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Linear and Nonlinear Finite Element Analysis of Active Composite Lam-inates
Dragan Marinkovic∗1, Heinz Koppe1, and Ulrich Gabbert1
1 Institut fur Mechanik, Otto-von-Guericke Univerisitat, Universitatsplatz 2, 39106 Magdeburg
The paper presents a finite element concept for analysis of thin-walled active structures featuring fiber reinforced compositelaminate as a passive structural material. The structure is rendered active by embedding piezoelectric material as a multi-functional material. A 9-node degenerated shell element based on the first order shear deformation theory is developed asa modelling tool capable of predicting the general behavior of the structure for controlling purposes. The von-Karman typenonlinearities are considered. The solution strategy of the geometrical nonlinear analysis is based on the incremental approachusing the updated Lagrangian formulation. Some numerical results are given to demonstrate the behavior of the element.
1 Introduction
The potential benefits of active structures in various applications have attracted numerous researchers to devote their workto this area, particularly in the field of modelling. On the one hand, a model is supposed to offer a satisfying accuracy, buton the other hand, the required numerical effort should also be acceptable. Hence, a compromise is to be made. At thepresent stadium of development, a great number of active structures belong to the group of thin-walled structures. A specialgroup comprise the structures featuring composite laminate as a passive structural material with embedded piezoelectric activeelements. Although a great deal of work is devoted to this field, the development of a reliable, accurate and numerically nottoo demanding tool for modelling of mentioned structures is still a challenging and attractive task. Nowadays the numericallyefficient formulations are practically by default addressed to the finite element method. This fact implies the developmentof a shell type of finite element for modelling of generally double-curved thin-walled active structures. Structures made ofcomposite laminates are susceptible to relatively large transverse deflections and, hence, the inclusion of the von-Karman typenonlinearities is of interest as well.
2 Degenerated shell element
The authors of the paper have developed a full biquadratic degenerated shell element capable of modelling arbitrarily shapedthin-walled active structures made of composite multi-layered material [1]. The element is of Mindlin type allowing transverseshear strains and stresses. Hence, it has 9 nodes and 6 global degrees of freedom per node (3 translations and 3 rotations), butonly 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 layersacross the thickness of the element. The element is capable of taking into account the additional stiffening effect due to thelinear distribution of the electric field over the thickness of the piezoelectric layers, which is deduced from the 4th Maxwell’sequation and the linear distribution of the in-plane strains across the thickness of the laminate. The element is implemented inCOSAR, a general purpose finite element package developed at the Institute of Mechanics, Otto-von-Guericke University inMagdeburg.
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Mid-surfaceX’
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cut-out
Fig. 1 Degenerated shell element, cross sectional cut-out and local coordinate system
∗ Corresponding author: e-mail: [email protected], Phone: +00 49 391 6711724, Fax: +00 49 391 6712439
PAMM · Proc. Appl. Math. Mech. 5, 111–112 (2005) / DOI 10.1002/pamm.200510036
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
3 Finite element equations
The finite element equations for a dynamic case are obtained starting from the Hamilton’s principle. The geometricallynonlinear analysis recognizes that the structure undergoes changes in the configuration during the loading, whereby smallstrains 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:
MU t + CU t = Rt − F tt KΦU∆U t−∆t + KΦΦ∆Φt−∆t = Qt
ext − Qt−∆tint (1)
where M and C are the mass and damping matrices, U and Φ are the mechanical displacements and electric potentials, KΦU
and KΦΦ are the piezoelectric coupling and dielectric stiffness matrices, respectively, Rt are the external mechanical forces attime t, F t
t are the internal mechanical forces at time t with respect to the configuration at time t, and finally Q denotes electriccharges, 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 configura-tion is negligible. The assumption allows the calculation of the internal forces at time t through the linear mechanical stiffnessmatrix, KUU , the piezoelectric stiffness matrix, KUΦ and the generalized displacements at time t, in the following form:
Ft = KUUU t + KUΦΦt (2)
4 Numerical examples
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 piezo-
electric patches (fig 2). The same voltage of 100 V is applied to both piezopatches resulting in internal bending momentsuniformly distributed over their edges. In this case the results obtained from three different elements are compared - 3Dhexahedron, Semi-Loof shell and degenerated shell element presented in this paper. The 3D hexahedron element is used witha fine mesh and it gives a referent solution (blue line, top deflection of -0,611 mm). The Semi-Loof and the degenerated shellelement 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. TheSemi-Loof element is represented with the red line (top deflection of -0,586 mm).
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3D-Hexaeder Semiloof Shell9
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 solvestatic case. A plate type of structure clamped on one of the edges (fig. 3) and exposed to the uniformly distributed transverseforce (magnitude of 3 · 103N ) on the opposite, free edge is considered. The linear and geometrically nonlinear result obtainedwith the here presented degenerated shell element is compared with the same results yielded by the shell element from thefinite element package ABAQUS.
-5,54913
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ABAQUS - nonl. COSAR - nonl.
ABAQUS - lin. COSAR - lin.
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ABAQUS - nonl. COSAR - nonl.
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Transverse deflection Displacement in length direction
mmmm
Fig. 3 Clamped thick plate under transverse force, linear and geometrically nonlinear results - transverse deflection anddisplacement in length direction obtained with degenerated shell and ABAQUS shell element
References
[1] D. Marinkovic, H. Koppe, U. Gabbert, in: Facta Universitatis, series: Mechanical Engineering 2, 11-24 (2004)
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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