FINITE ELEMENT FOR SANDWICH PANELS BASED ON ANALYTICAL
SOLUTION OF CONSTITUTIVE EQUATIONS
M. Linke and H.-G. Reimerdes
Department of Aerospace and Lightweight Structures
RWTH Aachen University
Wüllnerstraße 7, 52062 Aachen, Germany
Introduction
Sandwich structures with low strength honeycomb or foam cores are being used due to their high bending stiffness to weight
ratio in many structural applications. Because of their high flexibility, the deformation field of these core materials show a
nonlinear pattern in the through-thickness direction as well as in the core in-plane which can affect significantly the local as
well as the global structural behaviour of sandwich panels.
In principle, commercial finite element programs enable a computation of the nonlinear deformation pattern in the sandwich
core if a detailed idealisation is used in the through-thickness direction of the sandwich, in particular, in the core. However, the
application of standard finite elements during the analysis of such sandwich structures results in a very high computational
effort not only during the design of large sandwich structures but also in early design stages. Fast finite elements for the
analysis of sandwich structures accounting for the nonlinear deformation pattern in low strength cores are consequently
preferable.
In order to represent the mentioned nonlinear core behaviour appropriately for the static and stability analysis of sandwich
plates, a finite element is developed using a three-layer sandwich model as it is utilised in various finite elements formulations
(cp. [1], [2] or [3]). The face sheets are modelled as plate elements based on the Reissner-Mindlin theory considering
transverse shear strains according to an MITC4-plate element (Mixed Interpolation of Tensorial Components with 4 element
nodes according to [4]). The core displacement field is developed based on the solution of the underlying differential equations
of the core in the through-thickness direction z and the assumed displacement field of the face sheets as boundary conditions.
The differential equations are set up using a specific three-dimensional material law that neglects the in-plane core stiffnesses.
As a result, the core displacement pattern can be computed leading to in-plane core deformations which are cubic functions of
the trough-thickness direction z and to an out-of-plane deflection which is a quadratic function of z (cp. [2] and [3]). Based on
the interpolation functions, the linear and the geometric stiffness matrix for static and stability problems are derived. The
geometric stiffness matrix is set up by considering geometric nonlinearities in the v. Kármán sense in each sandwich layer.
Static and stability analyses based on the proposed element formulation are performed by integrating it in the finite element
(FE-) program Nastran (MSC.Software Corporation, USA).The results are compared to the ones of computations carried out
by the FE-program Marc (MSC.Software Corporation, USA) which are based on a detailed discretisation of the face sheets as
well as the core. The aim of this investigation is to demonstrate the capabilities of the presented element formulation.
Interpolation Functions
The finite sandwich element is defined on the basis of a three-layer model subdividing the sandwich into the face sheets and
the core as individual components. The idealised geometry of the finite element is illustrated in Figure 1. The values of the
face sheets are marked by the indices f1 and f2 for the bottom and top face sheet and the ones of the core by the index c.
The face sheets are idealised as plate elements based on the Reissner-Mindlin theory considering transverse shear strains
according to the above mentioned plate element [4]. For the in-plane and out-of-plane displacement field as well as for the
rotations bilinear interpolation functions are chosen. The introduced degrees of freedom per face sheet are shown in Figure 1
and comprise the displacements in the x-, y- and z-direction as well as the rotations about the x- and y-axis. In the usual way,
the coefficients of the assumed interpolation functions of the deformation pattern are determined by the selected degrees of
freedom. In order to prevent shear locking effects, the transverse shear strains are independently interpolated from the
displacement field (cp. for detailed description [5]).
Usually, the in-plane stiffnesses of the core (here E1c, E2c and G12c) are much smaller than the counterparts of the face sheets
so that the resulting strain energies of the core are also much smaller than the ones of the face sheets. This is in particular the
case for honeycomb cores. The in-plane core stiffnesses are consequently neglected in setting up a mathematical model of a
sandwich. The above sketched assumption (neglecting the stiffnesses E1c, E2c and G12c) leads to the simplification of the
material law of the core (cp. stress state in the core in Figure 2), i.e. the in-plane core stresses equal zero [6]:
c
! 1c = ! 2c = " 12
=0
However, in order to consider an inclination between the material coordinate system of the core and the local finite element
coordinate system, a rotation ! zc is introduced in the element formulation leading to (cp. Figure 2):
c
! xc = ! yc = " xy
=0
(1)
Figure 1. Geometry and degrees-of-freedom of finite sandwich element
The differential equations of the core are formulated on the basis of the equilibrium of forces at an infinitesimal core element
and by taking into account the displacement-strain relation. Defining a vector consisting of the displacements and the stresses
of the core, the following system of partial differential equations can be obtained (cp. [6] and [7]):
$ uc '
& c)
&v )
& c)
&w )
!z & c )
&"z )
&# c )
& xz )
&% # c )(
yz
where
+
=-,
0
0
*! x
0
c2
s2
+
G13 G23
0
0
*! y
0
s2
s
* 2
2G23 2G13
0
0
0
1
E3
0
0
0
0
0
*! x
0
0
0
0
0
0
0
0
0
0
!i =
!
= partial differentiation in i-direction
!i
s2
s .
* 2 0
2G23 2G13 0
c2
s2 0
0
+
G23 G13 0
0
0
0
0
0
0
*! y
0
0
0
0
0
0
0
0
0
0/
c
$ uc '
& c)
&v )
& c)
&w )
& c)
&"z )
&# c )
& xz )
&% # c )(
yz
u c , v c , w c = core displacements in x-, y- and z-direction
c
c
! zc , " xz
, " yz
= core peeling stress, core shear stress in x-z- resp. y-z-plane
c
c
E 3c , G13
, G23
= core material characteristics
The trigonometric functions are abbreviated by
( )
c = cos ! zc ,
( )
s = sin ! zc
and
( ) ( )
s2 = sin ! zc cos ! zc .
Figure 2. Stress state in sandwich core
The system of partial differential equations is analytically integrated in the through-thickness direction leading to polynomials of
the state variables in the z-direction. The following state variables are introduced (the indices b and t refer to the contact
planes between the bottom respectively top face sheet):
(
)
v ( x, y, z = !h / 2 ) = v ( x, y ) ,
w ( x, y, z = !h / 2 ) = w ( x, y ) ,
(
)
v ( x, y, z = h / 2 ) = v ( x, y )
w ( x, y, z = h / 2 ) = w ( x, y )
u c x, y, z = !h c / 2 = ubc ( x, y ) ,
c
c
c
c
u c x, y, z = h c / 2 = utc ( x, y )
c
b
c
b
c
c
c
c
c
t
c
t
and
c
w c ( x, y, z = 0 ) = wm
( x, y )
whereas the core mid-plane is referred to by the index m. If the out-of-plane deformation of the core mid-plane is assumed to
be known, the core displacement pattern is consequently determined by:
u c ( x, y, z ) = a0 ( x, y ) + a1 ( x, y ) z + a2 ( x, y ) z 2 + a 3 ( x, y ) z 3
v c ( x, y, z ) = b0 ( x, y ) + b1 ( x, y ) z + b2 ( x, y ) z 2 + b3 ( x, y ) z 3
w c ( x, y, z ) = c0 ( x, y ) + c1 ( x, y ) z + c2 ( x, y ) z 2
The coefficients ai, bi and ci are dependent on the chosen state variables. These state variables are clearly expressed through
the face sheet displacement field and the core mid-plane displacement wm if the displacement continuity conditions between
the face sheets and the core are taken into consideration. Assuming perfect displacement continuity conditions between the
core and the face sheets, the state variables of the core in the contact planes are expressed as:
ubc = u f 1 +
t f1 f1
!y ,
2
vtc = v f 2 +
utc = u f 2 !
t f2 f2
!x ,
2
t f2 f2
"y ,
2
wbc = w f 1
vbc = v f 1 !
and
t f1 f1
"x ,
2
wtc = w f 2
The out-of-plane deformation assumption in the core mid-plane is here defined by twelve additional nodal displacements, each
node with three degrees of freedom (cp. Figure 1) – one out-of-plane displacement and two rotational degrees of freedom – so
that the following general deformation pattern can be used:
w c ( x, y, z ) = d 0 + d1 x + d 2 y + d 3 x 2 + d 4 xy + d 5 y 2 + d 6 x 3 + d 7 x 2 y + d 8 xy 2 + d 9 y 3 + d10 x 2 y + d11 xy 2
The coefficients di are determined based on the nodal degrees of freedom in the usual way. As a consequence, the
interpolation functions of the face sheets as well as of the core are available, in particular, the ones of the core are fulfilling the
solution of the underlying partial differential equations of the core in the through-thickness direction.
