ELECTROMAGNETO-MECHANICS OFMATERIAL SYSTEMSAND STRUCTURES

ELECTROMAGNETO-MECHANICS OFMATERIAL SYSTEMSAND STRUCTURES

Yasuhide ShindoTohoku University, Japan

This edition first published 2015© 2015 John Wiley & Sons Singapore Pte. Ltd.

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Contents

About the Author ix

Preface xi

Acknowledgments xiii

1 Introduction 1References 2

2 Conducting Material Systems and Structures 52.1 Basic Equations of Dynamic Magnetoelasticity 52.2 Magnetoelastic Plate Vibrations and Waves 7

2.2.1 Classical Plate Bending Theory 92.2.2 Mindlin’s Theory of Plate Bending 132.2.3 Classical Plate Bending Solutions 162.2.4 Mindlin Plate Bending Solutions 232.2.5 Plane Strain Plate Solutions 26

2.3 Dynamic Magnetoelastic Crack Mechanics 322.4 Cracked Materials Under Electromagnetic Force 402.5 Summary 45

References 45

3 Dielectric/Ferroelectric Material Systems and Structures 47

Part 3.1 Dielectrics 473.1 Basic Equations of Electroelasticity 483.2 Static Electroelastic Crack Mechanics 49

3.2.1 Infinite Dielectric Materials 493.2.2 Dielectric Strip 57

3.3 Electroelastic Vibrations and Waves 603.4 Dynamic Electroelastic Crack Mechanics 683.5 Summary 72

vi Contents

Part 3.2 Piezoelectricity 723.6 Piezomechanics and Basic Equations 73

3.6.1 Linear Theory 733.6.2 Model of Polarization Switching 773.6.3 Model of Domain Wall Motion 803.6.4 Classical Lamination Theory 82

3.7 Bending of Piezoelectric Laminates 903.7.1 Bimorphs 903.7.2 Functionally Graded Bimorphs 1003.7.3 Laminated Plates 111

3.8 Electromechanical Field Concentrations 1133.8.1 Laminates 1133.8.2 Disk Composites 1233.8.3 Fiber Composites 1263.8.4 MEMS Mirrors 136

3.9 Cryogenic and High-Temperature Electromechanical Responses 1403.9.1 Cryogenic Electromechanical Response 1403.9.2 High-Temperature Electromechanical Response 147

3.10 Electric Fracture and Fatigue 1493.10.1 Fracture Mechanics Parameters 1503.10.2 Cracked Rectangular Piezoelectric Material 1733.10.3 Indentation Fracture Test 1853.10.4 Modified Small Punch Test 1893.10.5 Single-Edge Precracked Beam Test 1933.10.6 Double Torsion Test 2013.10.7 Fatigue of SEPB Specimens 203

3.11 Summary 212References 213

4 Ferromagnetic Material Systems and Structures 219

Part 4.1 Ferromagnetics 2194.1 Basic Equations of Magnetoelasticity 220

4.1.1 Soft Ferromagnetic Materials 2204.1.2 Magnetically Saturated Materials 2214.1.3 Electromagnetic Materials 222

4.2 Magnetoelastic Instability 2244.2.1 Buckling of Soft Ferromagnetic Material 2254.2.2 Buckling of Magnetically Saturated Material 2284.2.3 Bending of Soft Ferromagnetic Material 231

4.3 Magnetoelastic Vibrations and Waves 2334.3.1 Vibrations and Waves of Soft Ferromagnetic Material 2334.3.2 Vibrations and Waves of Magnetically Saturated Material 243

4.4 Magnetic Moment Intensity Factor 2504.4.1 Simply Supported Plate Under Static Bending 2514.4.2 Fixed-End Plate Under Static Bending 2524.4.3 Infinite Plate Under Dynamic Bending 255

Contents vii

4.5 Tensile Fracture and Fatigue 2564.5.1 Cracked Rectangular Soft Ferromagnetic Material 2574.5.2 Fracture Test 2614.5.3 Fatigue Crack Growth Test 263

4.6 Summary 265

Part 4.2 Magnetostriction 2654.7 Basic Equations of Magnetostriction 2654.8 Nonlinear Magneto-Mechanical Response 267

