpolates natural (convective) strain com-
ponents. In this case we intend to improve directly the
Cartesian componentC33 so a slightly different approach will be
used as explained next.
16
-
At the element center( = = 1
3, = 0
)the Cartesian deformation gra-
dient component along y3 can be computed from the isoparametric
interpo-lation
fC3 =6I=1
N IC3 xI (59)
The enhanced gradient in direction y3 is defined as
f3 = fC3 e
(60)
thus the interesting component of the right Cauchy-Green tensor
results in
C33 = fC3 fC3 e2 = CC33e2 (61)
Note that for the standard element the normal strain in
transverse direc-tions results in a first approximation
3 =C33 =
f3 f3 (62)
e33 = ln (3) =1
2ln (C33) (63)
that is practically constant in y3 direction for the linear
element, whereas ifwe now use the enhanced version
e33 =1
2ln(C33)
=1
2ln(CC33e
2)
=1
2ln(CC33)
+
= eC33 + (64)
In this EAS approach the changes that will be produced by the
enhancedf3 on the other components C13 and C23 are disregarded, as
they are computedas explained in previous sections. Thus here only
the influence on componentC33 is considered.
3.3.2. Balance equation and implicit solutionThe variation of
the Green strain now involves the internal DOF and
can be written
E33 =1
2C33 = f
C3 fC3 e2 + C33
=
(6I=1
N IC3 uI
) fC3 e2 + C33
= e2BC3 ue + C33 (65)
17
-
In the last expression the first term replaces the corresponding
part of thestandard displacement approach (the difference is the
factor e2 on the com-ponents associated to E33).
The balance equation (17) associated to variable is
11S33C33Jd = 0 (66)
thus must be solved iteratively. Denoting the residue by
11S33C33Jd = r (67)
the Newton-Raphson technique allows to approximate the value of
nullify-ing the residue (67)
11
[S33C33u
u +S33C33
]Jd + r = 0 (68)
with
S33C33u
= C33D3B + 2S33e2BC3 = C33D3B + 2S33B3 (69)
S33C33
= C33D33C33 + 2S33C33 = C33(D33C33 + 2S33
)(70)
where D is the tangent constitutive matrix (Voigts notation ),
D3 is itsthird row (1 6) and D33 its diagonal component. B is the
matrix relatingincremental Green strains with incremental
displacement when the assumedstrain approximations are used for
tensor C and B3 is the first term of (65).
When implicit techniques based on Newton-Raphson method are used
toobtain the equilibrium path, the DOF is locally condensed at
element levelfrom equation (68) 11
[(C33D3B + 2S33B3
)u + C33
(D33C33 + 2S33
)]Jd + r = 0
(71)H118u + k + r = 0
then = r
k 1k
Hu (72)
18
-
that is replaced in the corresponding balance equation
associated to displace-ment variation (16)
uTV
BTSdV uTGext = uT r (u, ) (73)
once linearized 11
[BT(S
uu +
S
)+
(BT
uu +
BT
)S
]Jd +r (u) = 0
11
[BT(DBu + DT3 C33
)+ SGu + 2S33B
T3
]Jd +r (u) = 0
11
[(BTDB + SG
)u +
(BTDT3 C33 + 2S33B
T3
)
]Jd +r (u) = 0(
KTu + HT
)+r (u) = 0
KTuHT(rk
+1
kHu
)+r (u) = 0(
KT HT 1k
H
)uHT r
k+r (u) = 0
(74)
The contributions of the last expression to the Newton-Raphson
schemeare the modified elemental stiffness matrix and equivalent
nodal force vector:
KT = KT HT 1k
H (75)
r = rHT rk
(76)
Note that above
k =
11C33
(D33C33 + 2S33
)2Jd (77)
KT = KM + KG =
11
BTDBJd +
11
SGJd (78)
H =
11
(C33D3B + 2S33B3
)Jd (79)
19
-
3.3.3. Geometric Stiffness MatrixThe contribution of the normal
transverse component S33 to equilibrium
is
rT3 =
V
B3S33dV
=
11e2BC3 S33Jd
= BC3 S33 (80)
where
S33 =
11e2S33Jd (81)
so the contribution to the geometric stiffness matrix is:
uTKG3u = fC3 fC3 S33
=6I=1
(uI)T 6
J=1
N IC3 NJC3 S33
1 11
uJ (82)3.3.4. Explicit solution
A straightforward way to address the problem when using explicit
inte-gration of the momentum equations is to use the condition
(72). The possiblesteps are:
1. Modify the equivalent nodal forces
r = r + HTrk
(83)
and compute a first update of the EAS parameter
n+1 = n rk
(84)
2. Use the standard central difference scheme to compute the
incrementaldisplacements u
3. Then compute a second update of the internal DOF
n+1 = n+1 Huk
(85)
Such a scheme requires (besides keeping that is
indispensable):
20
-
Keep k between steps, although it can be recomputed at the
beginningof the step the increment in the elemental data base is
very low so it isnot worthwhile
Store row vector H to avoid its re-computation, this implies an
impor-tant increase in the elemental data base
To know D3 (elastic-plastic relating increment in S33 (second
Piola-Kirchhoff) with increments in Green-Lagrange strains), that
imply anotably increase of floating point operations at each
integration point.This is because D3 is not available in explicit
integrators (the sameproblem appears for the stabilization of
under-integrated elements)
To avoid the increase of the elemental data base [23] suggest
(in implicitcodes) for models with a large number of internal DOFs
(12) not to store thecorresponding submatrices (that here have been
reduced to vector H and thescalar k) and to use an iterative scheme
to update the internal DOFs in theroutine that compute stresses and
equivalent nodal forces. These authors saythat with a few
iterations (they suggest using just 2) accurate enough resultsare
obtained.
On the one hand, if the iterative process is reasonable accurate
in a finiteelement code with implicit integration of the momentum
equations, it will beeven more accurate if the code is explicit. On
the other hand each iterationin the computation of the stresses is
proportionally very costly in an explicitcode, so it will be
assumed that just one iteration is enough to obtain
accurateresults. The numerical experiments on elastic-plastic
models indicate thatthis assumption is correct even using the
elastic component D33 (the iterativeprocess just requires the
diagonal component not the entire row D3).
4. Numerical examples
In the set of examples shown below we denote by SPr the
solid-shell ele-ment described above. A suffix Q indicates that the
improvement in the mem-brane behavior has been activated. The
original solid element on which thesolid-shell element is based is
denoted by Prism (developed in [7]). With com-parative purposes
results obtained with other elements are includes: Q1SPsis a
reduced integration solid-shell hexahedral element with an
excellent be-havior [20, 21, 22] while LBST and BBST are
rotation-free thin shell triangularelements, the former [8] uses
the standard constant strain triangle for themembrane part, while
the later includes an assumed strain approach for themembrane part
[10] almost identical to the formulation presented above.These
rotation-free shell elements compute bending strains resorting to
the
21
-
3X
Y1
2
4
Coordinates [mm]1: (0.04, 0.02)2: (0.18, 0.03)3: (0.16, 0.08)4:
(0.08, 0.08)
Figure 3: Patch test
geometrical configuration of a four-element patch, the element
and the threeadjacent ones, leading to a non-conforming approach
(see [18] for a compre-hensive treatment).
4.1. Patch testThe patch test is understood as a necessary
condition for the convergence
of the element. In the case of solid elements it is expected
that when nodaldisplacements corresponding to a constant strain
gradient (membrane patchtest) are imposed, constant efforts are
obtained in all the elements. In thecase of standard shell elements
displacements and rotations are imposed atnodes corresponding to a
constant curvature tensor (bending patch test) anda constant moment
tensor is expected to in all elements. In the case of asolid-shell
element clearly this must satisfy at least the membrane patch
testand, although it may not be necessary, it is highly desirable
that the elementsatisfies the bending patch test as this will lead
to a more robust and reliableelement.
The Figure 3 shows a patch of elements that has been widely used
toaccess quadrilateral shell elements and hexahedral solid shell
elements. Hereeach hexahedral has been replaced by two prismatic
elements. The size of thelargest sides is es a = 0.24mm and the
size of the shortest side is b = 0.12mm,while the thickness
considered is t = 0.001mm. The lower surface has beenlocated at
coordinate z = t/2. The mechanical properties of the materialare:
Youngs modulus E = 106MPa and Poisson ratio = 0.25. Because
theproblem considered is linear just 2 integration points across
the thickness areused located in the usual Gauss quadrature
positions ( = 1/3).
