Static and dynamic analysis of smart cylindrical shell

Shell type components and structures are very common in many mechanical and structural systems. In smart structural applications, piezolaminated plates and shells are commonly used. In this paper a finite element formulation is presented to model the static and dynamic response of laminated composite shells containing integrated piezoelectric sensors and actuators subjected to electrical, mechanical and thermal loadings. The formulation is based on the first order shear deformation theory and Hamilton's principle. In this formulation, the mass and stiffness of the piezo-layers have been taken into account. A nine-noded degenerated shell element is implemented for the analysis. The model is validated by comparing with existing results documented in the literature. A simple negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to actively control the dynamic response of an integrated structure through a closed control loop. The influence of the stacking sequence and position of sensors/actuators on the response of the laminated cylindrical shell is evaluated. Numerical results show that piezoelectric sensors/actuators can be used to control the shape and vibration of laminated composite cylindrical shell.     

A mixed finite element formulation for piezoelectric shells

A finite element formulation to analyze piezoelectric shell problems is presented. A reference surface of the shell is modelled with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electric degree of freedom, which is the difference of the electric potential in thickness direction. The formulation is based on the mixed field variational principle of Hu-Washizu. The independent fields are displacements u , electric potential φ, strains E , electric field E , stresses S and dielectric displacements D . The mixed formulation allows an interpolation of the strains and the electric field in thickness direction. Accordingly a three-dimensional material law is incorporated in the variational formulation. It is remarked that no simplification regarding the constitutive law is assumed. The formulation allows the consideration of arbitrary constitutive relations. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent stress and dielectric displacement fields. They are defined as zero in thickness direction. The present shell element fulfills the important patch tests: the in-plane, bending and shear test. Some numerical examples demonstrate the applicability of the present piezoelectric shell element. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Piezoelectric Finite Shell Element for the Geometrically Nonlinear Analysis of Sensors

A geometrically nonlinear finite element formulation to analyze piezoelectric shell structures is presented. The formulation is based on the mixed field variational functional of Hu–Washizu. Within this variational principle the independent fields are displacements, electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when using a three dimensional material law. It is remarked that no simplification regarding the constitutive relation is assumed. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent resultant stress and resultant dielectric displacement fields. The shell structure is modeled by a reference surface with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electrical degree of freedom, which is the difference of the electric potential through the shell thickness. The developed mixed hybrid shell element fulfills the in–plane, bending and shear patch tests, which have been adopted for coupled field problems. A numerical investigation of a smart antenna demonstrates the applicability of the piezoelectric shell element under the consideration of geometrical nonlinearity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite shell element with well balanced approximation functions for piezoelectric coupling

This contribution is concerned with a piezoelectric shell formulation. The present shell element has four nodes and bilinear interpolation functions. The nodal degrees of freedom are displacements, rotations and the electric potential on top and bottom of the shell. A 3D–material law is incorporated. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. This effect occurs in standard element formulations, where the mechanical and electrical degrees of freedom are approximated with lowest order interpolation functions. In order to overcome this problem a mixed multi–field variational approach is introduced. It allows for approximations of the electric field and the strains independent of the bilinear interpolation functions. A quadratic approach for the shear strains and the electric field is proposed through the shell thickness. This leads to well balanced approximation functions regarding coupling of electrical and mechanical fields. A numerical example illustrates the more precise results in contrast to standard elements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear transient FE analysis of piezolaminated smart structures

The presented paper focuses on the modeling and finite element simulation of thin composite structures with integrated piezoelectric layers. Geometrically nonlinear piezolaminated finite elements are based on the Reissner-Mindlin or third-order transverse shear deformation hypotheses and the assumptions of small strains but moderate rotations. The numerical results are compared to results in literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical solution of piezolaminated rectangular plates with arbitrary clamped/simply-supported boundary conditions under thermo-electro-mechanical loadings

This paper extends an analytical method for static analysis of general cross-ply piezolaminated rectangular plates with any combination of clamped/simply-supported boundary conditions under uncoupled thermo-electro-mechanical loadings. This method is based on the novel superposition method and the first-order shear deformation theory (FSDT). The FSDT enables this expanded method to consider the effect of shear deformation of the plate. The process of applying electrical and thermal resultants causes some advantages due to its simplicity and less computational process. In this analysis displacement components are written in terms of unknown force and moment resultants. Using Fourier series for displacement components, mechanical, thermal, and/or electrical stress resultants, the complex governing differential equations of the plate are reduced to a set of linear algebraic equations with non-trivial solution. The obtained equations may be solved analytically to determine the unknown stress resultants. Several examples are proposed, and their obtained numerical results are compared with those available in the literature to verify the convergence, high accuracy, and the capability of the present method to analyze the static behavior of piezolaminated plates. It is found that there is high agreement between the present results with those obtained by other investigators.     