4.8.1 Terfenol-D/Metal Laminates 2674.8.2 Terfenol-D/PZT Laminates 270

4.9 Magnetoelectric Response 2724.10 Summary 273

References 273

Index 277

About the Author

Dr. Shindo received his Doctorate of Engineering from Tohoku University in 1977. He iscurrently a professor in the Department of Materials Processing in the Graduate School ofEngineering at Tohoku University. Dr. Shindo also served on the Board of Editors of theInternational Journal of Solids and Structures and is currently serving on the editorial boardof Journal of Mechanics of Materials and Structures, the Advisory Board of Acta Mechanica,the International Editorial Board of AES Technical Reviews International Journal (Part A:International Journal of Nano and Advanced Engineering Materials (IJNAEM), Part B:International Journal of Advances in Mechanics and Applications of Industrial Materials(IJAMAIM), Part C: International Journal of Advances and Trends in Engineering Materialsand their Applications (IJATEMA), Part D: International Journal of Reliability and Safetyof Engineering Systems and Structures (IJRSESS)), the Editorial Advisory Board of TheOpen Civil Engineering Journal/The Open Textile Journal/The Open Conference ProceedingJournal/The Open Physics Journal (formerly The Open Mechanics Journal), the EditorialBoard of Advances in Theoretical and Applied Mechanics, Chemical Engineering andProcess Techniques, International Scholarly Research Notices (Mechanical Engineering),Journal of Applied Mathematics, the editorial board of Strength, Fracture and Complexity,An International Journal, and the Editor-in-Chief of The Open Mechanical EngineeringJournal and International Journal of Metallurgical & Materials Engineering. His primaryresearch interests are in the areas of mesomechanics of material systems and structures,electromagnetic fracture and damage mechanics, dynamics and cryomechanics of advancedcomposite materials/structural alloys, and reliability and durability of micro-/nanocomponentsand devices.

Preface

The science of electromagneto-mechanics, which is concerned with the interaction of electro-magnetic fields and deformation in material systems and structures, has developed because ofthe possibility of its practical applications in various fields such as electronic and electrome-chanical devices. As the area of science and technology expands, it becomes important thatnewly acquired knowledge and expertise are communicated effectively to those who can gainmost by applying them in practice. This book covers a very wide and varied range of subjectareas that fall under its subject and all aspects (theoretical, experimental, computationalstudies, and/or industrial applications) of electromagneto-mechanics from state-of-the-artfundamental research to applied research and applications in emerging technologies.

Yasuhide ShindoSendai, Japan

September, 2014

Acknowledgments

I am indebted to many authors whose writings are classics in the field of electromagneto-mechanics. It is also a pleasure to acknowledge the help received from my students and col-leagues. Special thanks go to Professor Fumio Narita of Tohoku University, who read the entiremanuscript and gave me many valuable suggestions for improvement. Finally, I would like tothank the publisher John Wiley & Sons for their continuous support for this project.

1Introduction

The electromagneto-mechanics of material systems and structures has been developing rapidlywith extensive applications in, for example, electronic industry, magnetic fusion engineering,superconducting devices, and smart materials and microelectromechanical systems (MEMS).Researchers in this interdisciplinary field are with diverse background and motivation. Thisbook reflects a cross section of recent activities in the electromagneto-mechanics of conductingmaterials, dielectric materials, piezoelectric materials and devices, ferromagnetic materials,magnetostrictive material systems, and so on.

Chapter 2 deals with the magneto-mechanics of conducting material systems and structures.Here, the theory of dynamic magnetoelasticity is presented. Vibrations and waves of conduct-ing plates are then considered, and the effect of the magnetic field on the flexural waves isexamined. The theory is also applied to various problems for cracked conducting plates, andthe influence of the magnetic field on the dynamic singular stresses is displayed graphically anddiscussed. In addition, the results for the cracked plates under large electric current and strongmagnetic field are presented, and the effect of the electromagnetic force on the mechanicalbehavior is shown.

Chapter 3 provides the electromechanical interactions of dielectric/ferroelectric materialsystems and structures. In Part 3.1, we present the theory of dielectrics. Basic equations ofelectroelasticity are given. Applications are then made to static electroelastic crack mechanics,electroelastic vibrations and waves, and dynamic electroelastic crack mechanics of dielectricmaterials. Part 3.2 is devoted to the discussion of linear and nonlinear piezoelectricity. For aliterature on this topic, we refer readers to Tiersten [1]. Piezomechanics and basic equationsare presented. Theory is then applied to various problems, including bending behavior, elec-tromechanical field concentrations, and cryogenic electromechanical response. Experimentaldata are also shown to validate the theoretical model. Furthermore, the theoretical and experi-mental results on the electric field dependence of fracture and fatigue of piezoelectric materialsystems are presented.

