I1N I DI t1P ,
•1 • :* ii ai
OF HYBRID PIEZOELECTRIC BEAMS
by
AKIL AUMED
Department of Applied Mechanics
Submitted
in fulfilment of the requirements of the degree of
Doctor of Philosophy
to the
INDIAN INSTITUTE OF TECHNOLOGY, I)ELJII
December, 2003
'['his is to certify that the thesis entitled "Coupled Zigzag and Third Order IVlodels for I'licr.,w) electromechanical Analysis of Hybrid Piezoelectric Beams" being .submitted by Akil Alinied to t,lu Indian Institute of 'Technology, Delhi for the award of die degree of Doctor of Philosophy in Applied Mechanics is a record of original bonafide research work carried out by him under our supervision and guidance. 'T'he thesis work, in our opinion, has reached the requisite standard fulfilling the rce uirenlents for the degree of Doctor of Philosophy.
The results contained in this thesis have not been submitted in part; or full to any other University or Institute for the award of any degree or diploma.
P'(21 N C 41t_1~c~er 3Y Y
(Prof. P. C. Duxnir) (Dr. S. Kapuria)
Professor, Associate Professor,
Deptt. of .A.ppliccl Mechanics Dept,t;. of Applied Mechanics Indian Institute of Technology Indian Institute of Technology
New Delhi - 110016 New Delhi - 110016
ITS USIAj ($11 I) OtU1
I am very illuch pleased to express my gratefulness and indebtedness to illy esteemed Supervisors Prof. P. C. Durnir and Dr. S. Kapuria for having spent the valuable t;irne from their busy schedule for consistent guidance and supervision. It would not have been possible to shape this work ill this form without their help, suggestions and constructive criticism, I am also thankful to Prof. G. P.
Dube for his profound help throughout the work.
I would like to thank Dr. Mohd. Arif, Dr. Nadeem Ahsan Siddiqui and Dr. Arshad Umar for their invaluable suggestions and encouragement.
I owe to my friends Mohd. Noor Desinukh, Molid. Alimned, Asif Hussein, Shahid Ahmad Sid- diqui Probir Saha, Snehasis and Krishna Jana, Md. Ahrnamuzzaman, Aminul Islam, 5k .Jaliarigir Hussein, Nausliad Alain and Molld. Umair for their help and cricoirageinent.
I wish to give thanks to Nazrul Islam, Mirquasim Hussein, Dr. Habibur Ralunan, iiuinayuu Kabir, Khalilur Rahnian, Anisur Rahman and Mafizul Plaque for their support and motivation to reach this level of education. I wish to thank Najrrma Ismail and her family for their initial assistance.
I want to thank all who have directly or indirectly helped me during the course of tlus work, and whose names have been missed here to record.
Finally, the inspiration, support and encouragement are also duly acknowledged to my parents, brothers, sisters and relatives.
Akil Ahmed
II
Two new efficient one-dimensional coupled zigzag theories are presented for piezoelectric h,y brid composite and sandwich beams with any lay-up. The axial displacement is approximated as a combination of global third order variation across the thickness with additional layer-wise lin- ear variation. The temperature-rise and the electric potential are approximated sub-layer-wise as piecewise linear across the thickness. The deflection is approximated to account for the transverse normal strain clue to the electric field through the transverse piezoelectric coefficient and due to the temperature-rise through the thermal expansion coefficient. The models consider both the axial and transverse electric fields. In model 1, the conditions of zero transverse shear stress at the bottom and the top surfaces and the conditions of its continuity at the layer interfaces are enforced approximately by neglecting the explicit contribution of the electric potential, whereas in model 2 these conditions are enforced exactly. The displacement field is expressed in terms of three primary displacement variables, electric potential variables and the thermal field. The number of the pri- nzary displacement variables is independent of the number of layers and identical to those used in the first order shear deformation theory (FSDT) and the third order theory (`1'OT). The governing coupled equations of motion, charge balance equations and boundary conditions are derived from the extended Hamilton's principle. These theories can accurately model open and closed circuit boundary conditions. The zigzag theory model 2, which incorporates the effect of the transverse thermal expansion coef-licient, in the approximation of deflection, is a new theory even for the drastic laminated beam for thermal load.
