PAMM · Proc. Appl. Math. Mech. 5 ,
111
–
112
(2005) / DOI 10.1002/pamm.2005100
36
Dragan Marinkovi´c
∗ 1
, Heinz K¨oppe
1
, and Ulrich Gabbert
1
1
Institut f¨ur Mechanik, Otto-von-Guericke Univerisit¨at, Universit¨atsplatz 2, 39106 Magdeburg
The paper presents a finite element concept for analysis of thin-walled active structures featuring fiber reinforced composite laminate as a passive structural material. The structure is rendered active by embedding piezoelectric material as a multifunctional material. A 9-node degenerated shell element based on the first order shear deformation theory is developed as a modelling tool capable of predicting the general behavior of the structure for controlling purposes. The von-K´arm´an type nonlinearities are considered. The solution strategy of the geometrical nonlinear analysis is based on the incremental approach using the updated Lagrangian formulation. Some numerical results are given to demonstrate the behavior of the element.
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5
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The potential benefits of active structures in various applications have attracted numerous researchers to devote their work to this area, particularly in the field of modelling. On the one hand, a model is supposed to offer a satisfying accuracy, but on the other hand, the required numerical effort should also be acceptable. Hence, a compromise is to be made. At the present stadium of development, a great number of active structures belong to the group of thin-walled structures. A special group comprise the structures featuring composite laminate as a passive structural material with embedded piezoelectric active elements. Although a great deal of work is devoted to this field, the development of a reliable, accurate and numerically not too demanding tool for modelling of mentioned structures is still a challenging and attractive task. Nowadays the numerically efficient formulations are practically by default addressed to the finite element method. This fact implies the development of a shell type of finite element for modelling of generally double-curved thin-walled active structures. Structures made of composite laminates are susceptible to relatively large transverse deflections and, hence, the inclusion of the von-K´arm´an type nonlinearities is of interest as well.
The authors of the paper have developed a full biquadratic degenerated shell element capable of modelling arbitrarily shaped thin-walled active structures made of composite multi-layered material [1]. The element is of Mindlin type allowing transverse shear strains and stresses. Hence, it has 9 nodes and 6 global degrees of freedom per node (3 translations and 3 rotations), but only 5 degrees of freedom in the local-running coordinate system (no rotation about local axis in the thickness direction, z’) and it has additionally as many electric degrees of freedom (difference of electric potential) as there are piezoelectric layers across the thickness of the element. The element is capable of taking into account the additional stiffening effect due to the linear distribution of the electric field over the thickness of the piezoelectric layers, which is deduced from the 4 th
Maxwell’s equation and the linear distribution of the in-plane strains across the thickness of the laminate. The element is implemented in
COSAR, a general purpose finite element package developed at the Institute of Mechanics, Otto-von-Guericke University in
Magdeburg.
z
1 x
3 x
2
1 y
1
4 t
9 r s z’
3 y’ z’ x
1
Z’
Y’
X’
Mid-surface
Piezoelectric layers cut-out x z y
2 x’ x’ y’ x
1 predefined structure reference direction
Fig. 1 Degenerated shell element, cross sectional cut-out and local coordinate system
∗
Corresponding author: e-mail: dragan.marinkovic@mb.uni-magdeburg.de
, Phone: +00 49 391 6711724, Fax: +00 49 391 6712439
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WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Section 1
112
The finite element equations for a dynamic case are obtained starting from the Hamilton’s principle. The geometrically nonlinear analysis recognizes that the structure undergoes changes in the configuration during the loading, whereby small strains are assumed. The incremental step-by-step approach represents a usual solution strategy in the nonlinear analysis.
Using the updated Lagrangian formulation, the discretization of the structure results in the following set of equations:
M t + C U
˙ t = R t − F t t K
Φ U
∆ U t − ∆ t + K
ΦΦ
∆Φ t − ∆ t = Q t ext
− Q t − ∆ t int
(1) where M and C are the mass and damping matrices, U and
Φ are the mechanical displacements and electric potentials,
K
Φ U and
K time t,
ΦΦ
F t t are the piezoelectric coupling and dielectric stiffness matrices, respectively,
R t are the external mechanical forces at are the internal mechanical forces at time t with respect to the configuration at time t, and finally Q denotes electric charges, the subscript ext stands for external and int for internal. All the quantities are defined on the element level.
Within the framework of the linear analysis small displacements are assumed so that the change in the structure configuration is negligible. The assumption allows the calculation of the internal forces at time t through the linear mechanical stiffness matrix,
K
UU , the piezoelectric stiffness matrix,
K
U Φ and the generalized displacements at time t, in the following form:
F t
= K
UU
U t + K
U Φ
Φ t
(2)
Two simple examples are chosen in order to give a brief demonstration of the behavior of the developed element.
The first example represents a static linear analysis of a beam structure actuated by a pair of oppositely polarized piezoelectric patches (fig 2). The same voltage of 100 V is applied to both piezopatches resulting in internal bending moments uniformly distributed over their edges. In this case the results obtained from three different elements are compared - 3D hexahedron, Semi-Loof shell and degenerated shell element presented in this paper. The 3D hexahedron element is used with a fine mesh and it gives a referent solution (blue line, top deflection of -0,611 mm). The Semi-Loof and the degenerated shell element both use the same mesh. A very fine agreement of the results achieved by the degenerated shell element (green line, top deflection of -0,605 mm) with the reference results from the 3D hexahedron element can be observed on the fig. 2. The
Semi-Loof element is represented with the red line (top deflection of -0,586 mm).
-6,1074E-01
-5,8550E-01
-6,0504E-01
E 1,00E
+01
2,00E 3,00E 4,00E 5,00E 6,00E 7,00E 8,00E
+01 +01 +01 +01 +01 +01 +01
3D-Hexaeder Semiloof
9,00E 1,00E
+01 +02
Shell9
1,10E
+02
1,20E
+02
Transverse deflection mm
Fig. 2 Thin clamped piezoelectric beam and results from hexahedron solid, degenerated shell and semi-loof element
The second example demonstrates the usage of the dynamic solver and the dynamic relaxation technique in order to solve static case. A plate type of structure clamped on one of the edges (fig. 3) and exposed to the uniformly distributed transverse force (magnitude of
3 · 10 3 N
) on the opposite, free edge is considered. The linear and geometrically nonlinear result obtained with the here presented degenerated shell element is compared with the same results yielded by the shell element from the finite element package ABAQUS.
10 20 30 40 50
-5, 54913
-5, 55500
-5, 71731
-5, 77690
ABAQUS - nonl.
ABAQUS - lin.
COSAR - nonl.
COSAR - lin.
Transverse deflection mm
4,00E-01
3,00E-01
2,00E-01
1,00E-01
0,00E+00
-1,00E-01
0
-2,00E-01 mm
10 20 30 40
0 , 2 9 6 0 5
0 , 2 9 0 11
50
ABAQUS - nonl.
ABAQUS - lin.
COSAR - nonl.
COSAR - lin.
Displacement in length direction
Fig. 3 Clamped thick plate under transverse force, linear and geometrically nonlinear results - transverse deflection and displacement in length direction obtained with degenerated shell and ABAQUS shell element
[1] D. Marinkovi´c, H. K¨oppe, U. Gabbert, in: Facta Universitatis, series: Mechanical Engineering 2 , 11-24 (2004)
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WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim