Geometrically nonlinear higher order shear deformation fe analysis of thin walled smart structures doctor of philosophy

Geometrically Nonlinear Higher Order Shear Deformation FE Analysis of Thin Walled Smart Structures Von der Fakultät für Maschinenwesen der Rheinisch–Westfälischen Technischen Hochschule Aachen zur Erl[.]

Geometrically Nonlinear Higher-Order Shear Deformation

FE Analysis of Thin-Walled Smart Structures

Von der Fakultät für Maschinenwesender Rheinisch–Westfälischen Technischen Hochschule Aachen

zur Erlangung des akademischen Grades einesDoktors der Ingenieurwissenschaftengenehmigte Dissertation

vorgelegt von

Duy Thang Vu

Berichter: Univ.-Prof Dr.-Ing D Weichert

apl Prof Dr.-Ing R Schmidt

Tag der mündlichen Prüfung: 05 September 2011

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar

D 82 (Diss RWTH Aachen University, 2011)

Geometrisch nichtlineare FE Berechnung von

dünnwandigen Intelligenten Strukturen auf Grundlage von

Schubdeformationstheorien höherer Ordnung

Von der Fakultät für Maschinenwesender Rheinisch–Westfälischen Technischen Hochschule Aachen

zur Erlangung des akademischen Grades einesDoktors der Ingenieurwissenschaftengenehmigte Dissertation

vorgelegt von

Duy Thang Vu

Berichter: Univ.-Prof Dr.-Ing D Weichert

apl Prof Dr.-Ing R Schmidt

Tag der mündlichen Prüfung: 05 September 2011

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar

D 82 (Diss RWTH Aachen University, 2011)

To my family

The work documented in this thesis has been carried out during the years I worked as a ing and research assistant at the Institute of General Mechanics of RWTH Aachen Unviversity.This work was made possible by a fellowship from Vietnamese Government, Ministry of Edu-cation and Training, and by financial support from the Institute of General Mechanics

teach-I would like to express my deepest appreciation and gratitude to my supervisor, apl Prof.Dr.-Ing Rüdiger Schmidt for his motivating support and guidance throughout this study, andfor several suggestions for improvements of the final form of the thesis My special thanks arealso directed to Univ.-Prof Dr.-Ing Dieter Weichert for his support, for accepting to review

my work and to be part of the examination board I also thank Univ.-Prof Dr.-Ing Dr.-Ing.E.h Walter Michaeli, Head of the examination board, and Univ.-Prof Dr.-Ing Jörg Feld-husen, member of the examination board

I am also very grateful to Priv.-Doz Dr.-Ing Marcus Stoffel, Ms Inge Steinert, Ms MariaUmlauft, Ms Julia Blumenthal, Ms Dijana Muminovic and the other colleagues, who helpedand supported me during these years I would especially like to thank Dr.-Ing Sven Lentzenfor his support and the numerous valuable discussions

Lastly, I cannot leave behind all the people who helped me in my academic work Specialthanks to my parents, my wife, my brothers and all the friends who encouraged and helped meachieve this goal

In this thesis the influence of geometrical nonlinearity is studied in the finite element analysis

of quasi-static and transient dynamic response of shape and vibration control of thin-walledstructures with integrated layers or patches of piezoelectric materials The thesis addressesthe kinematic hypotheses on which linear and nonlinear theories of such smart structures arebased Finite plate elements are developed, which employ strain-displacement relations based

on either first- or refined third-order transverse shear deformation hypothesis Using these matic models, comparative finite element simulations are performed for the transverse stressdistribution analysis, the nonlinear shape control and the time histories of nonlinear vibrationsand sensor output voltage due to a step force acting on thin beams and plates, respectively,with a piezoelectric patch bonded to the surface Furthermore, an experiment reported in liter-ature for vibration control of a clamped beam using a piezoelectric layer bonded to the surface

kine-is simulated The comparative studies are performed based on linear theory, von Kármán-typenonlinear theory, and nonlinear moderate rotation theory

