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WP3. Adaptive Composite Modeling

WP3. Adaptive Composite Modeling. FP6- STREP project contract N°013517 NMP3-CT-2005-013517 Bardonecchia (To), 13-14 July 2006 E. CARRERA - POLITO WPLeader. SUMMARY. 1 - WP3 Overview

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WP3. Adaptive Composite Modeling

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  1. WP3. Adaptive Composite Modeling FP6- STREP project contract N°013517NMP3-CT-2005-013517 Bardonecchia (To), 13-14 July 2006 E. CARRERA - POLITO WPLeader

  2. SUMMARY 1 - WP3Overview 2 - Task 3.1:Modeling composites with piezoelectric sensors/actuators (POLITO, IST, LPMM) 3-Task 3.2:Modeling thermo-piezoelectric composites (POLITO, ISMEP)

  3. WP3 Overview Participants: LPMM(4) ISMEP (2) IST (8) POLITO(3) ULB(9) WP Leader: POLITO Start: 1 (4), End: 21 (25) Interaction: WP1,WP4,WP5

  4. WP3 Overview Description of work: Task 3.1: Modeling composites with piezoelectric sensors/actuators ( POLITO, IST, LPMM) ACTIVE (In progress) Task 3.2: Modeling thermo-piezoelectric composite composites with piezoelectric sensors/actuators (POLITO, ISMEP) ACTIVE (In progress) Task 3.3: Piezoceramic shunted damping concepts (ISMEP, ULB)ACTIVE (In progress) Task 3.4: Models and concepts validation (ALL) ACTIVE (In progress)

  5. WP3 Overview Objectives: • Analytical and numerical (finite element) modeling of sandwich and laminated composites with piezoelectric layers. • Analytical and numerical modeling of thermal and pyroelectric effects in piezoelectric composites • Finite element modeling 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

  6. Task 3.1: Declared Topics (POLITO,IST,LPMM) 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. • New advanced methods based on Mixed Variational Theorem will be discussed to fulfil interlaminar continuity of transverse electro-mechanical variables such as normal stresses and normal electric displacement 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.

  7. Task 3.1:Multifield Problems(POLITO) • Physical Fields: Mechanical field Electrical field Thermal field • Interactions: • Mechanical problem • Electro-mechanical problem • Thermo-mechanical problem • Electro-thermo-mechanical problem Task 3.1 Task 3.2 • Variational Statements: • Principle of Virtual Displacement (PVD) • Reissner Mixed Variational Theorem (RMVT)

  8. Task 3.1:Variational Statements(POLITO) Mechanical Problem M : Modeled Variables (UF) G : Geometrical Relations C : Constitutive Equations PVD RMVT Electro-Mechanical Problem PVD Partial-RMVT Full-RMVT where

  9. Task 3.1:Unified Formulation (UF)(POLITO) • The key point of the Unified Formulation is the use of generalized assumptions for the variables of the problem For a generic variable the assumption is: The generic variable is separated into a set of thickness functions and the correspondent variable depending on in-plane coordinates Taylor Expansion Equivalent Single Layer Model Legendre Expansion Layer Wise Model

  10. Task 3.1:Unified Formulation (UF)(POLITO) • Layer Wise The variables are assumed independently for each layer k • Equivalent Single Layer This description requires an assumption of the variables for the whole multilayered Zig-Zag form with Murakami function

  11. Task 3.1:Unified Formulation (UF)(POLITO) Displacements ESL LW Normal Stresses Only LW Interlaminar Continuity : Imposed Values: Electrical Potential Only LW Normal Electric Displacement Only LW where

  12. Task 3.1: Geometrical Relations(POLITO) with is a scalar • These are the geometrical relations for curved shells where the metric coefficients are: • In case of plates the geometrical relations are a particular case, when the radii of curvature are infinite and the metric coefficients become one • The explicit form of the introduced arrays follows :

  13. Task 3.1:Constitutive Equations(POLITO) Electric Gibbs Energy : Electro-mechanical PVD Pure mechanical problem is a particular case of electro-mechanical problem

  14. Task 3.1:Constitutive Equations(POLITO) Partial extension of RMVT to electro-mechanical case Mechanical RMVT can be considered as a particular case of the partial extension Full extension of RMVT to electro-mechanical case

  15. Task 3.1:Acronyms (POLITO)

  16. Task 3.1:Steps to obtain the model(POLITO) Opportune Constitutive Equations Geometrical Relations for Shell or Plate Variational Statement Analytic Solution Unified Formulation (assumptions for the variables) Finite Element Method (FEM) Analytic Solution Algebraic Governing Equations and Boundary Conditions Integration by parts Closed form solution Finite Element Method (FEM) FEM Governing Equations Matrix Product Shape Functions