4.1.1. Membrane patch testThe prescribed nodal displacements (on
the boundary nodes) are defined
by the linear functions
ux = (x+y
2) 103 uy = (y + x
2) 103 (86)
22
-
and uz = 0 only on the nodes in the lower face to allow
contraction due toPoisson effect. Using present element SPr the
correct results are obtainedfor both the displacements of the
interior nodes according to (86) and theelement stresses (xx = yy =
1333, 3 MPa y xy = 400 Mpa). The internalDOF is null at all the
elements. The same results are obtained with theversion SPrQ as the
deformation gradient is constant.
4.1.2. Bending patch testIn this case the displacement field
associated with a constant bending
stress state is given by (103):
ux =(x+
y
2
) z2
uy =(y +
x
2
) z2
uz =(x2 + xy + y2
) 12
(87)
that is prescribed on the exterior nodes of both shell faces.
Again the resultsobtained with both element versions are correct.
The bending stresses at theintegration points are xx = yy = 0.3849
MPa y xy = 0.1155 MPa whilethe displacements at the interior nodes
correspond exactly with expression(87). The internal DOF results =
0.3333 108. Thus the element satisfythe bending patch test
also.
4.2. Cooks membrane problemThis example (see Figure 4) involves
a large amount of shear energy
and is commonly used to assess in-plane bending performance.
Plane straincondition will be considered here with two different
material behavior: a) aquasi-incompressible elastic material with G
= 80.1938GPa and K = 40.1104GPa corresponding with a Poisson ratio
= 0.4999 and b) an elastic-plastic material with elastic properties
G = 80.1938GPa andK = 164.21GPaimplying a Poisson ratio = 0.29 and
J2 plasticity with isotropic hardeningas a function of the
effective plastic strain ep defined by
y = 0.45 + 0.12924ep + (0.715 0.45)(1 e16.93ep) [GPa].
The applied load is 100kN for the elastic case and 5kN for the
elastic-plasticmaterial.
The plane strain condition implies coefficient C33 = 1 at all
points ( =0), thus the version without ANS for the in-plane
components locks due tothe almost incompressibility constraint in
the same way that a constant straintriangle does. Because of that
this example is intended to assess how the im-provement in the
membrane field collaborates to cure the volumetric locking.The
Figure 5 shows a convergence analysis as the mesh is refined. The
verti-cal displacement of point C has been plotted versus the
number of divisions
23
-
F48
4416
C
Figure 4: Cooks membrane problem. Geometry, units in mm.
per side. For comparison results obtained with three
two-dimensional solidelements have been included, a linear triangle
with a F formulation ([26])that averages the volumetric component
over two adjacent elements (onlyresults for the elastic case are
available) and two 4-node quadrilaterals: Q1P0entirely formulated
in displacement with 22 integration for the shear com-ponents and
the volumetric strain averaged over the element and Q1EA
([13])based on the EAS technique including four internal degrees of
freedom. Inthe Figure 5.a the plots identified by 0.4999 and 0.499
indicates the Poissonratio used to obtained those results with
element SPrQ. The first value is theproposed Poisson ratio for the
benchmark and the second a value slightlylower that allows to
assess the element sensitivity to the incompressibilityconstraint.
On one hand it can be seen that for the proposed Poisson ratioboth
quadrilaterals show a clearly better behavior and that a large
meshdensity is required to reach convergence with present element.
On the otherhand the curve = 0.499 shows that in that case
convergence is good enoughand similar to the triangle by [26]. For
the elastic-plastic model (Figure 5.b)where although plastic flow
is isochoric the elastic behavior is compressible,the results
plotted indicate that present element has a better
convergenceproperties than both quadrilaterals..
4.3. Cantilever beam with a point loadThis problem has been
analyzed by a numerous of authors (see for exam-
ple [25, 12, 21]). A cantilever plate strip of length L = 10mm
width B = 1mmand thickness t = 0.1mm is subjected to a transverse
load F = 40N. For the
24
-
Elements per side
Vert.