In Chapter 4, we deal with the magneto-mechanics of ferromagnetic material systems andstructures. Part 4.1 presents the theory and test of ferromagnetics. Reference on this topicmay be made to Brown [2]. Basic equations of magnetoelasticity are developed. Theory isthen applied to various problems, including magnetoelastic instability, magnetoelastic vibra-tions, and waves of soft ferromagnetic and magnetically saturated materials under magnetic

Electromagneto-Mechanics of Material Systems and Structures, First Edition. Yasuhide Shindo.© 2015 John Wiley & Sons Singapore Pte Ltd. Published 2015 by John Wiley & Sons Singapore Pte Ltd.

2 Electromagneto-Mechanics of Material Systems and Structures

fields, and some experiments are performed to validate the theoretical predictions. The mag-netoelastic analysis and experimental evidence are also presented for cracked plates underbending, and the effect of magnetic fields on the moment intensity factor is shown. Moreover,the tensile fracture and fatigue of soft ferromagnetic materials under magnetic fields are ded-icated. Part 4.2 is concerned with a discussion of magnetostriction. Works on the subject arefound to be in du Tremolet de Lacheisserie [3]. Basic equations of magnetostriction are given.Theoretical and experimental treatments of the nonlinear magneto-mechanical response inmagnetostrictive material systems are then presented. Here, the material systems consist of themagnetostrictive and elastic layers, and later, we consider the magnetostrictive layer bondedto the piezoelectric layer. In addition, the piezomagnetoelectric effect of particle-reinforcedcomposites is discussed.

There are extensive literatures on this subject. Some books are listed as follows. That is,Moon [4] organized the existing literatures on magneto-solid mechanics and gave a presen-tation of the basic principles and some useful method of analysis. Parton and Kudryavtsev[5] analyzed the behavior of piezoelectric materials and considered strength and failure prob-lems for piezoelectric and electrically conducting materials. In addition, Eringen and Maugin[6, 7] presented a unified approach to the nonlinear continuum theory of deformable and fluentmaterials subjected to electromagnetic and thermal loads. Also, there are the following con-ference proceedings books of IUTAM symposium: Maugin [8], Yamamoto and Miya [9], andHsieh [10], and of other mini-symposiums: Lee et al. [11], Yang and Maugin [12], and Shindo[13]. Moreover, the following monographs present a good discussion of this subject: Paria[14], Parkus [15, 16], Alblas [17], Moon [18], Hutter and van de Ven [19], Pao [20], Hsieh[21], Ambartsumian [22], and the set of chapters edited by Parkus [23]. In the above-listedliteratures, references to other papers can be found.

References[1] H. F. Tiersten, Linear Piezoelectric Plate Vibration, Plenum Press, New York, 1969.[2] W. F. Brown, Jr., Magnetoelastic Interactions, Springer-Verlag, Berlin, 1966.[3] E. du Tremolet de Lacheisserie, Magnetostriction: Theory and Applications of Magnetoelasticity, CRC Press,

Boca Raton, FL, 1993.[4] F. C. Moon, Magneto-Solid Mechanics, John Wiley & Sons, Inc., New York, 1984.[5] V. Z. Parton and B. A. Kudryavtsev, Electromagnetoelasticity, Gordon and Breach Science Publishers, New

York, 1988.[6] A. C. Eringen and G. A. Maugin, Electrodynamics of Continua I, Springer-Verlag, New York, 1989.[7] A. C. Eringen and G. A. Maugin, Electrodynamics of Continua II, Springer-Verlag, New York, 1990.[8] G. A. Maugin (ed.), Proceedings of the IUTAM/IUPAP Symposium on the Mechanical Behavior of Electromag-

netic Solid Continua, North-Holland, Amsterdam, 1984.[9] Y. Yamamoto and K. Miya (eds.), Proceedings of the IUTAM Symposium on Electromagnetomechanical Inter-

actions in Deformable Solids and Structures, North-Holland, Amsterdam, 1987.[10] R. K. T. Hsieh (ed.), Proceedings of the IUTAM Symposium on Mechanical Modeling of New Electromagnetic

Materials, Elsevier, Amsterdam, 1990.[11] J. S. Lee, G. A. Maugin and Y. Shindo (eds.), Mechanics of Electromagnetic Materials and Structures, AMD-Vol.

161, MD-Vol. 42, ASME, New York, 1993.[12] J. S. Yang and G. A. Maugin (eds.), Mechanics of Electromagnetic Materials and Structures, IOS Press, Ams-

terdam, 2000.[13] Y. Shindo (ed.), Mechanics of Electromagnetic Material Systems and Structures, WIT Press, Southampton, 2003.[14] G. Paria, “Magneto-elasticity and magneto-thermo-elasticity,” Adv. Appl. Mech. 10, 73 (1967).

Introduction 3

[15] H. Parkus, “Variational principles in thermo- and magneto-elasticity,” CISM Courses and Lectures Vol. 58,Springer-Verlag, Wien, 1970.

