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CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 275
Analysis of the FreeEdge Effect in Piezoelectric Laminated Plates by the Scaled Boundary FiniteElement Method J. Artel and W. Becker
Department of Mechanical Engineering, Darmstadt University of Technology, Germany J. Artel, W. Becker, "Analysis of the FreeEdge Effect in Piezoelectric Laminated Plates by the Scaled Boundary FiniteElement Method", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 275, 2006. doi:10.4203/ccp.83.275
Keywords: scaled boundary finiteelement method, freeedge effect, piezoelectricity.
Summary
The paper deals with the analysis of the freeedge effect in
laminated plates composed of piezoelectric materials by using the
scaled boundary finiteelement method. This novel method combines
the advantages of the finite element method and the boundary
element method as the method is finite element based although a
discretization of the boundary of the considered domain is
sufficient. Applying the scaled boundary finiteelement method to
the analysis of freeedge effect situations necessitates the
implementation of kinematic coupling equations, which is shown in
detail. Finally, a symmetric crossply laminate under uniform
strain is investigated exemplarily to show the efficiency of the
new method.
The occurrence of interlaminar stresses in the vicinity of free edges in laminated plates is of significant practical importance and is the aim of many analytical and numerical investigations within many years [1,2]. Generally, some interlaminar stress components become singular within the interface between two dissimilar adjacent layers. This leads to the assumption, that laminated plates consisting out of piezoelectric materials behave similar and that some components of the electric field strength vector also may become singular. Davi and Milazzo investigated a symmetric angleply laminate under several loading conditions by using the boundary element method [3]. Within this paper, the investigation of this effect is performed by using the scaled boundary finiteelement method and the results are compared with those generated by a finite element analysis. Both methods use full electromechanical coupling to simulate the piezoelectric material behaviour. The scaled boundary finiteelement method is a novel semianalytical method, which has been developed by Wolf and Song [4]. The method is based on the introduction of a local coordinate system, where the governing equations are employed in the weak form in two local directions and in the strong form in the so called scaling direction. The solution is achieved by approximating the weak form by a finite element approach and by solving the strong form analytically as it is given by a system of ordinary differential equations. Although a discretization of the boundary of the domain is sufficient, the method is finite element based and no fundamental solution is required within the analysis procedure. The method is applied to the analysis of the freeedge effect in piezoelectric laminated plates. A laminated plate is considered, which is extended to infinity in the scaling direction perpendicular to the free edge and in the direction parallel to the free edge. Within the scaled boundary finiteelement method the corresponding model is chosen to be represented by a slat in the scaling direction, where the free edge is discretized with elements. The loading is applied by a given strain field, which is taken into account by corresponding kinematic coupling equations. In order to fulfill the condition of an infinite extended laminate in the direction parallel to the free edge, special degrees of freedom have to be introduced on the sidefaces of the slat to hold this condition for the electric potential. The coupling of degrees of freedom is taken into account by a general linear relation between dependent and independent coupled degrees of freedom and a constant vector, which can be considered as a given displacement field. Generally, the coupling equations also have to be fulfilled in the scaling direction, which consequently leads to an extended formulation of the method. The implementation of the kinematic coupling equations within the method is presented. Furthermore, degrees of freedom are introduced, that are no functions of the scaling coordinate, which is necessary for the coupling of the electric potential. As a special problem, rigid body modes are handled as they occur in any freeedge analysis using the scaled boundary finiteelement method. The condition of zero resultant forces in the direction of the rigid body modes can be fulfilled in order to obtain a valid solution. By using the theory of generalized inverse matrices, the solution procedure is completed. A symmetric crossply laminate is analyzed under a given uniaxial stress field parallel to the free edge and the interlaminar stresses and electric field strengths are investigated. It is shown, that some electric field strength components become singular in the vicinity of the free edge. The comparison of the results generated by the scaled boundary finiteelement method and the finite element methods shows very good accordance, but the scaled boundary finiteelement method requires considerably less numerical effort. References
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