Carrera, Erasmo

Cinefra, Maria

Valvano, Stefano

CINEFRA, MARIA

VALVANO, STEFANO

CARRERA, Erasmo

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEA Finite Elements with Continue Transverse Electric Displacement for the Electro-Mechanical Analysis of ShellStructures / Cinefra M.; Valvano S.; Carrera E.. - (2014). ((Intervento presentato al convegno DeMEASS VI tenutosi aEde, Netherlands nel 25-28 May 2014.OriginalA Finite Elements with Continue Transverse Electric Displacement for the Electro-Mechanical Analysisof Shell StructuresPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2547146 since: 2016-07-27T11:46:39ZA FINITE ELEMENTS WITH CONTINUETRANSVERSE ELECTRIC DISPLACEMENT FORTHE ELECTRO-MECHANICAL ANALYSIS OFSHELL STRUCTURESM. Cinefra ∗, S. Valvano †, E. Carrera ‡§Keywords:Piezoelectric materials, Multifield problems, Shell and Plate, Unified Formulation,Mixed Interpolated Tensorial Components.AbstractBecause of their capability to adapt and/or sense to the external environment, so-called smartstructures have attracted considerable research over the past few years. This capability leads toa wide range of applications, in particular in the aerospace field. Among materials that can beused, only piezoelectric ones have shown the capability to perform effectively both as actuators andsensors, and the possibility of a simple integration with composite structures. The present workdeals with the analysis of multilayered plates and shells embedding piezoelectric layers as actuatorsand sensors, see Fig. 1. Finite elements with layer-wise capabilities are employed to ensure anaccurate description of the mechanical and electric fields in the layers. It is essential to take intoaccount the discontinuity of the mechanical and electrical properties at the layer interfaces. For thesereasons, the use of classical plate theories based on Kirchhoff and Reissner-Mindlin hypotheses canlead to inaccurate results. Even if analytical models are available, the solution of practical problemsoften demand the use of computational methods such as the finite element method. To this aim, therefined shell elements, recently formulated by the authors on the basis of Carrera Unified Formulation(CUF), have been extended to the electro-mechanical problem. The governing equations are derivedusing the Reissner’s Mixed Varational Theorem (RMVT) extended to the electro-mechanical case,the original RMVT has been modified to account for ”only” interlaminar continous D3. Continuityof mechanical variables, such as transverse shear and normal stress components, has been discardedto provide a simple ”electrical” modified RMVT, here called (RMV T −D3) .∗Research Assistant, Aerospace Department, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy,e-mail: maria.cinefra@polito.it, website: www.mul2.com.†Aerospace Department, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy, e-mail: ste-fano.valvano@polito.it, website: www.mul2.com.‡Professor, Aerospace Department, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy, e-mail:erasmo.carrera@polito.it, website: www.mul2.com.§currently at School of Aerospace, Mechanical and Manufacturing Engineering College, RMIT University, Melbourne,Australia.1∫V(δTpG σpC + δTnG σnC − δETpGDpC − δETnGDnM − δDTnM (EnG − EnC))dV = δLe − δLinThis model has already shown good results in the electro-mechanical analysis of multilayered platesand shells with analytical code, and only plate with FEM code [1], [2]. The Mixed Interpolated Ten-sorial Components (MITC) method is employed to contrast the membrane-shear locking phenomenonthat usually affects shell finite elements. This formulation has already shown all its potentiality asa base for finite elements in the mechanical analysis of multilayered shells [3]. Moreover, plate finiteelements based on CUF for the analysis of electro-mechanical problem have been already presentedin [4]. One of the most interesting features of the unified formulation consists in the possibility tokeep the order of the expansion of the state variables along the thickness of the plate as a parameterof the model. In so doing, both equivalent single layer (ESL) and layer-wise (LW) descriptions of thevariables are allowed. The electrical potential assumption for the layered actuators and sensors hasbeen limited to a LW description while the displacements assumption on the composite layers bothequivalent single layer (ESL) and layer-wise (LW) descriptions are allowed. This feature is partic-ularly suitable since electric degrees of freedom (dofs) are often avoided for plate/shell elements ora simple through- thickness linear variation is assumed for the electric potential. Some results fromthe static and dynamic analysis of plates and shells under electro-mechanical loads will be provided,in order to show the efficiency of models presented.zβR RβFigure 1: Double curvature shell with piezoelectric layers.References[1] E. Carrera, P. Nali., “Mixed piezoelectric plate elements with direct evaluation of transverse electricdisplacement,” International Journal for Numerical Methods in Engineering, 80:403-424; (2009).[2] E. Carrera, S. Brischetto, P. Nali., “Plates and shells for smart structures : classical and advancedtheories for modeling and analysis,” John Wiley & Sons, Ltd, (2011).[3] M. Cinefra and E. Carrera., “Shell finite elements with different though-the-thickness kinematicsfor the linear analysis of cylindrical multilayered structures,” International Journal for NumericalMethods in Engineering, 93, 160 182, (2013).[4] E. Carrera, S. Brischetto, M. Cinefra “Variable kinematics and advanced variational statementsfor free vibrations analysis of piezoelectric plates and shells.,” Computer Modeling in Engineering& Sciences, 65(3), 259-341, (2010).2