[16] H. Parkus, “Magneto-thermoelasticity,” CISM Courses and Lectures Vol. 118, Springer-Verlag, Wien, 1972.[17] J. B. Alblas, “Electro-magneto-elasticity,” Topics in Applied Continuum Mechanics, J. L. Zeman and F. Ziegler

(eds.), Springer-Verlag, Wien, p. 71 (1974).[18] F. C. Moon, “Problems in magneto-solid mechanics,” Mech. Today 4, 307 (1978).[19] K. Hutter and A. A. F. van de Ven, “Field matter interactions in thermoelastic solids: a unification of existing

theories of electro-magneto-mechanical interactions,” Lecture Notes in Physics Vol. 88, Springer-Verlag, Berlin,1978.

[20] Y.-H. Pao, “Electromagnetic forces in deformable continua,” Mech. Today 4, 209 (1978).[21] R. K. T. Hsieh, “Micropolarized and magnetized media,” Mechanics of Micropolar Media, O. Brulin and R. K.

T. Hsieh (eds.), World Scientific, Singapore, p. 187 (1982).[22] S. A. Ambartsumian, “Magneto-elasticity of thin plates and shells,” Appl. Mech. Rev. 35(1), 1 (1982).[23] H. Parkus (ed.), “Electromagnetic interactions in elastic solids,” CISM Courses and Lectures Vol. 257,

Springer-Verlag, Wien, 1979.

2Conducting Material Systems andStructures

If electrically conducting materials are used in strong magnetic field, we must consider theeffect of induced current. Figure 2.1 shows the dynamic magnetoelastic interactions of con-ducting materials. In this chapter, first, magnetoelastic vibrations and waves of conductingmaterials are discussed. Next, the influence of magnetic field on the dynamic singular stressesin cracked conducting materials is described.

The components of the superconducting structures are most often used in environments withlarge electric currents and strong magnetic fields. The singular stresses in cracked conductingmaterials under electromagnetic force are also examined in this chapter.

2.1 Basic Equations of Dynamic Magnetoelasticity

Let us now consider the rectangular Cartesian coordinates xi(O-x1, x2, x3). Electrically con-ducting materials are permeated by a static uniform magnetic field 𝐇0. We consider smallperturbations characterized by the displacement vector 𝐮 produced in the material.

The magnetic and electric fields may be expressed in the form

𝐇 = 𝐇0 + 𝐡, 𝐄 = 0 + 𝐞 (2.1)

where 𝐇 and 𝐄 are magnetic and electric field intensity vectors, and 𝐡 and 𝐞 are the fluctuatingfields and are assumed to be of the same order of magnitude as the particle displacement u.The extension of Maxwell’s theory from materials at rest to those in motion was performed byMinkowski in 1908 [1].

The linearized field equations are listed as follows [2]:

𝜎ji,j + 𝜀ijkjjB0k = 𝜌ui,tt (2.2)

𝜀ijkH0k,j = 0, B0i,i = 0 (2.3)

𝜀ijkhk,j − di,t = ji, 𝜀ijkek,j + bi,t = 0, bi,i = 0, di,i = 𝜌e (2.4)

Electromagneto-Mechanics of Material Systems and Structures, First Edition. Yasuhide Shindo.© 2015 John Wiley & Sons Singapore Pte Ltd. Published 2015 by John Wiley & Sons Singapore Pte Ltd.

6 Electromagneto-Mechanics of Material Systems and Structures

Magneticfield

Conducting materials

Induced current

Dynamic deformation

Dynamicmechanical load

Electromagneticbody force

Figure 2.1 Dynamic magnetoelastic interactions of conducting materials

where ui is the displacement vector component, 𝜎ij is the stress tensor component, H0i, B0i,hi, ei, bi, di, ji are the components of 𝐇0, magnetic induction vector 𝐁0, 𝐡, 𝐞, magnetic induc-tion vector 𝐛, electric displacement vector 𝐝, current density vector 𝐣, respectively, 𝜌 is themass density, 𝜌e is the free electric charge density, a comma followed by an index denotespartial differentiation with respect to the space coordinate xi or the time t, and the summationconvention over repeated indices is used. The permutation symbol 𝜀ijk is defined by

𝜀ijk = +1 if ijk is a cyclic permutation of 1, 2, 30 if any two indices are equal

−1 if ijk is an anticyclic permutation(2.5)

The equation of conservation of charge follows from the first and fourth of Eqs. (2.4)

𝜌e,t + ji,i = 0 (2.6)

