A complete direct approach to nonlinear modeling of dielectric elastomer plates
 92 Downloads
Abstract
In this paper, we present a complete direct approach to nonlinear modeling of thin plates, which are made of incompressible dielectric elastomers. In particular, the dielectric elastomers are assumed to exhibit a neoHookean elastic behavior, and the effect of electrostatic forces is incorporated by the purely electrical contribution to the augmented Helmholtz free energy. Our approach does not involve any extractiontype procedure from the threedimensional energy to derive the plate augmented free energy, but directly postulates the form of this energy for the structural plate problem treated in this paper. Results computed within the framework of this novel approach are compared to results available in the literature as well as to our own threedimensional finite element solutions. A very good agreement is found.
1 Introduction
The present paper is dedicated to the memory of Vladimir Vasilyevich Eliseev and his pioneering work on modern versions of the linear and nonlinear theories for thin elastic rods, plates and shells, for which he developed geometrically nonlinear equations in a compact tensorial form based on the principle of virtual work applied to material lines and surfaces. His most essential contributions to the topic of this paper can be found in [1, 2, 3, 4, 5, 6]. The present paper is based upon Eliseev’s work on modeling of thin plates and shells as material surfaces. In particular, we extend the direct approach for elastic shells he presented in [6] and that was further developed by Vetyukov [7] to the case of electroactive plates modeled as electroelastic material surfaces.
The general theory of elastic dielectrics, of which dielectric elastomers are a subclass, dates back to Toupin [8], and it has been further developed in, e.g., [9, 10, 11, 12]. Elastic dielectrics belong to the class of socalled smart or intelligent materials, with piezoelectric materials and electroactive polymers as prominent examples. Concerning the latter, we refer to, e.g., [13] or [14]. A practically important subclass of electroactive polymers are dielectric elastomers, which are rubbertype materials that exhibit a polarization when an external electric field is applied at electrodes mounted to its top and bottom surfaces. By this polarization, the electrodes get attracted due to the corresponding electrostatic forces, such that the resulting squeezing yields large inplane deformations. This property is used for actuation, see, e.g., [15, 16, 17, 18, 19] for a survey on soft robotics, as well as for sensing applications. This makes dielectric elastomers a promising technology, posing a soft alternative to piezoelectric ceramics. For threedimensional Eulerian and Lagrangian formulations, we refer to [20, 21]. Given the typical applications, devices made of dielectric elastomers are mostly put into practice in the form of thin films, such that a structural mechanics approach is well suited, which motivates our modeling approach for dielectric elastomer plates as electroelastic material surfaces with mechanical and electrical degrees of freedom. In general problems of dielectric elastomer actuators, numerical methods, such as the finite element method, are applied implementing solid elements for general threedimensional problems [21, 22, 23, 24] or solid shell elements to account for the typical thinness of the dielectric elastomer actuators, as developed in [25]. Within the threedimensional modeling framework for dielectric elastomers, the works [26, 27] are just a few examples of the exhaustive literature. Multiple extensions—e.g., to electroviscoelasticity treated in [28] or electrostriction in [29]—have been reported as well.
Concerning the modeling of elastic plates and shells in a geometrically nonlinear framework, we refer the reader to the above listed papers by Eliseev as well as to [30, 31, 32]. Shelltype finite element models for electroactive polymers were investigated in [33], electroelastic coupling of dielectric elastomers in combination with studies on diverse failure phenomena, e.g., pullin instability and the formation of wrinkles, was addressed in [34], and further contributions to these topics can be found in [35, 36]. With respect to our own contributions to the field, we only mention recent works related to modeling of electroactive plates and shells as electroelastic material surfaces. In [37] we have studied piezoelectric shells, and in [38] dielectric elastomer plates were investigated. Twodimensional constitutive relations for the plate were derived by numerical integration of a threedimensional augmented free energy through the plate thickness imposing a plane stress assumption and an a priori assumption concerning the distribution of the strain through the thickness of the plate in [38]; such an approach has also been used in [39] for hyperelastic shells.