Derivation of Stiffness Matrices
The linear and the geometric stiffness matrices are derived based on the principle of virtual displacements separating the
virtual strain vector into a linear and a geometrically nonlinear part (indicated by the subscript l and n respectively) according
to:
!
!
!
! !
!U " ! W = % # T (!$ l + !$ n ) dV " ! u T F = 0
(2)
!U, ! W = virtual strain energy, virtual work of external forces
! ! ! !
! , " , F, u = strain, stress, force and displacement vector
! !
!" l , !" n = linear and nonlinear virtual strain vector
V = element volume
where
The linear as well as the geometric stiffness matrix will be derived by solving separately the integral according to Equation (2)
for the linear and the geometrically nonlinear virtual strain vector.
Linear stiffness matrices
The principle of virtual work yields in the linear range for the strain energy:
!T !
$ ! "# l dV =
h c /2
2
+ 2 !
! Tc % c ' ! c .
! T + % fi ' % c '. !
Tfi % fi ' ! fi
"
#
Q
#
+
"
#
Q
#
dz
dydx
=
"
u
0
)
l
l
)
- & K ( + & K (0 u
l &
$$
( l
& (
0/
, i =1
/
0 , i =1
*h c /2
lx ly
$
0
!Q fi # , !Q c # = laminate and core elasticity matrix
"
$ " $
fi
! K # , ! K c # = linear stiffness matrix of face sheet and core
"
$ " $
!
u = element displacement vector containing the element degrees of freedom
where
The face sheets are idealised based on Reissner-Mindlin-theory. According to the four-node plate element with an assumed
transverse strain interpolation [4], the linear strain vector is set up as follows:
fi
!
!
!
! lTfi = $%! x0 , ! y0 , " xy0 , # x , # y , # xy , " yz0 , " xz0 &' = $% ( x u, ( y v, ( y u + ( x v, )( x* y , ( y* x , ( x* x ) ( y* y , N1T u, N 2T u &'
where N1 and N2 represent the interpolation functions in vector notation of the assumed transverse strain distribution.
Furthermore, the index 0 refers to the reference plane of the plate.
The laminate elasticity matrix is of order 8. The components corresponding to the classical laminate theory of the elasticity
matrix are computed according to [8] whereas for the transverse shear stiffnesses it is assumed that the shear stresses vary
linearly vanishing at the free edges of the sandwich.
The strain vector based on the specific orthotropic core material law with neglected in-plane core stiffnesses is defined as:
(
!
! !c !c
! c = ! zc , " yz
, " xz
) = (#
T
zw
c
, # z vc + # y wc , # z u c + # x wc
)
T
The elasticity matrix of the core is:
0
!1
'
!Q c # = ' 0 cos % zc
" $
'
c
" 0 & sin % z
#
c(
sin % z (
(
cos % zc $
0
T
0
0
0 # !1
! E3
'
'0 G
0 (( ' 0 cos % zc
23
'
'" 0
0 G13 ($ ' 0 & sin % zc
"
0
#
(
sin % zc (
(
cos % zc $
Geometric Stiffness Matrices
Considering the nonlinear virtual strain energies, the principle of virtual work yields:
!T !
$ ! "# n dV =
h c /2
2
' 2 t fi /2 !
*!
! Tc ! c *
!T '
Tfi ! fi
fi
c
!
"
#
+
)
n
$ ) & $fi
$c ! "# n ,, dzdydx = " u )( & -. K! /0 + -. K! /0,+ u
i =1
+
0 ( i =1 %t /2
%h /2
lx ly
$
0
" K !fi $ , " K !c $ = geometric stiffness matrix of face sheet and core
#
% # %
where
If the v. Kármán nonlinearities are considered in the face sheets, each nonlinear strain vector is defined by (cp. [9]):
(
!
