WP3' Adaptive Composite Modeling

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WP3. Adaptive Composite Modeling

• FP6- STREP project contract N013517NMP3-CT-2005-
  013517
• Paris, 12-13 December 2006
• E. CARRERA - POLITO WPLeader

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SUMMARY

• 1 - WP3 Overview
• 2 - Task 3.1 Modelling composites with
  piezoelectric
  sensors/actuators
  (POLITO, IST, LPMM)
• 3 - Task 3.2 Modelling thermo-piezoelectric
  composites (POLITO,ISMEP)

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WP3 Overview

• Partecipants
• LPMM(4) ISMEP (2) IST (8) POLITO(3),ULB(9)
• WP Leader
• POLITO
• Start
• 1 (4), End 21 (25)
• Interaction
• WP1,WP4,WP5

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WP3 Overview

• Description of work
• Task 3.1 Modelling composites with piezoelectric
  sensors/actuators ( POLITO,
  IST,LPMM) ACTIVE (Complete)
• Task 3.2 Modelling thermo-piezoelectric
  composite composites with piezoelectric
  sensors/actuators
  (POLITO, ISMEP) ACTIVE(In
  progress)
• Task 3.3 Piezoceramic shunted damping concepts
  
  (ISMEP, ULB) NOT Active
• Task 3.4 Models and concepts validation
  
  (ALL) NOT Active

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WP3 Overview

• Objectives
• Analytical and numerical (finite element)
  modelling of sandwich and laminated composites
  with piezoelectric layers.
• Analytical and numerical modelling of thermal and
  pyroelectric effects in piezoelectric composites
• Finite element modelling and analysis of new
  passive damping concepts using shunted
  piezoceramics
• Application and validation of the above advanced
  models and associated FE for various problems,
  such as vibration suppression of simple beams and
  plates due to mechanical or/and thermal loads by
  means piezoelectric sensors and actuators

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Task 3.1 Declared Topics (POLITO,IST,LPMM)
Comprehensive coupled piezoelectric models for
beam, plate and shell geometries will be
developed. The model has hierarchic capabilities
in the sense that accuracy can be increased by
augmenting computational efforts.

• The following main cases will be available
• Classical model based on known theories for
  laminates, such as
  CLT (Classical Laminated
  Theories) and FSDT (First order Shear Deformation
  Theory)
• Layer-wise models that have independent variables
  in each layer will be used to describe zig-zag
  fields for the displacement
• Classical methods with only displacement
  variables and advanced methods based on Mixed
  Variational Theorem will be discussed to fulfil
  interlaminar continuity of normal stresses.

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Task 3.1 Variational statements (POLITO)
Models have been developed in the framework of
the three following variational tools.
PVD Principle of Virtual Displacements
RMVT Reissner Mixed Variational Theorem
E RMVT Electric Reissner Mixed Variational
Theorem, four fields
The developments have been made according Unified
Formulation introduced by Carrera.
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Task 3.1 Constitutive and geometric relaction
(POLITO)
Plates
Shells
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Task 3.1 Formulation (POLITO)
Unified Formulation
Classic Formulation (PVD)
Mixed Formulation (RMVT)
Mixed formulation four field (RMVT)
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Task 3.1 Different theories(POLITO)
Equivalent Single Layer (ESL)
Funzione di Murakami (zig-zag)
Layer Wise (LW)
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Task 3.1 Acronyms (POLITO)
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Task 3.1Examples (POLITO)
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Task 3.1 Principle of virtual displacements
(PVD) (POLITO)
Constitutive relactions
Geometric relations
Thickness function
Shape function
Matrix product
FEM governing equations
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Task 3.1 Constitutive relations RMVT(POLITO)
Constitutive relations PVD
Constitutive relations RMVT
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Task 3.1 Reissner Mixed Variation Theory(RMVT)
(POLITO)
Substituting constitutive relations
Substituting geometric reltions
Substituting thickness functions
Substituting shape functions
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Task 3.1 Constitutive relations RMVT four
field(POLITO)

Constitutive relations PVD
Constitutive relations RMVT four fields
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Task 3.1 Reissner Mixed Variation Theory four
field(POLITO)
Substituting constitutive relations
Substituting geometric reltions
Substituting thickness functions
Substituting shape functions
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Task 3.1 Assembling (POLITO)
Assemblaggio nodo-layer
Variazione indice k
Assemblaggio nodo-multilayer
Variazione indici i,j
Equivalent single layer
Lawer wise
Eliminazione gradi di libertà vincolati
Assemblaggio struttura
Applicazione Penalty
Applicazione dei vincoli
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Task 3.1 Numerical Results (POLITO)