The present paper is structured as follows. After a brief summary of the threedimensional theory of dielectric elastomers, the concept of electroelastic plates as material surfaces is developed in detail. We start with the material independent equations and focus on the direct derivation of a plate augmented free energy for the electroelastic material surface afterward. This derivation is based on the polar decomposition of the surface deformation gradient tensor, which enables an additive decomposition of the augmented free energy into a membrane part, a bending part, and an electrical part. The latter is shown to account for the electrostatic forces responsible for the actuation behavior of the plate. A numerical implementation completes the theoretical part of the paper. Results computed within the framework of this novel approach are compared to results available in the literature as well as to our own threedimensional finite element solutions. A very good agreement is found.
2 Threedimensional formulation
2.1 Constitutive modeling for isotropic dielectric elastomers
3 Electroelastic plates as material surfaces
In this Section, we discuss the governing equations of a thin plate modeled as a twodimensional electroelastic material surface with mechanical and electrical degrees of freedom. For details concerning these equations, we refer the reader to [37, 38]. In particular, we consider the plate as a twodimensional continuum with five mechanical degrees of freedom, three translations and two rotations. This resembles the notion of a single rigid director \({\varvec{d}}\) with \({\varvec{d}} \cdot {\varvec{d}} = 1\) attached to each particle of the plate, which results in a Reissnertype theory; see [40]. Furthermore, we take the director to correspond to the unit normal vector of the material surface, \({\varvec{d}} = {\varvec{n}}\); then, we obtain a Kirchhoff–Love theory, see [41]. Concerning the electrical degrees of freedom, we use only the dominant one, i.e., the electric potential difference V.
3.1 Strain measures
3.2 Material independent equations
3.3 Constitutive modeling for isotropic dielectric elastomer plates
3.3.1 Augmented free energy
3.3.2 Constitutive relations revisited

Reference configuration: In the undeformed reference configuration, we have \({\mathbf {C}} = {\mathbf {I}}\) and \({\mathbf {K}} = {\mathbf {0}}\), such that \({\mathbf {n}} = {\mathbf {n}}_{\mathrm{es},0} =  q_0 V {\mathbf {I}}\), \({\mathbf {m}} = {\mathbf {m}}_{\mathrm{es},0} =  m^* {\mathbf {I}}\), and \(q = q_0 = c V\) hold. The elastic stress measures are zero in the reference configuration.
 Intermediate configuration: The intermediate configuration results from \({\mathbf {U}}\), and the strain measures are \({\mathbf {C}} = {\mathbf {U}}^2\) and \({\mathbf {K}} = {\mathbf {0}}\). In this scenario we have \({\mathbf {n}} = {\mathbf {n}}_{e} + {\mathbf {n}}_{\mathrm{es},m} = {\mathbf {n}}_{e}  q_m V {\mathbf {C}}^{1}\), \({\mathbf {m}} = {\mathbf {m}}_{\mathrm{es},m} =  m^* {\mathbf {I}} (\text {det} {\mathbf {C}})\), and \(q = q_m = q_0 \text {det} {\mathbf {C}}\) with the elastic stress measure \({\mathbf {m}}_{e} = {\mathbf {0}}\) and$$\begin{aligned} {\mathbf {n}}_{e} = {\mathbf {n}}_{e,m} = \frac{A}{4} \left( {\mathbf {I}}  (\text {det} {\mathbf {C}})^{1} {\mathbf {C}}^{1} \right) = 2 \frac{\partial W_m}{\partial {\mathbf {C}}}. \end{aligned}$$(71)
