Smart
Material
Systems
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slam
F R O N T I E R S IN
APPLIED
MATHEMATICS
The SIAM series on Frontiers in Applied Mathematics publishes monographs dealing with creative work in a substantive field involving applied mathematics or scientific computation. All works focus on emerging or rapidly developing research areas that report on new techniques to solve mainstream problems in science or engineering. The goal of the series is to promote, through short, inexpensive, expertly written monographs, cutting edge research poised to have a substantial impact on the solutions of problems that advance science and technology. The volumes encompass a broad spectrum of topics important to the applied mathematical areas of education, government, and industry.
EDITORIAL BOARD H.T. Banks, Editor-in-Chief, North Carolina State University Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB Carlos Castillo-Chavez, Arizona State University Doina Cioranescu, Université Pierre et Marie Curie (Paris VI) Lisa Fauci, Tulane University Pat Hagan, Bear Stearns and Co., Inc. Belinda King, Oregon State University Jeffrey Sachs, Merck Research Laboratories, Merck and Co., Inc. Ralph C. Smith, North Carolina State University AnnaTsao, AlgoTek, Inc.
BOOKS PUBLISHED IN FRONTIERS IN A P P L I E D MATHEMATICS Smith, Ralph C, Smart Material Systems: Model Development lannelli, M.; Martcheva, M.; and Milner, F. A., Gender-Structured Population Modeling: Mathematical Methods, Numerics, and Simulations Pironneau, O. and Achdou,Y., Computational Methods in Option Pricing Day, William H. E. and McMorris, F. R., Axiomatic Consensus Theory in Group Choice and Biomathematics Banks, H.T. and Castillo-Chavez, Carlos, editors, Bioterrorism: Mathematical Modeling Applications in Homeland Security Smith, Ralph C. and Demetriou, Michael, editors, Research Directions in Distributed Parameter Systems Hollig, Klaus, Finite Element Methods with B-Splines Stanley, Lisa G. and Stewart, Dawn L., Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods Vogel, Curtis R., Computational Methods for Inverse Problems Lewis, F. L.; Campos, J.; and Selmic, R., Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities Bao, Gang; Cowsar, Lawrence; and Masters, Wen, editors, Mathematical Modeling in Optical Science Banks, H.T.; Buksas, M. W; and Lin.T., Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts Oostveen.Job, Strongly Stabilizable Distributed Parameter Systems Griewank, Andreas, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation Kelley, C.T., Iterative Methods for Optimization Greenbaum, Anne, Iterative Methods for Solving Linear Systems Kelley, C.T., Iterative Methods for Linear and Nonlinear Equations Bank, Randolph E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 7.0 More, Jorge J. and Wright, Stephen J., Optimization Software Guide Rude, Ulrich, Mathematical and Computational Techniques for Multilevel Adaptive Methods Cook, L. Pamela, Transonic Aerodynamics: Problems in Asymptotic Theory Banks, H.T., Control and Estimation in Distributed Parameter Systems Van Loan, Charles, Computational Frameworks for the Fast Fourier Transform Van Huffel, Sabine and Vandewalle, Joos, The Total Least Squares Problem: Computational Aspects and Analysis Castillo, Jose E., Mathematical Aspects of Numerical Grid Generation Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 6.0 McCormick, Stephen F., Multilevel Adaptive Methods for Partial Differential Equations Grossman, Robert, Symbolic Computation: Applications to Scientific Computing Coleman,Thomas F. and Van Loan, Charles, Handbook for Matrix Computations McCormick, Stephen F.,Multigrid Methods Buckmaster, John D., The Mathematics of Combustion Ewing, Richard E., The Mathematics of Reservoir Simulation
Smart Material Systems Model Development Ralph C Smith North Carolina State University Raleigh, North Carolina
siam.
Society for Industrial and Applied Mathematics Philadelphia
Copyright © 2005 by the Society for Industrial and Applied Mathematics. 1098765432 I All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact: The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 017602098 USA, 508-647-7000 Fax: 508-647-7101,
[email protected], www.mathworks.coml
Library of Congress Control Number: 2005920039 ISBN 0-89871-583-0
siam.
is a registered trademark.