• Modal Analysis of an adaptive plate by Mixed and
  Classical FEs
• Static Analysis of an adaptive plate by Mixed and
  Classical FEs sensor
• Static Analysis of an adaptive plate by Mixed and
  Classical models actuator
• Analytic models for piezo-mechanics ring modal
  analysis

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Task 3.1 Modal Analysis(POLITO)
6x6 Q9 Integrazione selettiva
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Task 3.1Modal Analysis (POLITO)

• Every order of LW theories is able to obtain the
  frequencies with a maximum error of 2
• Mixed theories make us obtain better results
  using lower expansion order
• Murakamis function improves an ESL theory

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Task 3.1 Sensor (POLITO)
6x6 Q9 Integrazione selettiva
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Task 3.1 Sensor (POLITO)

• The exact value is 6.11 for a a/h4 plate
• Mixed element obtains better result than classic
  elements using lower expansion order
• ESL elements do not obtain correct values using
  low expansion order
• Murakamis function improves ELS models
• Mixed theories respect CZ0 requirements and
  obtain continuous normal stresses
• To obtain the normal electric displacement
  continuous is necessary to use the RMVT four
  fields

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Task 3.1 Actuator (POLITO)
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Task 3.1 Actuator (POLITO)

• The exact value is 0.4476 for a a/h4 plate
• Mixed theories obtains better result than classic
  theories using lower expansion order
• ESL theories do not obtain correct values using
  low expansion order
• Murakamis function improves ELS models
• Mixed theories respect CZ0 requirements and
  obtain continuous normal stresses
• To obtain the normal electric displacement
  continuous is necessary to use the RMVT four
  fields

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Task 3.1 Modal analysis for piezo-mechanic ring
(POLITO)
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Task 3.1 Modal Analysis (POLITO)

• Mixed theories obtains better result than classic
  theories using lower expansion order
• ESL theories do not obtain correct values using
  low expansion order
• Mixed theories respect CZ0 requirements and
  obtain continuous normal stresses
• To obtain the normal electric displacement
  continuous is necessary to use the RMVT four
  fields

Frequencies Hz. Compare PVD and RMVT models
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Task 3.2 Declared Topics (POLITO,IST,LPMM)
Development and implementation of computationally
finite elements of thermo-piezo-elastic model for
the analysis of smart structures. Accurate
hierarchical formulation based on classical
variational statement (PVD) and advanced
partial-mixed variational principles will be
proposed.

• The following main cases will be available
• Layer-wise models that have independent variables
  in each layer will be used to describe zig-zag
  fields for the displacement, electric potential,
  normal stresses, normal potential and
  temperature
• Temperature can be modelled like a load and the
  coupling temperature-mechanic field will be
  neglected
• Temperature can be considered like an unknown and
  the coupling temperature-mechanic field will be
  considered

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Task 3.2 Variational statements(POLITO)
Models have been developed in the framework of
the three following variational tools.

• PVD piezo-thermo-mechanic partially coupled
  (two fields u,E)
• PVD piezo-thermo-mechanic fully coupled (three
  fields u,E,q)
• RMVT piezo-thermo-mechanic partially coupled
  (three fields u,E,sn)
• RMVT piezo-thermo-mechanic fully coupled (four
  fields u,E,sn,q)
• ERMVT piezo-thermo-mechanic fully coupled (five
  fields u,E,sn, Dn,q)

Numerical results will be obtained soon. At the
moment just partially coupled methods have been
implemented.
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Task 3 POLITO Research Work

• Details can be read in
• E. Carrera, M. Boscolo, C. Fagiano, A. Robaldo,
  Unified Formulation to Assess Multilayered Plate
  Analysis of Thermo-Mechanical Problems, XVIII
  CONGRESSO NAZIONALE AIDAA, Volterra(PI), Italy,
  September 19-22, 2005
• E. Carrera, M. Boscolo, C. Fagiano, A. Robaldo,
  Mixed Finite Elements for Piezoelectric Plates
  based on Unified Formulation, XVII CONGRESSO
  AIMeTA DI MECCANICA TEORICA E APPLICATA,
  Florence, Italy, September 11-15, 2005
• E. Carrera, M. Boscolo, C. Fagiano, A. Robaldo,
  Mixed Elements for Accurate Vibration of
  Piezo-electric Plates, II ECCOMAS THEMATIC
  CONFERENCE ON SMART STRUCTURES AND MATERIALS,
  Lisbon , Portugal, July18-21,2005