 Actual configuration: Eventually the actual configuration involves the two strain measures \({\mathbf {C}} = {\mathbf {U}}^2\) and \({\mathbf {K}} =  {\mathbf {R}}^T \cdot {\mathbf {b}} \cdot {\mathbf {R}}\), with \({\mathbf {n}} = {\mathbf {n}}_{e} + {\mathbf {n}}_{\mathrm{es},m} + {\mathbf {n}}_{\mathrm{es},b} = {\mathbf {n}}_{e}  q V {\mathbf {C}}^{1}\), \({\mathbf {m}} = {\mathbf {m}}_{e} + {\mathbf {m}}_{\mathrm{es},m} = {\mathbf {m}}_{e}  m^* {\mathbf {I}} (\text {det} {\mathbf {C}})\) and \(q = q_m + q_b = q_0 \left( 1 + 2 \alpha \text {tr} {\mathbf {K}} \right) \text {det} {\mathbf {C}}\). The elastic stress measures are$$\begin{aligned} {\mathbf {n}}_{e}&= {\mathbf {n}}_{e,m} + {\mathbf {n}}_{e,b} = {\mathbf {n}}_{e,m}  (\text {det} {\mathbf {C}})^{1} D \left( (\text {tr} {\mathbf {K}})^2 \text {det} {\mathbf {K}} \right) = {\mathbf {n}}_{e,m} + 2 \frac{\partial W_b}{\partial {\mathbf {C}}}, \nonumber \\ {\mathbf {m}}_{e}&= {\mathbf {m}}_{e,b} = (\text {det} {\mathbf {C}})^{1} \frac{1}{2} D\left( {\mathbf {I}} \text {tr} {\mathbf {K}} + {\mathbf {K}} \right) = \frac{\partial W_b}{\partial {\mathbf {K}}}. \end{aligned}$$(72)
3.3.3 Small strain regime
4 Finite element implementations
4.1 Plate finite elements
In this Section, we briefly introduce the implementation of finite elements for the numerical solution of the electromechanically coupled plate problem. The present version of the classical Kirchhoff–Love theory of plates requires in general \(C^1\) continuity in the approximation of the deformed surface, which we achieve using a fournode finite element with the following approximation scheme: 16 shape functions for each spatial component of the position vector exactly represent any bicubic polynomial; the element thus has 48 mechanical degrees of freedom. Despite inherent restrictions concerning the topology of the mesh and connections between shell segments, the present finite element has a relatively broad spectrum of potential applications with respect to both, research and development. For more details concerning this shell element, we refer to [32].
4.1.1 Implementation
Finally, we seek for a stationary value of the total energy functional, \(\varSigma = \varSigma ^{\varOmega } + \varSigma ^\mathrm{ext}\). Here, \(\varSigma = \varSigma ({\varvec{r}}(q^\alpha ))\) depends on the field \({\varvec{r}}(q^\alpha )\), which is approximated using Eq. (85). To compute the total energy functional numerical integration over the elements, with \(3 \times 3\) integration points per element, is used. This results in nonlinear algebraic equations, from which equilibrium solutions are computed numerically by employing Newton’s method for seeking the stationary points of \(\varSigma \).
4.1.2 Boundary conditions
If an edge is free from kinematic constraints, then the external force factors acting on that edge need to be accounted for; as we are only considering problems without external loadings at such edges this is trivial. At a simply supported edge, the material points are fixed by appropriate penalty terms for the nodal positions \({\varvec{r}}^m\) and the derivatives \({\varvec{r}}^m_\alpha \) (\(\alpha \) corresponds to the direction along the edge). If the edge is clamped, then the direction of the normal vector \({\varvec{n}}\) needs to be additionally constrained. For a straight edge \({\varvec{n}} = \text {const}\), and the constraint will be fulfilled exactly, if we demand \({\varvec{N}} \cdot {\varvec{r}}^m_\beta =0\) and \({\varvec{N}} \cdot {\varvec{r}}^m_{12}=0\), in which \(\beta \) corresponds to the direction of the outer normal to the boundary of the domain.
4.2 Solid finite elements
As we will be using the threedimensional formulation to validate the plate formulation, solid finite elements are implemented for the incompressible neoHookean dielectric elastomer with the augmented free energy as given in Eq. (21). To account for the incompressibility of the material response, a mixed formulation, in which the displacement field \({\varvec{u}}_3\) from the referential position \({\varvec{R}}_3\) to the actual position \({\varvec{r}}_3\), i.e., \({\varvec{u}}_3 = {\varvec{r}}_3  {\varvec{R}}_3\), and the pressure p serve as variables (\(\varvec{u}\)p form), see, e.g., Zienkiewicz et al. [48]. In the electric domain, we employ the electric potential \(\varphi _3\) as independent variable, from which the material electric field is obtained as its material gradient, \({\varvec{\mathcal {E}}}_3 =  \nabla _3 \varphi _3\).
