A finite element formulation for piezoelectric shell structures considering geometrical and material non-linearities
Abstract: In this paper, we present a non-linear finite element formulation for piezoelectric shell structures. Based on a mixed multi-field variational formulation, an electro-mechanical coupled shell element is developed considering geometrically and materially non-linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non-linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three-dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non-linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature-dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature-dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.
Summary (5 min read)
- Piezoelectric material plays an important role for sensor and actuator devices.
- In order to consider the nonlinear material behavior in a classical shell formulation, the strain and the electric field in thickness direction have to be comprised.
- Thus, temperature can influence the performance of piezoelectric shell structures due to a change of the temperature dependent material parameters.
- Here the change of the saturation parameters of the polarization and the strain due to temperature is phenomenologically included, thus temperature-dependent hysteresis curves can be determined.
- The authors obtain the corresponding inextensible director vector d of the current configuration with the rotation tensor R by the orthogonal transformation d = RD.
- Commas denote a partial differentiation with respect to the coordinates ξα.
- Due to the shell geometry, the authors assume that the piezoelectric material is poled in thickness direction and the electrodes are arranged at the lower and upper surface.
3 Constitutive equations
- The authors introduce linear constitutive equations with the Green-Lagrangean strain E, the Lagrangean electric field E, the second Piola-Kirchhoff stresses S, and the dielectric displacements D. Focusing on their material model, here they neglect thermal stresses and pyroelectric effects.
- (8) The strains and the electric fields are summarized in the vector ε.
- The three dimensional elasticity matrix , the permittivity matrix , and the piezoelectric coupling modulus are arranged in ̄.
- In , the authors assume transversal isotropic material behavior with isotropy in the 23-plane, which can be specified with five independent parameters, see .
- The stress and dielectric displacement in thickness direction are defined as zero, thus the authors fulfill the normal zero stress condition of shells.
4.1 Nonlinear constitutive equations
- Ferroelectric ceramics show strong nonlinear behavior under high electric fields.
- The imprinted initial polarization changes its direction under high electric loading and shows the dielectric hysteresis.
- Thus, the linear constitutive behavior according to (8) has to be set up under consideration of the current state of polarization.
- P i,rel characterizes the part of the piezoelectric material that shows a macroscopic polarization.
- Since the switching effects of the ferroelectric domains can be treated as a volume conserving process, the irreversible strain Ei can be determined as proposed by Reference .
4.3 Interpretation for ferroelectric materials
- To interpret the Preisach model for ferroelectric hysteresis phenomena, the input and output variables have to be identified.
- The loading parameter for piezoelectric devices is the electric field E. With the parameter of the material specific saturation value of the electric field Esat the normalized value Erel is chosen as x(t) = Erel = ‖ E‖ Esat . (24) The corresponding output quantity is chosen as the normalized polarization P i,rel y(t) =.
- As a phenomenological model, the Preisach concept adjusts the final hysteresis form by means of an experimental determined function.
- For a detailed description and a discussion regarding the choice of the Preisach function see e.g. References [49, 50, 47, 51].
- It is remarked that the polarization output value denotes the normalized irreversible polarization.
4.4 Temperature-dependent hysteresis
- The influence of the temperature on the saturation parameter of the polarization P sat and the electric coercitive field Ec has been experimentally studied by Reference  for Pb(Zn1/3Nb2/3).
- O3− PbT iO3 single crystals and by Reference  for PZT ceramics.
- Following the experimental investigations in Reference  the authors assume a linear relation between the natural logarithms of the temperature and the saturation polarization.
- As there do not exist sufficient test results, this influence has been neglected.
5 Variational formulation
- The generalized stress tensor σ containing the second Piola-Kirchhoff stresses reads σ = 2∂CŴ (C).