A new smeared beans coupled consistent third order shear deformation theory (CTOT) is devel- oped with cubic approximation for the axial displacement, uniform deflection across the thickness and piecewise linear approximation of the electric potential and the temperature-rise. Unlike the existing third order theory (TOT), the conditions of zero transverse shear stress at the bottom and the top are exactly satisfied including the contribution of the electric potential. The governing equations arc obtained from the extended Hamilton's principle.
Analytical Fourier series solutions are developed for the zigzag theory models 1, 2 and time CTOT ., for simply-supported beams for the static loads, for the natural frequencies and for the steady state
His
response under harmonic loads with damping. To assess the new models, an exact two-dimernsional (2D) piezoelasticity benchmark solution is developed for steady state response of simply-supported beams tinder harmonic load with and without damping. The results of the new models are com- pared with the exact 2D piezothermoelasticity solutions and the FSDT solutions. For this purpose a hybrid test beam with highly inhomogeneous lay-up; beams with symmetric cross-ply, antisyrn- metric cross-ply and angle-ply composite substrate; beams with sandwich substrate and a 2-layer piezoelectric beam are analysed for various Llrerruo-electrotuechatiical loads. The results of zigzag theory model 2 are generally more accurate than those of the FSDT for the global response parain- eters and for the through-the-thickness variation of displacements, electric field, and stresses. This accurate model would be ideal for the study of active control of smart beams.
For static analysis of hybrid beams under electromechanical loads, a finite element model (FEM) is developed for the zigzag theory model 2 using cubic Hermite interpolation for deflection and electric potential, and linear Lagrange interpolation for the axial displacement; and shear rotation.
'I"lrc expressions of the stiffness matrix and the load vector are derived and evaluated in closed form. The FEM program developed is validated by comparison with the analytical solution for the simply-supported beam. The finite element model does not exhibit any shear locking.
The major contribution of this work is the development of coupled zigzag theory model 2 for static and dynamic thermo-electromechanical analysis of beams which considers electric field along axial and transverse directions, and incorporates a novel idea of non-uniform variation of transverse displacement across the thickness taking into account; the piezoelectric and thermal strain in time thickness direction. This theory is as efficient as FSDT and TOT and has been shown to be generally very accurate and the finite element model based on it is free of shear locking.
ry
Certificate
Acknowledgements ii
Abstract iii
Contents V
List of Figures ix
List of Tables xiv
I Introduction 1.
1.1 PREFACE
1.2 CLASSIFICATION OF LAMINATE THEORIES ... 2
1.2.1 Uncoupled Equivalent; Single Layer Theories ... 2
1.2.2 Coupled Equivalent Single Layer Theories ... 3
1.2.3 Layer-wise Theories ... 3
1,2,4 Efficient Layer-wise Theories ... 4
1. 3D Theories ... 4
1.2.6 Exact, pic'i.otlicn neelastici ty Solutions ... t1 1.3 VARIOUS BEAM MODELS ... 5
1.4 OBJECTIVES OF THIS RESEARCH WORK ... 11