4.2.1 Implementation
For the numerical analysis, we use the opensource multipurpose finite element code Netgen/NGSolve.^{1} Due to their thinwalled nature, the structures in the subsequent problems are discretized with prism elements, which are naturally aligned with the thickness direction. Both the components of the displacement field and the electric potential are approximated by means of hierarchical quadratic shape functions, whereas the pressure field is linearly interpolated. To reduce the number of unknowns, we employ symmetry conditions wherever applicable. Throughout the subsequent examples, a discretization of the thickness direction into eight equidistant layers of elements has proven to be sufficient. Regarding the (unstructured) triangular inplane discretization, the maximum length of an element edge is restricted to be not larger than a tenth of the smaller inplane dimension of the respective structure (with symmetry already accounted for). We use Newton’s method to solve the nonlinear boundary value problem that is obtained by requiring stationarity of the augmented free energy in Eq. (21).
4.2.2 Boundary conditions
For the threedimensional model, simply supported boundary conditions are realized by constraining the displacements along the respective edge on the center plane of the structure. In the proposed plate formulation, the displacement of the center plane is prohibited at clamped boundaries, whereas a deformation in thickness direction is not constrained. For this reason, the displacement in the thickness direction is only constrained at the edge on the center plane of the structure. Any displacement parallel to the center plane, however, is prohibited unless otherwise stated.
5 Validation
Material parameters
\(\mu _1\)/Pa  \(\mu _2\)/Pa  \(\mu _3\)/Pa  \(\alpha _1\)  \(\alpha _2\)  \(\alpha _3\)  \(\mu \)/Pa  \(\varepsilon _r\) 

\(5.49 \times 10^{4}\)  \(9.1 \times 10^{2}\)  \(6.3\)  0.7  3.25  \(3.7\)  20, 698  4.7 
5.1 Stability of a singlelayer dielectric elastomer plate
Critical buckling voltage \(V_\mathrm{crit,buckling}\)
ShellFE3  Netgen/NGSolve  [25]  

\(V_\mathrm{crit,buckling}\)/V  2.700  2.709  2.829 
Deviation  –  \(0.33\%\)  \(4.77\%\) 
5.2 Bending of a bilayer dielectric elastomer plate
5.3 A nonsymmetric stability problem
First, we discuss the stability behavior for a constant voltage \(V>V_\mathrm{crit}\) and for a varying mechanical pressure in more detail; see also the right graph in the bottom row of Fig. 7. Due to the applied voltage, the plate deforms into a configuration with center point deflections \(w_\mathrm{mid}>0\). Applying a mechanical pressure in negative thickness direction \(w_\mathrm{mid}\) decreases continuously until snapping to a configuration with negative values of \(w_\mathrm{mid}\) occurs. Upon decreasing the pressure backsnapping occurs at a pressure value smaller than the value for snapping. This behavior occurs for any values \(V>V_\mathrm{crit}\) with the critical pressure values depending intrinsically on the magnitude of the applied voltage. Next, the pressure load factor \(\lambda \) is held constant while the voltage V is varied; in this case bistable solutions exist for \(\lambda >\lambda _\mathrm{crit}\), see the left graph in the bottom row of Fig. 7. Again snapping upon loading and backsnapping upon unloading is observed. Figure 7 shows results computed with ShellFE3 (solid lines) and Netgen/NGSolve (solid circular markers) with a very good agreement between them. A practical application of the bistable behavior might be, for example, a voltage sensitive switching device, where the necessary pressure level for snapping can be adjusted by choosing the thickness of the dielectric elastomer switch properly.
Finally, we note that a structural mechanics framework—as developed in this paper—is especially valuable when it comes to the analysis of stability problems using a semianalytical approach to account for the dominating nonlinear terms, for example, studying the effect of geometric nonlinearities by simplified kinematics, e.g., using a von Kármántype theory, combined with a loworder Ritz approximation. Examples of this approach for the case of piezoelectric or thermoelastic plates can be found in [49, 50] or [51] identifying that in similar examples not only buckling and snapthrough / snapback, but also snap buckling behavior can be observed.