$1
!" nTfi = ! & # x w fi
%2
) , 12 ( # w ) , ( # w ) ( # w ) , 0, 0, 0, 0, 0 ')(
2
y
fi 2
x
fi
y
fi
The geometric stiffness matrix of the face sheet can then be obtained by (cp. [9]):
l x l y 2 t fi /2
# #$ #
0 0 i =1 "t fi /2
!
!
! Tfi dz%& nfi dydx =
T
. 2 lx ly (
' x N wfi +
!T 0
*
= %u
0 i =1
*) ' y N wfi -,
0
0
/
$# #
( nx
*n
) xy
lx ly 2
! Tfi ! fi
%& n dydx
# # $n
0 0 i =1
1
2
n xy + ( ' x N wfi +
!
! T . ( fi +1 !
3
*
dydx
u
=
%
u
0 $ ) K ! ,3 u
n y -, * ' y N wfi 3
/ i =1
2
)
,
2
!
n = force vector per unit length according to laminate theory
n x , n y , n xy = membrane forces (per unit length)
where
The out-of-plane deformation is replaced by the following relation:
!
w fi = N wTfi u
The virtual nonlinear strain vector is formulated for a general three-dimensional orthotropic continuum based on the v. Kármán
nonlinearities (cp. [9]):
c
!
!" nTc = $%" x , " y , " z , # yz , # xz , # xy &'
n
= $% ( x w c ( x , ( y w c ( y , ( z w c ( z , ( z w c ( y + ( y w c ( z , ( z w c ( x + ( x w c ( z , ( x w c ( y + ( y w c ( x &'
If the in-plane core stresses are neglected according to Equation (1), the geometric core stiffness matrix yields:
T
.
'& x wc *
c
l
l
0 x y h /2 )
,
! Tc ! c
!T 0
c
)
,
!
"
#
dzdydx
=
"
u
&
w
n
y
% %c
0 % % %c )
,
0 $h /2
0 0 $h /2
c
0
& z w ,+
)
(
/
l x l y h c /2
%
0
' 0
)
) 0
)- xz
(
0
0
- yz
1
c
- xz * ' & x w *
3
)
,
,
!
!
!
- yz , ) & y w c , dzdydx 3 u = " u '( K !c *+ u
3
)
,
! z ,+ ) & z w c ,
3
(
+
2
Example Calculations
A static and a stability analysis of a sandwich plate are illustrated in the following. The results calculated by the proposed
element formulation are compared to results of FE-computations performed by the FE-program Marc. To simplify matters, only
one quarter of the sandwich plate is discretised for the analysed problems.
A solution procedure for the derived stiffness matrices has been implemented into the finite element program MSC.Nastran
with a DMAP-program according to [10]. Based on this approach, the FE-program calculates the element data from the
standard Nastran bulk data input, computes the stiffness matrices by calling an external FORTRAN program and solves the
systems of equations for linear as well as stability problems.
The utilised Marc-models consist of bilinear-thick shell elements with transverse shear effects and of trilinear solid elements
(type 75 and 7 according to [11]) for modelling the face sheets and the core respectively. The element meshes of the Marcmodel and of the here proposed idealisation agree in the x-y-plane. The latter idealisation is referred to as User-model.
With eight solid elements in the through-thickness direction of the core, a sufficient convergence was achieved for the Marcmodels. The here shown results are consequently obtained by such a discretisation in the through-thickness direction z. The
reduction in the degrees of freedom in the z-direction from the Marc-model to the User-idealisation amounts almost to 60 %.