- The virtual quantities of v and the independently assumed strains, electric field, stresses, and dielectric displace- ments summarized in the generalized electromechanic fields ε̂(Ê, ̂E) and σ̂(Ŝ, ̂D) arrive to δv(δu, δω, δΔϕ), δε̂(δÊ, δ̂E) and δσ̂(δŜ, δ ̂D).
6 Finite element approximation
- The finite element formulation models the shell structure by a reference surface.
- The nodal position vector XI and the local cartesian coordinate system [A1I ,A2I ,A3I ] are generated with the mesh input.
- Here, t3 represents the normal vector in the midpoint of the element.
- The authors assume that the shell structure only counts for an electric potential in thickness direction of the shell by means of electrodes on the upper and lower surface of the shell.
- The element has to fulfill the patch tests.
6.1 Interpolation of the assumed strains and electric field
- The independent fields of the strains and the electric field are interpolated by ˆ̄ε =.
- Here ˆ̄ε characterizes the complete vector of the assumed strains and the assumed electric fields, whereas ε̂ specifies the reduced vector without the components in thickness direction.
- The area element dA = j dξdη is given with j(ξ, η) = |Xh,ξ ×Xh,η |. The matrix Neas contains parameters that are set orthogonal to the interpolations of the stresses, which is similar to the enhanced strain formulation given by Reference .
6.3 Approximation of the weak form and linearization
- The authors incorporate the interpolations of the strains, the electric fields, the stress resultants, and the dielectric displacements in equation (58) and formulate the approximation of the variational formulation on element level as G(θ, δθ) = δvTe ∫ Ωe [ BTNσ β − fa ].
- Considering nonlinear structural and material behavior, this formulation has to be linearized.
- The authors simplify the formulation and define the following element matrices kg[18×18] = ∫.
6.4 Actuator formulation
- In order to deal with the actuator use of piezoelectric shell structures, the authors postulate a linear distribution of the electric potential in thickness direction.
- The authors write the corresponding electric field in thickness direction as the average value for every element.
- According to the gradient relation, see equation (4), the electric potential is divided through the thickness.
- The basic benchmark test of a finite element formulation is the well known patch test.
- The test is passed if the formulation is able to display a state of constant stresses and constant dielectric displacements along with constant strains and a constant electric field for distributed element geometry.
- The geometry of the quadratic patch with distorted elements inside the patch is shown in Figure 5.
- For the shear test the same nodes are subjected to load F3 in 3-direction.
- The loadings and the boundary conditions are depicted in Figure 5.
7.2 Piezoelectric bimorph
- The piezoelectric bimorph is a well known piezoelectric benchmark test in order to proof the numerical formulation to the general applicability for sensor and actuator systems.
- For the discretisation, five elements are chosen, which correspond to five pairs of electrodes that are put along the length of the cantilever.
- Due to the deflection, an electric potential arises.
- The element ”H8D” has independent variables for the displacement, the electric potential and the dielectric displacements, in ”H8DS” also the stresses are included.
- The present shell element does not show any shear locking, thus the results fit the analytical solution even for strong distorted meshes.
7.3 90◦ cylindrical shell
- The system, see Figure 9, consists of four graphite epoxy layers, for which the authors account the orientation angles ϕF with [0/90/90/0] referring to the x axis.
- The geometry and the material parameters for the graphite epoxy and PZT layers according to Balamurugan and Narayanan  are given in Table 2.
- The radial system displacement w is measured along the centerline at b/2.
- The authors compare the present shell formulation with data from Balamurugan and Narayanan  for a degenerated nine-node quadrilateral shell element with quadratic approach for the electric potential in thickness direction and with Saravanos  who provides a laminated eight-node shell element with lineare electric thickness potential.
- The good accordance of the results shows the reliable applicability of the present laminated four node formulation for layered piezo-mechanical structures.
7.4 Steering of an antenna
- The authors show two versions of an antenna that can be manipulated via piezoelectric devices.
- Two piezoelectric patches with the width b and the length l are arranged with the distance a from the small hole of 2◦ in the middle of the antenna shell.