Contents Foreword
xi
Preface
xiii
Notation
xvii
Elements and Compounds
xxv
Abbreviations for Units
xxvii
1
Smart Material Applications 1.1 Smart Material Systems 1.2 Piezoelectric and Electrostrictive Applications 1.3 Magnetostrictive Transducers 1.4 Shape Memory Alloys 1.5 Piezoelectric, Electrostrictive and Ionic Polymers 1.6 Electrorheological and Magnetorheological Compounds 1.7 Sensor Technologies — Fiber Optics
2
Model Development for Ferroelectric Compounds 2.1 Physical Properties of Ferroelectric Materials 2.2 Linear Piezoelectric Models 2.3 Higher-Order Energy Relations 2.4 Preisach Hysteresis Models 2.5 Domain Wall Model 2.6 Homogenized Energy Model 2.7 Ginzburg-Landau Relations 2.8 Work in the Polarization Process
43 44 55 69 80 84 98 133 135
3
Model Development for Relaxor Ferroelectric Compounds 3.1 Physical Properties of Relaxor Compounds 3.2 Temperature-Dependent Equilibrium Model 3.3 Temperature-Dependent Domain Wall Model 3.4 Temperature-Dependent Homogenized Energy Model
139 140 143 151 154
vii
1 1 6 15 21 28 34 36
viii
Contents
4
Model Development for Ferromagnetic Compounds 4.1 Physical Properties of Ferromagnetic Materials 4.2 Fundamental Energy Relations 4.3 Linear Models 4.4 Higher-Order Energy Models 4.5 Preisach Models 4.6 Jiles–Atherton Model 4.7 Homogenized Energy Model
159 160 179 181 183 189 209 215
5
Model Development for Shape Memory Alloys 5.1 Physical Properties of Shape Memory Alloys 5.2 Energy Relations 5.3 Preisach Models 5.4 Domain Wall Models 5.5 Homogenized Energy Framework
241 244 254 258 259 263
6
Unified Modeling Frameworks for Ferroic Compounds 6.1 Physical Properties 6.2 Preisach Representations 6.3 Domain Wall Theory 6.4 Homogenized Energy Framework
275 277 284 285 288
7
Rod, 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10
8
Numerical Techniques 8.1 Quadrature Techniques 8.2 Numerical Approximation 8.3 Numerical Approximation 8.4 Numerical Approximation 8.5 Numerical Approximation
A B
Beam, Plate and Shell Models 295 Linear and Nonlinear Constitutive Relations 302 Linear Structural Assumptions 306 Rod Models 307 Beam Models 315 Plate Models 325 Shell Models - General Development 341 Shell Models – Special Cases 348 Timoshenko, Mindlin–Reissner, and von Karman Models . . . . 354 THUNDER Models 359 Abstract Model Formulation 365
of the of the of the of the
Rod Model Beam Model Plate Model Shell Model
373 374 385 393 402 406
Glossary of Terms
411
Mathematical Theory
429
B.1 B.2 B.3
429 431 434
Dirac Sequences Compactness of the Polarization Operator Continuity of the Homogenized Energy Model
Contents
C
D
ix
Legendre Transforms, Calculus of Variations, Mechanics Principles C.1 Legendre Transforms C.2 Principles from the Calculus of Variations C.3 Classical, Lagrangian and Hamiltonian Mechanics
437 437 439 442
Inversion Algorithm
447
Bibliography
449
Index
488
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Foreword With this volume, the Frontiers series continues its mission of publishing "cutting edge" treatments on applied mathematical aspects of emerging scientific and technological issues. This is the sixth book to appear in the general area of smart materials and structures since 1992 when the field grew to the point where a book treatment of the subject was in order. Some of the early monographs were sparse in mathematical detail, consisting of either collections of the authors' papers or collections of chapters written by several different authors embodying several points of view. One exception was the 1996 volume co-authored by the current author in which a first attempt at a rigorous and detailed mathematical treatment was given. The field has now had an additional decade of maturation and innovation and the timing is right for a new "frontiers" account. In the present volume, Professor Smith discusses mathematical and modeling foundations for all of the basic types of material systems that make up the field of smart materials and structures. Notable features of the present volume include careful attention to nonlinear physically-based models exhibiting significant hysteresis (and the resulting damping) as well as treatment of numerical approximation techniques appropriate for both estimation and control design. The field of smart materials and structures is relatively new and is known by several names: adaptronics, adaptive structures, intelligent material systems