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Task 3 POLITO Research Work

• The following experiments could be of POLITO
  interest
• The considered multilayered (any configuration
  with piezo-electric layers and pathces)
  structures can be beams or flat or curved panels
  with any geometrical boundary conditions (simply
  supported is the favorite one).
• 1. Vibration testing
• 1.1 Closed circuit - Open Circuit.
• 1.2 Calculation of the first 1-5
  frequencies.
• 2. Static Electromechanical testing
  Actuators/Sensors
• 2.1 Case of applied potential
• 2.2 Case of applied pressure
• 2.3 Case of applied charge
• Measurements of displacements, some
  stresses, Electrical variables (potentential,
  charge, displacements)

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Task 3 POLITO Research Work
3. Thermo-Electromechanical testing
Actuators/Sensors 3.1 Case of applied
potential 3.2 Case od applied pressure
3.3 Case of applied charge 3.4 Case of
Uniform heating (temperature is the same in the
top and bootm surface) 3.5 Case of
non-uniform heating (temperature is different in
the top and bottom surface). 4. Control,
closed loop experiments 4.1
piezo-mechanical 4.2 thermo-electro-mechanica
l
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IST Contribution to CASSEM Project

• FP6 STREP Project Contract N 013517NMP3 - CT
  - 2005 - 013517
• Paris, 12-13 December 2005
• IST Lisbon Technical University, Portugal
• Participant nr. 8

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IST contribution to WP 3 WP 3 Adaptive
composites modelling (Leader TUT / Italy)

• Task 3.1 Modelling composites with
  piezoelectric sensors/actuators
• (TL TUT, IST, LPMM)
• Higher Order Shear Deformation Theory and First
  Order Shear Deformation Theory coupled
  piezoelectric F.E. models for plate and shell
  geometries (based on displacement approach).
• Mixed layerwise theory F.E. model for plate
  structures which includes piezoelectric behavior
  of layers or patches.
• Mixed FE formulation based on least-squares
  variational principles, which is an alternate
  approach to the mixed weak form FE models.

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WP3 cont.HSDT and FSDT coupled piezoelectric
FE models

• Higher Order Shear Deformation Theory and First
  Order Shear Deformation Theory coupled
  piezoelectric F.E. models for plate and shell
  geometries, based on the generic displacement
  field

Among the IST research team, there are a few
published articles devoted to the development and
validation of these models for the analysis and
optimization of plate structures with
piezoelectric layers or patches.
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WP3 cont.HSDT and FSDT coupled piezoelectric
FE models

• since the beginning of the project
• J.M. Moita, C.M. Mota Soares, C.A. Mota Soares,
  Active Control of Forced Vibrations in Adaptive
  Structures Using a Higher Order Model, Composite
  Structures, Elsevier, UK, Vol.71, pp. 349-355,
  2005.
• J.M. Moita, P.G. Martins, C.M. Mota Soares, C.A.
  Mota Soares, Optimal Dynamic Control of
  Laminated Adaptive Structures Using a Higher
  Order Model and a Genetic Algorithm, Computers
  and Structures, (submitted in 2005).
• J.M. Moita, V.M. Franco Correia, P.G. Martins,
  C.M. Mota Soares, C.A. Mota Soares, Optimal
  Design in Vibration Control of Adaptive Stuctures
  Using a Simulated Annealing Algorithm, Composite
  Structures, (submitted in 2005).

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WP3 cont.Mixed layerwise FE model

• Mixed layerwise theory for laminated plate
  structures which includes piezoelectric materials
  (layers or patches).
• Due to the mixed formulation, transverse stresses
  (?zz, ?xz, ?yz), displacements (u, v, w) and
  electric potential (?) are considered as degrees
  of freedom, and are calculated without the need
  of post-processing. The model guarantees the
  continuity requirements for these quantities.
  The in-plane stress (?xx, ?yy, ?xy) and
  electric displacement (Dx, Dy, Dz) are calculated
  by post-processing of the obtained solution.
• R.M. Garcia Lage, C.M. Mota Soares, C.A. Mota
  Soares, Static and Free Vibration Analysis of
  Magneto-Elastic Laminated Plates by a Layerwise
  Partial Mixed Finite Element Model, The 3rd
  International Conference on Structural Stability
  and Dynamics, June 19-22, 2005, Kissimmee,
  Florida, USA.