Dis
pl.C
[mm
]
0 5 10 15 20 25 30 352
3
4
5
6
7
8
SouzaNetoQ1POQ1E40.49990.499
Elements per sideVe
rt.D
ispl
.C
[mm
]0 5 10 15 20 25 30 352
3
4
5
6
7
8
Q1POQ1E4SPrQ
(a) (b)
Figure 5: Cooks membrane problem. a) elastic
quasi-incompressible. b) J2 plasticity
selected Youngs modulus E = 106MPa the behavior is one with
large dis-placements but small strains. Using different values of
Poisson ratio ( = 0.0, = 0.3 , = 0.4999) it can be assessed if the
proposed assumed strain tech-niques allow to avoid respectively the
transverse shear locking, the Poissoneffect locking and the
volumetric locking.
The final deformed configurations (vertical displacement is 70%
of thelength) is achieved in ten equal load steps. The
discretization includes 16divisions in length, one in the width and
one across the thickness with twointegration points. The Figure 6.a
shows the vertical displacement of the tipversus the load factor
[0:1] for 5 different values of the Poisson ratio. Thecase = 0
allows to compare with the reference value (uz = 7.08 mm) andto see
if the approach used to cure transverse shear locking is adequate.
Theresult obtained uz = 7.06 mm indicates that effectively the
element is freeof transverse shear locking. The second value of
Poisson ratio ( = 0.30)is used to assess if the EAS technique
avoids the appearance of locking dueto Poissons effect. In this
case the computed displacement is uz = 7.01mmthat although is not
exactly the same value obtained for = 0 shows thatthe proposed
method avoids the Poissons effect locking allowing a
propergradation of the transverse normal strain. Finally the last
three values ofPoisson ratio (0.49, 0.499 y 0.4999) allow to
observe if the performance of theelement deteriorates significantly
in the quasi-incompressible range. It can beseen that although
differences grow with Poisson ratio, this are below 4% forthe
higher value considered. Besides, Figure 6.b plots the tip
displacement as
25
-
Tip Displacement [mm]
Loa
dFa
cto
r
0 2 4 6 80
0.2
0.4
0.6
0.8
1
0.00.30.490.4990.4999
Number of elementsTi
pdi
spla
cem
en
t[mm
]0 8 16 24 326
6.25
6.5
6.75
7
0.00.30.4990.4999BBST 0.0
(a) (b)
Figure 6: Bending of a cantilever plate strip
a function of the mesh density (number of divisions along the
length) for fourdifferent Poissons ratio. Results obtained with
element BBST and = 0, thatfor this example converges quite rapidly,
are also plotted for comparison. Forthe present element it can be
seen that convergence deteriorates for Poissonsratio larger than
0.499.
4.4. Cylindrical roofThis third example is the linear analysis
of a cylindrical shell under self
weight. The roof is free along the straight sides and supported
by rigiddiaphragms at the curved sides. Using symmetry
considerations just one-quarter of the roof is modeled. Five
structured meshes were used to assessconvergence with the same
number of elements along each side. Of the twopossible mesh
orientations, the one shown in Figure 7.a was considered. Asthe
problem is linear just one element with two integration points is
usedacross the thickness.
As this is a membrane dominated problem, the membrane
approxima-tion is crucial for a fast convergence, then this example
allows to assess theimportance of the ANS for in-plane components
in non-isochoric problems.The results for the vertical displacement
at the mid-side point of the free side(reference value is wA =
3.610.) are plotted in Figure 7.b as a function ofthe number of
elements per side. The figure includes two curves correspond-ing to
both formulations of the membrane part of the solid-shell
element.Also the results obtained with shell elements LBST and
BBST, which main
26
-
XY
Z
diaphragm
300
symm
etry
A
symmetry
free
300=40
Elements per side
Dis
pla
cem
en
tofA
0 5 10 15 20 25 30 350
0.5
1
1.5
2
2.5
3
3.5
4
LBSTBBSTSPrSPrQ
(a) (b)
Figure 7: Scordelis cylindrical roof. (a) geometry (b)
displacement of point A
difference is the membrane approach, are included for comparison
as theirbehavior should be similar to the solid-shell element
presented here. The re-sults of present element SPr are almost
identical to those obtained with thethin shell element LBST except
for the coarse mesh where the shell element ismore flexible because
its bending approach is non-conforming. When usingthe version with
improved membrane behavior results quickly converge tothe reference
value in a manner similar to the shell element with
improvedmembrane behavior BBST .