In der vorliegenden Arbeit wird der Einfluss der geometrischen Nichtlinearität in Finite mente Simulationen des quasistatischen und dynamischen Verhaltens bei Form- und Schwin-gungskontrolle dünnwandiger Strukturen mit integrierten piezoelektrischen Schichten oder Pat-ches untersucht Ein weiterer Schwerpunkt der Arbeit ist die Untersuchung des Einflussesder kinematischen Hypothesen, die linearen und nichtlinearen Theorien derartiger Struktu-ren zugrundegelegt werden Finite Plattenelemente werden unter Verwendung von Dehnungs-Verschiebungs-Beziehungen entwickelt, die auf der Schubdeformationstheorie erster oder dritterOrdnung basieren Unter Verwendung dieser beiden Strukturhypothesen werden vergleichendeFinite Elemente Simulationen durchgeführt, unter anderem für die transversale Schubspan-nungsverteilung, für nichtlineare Formkontrolle sowie für das nichtlineare Schwingungsverhal-ten und die zeitliche Entwicklung der Sensorspannungen infolge impulsförmiger Belastungendünner Balken und Platten mit aufgeklebtem piezoelektrischen Patch Außerdem wird ein inder Literatur beschriebenes Experiment zur Schwingungskontrolle eines eingespannten Balkensmit piezoelektrischer Aktorschicht simuliert Die Vergleichsrechnungen werden sowohl mit derlinearen als auch mit nichtlinearen Plattentheorien durchgeführt Bei letzteren werden großeVerschiebungen im Sinne der von Kármánschen Theorie und der Theorie moderater Rotationenberücksichtigt

Ele-Table of Contents

2.1 Introduction 7

2.2 Convective coordinates, base vectors and metric tensor 8

2.3 Stresses and strains 11

2.4 Third-Order Transverse Shear Deformation Hypothesis 14

2.5 The equations of motion 16

3 Piezoelectric materials 23 3.1 Introduction 23

3.2 Smart materials 25

3.2.1 Piezoelectrics and other smart materials 25

3.2.2 Piezoelectric effect 26

3.3 Linear theory of piezoelectricity 32

3.3.1 Elastic body 32

3.3.2 Piezoelectric materials 33

4 Electromechanical finite plate element 39 4.1 Variational formulation 39

4.2 Kinematical relations 40

I

II TABLE OF CONTENTS

4.3 Constitutive relations 42

4.4 Total Lagrangian formulation 43

4.4.1 The stress resultant method 46

4.4.2 Finite element implementation 48

5 Numerical Examples 61 5.1 Piezoelectric bimorph beam 61

5.1.1 Cantilevered piezoelectric bimorph beam 62

5.1.2 Piezoelectric bimorph beam with different boundary conditions 63

5.2 Simply supported composite piezolaminated plate 65

5.2.1 Shape control of hinged composite piezolaminated plate 65

5.2.2 Hinged thick PZT plate 69

5.3 Cantilevered isotropic beam with PZT sensor patch 71

5.4 Clamped piezolaminated plate 74

5.5 Two side hinged composite plate 77

5.6 Cantilevered piezolaminated beam with mass 80

6 Summary and Outlook 91 Literature 92 A Convective coordinates, base vectors, metric and curvature tensors, Christof-fel symbols 101 A.1 Plate structure 101

A.2 Cylindrical structure 104

A.3 Spherical structure 108

B Strain displacement relations 115 B.1 Refined von Kármán FOSD Theory for shells 115

B.1.1 Strain displacement relations 115

TABLE OF CONTENTS III

B.1.2 Plate 116

B.1.3 Cylindrical shell 117

B.1.4 Spherical shell 118

B.1.5 Strain displacement relations for plates, cylindrical shells and spherical shells 120