Dynamic Stability Analysis of Shells Using a Finite Element Based Reduction Method

A finite element formulation of a reduction method for dynamic stability analysis of imperfection-sensitive shell structures is presented. The reduction method makes use of a perturbation approach, initially developed for static buckling and later extended to dynamic buckling analysis. The single mode dynamic buckling analysis and its extension to parametric excitation analysis are described. The approach is available within a general purpose finite element code. Characteristic results for the parametric excitation analysis of a composite cylindrical shell are shown. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A hybrid element formulation for electromechanical problems

The difficulty in the modeling of ferroelectric materials is the coverage of the complicated interactions between electrical and mechanical quantities on the macroscale, which are caused by switching processes on the microscale. In the present work we present an electric hybrid element formulation where the stresses and the electric fields are derived by constitutive relations as presented in [1]. Therefore the displacements, the electric potential and the electric displacements are approximated by bilinear ansatz functions. Applying a static condensation procedure we obtain a modified finite element formulation governed by the degrees of freedoms associated to the displacements and the electric potential. The anisotropic material behavior is modeled within a coordinate-invariant formulation [6] for an assumed transversely isotropic material [4]. In this context a general return algorithm is applied to compute the remanent quantities at the actual timestep. Resulting hysteresis loops for the ferroelectric ceramics are presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element model for laminated piezoelectric shell structures

Thin piezoelectric laminates are used for a wide range of technical applications. A four-node piezoelectric shell element is presented to analyse such structures effectively. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. In order to overcome this problem the finite element formulation is based on a mixed variational principle implying six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. This allows for an interpolation of the strains and the electric field in thickness direction independent of the bilinear interpolation functions. A piecewise quadratic approach for the shear strains in thickness direction and the corresponding electric field is proposed for arbitrarily layered shells. Regarding coupling of electrical and mechanical fields this yields to an appropriate balance of the approximation functions. Numerical examples show more precise results in contrast to standard elements with lowest order interpolation functions. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 2. Geometrically Nonlinear Statement

Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells subjected to large displacements and arbitrarily large rotations. As the sought-for functions, six displacements and eleven strains of the shell faces are chosen, which allows us to use nonlinear deformation relationships exactly representing arbitrarily large displacements of the shell as a rigid body. The stiffness matrix of a shell element has a proper rank and is calculated based on exact analytical integration. The bilinear element developed does not allow false rigid displacements and is not subjected to the membrane, shear, or Poisson locking phenomenon. The results of solving the well-known test problem on a nonsymmetrically fixed circular arch subjected to a concentrated load and the problem on a locally loaded toroidal multilayer rubber-cord shell are presented.     

A Four-node Finite Element for Piezoelectric Shell Structures in Convective Co-ordinates

In this contribution, an isoparametric piezoelectric shell element is presented which is based on convective coordinates and which allows for the analysis of arbitrary shell geometries. A two-field variation formulation [1, 2] is used in which the displacements and the electric potentials serve as independent variables. Especially, for thin-walled structures under certain boundary conditions and load cases, the displacement based element tend to shear and membrane locking. In order to avoid this poor behaviour, the Assumed Natural Strain (ANS) method [3] is introduced into the piezoelectric shell element. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A solid shell finite element formulation to simulate the behaviour of thin dielectric elastomer structures

To analyse the behaviour of thin structures of dielectric elastomer (DE) material a solid shell finite element is presented. The main characteristics of DEs are a non-linear hyper elastic behaviour, the quasi-incompressibility, and the ability to transform electric energy into mechanical work. Applying a voltage to thin DE structures may produce large elongation strains of 120-380%. These large strains, the efficient electro-mechanical coupling, and the light weight make DEs very attractive for the usage in actuators. Thus, there is a need for detailed research. With respect to the electro-mechanical coupling a constitutive model is presented. An electric stress tensor and a total stress tensor are introduced by considering the electrical body force and couple in the balance of linear momentum and angular momentum, respectively. The governing equations are derived and embedded in the solid shell formulation. The element formulation is based on a Hu-Washizu mixed variational principle using six independent fields: displacements, electric potential, strains, electric field, mechanical stresses, and dielectric displacements. It allows large deformations and accounts for physical nonlinearities to capture two of the main characteristics of DEs. The shell element could be applied for the modelling of arbitrary curved thin structures. The ability of the present element formulation is demonstrated in several examples. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact interaction of composite shells,subjected to follower loads,with a rigid convex foundation