  17. Task 3.1:Analytical Solution(POLITO) • Integration by parts: the following additional arrays have been introduced to perform integration by parts Governing equations admits Navier-type closed form solution if the adopted materials fulfill the following condition (transversely isotropic materials), the following harmonic assumptions can be made for the fields in case of simply supported plates and shells: Upon substitution of these assumptions the governing equations on domain assume the form of a linear system of algebraic equations in the domain, while boundary conditions are exactly fulfilled

  18. Task 3.1:Analytical Solution(POLITO) PVD Full-RMVT The equilibrium equations on the domain are: The equilibrium equations on the domain are: The possible Boundary Conditions are: The possible Boundary Conditions are: System of algebraic equations: System of algebraic equations: Algebraic system for RMVT for pure mechanical case Algebraic system for PVD for pure mechanical case Algebraic system for partial extension of RMVT to electro-mechanical case Algebraic system for PVD for electro-mechanical case Mechanical PVD is a particular case of electro-mechanical PVD Nuclei for Full-RMVT are completely different from the others

  19. Index variation Task 3.1:Finite Element Method(POLITO) Assembling node-layer Index variation k Assembling node-multilayer Index variation i,j Equivalent single layer Layer wise Remove constrained degree of freedom Assembling Structure Penalty Applying Constrain applying

  20. Task 3.1:Finite Element Method(POLITO) PVD Partial-RMVT Full-RMVT

  21. Task 3.1:Numerical results(POLITO) Modal Analysis of an adaptive plate by Classical and Partial-Mixed Theories (Analytical Method and FEM) Static Analysis of an adaptive plate by Classical, Partial-Mixed and Full-Mixed theories (Analytical Method and FEM): sensor Static Analysis of an adaptive plate by Classical, Partial-Mixed and Full-Mixed theories (Analytical Method and FEM): actuator Modal Analysis of an adaptive shell by Classical and Partial-Mixed theories (Analytical Method) Static Analysis of an adaptive shell by Classical, Partial-Mixed and Full-Mixed theories (Analytical Method): sensor and actuator

  22. Task 3.1: Numerical Results for Plates(POLITO) Materials Properties Modal Analysis Sensor case Actuator case

  23. Task 3.1:Modal Analysis(POLITO) • Every order of LW theories is able to obtain the frequencies with a minimum error • Mixed theories make us obtain better results using lower expansion order • Murakami’s function improves an ESL theory • Fem results are reasonable if compared with analytical ones

  24. Task 3.1:Sensor(POLITO) • Full-RMVT permits to obtain the normal electric displacement continue through the thickness, in particular the interface • The other theories give good results but normal electric displacement is discontinue • This happens, because in Full-RMVT normal electric displacement is ‘a priori’ variable • 4th order of expansion gives better results Analytical Solutions

  25. Task 3.1:Sensor(POLITO) Finite Element Method To obtain the normal electric displacement, Full-RMVT with ‘a priori’ transverse electro-mechanical variables is required

  26. Task 3.1:Sensor(POLITO) • To obtain normal stresses partial or full RMVT is required, because in them transverse stresses are modeled variables • Layer Wise models are required • Zig-Zag function improves the Equivalent Single Layer models Finite Element Method

  27. Task 3.1:Actuator(POLITO) Analytical Solutions • All theories give good results • Electric potential is ‘a priori’ variable in PVD, RMVT and F-RMVT model

  28. Task 3.1:Actuator(POLITO) Finite Element Method • Displacement v through the thickness is reported • To obtain 3D solution Layer Wise theory and high order of expansion are required • For mechanical displacement PVD or RMVT are enough, but for normal electric displacement Full-RMVT is required

  29. Task 3.1:Numerical results for Shells(POLITO) Modal Analysis for piezo-mechanical ring Materials Properties Closed Circuit

  30. Task 3.1: Numerical Results for Shells Ren Shell Varadan-Bhaskar Cylindrical Shell • One piezoelectric layer • Four layer in piezoelectric and anisotropic material • One piezoelectric layer • Four layer in piezoelectric and anisotropic material • Actuator case • Actuator case • Sensor case • Sensor case The same materials of plate case

  31. Task 3.1: Modal Analysis(POLITO) Classical Theories • Mixed theories obtain better results than classical theories using lower expansion order • For high order of expansion Classical theories and Mixed theories give the same results • ESL theories do not obtain correct values using low expansion order , to obtain correct results high order of expansion are required • Full-RMVT could improve the results Mixed Theories Frequencies [Hz]. Compare PVD and RMVT models

  32. Task 3.1:Static Analysis(POLITO) Multilayered Varadan-Bhaskar shell. Sensor case Multilayered Ren shell. Actuator case • For electric potential, displacement w and in-plane stress there are no differences between the three model, high order of expansion are better • For normal electric displacement the Full-RMVT gives different results, thanks the diagrams we understand that they are correct • For normal stress only high order of expansion (RMVT or F-RMVT) gives correct results