B.2 Refined von Kármán TOSD Theory for plates 122

B.2.1 Strain displacement relations 122

B.2.2 Expansion 122

C System matrices 125 C.1 System matrices of FOSD shell elements 125

C.1.1 Strain-Displacement matrices 125

C.1.2 Stress resultant matrix 128

C.1.3 The electric field 128

C.1.4 Mass matrix 129

C.2 System matrices of TOSD plate elements 131

C.2.1 The electric field 133

C.2.2 Mass matrix 133

IV TABLE OF CONTENTS

Nomenclature

¯

 ,i partial derivative with respect to Θi

 ; i covariant derivative with respect to Θi in the undeformed configuration

 | i covariant derivative with respect to Θi on the undeformed reference surface

δi

ǫijk permutation tensor components equal to 0, −√g or √g

ν unit outward normal vector on the lateral boundary surface

VI Nomenclature

ti stress vector acting on the surface Ωi

¨

x position vector to a point on the reference surface

[Hi] transversely integrated material coefficient matrix

[Sm] mechanical stress resultant matrix

[Se] electrical stress resultant matrix

Chapter 1

Introduction

In recent years, the subject of smart materials and structures has experienced tremendousgrowth in terms of research and development This is due to the technological implications ofthese novel devices that make smart structures technology one of the most important emergingtechnologies for the future The development of smart materials and structural systems involvesaerospace, chemical, civil, electrical, material and mechanical engineering scientists One of themain objectives is to design certain types of structures and systems capable of adapting to orcorrecting for changing operating conditions Furthermore, in modern engineering one of themain concerns is to make structures as light as possible in order to save weight The disad-vantage that comes along is that these structures tend to suffer from static as well as dynamicinstabilities, which can be overcome by integrating smart materials which can sense as well

as be actuated The use of materials with piezoelectric behavior is one promising possibility.Especially for thin-walled structures, integration of smart materials, like piezoelectric layers orpatches that can sense as well as be actuated, is a promising possibility for shape and vibrationcontrol

In order to understand piezointegrated thin-walled structures better, modeling and numericalsimulation of the geometrically linear and nonlinear static and dynamic response of adaptivestructures with integrated distributed control capabilities has attracted considerable researchinterest in recent years [15, 34, 14, 57]

The following literature review is divided into two parts, an introduction to the development

of general shell theory and its application in the field of smart structures Firstly, we give ashort review of the geometrically nonlinear plate and shell theories based on various kinematicalhypotheses Librescu [50, 51] and Librescu and Schmidt [52] gave the shell theory taking into ac-count the higher-order effects of the transverse displacement field for unrestricted rotations andmoderate rotations For third-order shear deformation (TOSD) theories, Reddy [68] proposed

1

2 CHAPTER 1 INTRODUCTION

a model for refined von Kármán-type (RVK) nonlinear plate theory and Başar [2] developed

a finite rotation shell theory For first-order shear deformation (FOSD) theory, Habip [22]and Habip and Ebcioglu [23] derived plate and shell theories for large rotations, Schmidt andReddy [73] gave a shell theory for moderate rotations, and Wempner [86] developed a vonKármán-type thin shell theory Librescu and Schmidt [53] and Schmidt and Librescu [72]developed the layerwise FOSD (zig-zag) theory and Başar [2] developed the layerwise TOSD(zig-zag) theory In the work of Reddy [68], the refined von Kármán TOSD finite elementwas applied for large deformation analysis of composite plates Palmerio et al [61] and Kreja et

al [31] implemented and extended the finite element FOSD model within the moderate rotation(MRT) theory In [30, 29], Kreja and Schmidt developed the finite element FOSD model for thelarge rotation theory of Habip [22] Recently, Lentzen [37] also presented an implementation

of the fully nonlinear FOSD shell theory [22] based on refined finite elements developed byGruttmann and Wagner [20, 21]

In the following we give a review of the models available in literature for thin-walled smartstructures The work of Allik and Hughes [1] is a pioneer work in the field of piezoelectricstructures They have analyzed the interactions between electricity and elasticity by develop-ing a linear finite element analysis using a tetrahedral element Crawley and Luis [15], Robbinsand Reddy [70] have applied this model to linear piezolaminated beams, while Batra et al [9],Lammering [34], Piefort [64] and Tzou et al [81] have treated linear plates and shells Tzou [80]has suggested a modified piezoelectric 8-node hexahedron element Three additional internaldegrees of freedom are introduced to avoid the shear looking for modeling thin structures.Yi et