Based on a 7-parameter shell model, a numerical algorithm is developed for solving the contact problem for a multilayered composite shell lying on a rigid convex foundation, which is subjected to a follower pressure and undergoes arbitrarily large rotations. A new geometrically exact solid shell element is formulated, which permits one to solve the nonlinear deformation problem for thin-walled composite structures under unilateral contact constraints by using a small number of load steps. The calculation of a homogeneous ring and an angle-ply toroidal shell interacting with plane and cylindrical foundations is considered.     

Quantized planary dynamics in orthogonal magneto-electrostatic fields

At the first quantization level, we deal with the planary dynamics of a charged scalar evolving in static orthogonal magnetic and electric fields. Working in a relativistic approach, we get the quantum eigenstates and the energy spectrum exhibiting a non-linear dependence on the exterior fields and the particle momentum parameter. Analyzing the generalized Landau-type energy levels, we point out a shift of the Larmor pulsation, due to the electrostatic field and derive a critical induction-energy spectrum. The same has been done for strong magnetic fields and a compulsory relation between the particle momentum and the electric field intensity has been obtained. For quasi-on-shell particles, moving in either strong or weak magnetic field, we derive the completely possible momentum spectrum. It turns out that, in extremely faint electrostatic fields, it yields the same momentum quantization.     

Formulation and evaluation of a finite element model for the linear analysis of stiffened composite cylindrical panels

A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-roder shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilibrium equations are satisfied. The elements are free of membrane and shear locking with correct satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed showing the validity of the models.     

Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations

We consider a bilinear reduced-strain finite element formulation for a shallow shell model of Reissner-Naghdi type. The formulation is closely related to the facet models used in engineering practice. We estimate the error of this scheme when approximating an inextensional displacement field. We make the strong assumptions that the domain and the finite element mesh are rectangular and that the boundary conditions are periodic and the mesh uniform in one of the coordinate directions. We prove then that for sufficiently smooth fields, the convergence rate in the energy norm is of optimal order uniformly with respect to the shell thickness. In case of elliptic shell geometry the error bound is furthermore quasioptimal, whereas in parabolic and hyperbolic geometries slightly enhanced smoothness is required, except for the degenerate cases where the characteristic lines are parallel with the mesh lines. The error bound is shown to be sharp.

     

3D stress analysis of generally laminated piezoelectric plates with electromechanical coupling effects

In this paper, the interlaminar stresses of generally laminated piezoelectric (PZT) plates are presented. The electromechanical coupling effect of the piezoelectric plate is considered and the governing equations and boundary conditions are derived using the principle of minimum total potential energy. The solution procedure is a three-dimensional multi-term extended Kantorovich method (3DMTEKM). The objective of this paper is to study coupling influence on the edge effects of piezolaminated plates with finite dimensions and arbitrary lay-ups under uniform axial strain. These results can provide a benchmark for checking the accuracy of the other numerical methods or two-dimensional laminate theories. To verify the accuracy of the 3DMTEKM, special cases such as cross-ply or symmetric laminates are investigated and the results are compared with other analytical solutions available in the literature. Excellent agreement is achieved and then other numerical results are presented for general cases. Numerical examples imply on the singular behavior of interlaminar normal/shear stresses and electric field strength components near the edges of the piezolaminated plates. The coupling influence on the free edge effect with respect to the lay-ups of piezoelectric plate is studied in several examples.     

Dynamics of Layered Structures with Bonded Piezoelectric Material

For the proper design of plate and shell composite structures with piezoelectric patches bonded to the surfaces of a host structure, appropriate finite elements are presented. The linear piezoelectric effect is taken into account so that the displacements and the electric potential serve as independent variables. The electric potential can be condensed on the element level and does not enter the system of equations. This condensation has the advantage of a more efficient calculation especially for dynamic problems. With dynamic examples the two formulations are compared with each other and with solutions from the literature, which prove the accuracy and the efficiency of the formulations.