  33. Task 3.1: Static Analysis(POLITO) One piezoelectric Layer

  34. Task 3.1:Static Analysis(POLITO) Multilayered piezoelectric Shell

  35. Task 3.1:Static Analysis(POLITO) Multilayered piezoelectric Shell

  36. Task 3.2:Declared Topics (POLITO,ISMEP) 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 (RMVT) 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

  37. Task 3.2:Variational Statements (POLITO) Thermo-mechanical case Numerical results via FEM Partial-PVD Displacement u as variable (temperature q as loading) Full-PVD Displacement u and temperature q as variables Numerical results via FEM Displacement u, normal stresses sn as variables (temperature q as loading) Partial-RMVT Full-RMVT Displacement u, normal stresses sn and temperature q as variables Piezo-thermo-mechanical case Partial-PVD Displacement u and electric potential f as variables (temperature q as loading) Full-PVD Displacement u, electric potential f and temperature q as variables Partial-RMVT Displacement u, electric potential f and normal stresses sn as variables (temperature q as loading) Full-RMVT Displacement u, electric potential f, normal stresses sn and temperature q as variables Full-RMVT Displacement u, electric potential f, normal stresses sn, normal electric displacement Dn and temperature q as variables These cases are possible developments for the future (Analytical and Finite Element methods)

  38. Task 3.2:Partial RMVT for thermo-elastic case(POLITO) Finite Element Method for Plates Variational Statement Geometrical Relations Don’t change respect to electro-mechanical case, see the case for plates Constitutive Equations where

  39. Task 3.2:Partial RMVT for thermo-elastic case (POLITO) Unified Formulation (UF) Finite Element Discretization Governing Equations

  40. Task 3.2: Numerical Evaluation (POLITO) A simply supported orthotropic plate loaded by harmonic distribution of in plane temperature fields have been analysed . A simple linear trough the thickness distribution of temperature has been considered: where is the value of temperature at the top surface of the plate. The orientation is [0°/90°/0°], and each layer has the same thickness. The mechanical and thermal properties of material are: Results are in terms of: The mesh size has been limited to [6X6] and a 9 node element has been used

  41. Task 3.2:Numerical Evaluation (POLITO) • Layer Wise theories and high order of expansion are required to obtain the exact solution, in this case there aren’t any differences between PVD and RMVT model • For RMVT model the results are better than PVD in case of ESL • Zig-Zag function improve the results in case of ESL theory

  42. Task 3.2:Numerical Evaluation(POLITO) • High order of expansion gives better results • For low order of expansion RMVT results are better than PVD results, and LW theories are required

  43. Task 3.2:Numerical Evaluation (POLITO) • Layer wise theories are required to describe the evaluation through the thickness • Zig-zag function permits to use Equivalent Single Layer theories, so the evaluation through the thickness is similar to Layer Wise theory

  44. Task 3:POLITO- Research Work Details can be read in: • E. Carrera, M. Boscolo, Classical and Mixed Finite Elements for Static and Dynamics Analysis of Piezoelectric Plates , to be published, 2006 • E. Carrera, C. Fagiano, Mixed Piezoelectric Plate Elements with Continuous Transverse Electric Displacement, to be published, 2006 • A. Robaldo, E. Carrera, Mixed Finite Elements for Thermoelastic Analysis of Multilayered Anisotropic Plates, to be published, 2006 • 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 • 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, Unified Formulation to Assess Multilayered Plate Analysis of Thermo-Mechanical Problems, XVIII CONGRESSO NAZIONALE AIDAA, Volterra(PI), Italy, September 19-22, 2005 • E. Carrera, S. Brischetto, Reissner Mixed Theorem Applied to Bending Analysis of Piezoelectric Shells, submitted to Journal of Intelligent Material and Structures, 2006 • E. Carrera, S. Brischetto, M. D’Ottavio Vibration of Piezoelectric Shells by Unified Formulation and Reissner’s Mixed Theorem, II ECCOMAS THEMATIC CONFERENCE ON SMART STRUCTURES AND MATERIALS, Lisbon, Portugal, July 18-21, 2005 • E. Carrera, S. Brischetto, Piezoelectric Shells Theories with ‘a priori’ Continue Transverse Electro-Mechanical Variables, to be published, 2006

  45. 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, stresses, Electrical variables (potential, charge, displacements)

  46. Task 3:POLITO-Research Work 3. Thermo-Electromechanical testing Actuators/Sensors 3.1 Case of applied potential 3.2 Case of applied pressure 3.3 Case of applied charge 3.4 Case of uniform heating (temperature is the same at the top and bottom surface) 3.5 Case of non-uniform heating (temperature is different at the top and bottom surface). 4. Control, closed loop experiments 4.1 piezo-mechanical 4.2 thermo-electro-mechanical

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