al [87] used twenty-node solid elements including electrical degrees of freedom for geometricallynonlinear analysis of structures integrated with piezoelectric materials Sze et al [78] developed

an eight-node piezoelectric solid-shell element for modeling smart structures with segmentedpiezoelectric sensors and actuators In their model, the electric potential is approximated by

a linear transverse distribution In the framework of Klinkel and Wagner [28], a cally nonlinear eight-node solid shell element is developed to analyze piezoelectric structuresfor large deformations The finite element formulation is based on a variational principle of theHu-Washizu type and includes six independent fields: displacements, electric potential, strains,electric field, mechanical stresses and dielectric displacements This model applied to a cubictransverse distribution of the electric potential Tzou and Ye [82] have developed a triangularsolid shell element using the layerwise constant shear angle theory for modeling and analyzingvibration control of laminated piezoelectric structures The shape functions for the in-planemechanical degrees of freedom are biquadratic polynomials and the shape functions for bothmechanical and electrical degrees of freedom in the thickness direction are linear For furtherreferences we refer to the review paper of Benjeddou [10]

There is a vast literature on modeling and simulation of smart structures using linear finiteelements for static and dynamic problem in the range of small deflections For geometri-cally nonlinear problems, however, the literature is rather scarce Icardi and Di Sciuva [24]studied static geometrically nonlinear multilayered plates with surface-bonded induced-strainactuators using a third-order zigzag layerwise plate model and the von Kármán nonlinear-ity Mukherjee and Chaudhuri [57] analyzed the sensed voltage of a PVDF bimorph cantileverbeam using the first-order shear deformation hypothesis and the von Kármán nonlinearity.Marinković [55] developed a degenerated shell element to analyze piezolaminated compositestructures His model uses an updated Lagrangian scheme for finite deformations Lentzenand Schmidt [41, 43, 42, 40, 48], Vu et al [84] and Nguyen et al [60] performed sensor outputvoltage analysis, deflection analysis and shape control of smart isotropic or composite lami-nated beams, plates and shells with integrated piezoelectric actuator and sensor layers in theframework of first- and third-order shear deformation theories in range of moderate rotations.Krishna and Mei [32], Chandrashekhara and Bhatia [11], Wang and Varadan [85], Varelis andSaravanos [83], Klinkel and Wagner [28], Chróścielewski et al [14], and Lentzen and Schmidt [39]considered active buckling control and pre- and post-buckling analysis of smart structures.Shi and Atluri [77], Lee and Beale [36] studied nonlinear vibration control problems based onBernoulli-type beam theories Zhou and Wang [90] used the von Kármán nonlinearityand Chróścielewski et al [13, 12, 14] used finite rotation theory for nonlinear vibration controlproblems of beams Mukherjee and Chaudhuri [58] have analyzed the generation of sensedvoltage due to nonlinear vibrations of a PVDF bimorph cantilever beam in the framework ofvon Kármán first-order shear deformation theory Lentzen and Schmidt [46, 44, 45, 47] andRao et al [66] analyzed the sensor voltage output of surface bonded piezoelectric patches onbeams, plates and shells in the framework of a nonlinear moderate rotation first-order shear de-formation theory Based on the moderate rotation first-order shear deformation model, Lentzenand Schmidt [46, 44, 45, 47, 38] developed a finite element code for the simulation of nonlinearvibration control of smart isotropic or composite laminated beams, plates and shells with inte-grated piezoelectric actuator and sensor layers Batra et al [9, 8] studied shape and vibrationcontrol of plates at finite deformations considering nonlinear constitutive equations for piezo-electric patches

Lai et al [33] and Zhou et al [88, 89] considered nonlinear flutter suppression based on classicalvon Kármán large deflection plate theory and Shen and Sharpe [76] considered this problemusing discrete Kirchhoff theory finite elements

Vu et al [84] and Schmidt and Vu [74] developed a finite element code for the simulation

of nonlinear shape and vibration control of composite laminated beams and plates integratedpiezoelectric actuator and sensor layers based on third-order shear deformation theory The