Static Problem
The investigated static problem is illustrated in Figure 3. The plate is simply supported at all edges. Each face sheet consists
of a laminate with the layer set-up [ 0°, 90°]sym. The 0°-direction of the laminate coincides with the global x-direction according
to Figure 3. The single layers have the same thickness, and they possess orthotropic material characteristics:
E1 = 135000MPa
, E2 = 10000MPa , G12 = 3500MPa , G13 = G23 = 3800MPa and !13 = ! 23 = 0.27
The core consists of an aluminium honeycomb (Aluminium 5052, expanded hexagonal with density of 32 kg/m³ and cell size of
4.8 mm according to [12]). Its material properties and the angle between the material and the global coordinate system are
defined as follows:
E 3 = 230MPa , G13 = 180MPa , G23 = 98MPa and ! zc = 30°
Figure 3. Geometry and external loads of static problem
0
x-coordinate in mm
20
40
0
60
normal stress in x-direction in MPA .
-0,6
z-displacement in mm
x-coordinate in mm
20
30
40
50
60
-50
-0,5
-0,7
-0,8
-100
-150
-200
-250
Marc-P1
User-P1
User-P4
Marc-P4
Marc
User
-0,9
10
-300
Figure 4. Diagram on the left: z-displacement of upper face sheet for Marc- and User-model; diagram on the right: normal
stresses in x-direction of upper face sheet for Marc- and User-model (P1 refers to the free edge and P4 to the contact plane)
Here, the results are shown in the vicinity of the load introduction where localised effects exist leading to a complex core stress
state. In Figure 4 the z-deflections of the top face layer and in the normal stresses in the global x-direction in the top face sheet
are shown. The stresses are averaged values per node. The results are valid along the plate centerline in the positive xdirection starting from x=0 and y=0.
The results of the z-displacement as well as of the normal stresses agree very well between the User- and the Marc-model.
Due to the investigated honeycomb core with very low stiffnesses in the x-y-plane, the introduced assumptions during the
element formulation are fulfilled sufficiently so that a clear reduction of the element degrees of freedom are achieved without
almost no loss in accuracy.
Stability Problem
In order to demonstrate the element capabilities, the stability problem is chosen in such a way that a global as well as a local
buckling phenomena can be simultaneously analysed for the first buckling modes. The sandwich plate is simply supported
along all edges and at one edge in the global x-direction opposite to the external load (cp. Figure 6). The geometry as well as
the material properties agree with the ones of the before investigated static problem except that, first, no inclination between
the core material and the global coordinate system exits. Secondly, the face sheet thicknesses are equal, i.e.:
! zc = 0°
and
t f 1 = t f 2 = 1.5mm
Figure 6. External loads of stability problem
The computed buckling mode shapes are shown in the Figures 7 and 8 for the Marc- respectively for the User-model. The
critical buckling mode shape (each illustrated on the left in the mentioned figures) is a global one with one half wave in the
direction transverse to the external load (m) and one in the load direction (n). The deviation in the critical load between both
idealisations is less than 0.5 %. Furthermore, for the illustrated local buckling mode with m=1 and n=15, the deviation between
both models increases to about 5.8 %.
Figure 7. Buckling modes of Marc-model (on the left: critical buckling mode (global) with m=1 and n=1,
on the right: face sheet buckling mode with m=1 and n=15)
Figure 8. Buckling modes of User-model (on the left: critical buckling mode (global) with m=1 and n=1,
on the right: face sheet buckling mode with m=1 and n=15)
Conclusion
A displacement-based finite element formulation for the static and the stability analysis of sandwich plates is presented. The
finite element is based on a three-layer model consisting of the face sheets and the core as single subsystems. The face
sheets are described by Reissner-Mindlin plate elements using an independently assumed transverse shear interpolation. The
core deformation pattern is derived by setting up the underlying partial differential equations of the core as well as by
introducing the displacement continuity condition between the core and the face sheets. The linear as well as the geometric
stiffness matrices are then computed considering the v. Kármán strains in the geometrically nonlinear range. This approach
leads to an element formulation of 52 degrees of freedom.
The finite element formulation is implemented into the FE-program Nastran by a DMAP-program. The numerical computations
carried out are compared to Marc-computations in order to verify the element formulation and to illustrate the element
capabilities. In general, it could be demonstrated that a good correlation between the developed element and the classical
finite element idealisation based on Marc is achieved although a significant reduction in the degrees of freedom is reached.
This is the case for linear as well as classical stability problems if the core in-plane stiffnesses are small compared to the ones
of the face sheets.
References
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