1.5 EXACT 2D SOLUTION FOR HARMONIC LOAD. . ... 12
1.6 ZIGZAG THEORY MODEL 1 ... 13
1.7 ZIGZAG rjIJEOI,Y MODEL 2 ... 14
1.8 COUPLED CONSISTENT THIRD ORDER, THEORY ... is 1.9 NUMERICAL 1tESU1JI'S AND ASSESSMENT .. ... iS 1.10 CONSTITUTIVE EQUATIONS FOR AN ANGLE-PLY BEAM ... 10
V
2 Exact 2D Solution for Harmonic Excitation 20
2.1 INTRODUCTION ...20
2.2 GOVERNING EQUATIONS FOR 2D EXACT SOLUTION ...21
2.3 GENERAL SOLUTION OF 2D GOVERNING EQUATIONS ...25
2.4 NUMERICAL RESULTS ...32
2.5 CONCLUSIONS ...39
3 Coupled Zigzag Theory Model 1 40 3.1 INTRODUCTION ... 10
3.2 APPROXIMATIONS MADE IN TIIE ZIGZAG ThEORY MODEL 1 ]I 3.3 EXTENDED HAMILTON'S PRINCIPLE ... 46
3.4 I3EAM INERTIA TERMS ... 17
3.5 STRESS AND ELECTRIC DISPLACEMENT RESULTANrIS 47 3.6 DYNAMIC FIELD EQUATIONS AND I3OUNDARY CONDITIONS ... 19
3.7 STRESSES AND ELECTRIC DISPLACEMENTS ... 50
3.8 BEAM (X)NS1.'1TUTIVE EQUATIONS ... 51
3.9 COUPLED ELECTROMECHAN1CAL DYNAMIC EQUATIONS ... Si 3.10 ANALYTICAL SOLUTION FOR SIMPLY-SUPPORTEI) BEAM ... )2
3.11. STATICS RESULTS AND ASSESSMENT ... 56
3.12 DYNAMICS RESULTS AND ASSESSMENT ... 6 1 3.13 CONCLUSIONS ... 68
4 Coupled Zigzag Theory Model 2 70 4.1. INTRODUCTION ... 70
4.2 APPROXIMATIONS MADE IN THE ZIGZAG THEORY MODEL 2 ... 71
4.3 EXTENDED IIAMIUfON'S PRINCIPLE ... 78
4.4 INE1fl'IA MATRiCES ... 79
4.5 STRESS AND ELECTRIC DISPLACEMENT RESULTANTS ... 0
4.6 DYNAMIC FIELD EQUATIONS AND BOUNDARY CONDITIONS ... 81
4.7 STRESSES AND ELECTRIC DISPLACEMENTS ... 83
4.8 BEAM CONSTITUTIVE EQUATIONS ... 83
4.9 COUPLED ELI DYNAMIC EQUATIONS ... 85 4.10 ANALYTICAL SOLUTION FOR SIMPLY-SUPPOiITEI) 13EAM ... HG
vi
5 FEM for Zigzag Theory Model 2 for Static Electromechanical Load 91
5.1 INTRODUCTION ...91
5.2 INTERPOLATION AND VARIATIONAL EQUATION ...91
5.3 ELEMENT STRAINS ...93
5.4 ELEMENT STIFFNESS MATRIX AND LOAD VECTOR ...95
5.5 ELEMENTS OF STIFFNESS MATRIX AND LOAD VECTOR ...99
6 Coupled Consistent Third Order Theory 103 6.1 INTRODUCTION ...10:1 6.2 APPROXIMATIONS MADE IN CTOT ...104
6.3 EXTENDED HAMILTON'S PRINCIPLE ...106
6.4 DYNAMIC FIELD EQUATIONS AND BOUNI)AIIY CONDITIONS ...109
6.5 BEAM CONSTITUTIVE EQUATIONS ...0
6.6 COUPLED ELECTRO-MECHANICAL DYNAMIC EQUATIONS ...113
6.7 ANALYTICAL SOLUTION FOR SIMPLY-SUPPORTED BEAM ...114
7 Assessment for Static Electromechanical Loads 116 7.1 INTRODUCTION ... 116
7.2 ASSESSMENT OF ZIGZAG THEORY FOR SINUSOIDAL LOADS ... 116
7.3 ASSESSMENT OF ZIGZAG THEORY FOR UNIFORM LOADS ... 122
7.4 ASSESSMENT OF CTOT AND TOT ... 129
7.5 ASSESSMENT FOR ANGLE-PLY BEAMS ... 3
7.6 FEM RESULTS ... 37
7.7 CONCLUSIONS ... 140
8 Assessment, for Harmonic Electromechanical Loads 141 8.1 IN'l RODUCTION ...ii I 8.2 ASS 14SSMI4NJ,' OF ZIGZAG THEORY FOR SINUSOIDAl. LOADS ...1
8.3 ASSESSMENT OF CTOT FOR SINUSOIDAL LOADS ...154
8.4 ASSESSMENT FOR UNIFORM LOADS ...1
8.5 ASSESSMENT FOR ANGLE-PLY BEAMS ...168
8.6 CONCLUSIONS ...175
9 Assessment; for Thermal Loads 176
9.1 INTRODUCTION ...1 76
vii
9.2 ASSESSMENT FOR ELASTIC BEAMS ...ho
9.3 ASSESSMENT OF ZIGZAG MODEL 2 FOR HYBRID BEAMS ...188
9.4 ASSESSMENT OF CTOT AND TOT FOR THERMAL LOADS ...201
9.5 ASSESSMENT FOR, ANGLE-PLY BEAMS ...210
9.6 CONCLUSIONS ...211
10 Conclusions 21.5 10.1 CONCLUSIONS FROM THE PRESENT WORK ...215
10.2 FUTURE SCOPE OF WORK ...21$
Bibliography 220
Brief Biodata of the Author 231
List of Publications from the Thesis 232
viii