4.5. Semi-spherical shell with a 18o holeThe pinched hemisphere
is considered in order to introduce initially dou-
ble curved geometry. This is an extensively analyzed shell
problem in thecontext of large elastic displacements. The Figure
8.a shows the geometryconsidered once symmetry conditions are
introduced and the loads applied.This is mainly an inextensional
bending problem where the Poisson effect isimportant and the
membrane behavior is not. Because of the double curva-ture the
curvature-thickness locking and the membrane locking may appear.Two
meshes have been considered that include 16 and 24 elements per
side.The coarse one is usually used to determine if the element
suffers any lockingdue to the initial curvature, i.e. if the
results differs by more than 5% of thetarget values some degree of
locking exists. The middle surface radius is esR = 10mm and the
thickness is t = 0.04mm (R/t = 250). The mechanicalproperties
adopted are E = 6.825 104GPa and = 0.3.
27
-
XY
Z
(a) (b)
Figure 8: Semi-spherical shell with a hole. Original and
deformed geometry.
The Figure 8.b shows the deformed configuration for an inward
displace-ment of the loaded point equal to 60% of the shell radius.
Whilst the Figure9 plots the displacement (absolute values) of the
loaded points where thelargest displacement corresponds to the
inward load. The results presentedin Reference [24] using a shear
deformable shell element (mesh 16 16) andconverged results obtained
with element BBST using a 32 32 mesh are alsoincluded. Results
obtained with both formulations of present elements withboth meshes
are plotted. Those obtained with the coarsest mesh (16 ele-ments
per side) are 8% lower than the target values, in contrast with
thosepresented in Reference [21] where an excellent approximation
is obtainedwith the same mesh using element Q1SPs. The results
corresponding to thefiner mesh (24 elements per side) are in good
agreement with the convergedresults.
4.6. Slit annular plateAn annular plate with a radial cut is
clamped at one of the slit edges
and subjected to a transverse load in the other one. The Figure
10.a showsthe original geometry and indicates the size and
mechanical properties of thematerial. The Figure 10.b shows the
deformed configuration for the maxi-mum load factor considered.
This is a popular benchmark to assess shellelements under large
rotations. Initially proposed by [2] it has been consid-ered by
many authors and in Reference [27] converged results are
presentedin tabular form that have been used here for comparison.
The latter re-sults have been obtained using the element SR4
present in commercial codeAbaqus[1] employing a mesh with 10 80
elements.
Two discretizations have been considered here, the first with 5
36 di-visions and the second with 10 72 divisions. The vertical
displacements
28
-
Load
Dis
pla
cem
en
ts
0 20 40 60 80 1000
1
2
3
4
5
6
BBSTSIMOSPrQ_16SPrQ_24SPR_16SPR_24
(a) (b)
Figure 9: Semi-spherical shell with a hole. Displacements of the
loaded points.
X Y
Z
A
B
Ri
Re
Re=10
=0
Pmax=0.8
Ri=6
E=21x106
thick. = 0.03
(a) (b)
Figure 10: Slit Annular plate. (a) Initial geometry. (b)
deformed geometry
29
-
Displ. points A & B
P/P m
ax
0 5 10 150
0.2
0.4
0.6
0.8
1
SzeSPr-5x36SPrQ-5x36SPrQ-10x72
Figure 11: Slit annular plate. Displacements of the free
slit.
of two points on the loaded edge, denoted in Figure 10.b as A
and B areused for comparison. The Figure 11 plots the evolution of
displacementsmentioned above in terms of load factor. It includes
converged results fromReference [27], results for both meshes
considered when using SPrQ versionand results for the coarsest mesh
for element SPr. A comparison of the resultsfor the coarse mesh
allows to observe the influence of the ANS for membranepart that
for the maximum load factor indicate a difference in
displacementslarger than 4%. Comparing the results obtained with
the SPrQ version forboth meshes with the converged ones it can be
seen that for low values of theload factor the three curves almost
coincide but for higher load factor thedisplacements with the
coarse mesh separate from the reference values untilalmost a 4% for
the maximum load factor. The results for the fine mesh arein
excellent agreement with those provided in [27]
4.7. Hinged cylindrical panel under point loadThis example
considers a rectangular cylindrical panel simple supported
along the straight sides and free along the curved sides, that
is subjectedto a vertical point load in its center (see Figure 12).