The linearized constitutive equations can be written as

𝜎ij = 𝜆uk,k𝛿ij + 𝜇(ui,j + uj,i) (2.7)

𝜎Mij = H0jbi + hjB0i −

12(H0kbk + hkB0k)𝛿ij (2.8)

B0i = 𝜅H0i, bi = 𝜅hi, (2.9)

di = 𝜖ei + (𝜖𝜅 − 𝜖0𝜅0)𝜀ijkuj,tH0k (2.10)

ji = 𝜎(ei + 𝜀ijkuj,tB0k) (2.11)

where 𝜎Mij is the Maxwell stress tensor component, 𝜆 = 2G𝜈∕(1 − 2𝜈) and 𝜇 = G are the Lamé

constants, G = E∕2(1 + 𝜈) is the shear modulus, E and 𝜈 are Young’s modulus and Poisson’sratio, respectively, 𝜅 is the magnetic permeability, 𝜖 is the permittivity, 𝜅0 = 1.26 × 10−6 H/mis the magnetic permeability of free space, 𝜖0 = 8.85 × 10−12 C/Vm is the permittivity of freespace, and 𝜎 is electric conductivity. The Kronecker delta 𝛿ij is defined by

𝛿ij = 1 if i = j0 if i ≠ j

(2.12)

Conducting Material Systems and Structures 7

n

Figure 2.2 An arbitrary material volume element

The linearized boundary conditions are obtained as

[[𝜎ji + 𝜎Mji ]]nj = 0 (2.13)

𝜀ijknj[[hk]] = jsi , 𝜀ijknj[[ek + uk,tB0k]] = 0[[bi]]ni = 0, [[di]]ni = 𝜌s

e(2.14)

where 𝜌se is the surface charge density, jsi is the component of surface current density vector

𝐣s, ni is the component of outer unit vector 𝐧 normal to an undeformed material as shown inFig. 2.2, and [[fi]] means the jump in any field quantity fi across the boundary; that is, [[fi]] =f ei − fi. The superscript e denotes the quantity outside the material. The conservation of charge

applied to the region gives the following normal boundary condition:

[[ji]]ni = −𝜌se,t (2.15)

2.2 Magnetoelastic Plate Vibrations and Waves

In this section, the magnetoelastic vibrations and waves of a conducting material are discussed.Consider an electrically conducting elastic plate with thickness 2h in a rectangular Cartesiancoordinate system (x, y, z). The coordinate axes x and y are in the middle plane of the plate,and the z-axis is normal to this plane. It is assumed that the plate has the permittivity 𝜖 = 𝜖0and magnetic permeability 𝜅 = 𝜅0, respectively. A uniform magnetic field 𝐇0 is applied.

Using the first of Eqs. (2.9), Eqs. (2.3) can be expressed as

He0z,y − He

0y,z = 0, He0x,z − He

0z,x = 0, He0y,x − He

0x,y = 0,

He0x,x + He

0y,y + He0z,z = 0

H0z,y − H0y,z = 0, H0x,z − H0z,x = 0, H0y,x − H0x,y = 0,H0x,x + H0y,y + H0z,z = 0

(2.16)

The mechanical constitutive equations are taken to be the usual Hooke’s law

𝜎xx = 𝜆(ux,x + uy,y + uz,z) + 2𝜇ux,x

𝜎yy = 𝜆(ux,x + uy,y + uz,z) + 2𝜇uy,y

𝜎zz = 𝜆(ux,x + uy,y + uz,z) + 2𝜇uz,z

𝜎xy = 𝜎yx = 𝜇(ux,y + uy,x)𝜎yz = 𝜎zy = 𝜇(uy,z + uz,y)𝜎xz = 𝜎zx = 𝜇(uz,x + ux,z)

(2.17)

8 Electromagneto-Mechanics of Material Systems and Structures

and the Maxwell stresses are

𝜎Mxx = 𝜅0hxH0x − 𝜅0hyH0y − 𝜅0hzH0z

𝜎Myy = 𝜅0hyH0y − 𝜅0hzH0z − 𝜅0hxH0x

𝜎Mzz = 𝜅0hzH0z − 𝜅0hxH0x − 𝜅0hyH0y

𝜎Mxy = 𝜎M

yx = 𝜅0hxH0y + 𝜅0hyH0x

𝜎Myz = 𝜎M

zy = 𝜅0hyH0z + 𝜅0hzH0y

𝜎Mzx = 𝜎M

xz = 𝜅0hzH0x + 𝜅0hxH0z

(2.18)

The currents are determined by Ohm’s law, Eq. (2.11), and they are

jx = 𝜎{ex + 𝜅0(uy,tH0z − uz,tH0y)}jy = 𝜎{ey + 𝜅0(uz,tH0x − ux,tH0z)}jz = 𝜎{ez + 𝜅0(ux,tH0y − uy,tH0x)}