6 Conclusions
In this paper we presented a complete direct approach for modeling geometrically and physically nonlinear dielectric elastomer plates as twodimensional electroelastic material surfaces. In contrast to common approaches, our formulation does not rely on the threedimensional theory, but it directly develops the twodimensional one. In particular, the polar decomposition of the surface deformation gradient tensor was used to postulate the augmented free energy of the material surface.
Numerical results computed with plate finite elements for the present novel theory were compared to 3D finite elements implemented in Netgen/NGSolve, which are based on the threedimensional counterpart to our theory. A very good agreement between the results was found in all example problems.
Footnotes
 1.
Available for download at https://www.ngsolve.org.
Notes
Acknowledgements
Open access funding provided by TU Wien (TUW). A. Humer acknowledges the support by the COMETK2 Center of the Linz Center of Mechatronics (LCM) funded by the Austrian federal government and the federal state of Upper Austria. The authors also wish to thank Astrid Pechstein for providing numerical results published in [24].
References
 1.Eliseev, V.V.: The nonlinear dynamics of elastic rods. J. Appl. Math. Mech. 52, 493–498 (1988)MathSciNetCrossRefGoogle Scholar
 2.Eliseev, V.V.: Constitutive equations for elastic prismatic bars. Mech. Solids 24(1), 70–75 (1989)MathSciNetGoogle Scholar
 3.Eliseev, V.V.: Saint Venant problem and elastic moduli for bars with curvature and torsion. Mech. Solids 26(2), 167–176 (1991)Google Scholar
 4.Eliseev, V.V.: Mechanics of Elastic Bodies. Petersburg State Polytechnical University Publishing House, Saint Petersburg (1999). (in Russian)Google Scholar
 5.Eliseev, V.V.: Mechanics of Deformable Solid Bodies. St. Petersburg State Polytechnical University Publishing House, Saint Petersburg (2006). (in Russian)Google Scholar
 6.Eliseev, V.V., Vetyukov, Y.: Finite deformation of thin shells in the context of analytical mechanics of material surfaces. Acta Mech. 209, 43–57 (2010)CrossRefGoogle Scholar
 7.Vetyukov, Y.: Nonlinear Mechanics of ThinWalled Structures: Asymptotics, Direct Approach and Numerical Analysis. Springer, Vienna, NY (2014)CrossRefGoogle Scholar
 8.Toupin, R.A.: The elastic dielectric. J. Ration. Mech. Anal. 5(6), 849–915 (1956)MathSciNetzbMATHGoogle Scholar
 9.Pao, Y.H.: Electromagnetic Forces in Deformable Continua. In: NematNasser, S. (ed.) Mechanics Today 4, pp. 209–306. Pergamon Press, Oxford (1978)CrossRefGoogle Scholar
 10.Prechtl, A.: Eine Kontinuumstheorie elastischer Dielektrika. Teil 1: Grundgleichungen und allgemeine Materialbeziehungen. Archiv für Elektrotechnik 65(3), 167–177 (1982). (in German)MathSciNetCrossRefGoogle Scholar
 11.Prechtl, A.: Eine Kontinuumstheorie elastischer Dielektrika. Teil 2: Elektroelastische und elastooptische Erscheinungen. Archiv für Elektrotechnik 65(4), 185–194 (1982). (in German)MathSciNetCrossRefGoogle Scholar
 12.Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. NorthHolland, Amsterdam (1988)zbMATHGoogle Scholar
 13.Pelrine, R.E., Kornbluh, R.D., Joseph, J.P.: Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation. Sens. Actuators A 64, 77–85 (1998)CrossRefGoogle Scholar