The middle surfacegeometry is defined by the length of the panel L
= 508mm, the radius of thecylinder R = 2540mm and the half angle =
0.1rad. The behavior of thepanel presents a limit point, followed
by a strong loss of strength and a finalstiffening once the
curvature is inverted. Two different thicknesses for thesame
mid-surface geometry have been considered t = 12.7 and t = 6.35
that
30
-
LR
Simple
supp
orted
FreeY
X
Z
Figure 12: Cylindrical Panel under point load.
for the thin case leads to a snap back of the loaded point. This
example hasbeen widely used to assess the performance of shell
elements and non-linearpath-following techniques.
For this problem three meshes have been considered with 4, 6 and
8 ele-ments per side. In this case 2 elements in the thickness
direction have beenused that allows to introduce the hinge in the
middle surface and then tocompare with solutions obtained with
shell elements. The vertical displace-ment of the loaded point A
(for both thicknesses) and the mid point of thefree side B (only
for the thin case) are used to study convergence and forcomparison
with other results. Two sets of converged results are
included,those obtained from reference [27] using a shear
deformable shell element(meshes with 16 16 and 2424 elements for
the thick and thin case respec-tively) and our own results obtained
with the thin shell element BBST usinga mesh with 16 16 elements.
Also to compare with another solid-shell ele-ment results obtained
for the coarsest mesh (4 4 2) with element Q1STs[21] are plotted.
For the thick case (Figure 13.a) there are some discrepan-cies when
comparing the displacements obtained using present element
withshell elements but they are almost identical for the three
meshes considered.Also they are very similar to those obtained with
the hexahedral solid-shellelement on the postcritical path. For the
thin case (Figure 13.b) the resultsobtained with the three meshes
seem to converge to those obtained with shellelements. The coarsest
mesh is clearly inadequate to obtain reliable resultsfor the entire
path, particularly for the descending post-critical path wherethe
snap back of the loaded point occurs. This can also be seem in the
resultsobtained with element Q1STs [21].
4.8. Square thin film under in-plane shearThis example has been
previously analyzed in [28] using the commercial
code Abaqus[1]. Experimental data is also available [15]. The
problem con-
31
-
Center Displacement
Loa
d
0 5 10 15 20 25 300
500
1000
1500
2000
2500
30004el/s6el/s8el/sSzeQ1STsBBST
Point B
Point A
Displacement
Loa
d
0 5 10 15 20 25 30-600
-400
-200
0
200
400
600
800
4el6el8elSzeQ1STsBBST
(a) (b)
Figure 13: Cylindrical panel. a) t = 12.7 b)t = 6.35.
a
a
=1
Figure 14: Square thin film under in-plane shear
sists of a square membrane (see Figure 14) with side a = 229mm
made ofa thin film of Mylar with thickness t = 0.0762mm. The Mylar
mechanicalproperties are E = 3790MPa and = 0.38. The top and bottom
edgesare clamped and the lateral edges are free. The top edge is
subjected to auniform horizontal displacement = 1mm along the edge.
The Figure 14also show a perspective view of the deformed membrane
(scaled 5X in thetransverse direction).
Three uniform structured mesh with 2626, 5151 and 101101
nodes,with 1250, 5000 and 20000 elements respectively have been
considered. Inthe sequel these meshes will be referenced as mesh
25, 50 and 100, associatedto the number of subdivision along each
side
Figure 15 plots two out-of-plane displacement profiles along the
center ofthe square in both Cartesian directions. The plot on the
left corresponds to
32
-
Coord. x
Dis
pl.z
0 50 100 150 200
-1
0
1
2
3
2550100BBST
Coord. y
Dis
pl.z
0 50 100 150 200
-1
0
1
2550100BBST
(a) (b)
Figure 15: Square membrane under in-plane shear. Transverse
displacement profiles alongthe center of the square. a) y = a/2. b)
x = a/2.
y = 114.5 mm and the plot on the right to x = 114.5 mm. The
results for thethree meshed defined above are included. These
deformed configurations aresimilar to the experimental evidence[15]
and also coincide with the numericalresults presented in [28]
obtained with program Abaqus [1] using the quadri-lateral shell
element S4R5. In that work mesh 100 was used and reported asthe
minimum acceptable mesh according to their convergence studies.
