Nonlinear modelling and analysis of thin piezoelectric plates: Buckling and post-buckling behaviour

Krommer, Michael; Vetyukova, Yury; Staudigl, Elisabeth

doi:10.12989/sss.2016.18.1.155

Cited by 10 publications

(11 citation statements)

References 32 publications

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“…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 semi-analytical approach to account for the dominating nonlinear terms, for example, studying the effect of geometric nonlinearities by simplified kinematics, e.g., using a von Kármán-type theory, combined with a low-order 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 snap-through / snap-back, but also snap buckling behavior can be observed.…”

Section: A Non-symmetric Stability Problemmentioning

confidence: 99%

A complete direct approach to nonlinear modeling of dielectric elastomer plates

Hansy-Staudigl

Krommer

Humer³

2019

Acta Mech

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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 neo-Hookean 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 extraction-type procedure from the three-dimensional 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 three-dimensional finite element solutions. A very good agreement is found. IntroductionThe 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 electro-active plates modeled as electro-elastic material surfaces.The general theory of elastic dielectrics, of which dielectric elastomers are a sub-class, 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 so-called smart or intelligent materials, with piezoelectric materials and electro-active polymers as prominent examples. Concerning the latter, we refer to, e.g., [13] or [14]. A practically important sub-class of electro-active polymers are dielectric elastomers, which are rubber-type 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 in-plane 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 E. Hansy-Staudigl · M. Krommer (B) TU Wien,

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Section: A Non-symmetric Stability Problemmentioning

confidence: 99%

A complete direct approach to nonlinear modeling of dielectric elastomer plates

Hansy-Staudigl

Krommer

Humer³

2019

Acta Mech

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“…[3]) in cylindrical coordinates reads Section forces are denoted by n i and section moments by m i .…”

Section: Modelingmentioning

confidence: 99%

“…The balance of momentum for plate level section forces and section moments (see e.g. [3]) in cylindrical coordinates reads…”

Section: Modelingmentioning

confidence: 99%

Steady state dynamics of pre‐stressed, piezoelectrically excited circular plates – a harmonic balance approach

Tóth

2019

Proc Appl Math and Mech

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“…This coupling results in a non-local formulation over the domain i , as it has been discussed in detail in [4] for a linear plate. As in (50) c > 0 and the enthalpy H is not positive definite, we cannot speak about the minimality problem in the presence of electrical unknowns, but experience shows that small negative eigenvalues of the stiffness matrix do not influence the convergence of the quasi-Newton scheme for solving the equation in (54). Dividing the domain into finite elements, we keep in mind that the function H Σ is integrable due to the continuity of the approximation.…”

Section: Elastic Forces and Stiffness Of The Elementmentioning

confidence: 99%

“…The unloading branch (dashed line in Fig. 10) is not identical with the loading one, which can be attributed to phenomena such as snap-through and snapback or snap-buckling; see Krommer and Irschik [53] and Krommer et al [54] for the discussion of such phenomena for the simpler problem of a piezoelectric plate. In any case, we found multiple equilibria in the vicinity of χ * , and we observed a small hysteresis loop.…”

Section: Rotationally Symmetric Deformationmentioning

confidence: 99%

Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells

2017

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We present a novel multistage hybrid asymptotic-direct approach to the modeling of the nonlinear behavior of thin shells with piezoelectric patches or layers, which is formulated in a holistic form for the first time in this paper. The key points of the approach are as follows: (1) the asymptotic reduction in the threedimensional linear theory of piezoelasticity for a thin plate; (2) a direct approach to geometrically nonlinear piezoelectric shells as material surfaces, which is justified and completed by demanding the mathematical equivalence of its linearized form with the asymptotic solution for a plate; and (3) the numerical analysis by means of an FE scheme based on the developed model of the reduced electromechanically coupled continuum. Our approach is illustrated by examples of static and steady-state analysis and verified with three-dimensional solutions computed with the commercially available FE code ABAQUS as well as by comparison with results reported in the literature.

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Nonlinear modelling and analysis of thin piezoelectric plates: Buckling and post-buckling behaviour

Cited by 10 publications

References 32 publications

A complete direct approach to nonlinear modeling of dielectric elastomer plates

A complete direct approach to nonlinear modeling of dielectric elastomer plates

Steady state dynamics of pre‐stressed, piezoelectrically excited circular plates – a harmonic balance approach

Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells

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