 14.BarCohen, Y.: Electroactive Polymer (EAP) Actuators as Artificial Muscles: Reality, Potential, and Challenges. SPIE, Bellingham, WA (2004)Google Scholar
 15.Choi, H.R., Jung, K., Ryew, S., Nam, J.D., Jeon, J., Koo, J.C., Tanie, K.: Biomimetic soft actuator: design, modeling, control, and applications. IEEE/ASME Trans. Mechatron. 10, 581–593 (2005)CrossRefGoogle Scholar
 16.Carpi, F., Migliore, A., Serra, G., Rossi, D.D.: Helical dielectric elastomer actuators. Smart Mater. Struct. 14, 1–7 (2005)CrossRefGoogle Scholar
 17.Carpi, F., Salaris, C., Rossi, D.D.: Folded dielectric elastomer actuators. Smart Mater. Struct. 16, S300–S305 (2007)CrossRefGoogle Scholar
 18.Arora, S., Ghosh, T., Muth, J.: Dielectric elastomer based prototype fiber actuators. Sens. Actuators A 136, 321–328 (2007)CrossRefGoogle Scholar
 19.Gu, G.Y., Zhu, J., Zhu, L.M., Zhu, X.: A survey on dielectric elastomer actuators for soft robots. Bioinspir. Biomime. 12(1), 011003 (2017)CrossRefGoogle Scholar
 20.Dorfmann, A., Ogden, R.W.: Nonlinear electroelasticity. Acta Mech. 174, 167–183 (2005)CrossRefGoogle Scholar
 21.Vu, D.K., Steinmann, P., Possart, G.: Numerical modelling of nonlinear electroelasticity. Int. J. Numer. Meth. Eng. 70, 685–704 (2007)MathSciNetCrossRefGoogle Scholar
 22.Gao, Z., Tuncer, A., Cuitiño, A.: Modeling and simulation of the coupled mechanicalelectrical response of soft solids. Int. J. Plast. 27(10), 1459–1470 (2011)CrossRefGoogle Scholar
 23.Skatulla, S., Sansour, C., Arockiarajan, A.: A multiplicative approach for nonlinear electroelasticity. Comput. Methods Appl. Mech. Eng. 245–246, 243–255 (2012)MathSciNetCrossRefGoogle Scholar
 24.Pechstein, A.: Large deformation mixed finite elements for smart structures. Mech. Adv. Mater. Struct. (2019). https://doi.org/10.1080/15376494.2018.1536932
 25.Klinkel, S., Zwecker, S., Mueller, R.: A solid shell finite element formulation for dielectric elastomers. J. Appl. Mech. 80, 0210261–02102611 (2013)CrossRefGoogle Scholar
 26.McMeeking, R.M., Landis, C.M.: Electrostatic forces and stored energy for deformable dielectric materials. J. Appl. Mech. 72(4), 581–590 (2005)CrossRefGoogle Scholar
 27.Mehnert, M., Hossain, M., Steinmann, P.: On nonlinear thermoelectroelasticity. Proc. R. Soc. A Math. Phys. Eng. Sci. (2016). https://doi.org/10.1098/rspa.2016.0170 MathSciNetCrossRefGoogle Scholar
 28.Mehnert, M., Hossain, M., Steinmann, P.: Numerical modeling of thermoelectroviscoelasticity with fielddependent material parameters. Int. J. NonLinear Mech. 106, 13–24 (2018)CrossRefGoogle Scholar
 29.Zäh, D., Miehe, C.: Multiplicative electroelasticity of electroactive polymers accounting for micromechanicallybased network models. Comput. Methods Appl. Mech. Eng. 286, 394–421 (2015)MathSciNetCrossRefGoogle Scholar
 30.Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
 31.Berdichevsky, V.L.: Variational Principles of Continuum Mechanics: I. Springer, Berlin, Heidelberg (2009)CrossRefGoogle Scholar
 32.Vetyukov, Y.: Finite element modeling of Kirchhoff–Love shells as smooth material surfaces. ZAMM 94, 150–163 (2014)MathSciNetCrossRefGoogle Scholar
 33.Ortigosa, R., Gil, A.J.: A computational framework for incompressible electromechanics based on convex multivariable strain energies for geometrically exact shell theory. Comput. Methods Appl. Mech. Eng. 317, 792–816 (2017)MathSciNetCrossRefGoogle Scholar
