Geometrically Exact Four-Node Piezoelectric Solid-Shell Element

This paper presents the geometrically exact assumed stress-strain four-node piezoelectric solid-shell element with six displacement degrees of freedom per node on the basis of the first-order equivalent single-layer theory. The proposed piezoelectric laminated shell formulation is based on the objective strain-displacement relationships, written in curvilinear reference surface coordinates, and generalizes the authors' purely mechanical finite element formulation. The mechanical and piezoelectric degrees of freedom are coupled via constitutive equations and the electric potential is assumed to be linear through the thickness of the piezoelectric layer. To overcome shear and membrane locking and have no spurious zero energy modes, the assumed strain and stress resultant fields are invoked. In order to circumvent thickness locking, the ad hoc modified laminate stiffness matrix, corresponding to the generalized plane stress condition, is employed. The elemental stiffness matrix has six zero eigenvalues and requires only direct substitutions. Moreover, it is evaluated by applying the 3D analytical integration that is economical and allows using extremely coarse meshes.

[1]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .

[2]  Gennady M. Kulikov,et al.  Equivalent Single-Layer and Layerwise Shell Theories and Rigid-Body Motions—Part II: Computational Aspects , 2005 .

[3]  K. Y. Sze,et al.  Three‐dimensional continuum finite element models for plate/shell analysis , 2002 .

[4]  Theodore H. H. Pian,et al.  Finite elements based on consistently assumed stresses and displacements , 1985 .

[5]  E. Ramm,et al.  Three‐dimensional extension of non‐linear shell formulation based on the enhanced assumed strain concept , 1994 .

[6]  Shijie Zheng,et al.  The formulation of a refined hybrid enhanced assumed strain solid shell element and its application to model smart structures containing distributed piezoelectric sensors/actuators , 2004 .

[7]  Nam Seo Goo,et al.  A nine-node assumed strain shell element for analysis of a coupled electro-mechanical system , 2003 .

[8]  Gennady M. Kulikov,et al.  Equivalent Single-Layer and Layerwise Shell Theories and Rigid-Body Motions—Part I: Foundations , 2005 .

[9]  P. Vigoureux.  Piezoelectricity An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals , 1947, Nature.

[10]  H. Tzou.  Piezoelectric Shells: Distributed Sensing and Control of Continua , 1993 .

[11]  Hazem Kioua,et al.  Piezoelectric induced bending and twisting of laminated composite shallow shells , 2000 .

[12]  K. Washizu,et al.  Variational Methods in Elasticity and Plasticity, 3rd Ed. , 1982 .

[13]  K. Y. Sze,et al.  A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part II?smart structure modelling , 2000 .

[14]  Gennady M. Kulikov,et al.  Simple and effective elements based upon Timoshenko–Mindlin shell theory , 2002 .

[15]  S. Lee,et al.  An eighteen‐node solid element for thin shell analysis , 1988 .

[16]  Gennady M. Kulikov,et al.  Non-linear strain-displacement equations exactly representing large rigid-body motions. Part I Timoshenko-Mindlin shell theory , 2003 .

[17]  D. Talaslidis,et al.  A Simple and Efficient Approximation of Shells via Finite Quadrilateral Elements , 1982 .

[18]  Ayech Benjeddou,et al.  Advances in piezoelectric finite element modeling of adaptive structural elements: a survey , 2000 .

[19]  Gennady M. Kulikov,et al.  Geometrically exact assumed stress–strain multilayered solid-shell elements based on the 3D analytical integration , 2006 .

[20]  K. Sze,et al.  MODELLING SMART STRUCTURES WITH SEGMENTED PIEZOELECTRIC SENSORS AND ACTUATORS , 2000 .

[21]  Hoon Cheol Park,et al.  Analysis of multi-layered actuators using an assumed strain solid element , 2002 .