and structures, smart materials and structures and combinations of these words. Several conferences a year have been devoted to these topics since the first one was held approximately fifteen years ago. Professor Smith has been a significant and prolific contributor to the largest of these conferences, eventually becoming one of the Chairs. He also serves as an associate editor for one of the major journals serving this community. In this sense, his career has developed along with this discipline, placing him in an excellent position to write this text. In this volume, Smith discusses current and future applications of advanced smart material systems. The text provides a comprehensive development of both linear and nonlinear models required to characterize these materials in a manner that facilitates design and control development. While the focus is primarily on piezoelectric, magnetic and shape memory material systems, the text also includes applications exploiting ionic polymers, magnetorheological compounds and fiber optic sensors. Three classes of nonlinear models are discussed, all of which provide unified characterization frameworks for the broad class of combined smart material xi
xii
Foreword
systems. The book includes an extensive development of structural models based on the nonlinear constitutive models as well as a chapter on numerical techniques for approximating solutions to the structural systems. In summary, this monograph will surely be of significant use to researchers and students in the engineering and material sciences as well as those in applied mathematics who are interested in modeling smart materials and structures. Daniel J. Inman Virginia Tech Blacksburg, VA December 1, 2004
Preface Writing a book on smart materials bears a certain semblance to purchasing a high-end laptop — due to rapid advances in the field, it is tempting to postpone the decision every six months in order to include the latest technological development. At some point, however, one must simply focus on the present technology and realize that upgrades will be necessary in the future. The field of smart materials has advanced rapidly in the last 15 years due to an increasing awareness of material capabilities, the development of new materials and transducer designs, and increasingly stringent design and control specifications in aerospace, aeronautic, industrial, automotive, biomedical, and nano-systems. Equally important for the advancement of the field is the development of models, numerical approximation techniques, and control designs which accommodate the hysteresis and constitutive nonlinearities inherent to the materials. The majority of initial investigations focused on linear analysis based on the approximately linear material behavior observed in low to moderate drive regimes and obtained with certain amplifier and feedback designs. When applicable, linear models and control designs are certainly advantageous and should be considered first. However, for an increasing number of applications, the attributes which provide smart materials with unique actuator and sensor capabilities are inexorably due to physical mechanisms that produce hysteresis — hence these mechanisms must be incorporated in models and control designs to achieve the unique performance capabilities offered by the compounds. For example, the damping provided by shape memory alloy tendons in civil and aerospace structures is proportional to the area of the hysteresis loop. Hence optimal vibration attenuation requires optimal hysteresis which is diametrically opposite to the strategy of operating in approximately linear regimes. In this monograph, we provide a unified development of linear and nonlinear models for smart material systems as a prelude to model-based control design. We focus significant attention on the physical mechanisms that provide the materials with unique transducer capabilities but yield hysteretic and nonlinear behavior, and, to the extent possible, we use the underlying physics to guide the development of unified constitutive frameworks for quantifying the dynamics of a broad class of ferroic compounds. These constitutive relations are subsequently employed to construct structural models for a range of transducer constructs and geometries. Finally, we address the development of numerical approximation techniques which xiii
xiv
Preface