 34.Poya, R., Gil, A.J., Ortigosa, R., Sevilla, R., Bonet, J., Wall, W.A.: A curvilinear high order finite element framework for electromechanics: From linearised electroelasticity to massively deformable dielectric elastomers. Comput. Methods Appl. Mech. Eng. 329, 75–117 (2018)MathSciNetCrossRefGoogle Scholar
 35.Greaney, P., Meere, M., Zurlo, G.: The outofplane behaviour of dielectric membranes: Description of wrinkling and pullin instabilities. J. Mech. Phys. Solids 122, 84–97 (2019)MathSciNetCrossRefGoogle Scholar
 36.Su, Y., Conroy Broderick, H., Chen, W., Destrade, M.: Wrinkles in soft dielectric plates. J. Mech. Phys. Solids 119, 298–318 (2016)MathSciNetCrossRefGoogle Scholar
 37.Vetyukov, Y., Staudigl, E., Krommer, M.: Hybrid asymptoticdirect approach to finite deformations of electromechanically coupled piezoelectric shells. Acta Mech. 229(2), 953–974 (2018)MathSciNetCrossRefGoogle Scholar
 38.Staudigl, E., Krommer, M., Vetyukov, Y.: Finite deformations of thin plates made of dielectric elastomers: modeling, numerics and stability. J. Intell. Mater. Syst. Struct. 29(17), 3495–3513 (2018)CrossRefGoogle Scholar
 39.Kiendl, J., Hsu, M.C., Wu, M.C., Reali, A.: Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials. Comput. Methods Appl. Mech. Eng. 291, 280–303 (2015)MathSciNetCrossRefGoogle Scholar
 40.Naghdi, P.: The theory of shells and plates. In: Flügge, S., Truesdell, C. (eds.) Handbuch der Physik VIa/2, pp. 425–640. Springer, Berlin (1972)Google Scholar
 41.Altenbach, H., Eremeyev, V.A.: CosseratType Shells. In: Generalized Continua from the Theory to Engineering Applications, Altenbach, H, Eremeyev, V.A. (eds.), CISM International Centre for Mechanical Sciences (Courses and Lectures) 541, Springer, Vienna (2013)CrossRefGoogle Scholar
 42.Ciarlet, P.: An introduction to differential geometry with applications to elasticity. J. Elast. 1–3, 1–215 (2005)MathSciNetCrossRefGoogle Scholar
 43.Duong, T.X., Roohbakhshan, F., Sauer, R.A.: A new rotationfree isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries. Comput. Methods Appl. Mech. Eng. 316, 43–83 (2017)MathSciNetCrossRefGoogle Scholar
 44.Nemenyi, P.: Eigenspannungen und Eigenspannungsquellen. ZAMM 11, 1–8 (1931)CrossRefGoogle Scholar
 45.Reissner, H.: Selbstspannungen elastischer Gebilde. ZAMM 11, 59–70 (1931)CrossRefGoogle Scholar
 46.Koiter, W.T.: On the nonlinear theory of thin elastic shells. Proc. Koninklijke Nederlandse Akademie van Wetenschappen B 69, 1–54 (1966)MathSciNetGoogle Scholar
 47.Ziegler, F.: Mechanics of Solids and Fluids, 2nd edn. Springer, Vienna, New York (1998)zbMATHGoogle Scholar
 48.Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 6th edn. Elsevier, Amsterdam (2005)zbMATHGoogle Scholar
 49.Heuer, R., Irschik, H., Ziegler, F.: Nonlinear random vibrations of thermally buckled skew plates. Probab. Eng. Mech. 8, 265–271 (1993)CrossRefGoogle Scholar
 50.Irschik, H.: Large thermoelastic deflections and stability of simply supported polygonal panels. Acta Mech. 59, 31–46 (1986)CrossRefGoogle Scholar
 51.Krommer, M., Vetyukov, Y., Staudigl, E.: Nonlinear modelling and analysis of thin piezoelectric plates: buckling and postbuckling behaviour. Smart Struct. Syst. 18(1), 155–181 (2016)CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.