(2.19)

The stress equations of motion, Eq. (2.2), are given by

𝜎xx,x + 𝜎yx,y + 𝜎zx,z = 𝜌ux,tt − 𝜅0(jyH0z − jzH0y)𝜎xy,x + 𝜎yy,y + 𝜎zy,z = 𝜌uy,tt − 𝜅0(jzH0x − jxH0z)𝜎xz,x + 𝜎yz,y + 𝜎zz,z = 𝜌uz,tt − 𝜅0(jxH0y − jyH0x)

(2.20)

Neglecting displacement currents compared to the conduction currents, and using the secondof Eqs. (2.9) and Eq. (2.10), the Maxwell’s equations, Eqs. (2.4), are

hez,y − he

y,z = 0, hex,z − he

z,x = 0, hey,x − he

x,y = 0

hz,y − hy,z = jx, hx,z − hz,x = jy, hy,x − hx,y = jz(2.21)

eez,y − ee

y,z = −𝜅0hex,t, ee

x,z − eez,x = −𝜅0he

y,t, eey,x − ee

x,y = −𝜅0hez,t

ez,y − ey,z = −𝜅0hx,t, ex,z − ez,x = −𝜅0hy,t, ey,x − ex,y = −𝜅0hz,t

(2.22)

hex,x + he

y,y + hez,z = 0

hx,x + hy,y + hz,z = 0(2.23)

eex,x + ee

y,y + eez,z = 0

𝜖0(ex,x + ey,y + ez,z) = 𝜌e(2.24)

From Eqs. (2.13) – (2.15), we obtain the linearized boundary conditions

𝜎Mezz (x, y,±h, t) − {𝜎zz(x, y,±h, t) + 𝜎M

zz (x, y,±h, t)} = 0

𝜎Mezy (x, y,±h, t) − {𝜎zy(x, y,±h, t) + 𝜎M

zy (x, y,±h, t)} = 0

𝜎Mezx (x, y,±h, t) − {𝜎zx(x, y,±h, t) + 𝜎M

zx (x, y,±h, t)} = 0

(2.25)

Conducting Material Systems and Structures 9

hex(x, y,±h, t) − hx(x, y,±h, t) = jsy

hey(x, y,±h, t) − hy(x, y,±h, t) = −jsx

(2.26)

eex(x, y,±h, t) − ex(x, y,±h, t) = 0

eey(x, y,±h, t) − ey(x, y,±h, t) = 0

(2.27)

hez(x, y,±h, t) − hz(x, y,±h, t) = 0 (2.28)

𝜖0{eez(x, y,±h, t) − ez(x, y,±h, t)} = 𝜌s

e (2.29)

jz(x, y,±z, t) = 0 (2.30)

2.2.1 Classical Plate Bending Theory

Classical plate bending theory for magnetoelastic interactions in a conducting material isapplied. By using magnetoelastic thin plate theory [3, 4], the rectangular displacement, mag-netic field and electric field components can be expressed as follows:

ux = −zw,x, uy = −zw,y, uz = w(x, y, t) (2.31)

hz = f (x, y, t), ex = 𝜑(x, y, t), ey = 𝜓(x, y, t) (2.32)

where w(x, y, t) represents the deflection of the middle plane of the plate, f , 𝜑, 𝜓 are the func-tions of x, y, t. Substituting from Eqs. (2.31) and (2.32) into the fourth and fifth of Eqs. (2.21)and the second of Eqs. (2.24), using Eqs. (2.19), and assuming 𝜌e = 0, we obtain

hx,z = f,x + 𝜎{𝜓 + 𝜅0(H0xw,t + H0zzw,xt)}

hy,z = f,y − 𝜎{𝜑 − 𝜅0(H0yw,t + H0zzw,yt)}

ez,z = −𝜑,x − 𝜓,y

(2.33)

For conductors with finite electric conductivity, jsx and jsy can be neglected so that, from Eqs.(2.26)–(2.30), we get

hx(x, y,±h, t) = hex(x, y,±h, t)

hy(x, y,±h, t) = hey(x, y,±h, t)

(2.34)

𝜑(x, y,±h, t) = eex(x, y,±h, t)

𝜓(x, y,±h, t) = eey(x, y,±h, t)

(2.35)

f (x, y,±h, t) = hez(x, y,±h, t) (2.36)

ez(x, y,±h, t) + 𝜅0{H0yux,t(x, y,±h, t) − H0xuy,t(x, y,±h, t)} = 0 (2.37)