- The authors compare those results to the experimental data and a numerical calculation with a reduced eight-node element by Gupta et al. .
- Figure 12(b) shows the displacement curve dependent on the angle around the middle axis of the antenna.
- Here, the authors distinguish between the following loading cases 1.
7.5 Test of the Preisach model for ferroelectric hysteresis
- In order to validate the results of the temperature-dependent Preisach model for the ferro- electric hysteresis effects, a simple material cube made of soft PZT with an edge length a, see Figure 16, is chosen.
- Besides experimental results also a micro-mechanical model is introduced in .
- For the micro-mechanical calculation, the material paramters in  are derived from the single crystal parameters of barium titanate and are modified via a correction parameter to get the material parameter of soft PZT single crystal.
- Figure 17 presents the dielectric hysteresis and the butterfly hysteresis for a temperature of 25◦C.
- However above the saturation values the reversible part is underestimated.
7.6 Piezoelectric ceramic disc
- A thin piezoelectric PZT ceramic disc is introduced by Yimnirun et al.  who gives experimental results for the temperature-dependent polarization behavior, see Figure 19.
- For the numerical calculation the authors choose the three representative temperatures 298K, 373K and 453K.
- To simplify the calculation, the authors additionally introduce a small hole at the disc center characterized by the diameter d2, which lets the polarization change unaffected and thus does not influence the dielectric hysteresis curve.
- The missing parameters of the permittivity are added from experienced data.
- The reversible polarization is underestimated compared to the experiment and shows a nonlinear behavior.
7.7 Telescopic cylinder
- The quite small displacements of piezoelectric structures can be enlarged by special architectures.
- With several nested cylinders, which are alternately connected at the top and the bottom, actuators with much higher displacements and be composed.
- The radii of the cylinders numerated from the inner to the outer tube are r1, r2, r3, r4 and r5 and correspond to the middle surface of every cylinder.
- With respect to the material nonlinear behavior, it results a hysteresis curve for the maximal displacement w, which is displayed in Figure 22 together with the experimental data from .
- An open question is how the experimental results reach the initial value of the remanent displacement again after −1200V for the first subhysteresis ±300V .
- The authors have presented a piezoelectric finite shell element.
- The mixed hybrid formulation includes independent thickness strains, which allows a consideration of three-dimensional nonlinear constitutive equations.
- By means of a temperature-dependent Preisach model the authors consider the actual polarization state and thus they incorporate ferroelectric hysteresis phenomena.
- With only one electrical degree of freedom, the formulation simulates the behavior of both piezoelectric sensor and actuator systems appropriately.
- The presented examples show the influence of the temperature for the ferroelectric nonlinear behavior.
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Cites background or methods from "A finite element formulation for pi..."
...Although several macroscale formulations were developed for piezoelectric shell elements [27, 12, 13], computational homogenization of shells is only recently receiving major attention ....
...3, following , which in turn is largely based on Naghdi’s theory for the mechanical part [33, 36]....
...where 33, (1)33 are the constant and linear components of the thickness strain, while ~ E3, ~ E(1)3 represent the constant and linear parts of the electric field in the thickness direction ....
...For more details, see the original paper ....
...Herein, a linear approximation is adopted, which is sufficient to pass the out-of-plane bending patch test ....
Cites background from "A finite element formulation for pi..."
...Schulz et al. (2011) and Zemčı́k et al. (2007) proposed a high-performance four-node flat shell element with piezoelectric coupling for the dynamic analysis of lightweight smart shell structures....
"A finite element formulation for pi..." refers background in this paper
...Preisach , in order to analyze nonlinear behavior of ferromagnetic material....
...The matrix Neas contains parameters that are set orthogonal to the interpolations of the stresses, which is similar to the enhanced strain formulation given by Reference ....
"A finite element formulation for pi..." refers methods in this paper
...A well-known phenomenological model is the Preisach model, see Reference ....
...For ̄= ̄=0, the plane strain problem according to Reference  is solved....
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