Herethe profiles for mesh 100 obtained with shell element BBST [11]
have alsobeen included. For present element (SPrQ) the mesh 25 does
not capturethe correct number of waves, but meshes 50 and 100 are
almost coincidenton the wrinkled zones. It can be seen that the
results are coincident withthose obtained with shell element BBST
but on the free boundaries where themembrane is in slack state.
Note also that for mesh 50, that leads to verygood results, the
element aspect ratio is x/t = 60.
4.9. Elastic-plastic conical shellThis a second example that
allows to study the performance of present
element under large displacements and large elastic-plastic
strains. The ge-ometrical details are shown in Figure 16. The
elastic mechanical propertiesare E = 206.9MPa, = 0.29 whilst the
plastic behavior obeys von Misesyield function with isotropic
hardening ruled by function
y (ep) = 0.45 + 0.12924ep + 0.265
(1 e16.93ep) [MPa].
The reference load is P = 0.01N/mm that is scaled by the load
factor .
33
-
PH
R2
R1
t
R1= 1mm
R2= 2mm
H = 1mm
t = 0.1mm
Figure 16: Conical shell geometry
Due to the axisymmetric nature of the problem only a sector of =
0.1rad is included in the discretization applying adequate
kinematic constraints.Four uniform discretization along the
meridian direction has been consideredwith 4, 8, 16 and 32
divisions. In the hoop direction only one division isused and the
number of integration points across the thickness has been setto 5.
The Figure 17 plots the load factor versus the vertical
displacement ofthe loaded line. The converged results obtained by
[21] that discretize onequarter of the geometry using 32 32
elements have been included. Theresults obtained with element SPrQ
for the four meshes are plotted but onlythose for the coarsest mesh
using element SPr, again to assess the influence ofthe membrane
improvement. From the plot it can be said that the mesh with8
elements along the meridian gives results that are practically
converged.Element SPr is clearly stiffer than its counterpart
SPrQ.
4.10. Deep drawing and elastic springbackThis is one of the
Numisheet93 benchmarks ([16]) where a U-shaped
(one axis bending) deep drawing is performed and the elastic
springback ismeasured. The Figure 18 shows the forming tools
geometrical details. Theblank is made of an aluminum alloy with
elastic properties E = 71GPa and = 0.33. The plastic behavior is
defined by the Lankford ratios r0 = 0.71,r45 = 0.58 and r90 = 0.7
with isotropic hardening given by
y (ep) = 576.79 (0.01658 + ep)0.3593 .
For the friction between blank and forming tools a coefficient =
0.162 hasbeen adopted while the blank folder force is 2.45 kN. The
punch is first moved70mm downwards and all the tools are then
removed to allow the springback.
34
-
Displacement of the loaded line
Loa
dfa
cto
r
0 0.25 0.5 0.75 1 1.25 1.5 1.75 20
1
2
3
4
5
SPr-4SPrQ-4SPrQ-8SPrQ-16SPrQ-32Q1STs
Figure 17: Conical shell convergence study
F/2F/2
5055
R5
R5 R56
BLANK HOLDER
Blank size: 350
Z
X
STR
OKE
70 1 DIE
PUNCH
55
52
6
Figure 18: Deep drawing of a strip
35
-
AB
15
C
35
2
1
Figure 19: Elastic springback
This example assess the behavior of the element under moderate
to largeplastic strains (less than 20%) and the bending behavior
after such defor-mation. From symmetry consideration just one
quarter of the problem hasbeen discretized using 10 uniform
divisions in the half-width and 75 divi-sions in the half-length
with mesh size of 5mm in the zones that are neverin contact with
the the tools shoulder and mesh size of 1.5mm on the zonesthat are
plastically bent. The mesh size of 1.5 mm is the maximum that canbe
used to correctly capture the contact due to the low radius of the
tools(5mm), as a higher mesh size leads on the one hand to
erroneous contactforces and on the other to a wrong prediction of
the springback as discussedin [3]. The Figure 19 shows a profile of
the blank after the springback stage.The measured parameters used
for comparison with experimental values arealso shown in this
figure. Using 5 integration point the following values havebeen
obtained (the values in brackets are the reported experimental
range) = 87mm [81 99], 1 = 111o[110 116] and 2 = 69o[68 76]. It can
beseen that all the parameters computed are within the experimental
values.
