A hybrid-stress solid-shell element for non-linear analysis of piezoelectric structures

Abstract

This paper presents eight-node solid-shell elements for geometric non-linear analyze of piezoelectric structures. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger-Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid-stress solid-shell element is formulated. The presented finite shell element is able to model arbitrary curved shell structures. Non-linear numerical examples demonstrate the ability of the proposed model to analyze nonlinear piezoelectric devices.piezoelectric structures, solid-shell element, hybrid-stress, geometric non-linearThe non-linear characteristics can significantly influence the performance of piezoelectric structures. The nonlinear problems of piezoelectric structures comprise of geometrically nonlinear and material nonlinearity. Generally, only the strong electric field subject to piezoelectric material, just the material nonlinearity of piezoelectric structures is considered [1][2][3][4] . When the electric field is not too large, geometrically nonlinear analysis is only considered for nonlinear problem. The researcher has already carried out many beneficial researches concerning geometrically non-lineariy [5][6][7][8][9][10][11] . The nonlinear electric potential distribution in piezoelectric layer was described by introducing internal electric potential node in ref. [5]. A solid shell element for large deformable composite structures with piezoelectric layers and active vibration control was presented in ref. [6]. But, an optimal number of enhancing assumed strain (EAS) parameters was used for composite structures in this reference. A modified generalized laminate stiffness matrix is adopted for composite structures in this paper.In this paper, geometric non-linear is considered only, but piezoelectric non-linearity is not taken ino account. An eight-node solid-shell element is derived for geometric non-linear analysis of homogeneous and laminated elastic shells to analyze piezoelectric structures. Following the common practice of resolving shear and trapezoidal lockings in this particular element configuration, ANS is employed to interpolate the natural thickness and transverse shear strains [12][13][14][15][16] . To overcome the thickness locking, an ad hoc modified generalized laminate stiffness matrix is adopted. Unlike EAS which can only reproduce accurate thickness stress and strain for homogeneous shells, the matrix also works properly for laminated shells. As a low-cost vehicle to enhance the element inplane response, selectively reduced integration (SRI) is applied to the in-plane shear strain. By taking the generalized stresses defined with respect to the

Analysis of the second Piola‐Kirchhoff stress in nonlinear thick and thin structures using exact geometry SaS solid‐shell elements

“…To implement the efficient analytical integration throughout the shell element, the extended ANS method is utilized to interpolate the displacement‐dependent strains where are the strains of SaS at element nodes.Remark The idea of such approach can be traced back to the ANS method developed by many scientists to cure the isoparametric six‐ and seven‐parameter solid‐shell elements from shear and membrane locking. In contrast to a conventional formulation, we treat the term ANS in a broader sense.…”

“…To prevent thickness locking, the 3D constitutive equations should be modified using the generalized plane stress conditions . Alternatively, other finite element techniques can be utilized, namely a hybrid stress method, in which the transverse normal stress is assumed to be constant through the shell thickness and the most popular enhanced assumed strain method with the transverse normal strain enriched in the thickness direction by a linear term. The analysis of functionally graded shells can be found in works …”

“…To circumvent element locking, the hybrid‐mixed method pioneered by Pian can be applied efficiently. There are three types of geometrically nonlinear hybrid‐mixed solid‐shell elements in the literature, namely, the hybrid stress, hybrid strain and hybrid stress‐strain elements. These finite elements are based respectively on the Hellinger‐Reissner variational principle with displacements and stresses as independent variables, the modified Hellinger‐Reissner variational principle, in which the displacements and strains are used as primary variables, and the Hu‐Washizu variational principle depending upon displacements, strains, and stresses.…”

Modeling and analysis of spiral actuators by exact geometry piezoelectric solid-shell elements

“…More reliable results can be obtained through the isoparametric six-parameter piezoelectric solid-shell elements (Klinkel and Wagner, 2006, 2008; Lee et al, 2003; Lentzen, 2009; Sze et al, 2000; Sze and Yao, 2000; Tan and Vu-Quoc, 2005; Yao and Sze, 2009; Zheng et al, 2004). These elements are defined by two layers of nodes on outer surfaces of the shell with three displacement degrees of freedom (DOFs) and one electric potential DOF per node.…”

“…To avoid Poisson thickness locking, the 3D constitutive equations should be modified using the generalized plane stress conditions (Lee et al, 2003). The hybrid stress method (Sze et al, 2000; Sze and Yao, 2000; Yao and Sze, 2009), in which the transverse normal stress is assumed to be constant through the thickness, and the most popular enhanced assumed strain method (Klinkel and Wagner, 2006, 2008; Lentzen, 2009; Tan and Vu-Quoc, 2005; Zheng et al, 2004), in which the transverse normal strain is enriched in the thickness direction by a linear term, can be also applied. Still, the isoparametric solid-shell element formulation is computationally inefficient because stresses and strains are analyzed in the global and local orthogonal Cartesian coordinate systems, although the normalized element coordinates represent already convected curvilinear coordinates.…”