English

A Finite Elements with Continue Transverse Electric Displacement for the Electro-Mechanical Analysis of Shell Structures

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[Proceeding] A Finite Elements with Continue Transverse Electric
Displacement for the Electro-Mechanical Analysis of Shell Structures
Original Citation:
Cinefra M.; Valvano S.; Carrera E. (2014). A Finite Elements with Continue Transverse Electric
Displacement for the Electro-Mechanical Analysis of Shell Structures. In: DeMEASS VI, Ede,
Netherlands, 25-28 May 2014.
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A FINITE ELEMENTS WITH CONTINUE
TRANSVERSE ELECTRIC DISPLACEMENT FOR
THE ELECTRO-MECHANICAL ANALYSIS OF
SHELL STRUCTURES
M. Cinefra ∗, S. Valvano †, E. Carrera ‡§
Keywords:
Piezoelectric materials, Multifield problems, Shell and Plate, Unified Formulation,
Mixed Interpolated Tensorial Components.
Abstract
Because of their capability to adapt and/or sense to the external environment, so-called smart
structures have attracted considerable research over the past few years. This capability leads to
a wide range of applications, in particular in the aerospace field. Among materials that can be
used, only piezoelectric ones have shown the capability to perform effectively both as actuators and
sensors, and the possibility of a simple integration with composite structures. The present work
deals with the analysis of multilayered plates and shells embedding piezoelectric layers as actuators
and sensors, see Fig. 1. Finite elements with layer-wise capabilities are employed to ensure an
accurate description of the mechanical and electric fields in the layers. It is essential to take into
account the discontinuity of the mechanical and electrical properties at the layer interfaces. For these
reasons, the use of classical plate theories based on Kirchhoff and Reissner-Mindlin hypotheses can
lead to inaccurate results. Even if analytical models are available, the solution of practical problems
often demand the use of computational methods such as the finite element method. To this aim, the
refined shell elements, recently formulated by the authors on the basis of Carrera Unified Formulation
(CUF), have been extended to the electro-mechanical problem. The governing equations are derived
using the Reissner’s Mixed Varational Theorem (RMVT) extended to the electro-mechanical case,
the original RMVT has been modified to account for ”only” interlaminar continous D3. Continuity
of mechanical variables, such as transverse shear and normal stress components, has been discarded
to provide a simple ”electrical” modified RMVT, here called (RMV T −D3) .
∗Research Assistant, Aerospace Department, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy,
e-mail: maria.cinefra@polito.it, website: www.mul2.com.
†Aerospace Department, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy, e-mail: ste-
fano.valvano@polito.it, website: www.mul2.com.
‡Professor, Aerospace Department, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy, e-mail:
erasmo.carrera@polito.it, website: www.mul2.com.
§currently at School of Aerospace, Mechanical and Manufacturing Engineering College, RMIT University, Melbourne,
Australia.
1
∫
V
(
δTpG σpC + δ
T
nG σnC − δETpGDpC − δETnGDnM − δDTnM (EnG − EnC)
)
dV = δLe − δLin
This model has already shown good results in the electro-mechanical analysis of multilayered plates
and shells with analytical code, and only plate with FEM code [1], [2]. The Mixed Interpolated Ten-
sorial Components (MITC) method is employed to contrast the membrane-shear locking phenomenon
that usually affects shell finite elements. This formulation has already shown all its potentiality as
a base for finite elements in the mechanical analysis of multilayered shells [3]. Moreover, plate finite
elements based on CUF for the analysis of electro-mechanical problem have been already presented
in [4]. One of the most interesting features of the unified formulation consists in the possibility to
keep the order of the expansion of the state variables along the thickness of the plate as a parameter
of the model. In so doing, both equivalent single layer (ESL) and layer-wise (LW) descriptions of the
variables are allowed. The electrical potential assumption for the layered actuators and sensors has
been limited to a LW description while the displacements assumption on the composite layers both
equivalent single layer (ESL) and layer-wise (LW) descriptions are allowed. This feature is partic-
ularly suitable since electric degrees of freedom (dofs) are often avoided for plate/shell elements or
a simple through- thickness linear variation is assumed for the electric potential. Some results from
the static and dynamic analysis of plates and shells under electro-mechanical loads will be provided,
in order to show the efficiency of models presented.
z
β
R Rβ
Figure 1: Double curvature shell with piezoelectric layers.
References
[1] E. Carrera, P. Nali., “Mixed piezoelectric plate elements with direct evaluation of transverse electric
displacement,” International Journal for Numerical Methods in Engineering, 80:403-424; (2009).
[2] E. Carrera, S. Brischetto, P. Nali., “Plates and shells for smart structures : classical and advanced
theories for modeling and analysis,” John Wiley & Sons, Ltd, (2011).
[3] M. Cinefra and E. Carrera., “Shell finite elements with different though-the-thickness kinematics
for the linear analysis of cylindrical multilayered structures,” International Journal for Numerical
Methods in Engineering, 93, 160 182, (2013).
[4] E. Carrera, S. Brischetto, M. Cinefra “Variable kinematics and advanced variational statements
for free vibrations analysis of piezoelectric plates and shells.,” Computer Modeling in Engineering
& Sciences, 65(3), 259-341, (2010).
2


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A Finite Elements with Continue Transverse Electric Displacement for the Electro-Mechanical Analysis of Shell Structures

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