adhere to the physical principles used to construct models and are appropriate for transducer and control design. In Chapter 1, we summarize a range of present and projected smart material applications to illustrate both the scope of the field and issues that must be addressed in models. Chapters 2–5 address the development of linear and nonlinear models for ferroelectric, ferromagnetic and ferroelastic compounds. In each of these chapters, we discuss three classes of nonlinear models — Preisach models, domain wall models, and homogenized energy models — since they comprise unified frameworks for the combined class of ferroic compounds, as illustrated in Chapter 6, and are amenable to control design. Whereas all three frameworks encompass physical principles, we focus primarily on the third since the combination of lattice-level energy principles and stochastic homogenization to incorporate material nonhomogeneities provide it with extreme flexibility in a range of smart material applications and operating regimes. It is illustrated in Chapters 2 and 4 that the homogenized energy framework provides an energy basis for extended Preisach formulations. It is thus anticipated that the symbiotic investigation of energy-based and extended Preisach models will strongly contribute to the growth of unified characterization frameworks with the former providing strengths inherent to the energy basis and the latter providing a mature mathematical framework for model analysis and identification. The unified constitutive models developed in Chapters 2-6 provide the nucleus for the linear and nonlinear rod, beam, plate and shell models detailed in Chapter 7. In this manner, relevant physics is incorporated in the system models to augment accuracy and improve efficiency as required for real-time implementation. Chapter 8 summarizes numerical approximation techniques that are appropriate for both simulations and control design. The topic of model development for smart material systems is highlv interdisciplinary and this book was written with the goal of making it accessible to scientists from a broad range of disciplines with backgrounds ranging from students entering the field to experts within the constituent disciplines. To aid the understanding of both mathematical and physical terminology, we have included numerous definitions throughout the text and have provided an extensive glossary of terms in an appendix. We have also placed mathematical proofs in an appendix so they are available for those who are interested but do not deter readers focusing primarily on physical aspects of the theory. Finally, open research questions are indicated at various points to encourage investigations deemed necessary to advance the state of knowledge in this rapidly evolving field. Whereas the text does not include exercises, preliminary versions have been used in graduate courses on smart material systems and much of the detail provided throughout the book was motivated by feedback from students in those classes. Various resources will be maintained at the website http://www.siam.org/ books/f r32 to augment the text and provide a mechanism to update the material. To aid the reader in the implementation of certain models, we have included MATLAB® m-files for the homogenized energy framework used to characterize hysteresis in ferroelectric, ferromagnetic and shape memory alloy compounds as well as well as rod, beam and plate codes. Whereas these are research rather than
Preface
xv
commercial-level codes, they illustrate facets of the models and will provide a template for simulating the behavior of various smart material systems. We will also maintain a list of errata and certain updates to the material at this website. The reader will note that while a number of smart materials are discussed in Chapter 1, the model development in subsequent chapters focuses almost exclusively on ferroelectric, relaxor ferroelectric, ferromagnetic and ferroelastic compounds. Although this includes a large cross-section of presently employed materials, it neglects a number of established and emerging compounds such as ionic and amorphous polymers, MR and ER fluids and solids, and fiber optic sensors. This omission is dictated solely by the unified nature of ferroic compounds, in addition to length limitations, and should not be construed as an indication of material merit. Numerous