4.11. Deep drawing of a square sheetThe last example considered
is the deep drawing of a thin sheet cor-
responding to other of the benchmarks proposed in
NUMISHEET93[16].The Figure 20 shows the geometry of the tools. The
undeformed sheet issquare with side length 150 mm. The elastic
mechanical properties of themild steel considered are: elastic
modulus E = 206GPa and Poisson ratio
36
-
Punch
Stro
keM
in.50
BlankHolder
35 50437
482
XR5
Z
R10
X
Y
R12
8535 2
352
85
Punch
Figure 20: Geometry of the tools (dimensions in mm) for the
Numisheet 93 benchmark
= 0.3. For the plastic behavior the classical Hills yield
function withconstant coefficients F = 0.283 , G = 0.358 , H =
0.642 , L = 1.065 ,M = 1.179 , N = 1.289 was assumed. These
coefficients were computedfrom the Lankford ratios R0 = 1.79, R90 =
2.27 and R45 = 1.51. Isotropichardening is defined by the yield
stress along rolling direction (X direction)0 = 567.3 (0.00713 +
ep)
0.264 .The symmetry conditions shown in the figure have been
considered, then
just one quarter of the geometry has been included in the model.
The dis-cretization of the sheet includes 30 elements on each side
(1800 elements inthe plane) and 7 integration points in the
thickness direction. The blankholder force used is 19,6 kN and the
adopted friction coefficient is = 0.144.The simulation considered a
punch stroke of 40 mm.
The Figure 21 plots the punch force versus the punch travel.
This figureincludes the results obtained with the 3-node triangular
shell element BBSTwith 7 through the thickness integration points
and with the solid elementPrism with four layers, in both cases
with the same in-plane mesh, for com-parison. Results of element
version SPr are not reported as in constrainedproblem of this type
it suffers from a severe volumetric locking due to theisochoric
plastic flow. It can be seen that the differences between the
differentmodels are very small.
The Figure 22 shows the contour fills of the effective plastic
strain for thepresent solid-shell model (center), and those
obtained with the solid (Prism)and shell (BBST) models used for
comparison. For the shell model the zonewith the largest plastic
strain is less widespread and at the point with thelargest
thickness increase (mid-side points) the equivalent plastic strain
islower. The differences between the solid model and the
solid-shell model are
37
-
Punch Travel [mm]
Pun
chFo
rce
[kN]
0 10 20 30 400
20
40
60
80
SPrQNBSTPrism
Figure 21: Punch force versus punch travel.
quite small.
5. Conclusions
In this paper we have developed a triangular prism solid-shell
elementsuitable for nonlinear analysis with elastic-plastic large
strains. In the for-mulation assumed strain techniques have been
used to prevent transverseshear locking and to improve the membrane
behavior. To avoid the lock-ing due to the Poisson effect an
enhanced assumed strain method with justone internal degree of
freedom has been proposed. The volumetric locking is
Prism-4 layers
XY
Z
0.80.70.60.50.40.30.20.10
SPrQ BBST
Figure 22: Equivalent plastic strain for the final punch
travel
38
-
alleviated as the combination of the effects of the membrane and
the trans-verse approaches. The formulation is simple and can
effectively achieve theobjectives. The main conclusions are:
Transverse shear locking disappears completely in all cases
analyzed
The improvement in the membrane field is not only important in
mem-brane dominated problems (e.g. deep drawing simulations), it
resultscrucial to cure the volumetric locking.
The EAS approach for the transverse normal strain effectively
avoidsthe locking due the Poisson effect in bending and
collaborates in themitigation of the volumetric locking in
quasi-incompressible problems.
For explicit time integration, one local (elemental) iteration
for theupdate of the internal degree of freedom seems to be enough
to obtainreliable results. Even for elastic-plastic problems the
use of the elasticmechanical parameters leads to the correct
results.
For double curvature surfaced, the element converges to the
correct re-sults but the not with the same speed as reduced
integration hexahedralsolid-shell elements.
The element did not show any problem in large elastic-plastic
straincases.
Acknowledgments
The author acknowledges the financial support from CONICET
(Ar-gentina) and SeCyT-UNC.
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