10 Electromagneto-Mechanics of Material Systems and Structures

Integrating the representations (2.33) with respect to z, we obtain the remaining electromag-netic field components as

hx =12{hx(x, y, h, t) + hx(x, y,−h, t)} + z{f,x + 𝜎(𝜓 + 𝜅0H0xw,t)}

+𝜎𝜅0H0zz2 − h2

2w,xt

hy =12{hy(x, y, h, t) + hy(x, y,−h, t)} + z{f,y − 𝜎(𝜑 − 𝜅0H0yw,t)}

+𝜎𝜅0H0zz2 − h2

2w,yt

ez =12{ez(x, y, h, t) + ez(x, y,−h, t)} − z(𝜑,x + 𝜓,y)

(2.38)

Therefore, all the electromagnetic field components are represented by means of the fourdesired functions w, f , 𝜑, 𝜓 . Integrating the first and second of Eqs. (2.33) and the sixth ofEqs. (2.22) with respect to z from −h to h, we have

f,x + 𝜎(𝜓 + 𝜅0H0xw,t) =1

2h{hx(x, y, h, t) − hx(x, y,−h, t)}

f,y − 𝜎(𝜑 − 𝜅0H0yw,t) =1

2h{hy(x, y, h, t) − hy(x, y,−h, t)}

𝜓,x − 𝜑,y = −𝜅0f,t

(2.39)

The stress boundary conditions on the plate surfaces are

𝜎zx = 𝜎zy = 𝜎zz = 0 (z = ±h) (2.40)

The bending and twisting moments per unit length (Mxx,Myy,Mxy = Myx) and the verticalshear forces per unit length (Qx,Qy) can be expressed in terms of w as

Mxx = ∫h

−h𝜎xxz dz = −D(w,xx + 𝜈w,yy)

Myy = ∫h

−h𝜎yyz dz = −D(w,yy + 𝜈w,xx)

Mxy = Myx = ∫h

−h𝜎xyz dz = −D(1 − 𝜈)w,xy

(2.41)

Qx = ∫h

−h𝜎zx dz = −D(w,xx + w,yy),x

Qy = ∫h

−h𝜎zy dz = −D(w,xx + w,yy),y

(2.42)

Conducting Material Systems and Structures 11

where D = 4𝜇h3∕3(1 − 𝜈) is the flexural rigidity of the plate. Now, if we substitute Eqs. (2.19)into Eqs. (2.20), multiply the first and second of Eqs. (2.20) by z dz, and integrate from −h toh, taking into account the boundary condition in Eq. (2.40), we obtain the results

Mxx,x + Myx,y − Qx = −23𝜌h3w,xtt − mC

xx

Mxy,x + Myy,y − Qy = −23𝜌h3w,ytt − mC

yy

(2.43)

The moments mCxx and mC

yy are derived as

mCxx = ∫

h

−h(jyBz − jzBy)z dz

= 23𝜎𝜅0h3{𝜅0(H2

0y + H20z)w,xt − 𝜅0H0xH0yw,yt + H0y𝜑,x + H0y𝜓,y}

mCyy = ∫

h

−h(jzBx − jxBz)z dz

= −23𝜎𝜅0h3{𝜅0H0xH0yw,xt − 𝜅0(H2

0x + H20z)w,yt + H0x𝜑,x + H0x𝜓,y}

(2.44)

If the third of Eqs. (2.20) is multiplied by dz and integrated from −h to h, taking into accountthe boundary condition (2.40), we obtain

Qx,x + Qy,y = 2h𝜌w,tt − qC (2.45)

The load qC applied to the plate is derived as

qC = ∫h

−h(jxBy − jyBx)dz = 2h𝜎𝜅0{H0y𝜑 − H0x𝜓 − 𝜅0(H2

0x + H20y)w,t} (2.46)

Eliminating Qx,Qy from Eqs. (2.43) and Eq. (2.45) and taking into account Eq. (2.41), we havethe equation of motion for a thin plate under the influence of magnetic field

D(w,xxxx + 2w,xxyy + w,yyyy) −23𝜌h3(w,xx + w,yy),tt + 2𝜌hw,tt − mC

xx,x − mCyy,y − qC = 0 (2.47)

Equations (2.39) and (2.47) are the basic equations of linear bending theory for conductingthin plates.