references are provided in Chapter 1 to guide readers investigating applications which exploit alternative materials. A significant body of this research resulted from collaborations with students, postdocs and colleagues, and credit to their contributions is liberally given through the citations. Special thanks are also extended to collaborators whose comments on preliminary versions of the manuscript have significantly improved the exposition and reduced the number of typos by several orders of magnitude. Specifically, sincere thanks are given to Brian Ball, Lynn Boyles, Tom Braun, Marcelo Dapino, Andrew Hatch, Emily Lada and Stefan Seelecke for their attention to detail and honest feedback when reading parts of the manuscript. Andrew Hatch and Jordan Massad are credited with writing and editing much of the software that is included at the text website to render it in a form that is friendlier to users. The support provided by several federal funding agencies has been instrumental both for the research summarized here and the writing of this text. These agencies include the Air Force Office of Scientific Research (Dynamics and Control Program), the DARPA Mosaic Program, NASA Langley Research Center, and the National Science Foundation (Division of Civil and Mechanical Systems). Finally, I would like to thank Lisa Briggeman, Simon Dickey, Elizabeth Greenspan and April Schilpp from SIAM for their assistance and encouragement throughout the process of writing this book. Ralph C. Smith North Carolina State University Raleigh, NC December 1, 2004
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Notation Meaning
Symbol
Page
Material Properties and Models
a a a1,
a2,
a3
0 ßA, ßM 7 £ £
£R eo e
eo
n n na a a o Oc, Oi, Or A Ao, A s ,
A
AB
A s ||, AsT A100, A111 A
Coupling coefficient in effective field models Index for austenite or martensite Direction cosines between magnetization and crystallographic axes Standard deviation in Boltzmann probably Chemical free energies for austenite, martensite Shear strain rate for ER or MR fluids Strain Local average strain Remanerice strain Spontaneous strain Dielectric permittivity Permittivity of a vacuum (8.854 x 10–12 F/m) ER or MR fluid viscosity Local inverse susceptibility Phase-dependent entropy constants Incidence and reflection angles for an optical fiber Thermal conductivity of medium Magnetostriction Bragg wavelength for an optical fiber Spontaneous magnetostriction measured parallel and perpendicular to applied field Saturation magnetostrictions in the {100} and (111) directions Grating period for a Bragg optical sensor xvii
86 264 163 102 256 34 427 266 426 256
87 55 34 100 256 38 126 423
40
174 174 40
xviii Symbol
Notation Meaning
Page
Material Properties and Models (Continued)
u u u uo v, v1, v2 P Pe
a OA, OM
OAs, OMs Oc Oe 07
OR T
0
9 O
Oo
X XA, X M + , XM-
U W
o A A Af, As B,B Bs
CA,CM c c d31, d33
Measure in Preisach models Boltzmann probability function Magnetic permeability Permeability of free space (4n x 10–7 H/m) Densities in Preisach and energy models Material or charge density Average electrical resistivity Stress Critical austenite and martensite stresses Austenite and martensite starting stresses Coercive stress Effective stress Interaction stress Relative stress Shear stress Electric potential Conjugate field Average exchange energy Energy required to reorient single dipole or moment in an ordered lattice Dielectric or magnetic susceptibility Characteristic functions Helmholtz energy Frequency at which dipoles or moments attempt to switch Transducer surface area Austenite phase Cross-sectional area Austenite finish and start temperatures Magnetic induction or flux Saturation magnetic induction Phase-dependent specific heat capacities Average specific heat for a material Kelvin-Voigt damping coefficient Piezoelectric strain constants
82 102 166 167 112 135 126 427 266 246 412 260 260 269 34 135 282 75 75 423 267 62
219 126 22 126 22 422 165 254 268 68 55
xix
Notation Symbol
Meaning
Page
Material Properties and Models (Continued) D,D De
e E,E E Ec
Ec Ee
Eh E1
Er E0 g31, g33
G ha
hc H,H Hc
Hc He
Hh
HI J k
Kp Ks
k31
K, K1, K2 L(t) m,m M,M M