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WP3 cont.Mixed layerwise FE model(cont. 2)

• The problem fields, for each layer , are
• The shape functions F(z) describe the plate
  behaviour in the thickness direction, which can
  be linear, quadratic or cubic.
• The functions szz, sxz, syz, U, V, W, ? denote
  the nodal values of primary variables.

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WP3 cont.Mixed layerwise FE model(cont. 3)
Numerical example 1

• Ilustrative numerical example 1 simply supported
  PVDF piezolaminated plate with square shape (a?a)
  and thickness h.
• Plate lay up is taken to be 0º/90º/0º and the
  PVDF layers properties are
• Three geometric configurations with
  side-to-thickness ratios a/h4 10 50 (with
  h0.01m) and two load cases are analysed

LC1 LC2
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WP3 cont.Mixed layerwise FE model(cont.
4)Transverse stresses and transversal electrical
displacement LC1
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WP3 cont.Mixed layerwise FE model(cont.
5)Mechanical displacements and electric
potential LC1
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WP3 cont.Mixed layerwise FE model(cont.
6)Transverse stresses and transversal electrical
displacement LC2
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WP3 cont.Mixed layerwise FE model(cont. 7)
Mechanical displacements and electric potential
LC2
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WP3 cont.Mixed layerwise FE model(cont. 8)
Numerical example 2

• Ilustrative numerical example 2 simply supported
  piezolaminated single layer plate with square
  shape (a?a) and thickness (h0.01m). The plate is
  made of PZT-4 with stiffness and piezoelectric
  properties as follows
• Typical results, for free vibration problems, are
  presented below

Ref.1 Heyliger Saravanos, J. Acoust. Soc.
Am., 98 (1995) 1547-1557.
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WP3 cont.Mixed layerwise FE model(cont. 8)
Cross thickness modal distributions
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Mixed Least-Squares FE Model

• A mixed FE formulation based on least-squares
  variational principles, which is an alternate
  approach to the mixed weak form FE models, is
  being developed.
• The least-squares-based FE model is being
  developed for the static and dynamic analysis of
  laminated composite plates using the FSDT, with
  the generalized displacements and stress
  resultants as independent variables.

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Mixed Least-Squares FE Model Governing Equations

• Equilibrium Equations and Laminate Constitutive
  Equations

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Mixed Least-Squares FE Model Least-Squares
Formulation (1)

• In a more compact notation the governing
  equations become

where,

• Then, the associated least-squares functional is
  given by

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Mixed Least-Squares FE Model Least-Squares
Formulation (2)

• Leading to the following variational problem

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Mixed Least-Squares FE Model Numerical Examples
(1)

• Results for the static analysis of the square
  composite laminates (0/90), (0/90/0/90/0) and
  (-45/45)4 under uniformly distributed load, with
  different boundary conditions and
  side-to-thickness ratios.

Analytic solutions using FSDT by Navier series
(mn1,,30) and Lévy series (n1,,30).
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Mixed Least-Squares FE Model Numerical Examples
(2)

• Normalized deflections of the previous laminates
  with all boundaries simply supported (SSSS-1 or
  SSSS-2 as tabled).

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Mixed Least-Squares FE Model Numerical Examples
(3)

• Normalized deflections of the previous laminates
  with two opposite boundaries free and two simply
  supported (FFSS-1 or FFSS-2 as tabled).

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IST contribution to WP 7WP 7 Dissemination and
innovation related activities(Leader CRPHT)

• Publishing of Articles in selected Scientific
  International Journals
• Composite Structures
• Computers Structures
• Journal of Inverse Problems in Science and
  Engineering
• Participation in International Conferences
• ICCS-13th Intern. Conf. of Composite Structures
  (Australia)
• CST-8 (Tenerife)
• ECCM-2006 (Lisbon)
• SPIE - Smart Structures Materials/NDE 2006
  (San Diego, CA)