Electric D field Effective D field Order parameter Electric field Electric field data Coercive electric field Local average coercive field Effective electric field Bias electric field Interaction electric field Energy for 180° domain walls Depolarizing electric field Piezoelectric stress constants Gibbs energy Specific enthalpy Heat transfer coefficient Magnetic field Coercive magnetic field Local average coercive field Effective magnetic field Bias magnetic field Interaction magnetic field Exchange integral Boltzmann's constant (1.38 x 10–23 J/K) Irreversible loss constant in domain wall models Preisach kernel Electromechanical or magnetomechanical coupling factor Anisotropy constants Interface line in the Preisach plane between S+(t) and S-(t) Magnetic moment Magnetization Set of finite, signed Borel measures
414 91 423 47 119 412 113 86 76 111 90 46 28 62 271 126 166 412 199 180 186 161 179 75 91 81
67 180 194 185 422 82
Notation
XX
Symbol
Meaning
Page
Material Properties and Models (Continued)
M M A/an
Mf, Ms Mirr Mrew;
MR Ms
Mo <M+), (M_) M+, MMPB n1, n2 ne N+,N–
P+-.P-+ Po P,P
P P Pan Pirr Prev
PR Ps (P+), (P-) Po
Q Q R s
Actuator mass Local average magnetization Anhysteretic magnetization Martensite finish and start temperatures Irreversible magnetization Reversible magnetization Remanence magnetization Saturation magnetization Spontaneous magnetization Average magnetization due to positively and negatively oriented moments Martensite phases Morphotropic phase boundary Refractive indices for optical fibers Modal index for a Bragg optical sensor Number of positively and negatively oriented dipoles or moments Likelihood of dipole, moment switch from + to — or conversely Dipole strength Polarization Local average polarization Polarization data Switching polarization Irreversible polarization Reversible polarization Remanence polarization Saturation polarization Average polarization due to positively and negatively oriented dipoles Spontaneous polarization Heat Electric charge Correlation between cells of Mg cations in PMN Compliance
126 203 411 22 199 199 422 426 426 218 244 48 37 40
74 102 74 425 79 119 85 91 92 425 426 102 426 58 135 146 56
Notation
Symbol
xxi
Meaning
Page
Material Properties and Models (Continued) Si Sl, S2
s s
SA,SM S_ S+(t), S-(t)
SA S irr t ts
T Tc
Tc TE Tf Tg TR
T0 T
u
UA,UE,UX,UM
V V W XA, XM x+, x –
X YA,YM yP Y ijki
Dipole, spin or moment orientations Thresholds in Preisach models Entropy Preisach plane Specific entropies for austenite and martensite Index set of negative dipole states Sets in the Preisach plane of positively and negatively oriented moments Compact Preisach plane Entropy production due to irreversible processes Time Switching time Temperature Curie point (transition temperature) Local Curie point Temperature of surrounding environment Freezing temperature Glass transition temperature Temperature of a reference state Curie temperature Relaxation time Internal energy Anisotropy, exchange, magnetomechanical and magnetostatic energies Voltage Control volume Work Fractions of austenite, martensite variants Fractions of positively and negatively oriented dipoles or moments Density of neighboring B' sites containing Mg cations in PMN Young's moduli of austenite, martensite Young's modulus at constant polarization
74 191 58 81 254 106 194 193 59 82 82 58 413 148 126 142 30 254 414 104 58 163 68 102 58 273 102 146 256 57
xxii Symbol
Notation Page
Meaning Numerical Approximation
DPn
En K L1, L2, L3 M
Q
Degree of precision Quadrature error Stiffness matrix Area coordinates Mass matrix Damping matrix
376 376 386 383 386 386
Operators and Spaces
C Cn[a,b] C(0,T;X) Hp(a,b) Hpo(a, b)
1C Kt L 2 (a,6) L 2 (0,T;X)
Q
R(k) T(q)
T a (q)
y
Observation operator Space of functions that are n-times continuously differentiable on the interval [a, 6] Space of X-valued continuous functions on [0, T] Sobolev space of functions whose first p distributional derivatives exist in L 2 (a, 6) Subset of Hp(a,b) whose elements satisfy essential boundary conditions Parameter-to-observation map Moore-Penrose generalized inverse of K Space of square integrable functions Space of X-valued square integrable functions on (0, T) Parameter space Range of the operator K Least squares functional Regularized least squares functional Observation space
119 412 412 419 419 119 119 421 421 119 119 119 120 119
Structural Models 7 Ex
0a, 0ß, 0x, 0u
k
Air damping coefficient Strain at thickness coordinate z (also Eß, Eaß, £f3n where a,ß = x , y , 0 ) Neutral surface rotation Curvature
316 329 344 318
Notation Symbol
xxiii Meaning
Page
Structural Models (Continued)
P