If we consider a perfectly conducting plate (𝜎 → ∞), we get from Eqs. (2.19)

ex + 𝜅0(uy,tH0z − uz,tH0y) = 0

ey + 𝜅0(uz,tH0x − ux,tH0z) = 0

ez + 𝜅0(ux,tH0y − uy,tH0x) = 0

(2.48)

12 Electromagneto-Mechanics of Material Systems and Structures

Hence, from the fourth, fifth, and sixth of Eqs. (2.22) and Eqs. (2.48), with Eqs. (2.31), weobtain the magnetic field intensity components as

hx = H0zw,x +𝜈

1 − 𝜈H0xzw,xx − H0yzw,xy +

11 − 𝜈

H0xzw,yy

hy = −H0zw,y +1 − 2𝜈1 − 𝜈

H0yzw,xx − H0xzw,xy −𝜈

1 − 𝜈H0yzw,yy

hz = −H0xw,x + H0yw,y − H0zzw,xx + H0zzw,yy

(2.49)

From the fourth, fifth, and sixth of Eqs. (2.21), with Eqs. (2.49), we also have

jx = −1 − 2𝜈1 − 𝜈

H0yw,xx − 2H0xw,xy −1

1 − 𝜈H0yw,yy

jy =( 1

1 − 𝜈H0x + H0z

)w,xx − 2H0yw,xy +

( 11 − 𝜈

H0x − H0z

)w,yy

jz = −2H0zw,xy +1 − 2𝜈1 − 𝜈

H0yzw,xxx −1

1 − 𝜈H0xzw,xxy +

1 − 2𝜈1 − 𝜈

H0yzw,xyy

− 11 − 𝜈

H0xzw,yyy

(2.50)

Moreover, from Eqs. (2.20), with Eqs. (2.50), we have

Mxx,x + Myx,y − Qx = −23𝜌h3w,xtt − mCP

xx

Mxy,x + Myy,y − Qy = −23𝜌h3w,ytt − mCP

yy

(2.51)

Qx,x + Qy,y = 2h𝜌w,tt − qCP (2.52)

where

mCPxx = h{𝜎zx(x, y, h, t) − 𝜎zx(x, y,−h, t)}

+𝜅0

1 − 𝜈 ∫h

−h[{(1 − 𝜈)H2

0z + H0zH0x}w,xx + {−(1 − 𝜈)H20z + H0zH0x}w,yy

−(1 − 2𝜈)H20yzw,xxx + H0xH0yzw,xxy − (1 − 2𝜈)H2

0yzw,xyy

+H0xH0yzw,yyy]z dz

mCPyy = h{𝜎zy(x, y, h, t) − 𝜎zy(x, y,−h, t)}

+𝜅0

1 − 𝜈 ∫h

−h{−(1 − 2𝜈)H0yH0zw,xx − H0yH0zw,yy − (1 − 2𝜈)H0xH0yzw,xxx

+H20xzw,xxy − (1 − 2𝜈)H0xH0yzw,xyy + H2

0xzw,yyy}z dz(2.53)

qCP = 𝜎zz(x, y, h, t) − 𝜎zz(x, y,−h, t)

−𝜅0

1 − 𝜈 ∫h

−h

[{H2

0x + (1 − 2𝜈)H20y + (1 − 𝜈)H0zH0x}w,xx

+{H20x − H2

0y + (1 − 𝜈)H0zH0x}w,yy

]dz (2.54)

Conducting Material Systems and Structures 13

From Eqs. (2.51) and (2.52), with Eqs. (2.41), we obtain

D(w,xxxx + 2w,xxyy + w,yyyy) −23𝜌h3(w,xx + w,yy),tt

+2𝜌hw,tt − mCPxx,x − mCP

yy,y − qCP = 0 (2.55)

2.2.2 Mindlin’s Theory of Plate Bending

Mindlin’s theory of plate bending [5], which accounts for the rotatory inertia and shear effects,is applied for magnetoelastic interactions in a conducting material. The rectangular displace-ment components may assume the forms

ux = z𝛹x(x, y, t), uy = z𝛹y(x, y, t), uz = 𝛹z(x, y, t) (2.56)

where 𝛹z represents the normal displacement of the plate, and 𝛹x and 𝛹y denote the rotationsof the normals about the x- and y-axes. The magnetic and electric field components areexpressed by Eqs. (2.32). Substituting from Eqs. (2.56) and (2.32) into the fourth and fifthof Eqs. (2.21) and the second of Eqs. (2.24), using Eqs. (2.19), and assuming 𝜌e = 0, weobtain

hx,z = f,x + 𝜎{𝜓 + 𝜅0(H0x𝛹z,t − H0zz𝛹x,t)}

hy,z = f,y − 𝜎{𝜑 − 𝜅0(H0y𝛹z,t − H0zz𝛹y,t)}

ez,z = −𝜑,x − 𝜓,y

(2.57)

Integrating the representations (2.57) with respect to z, we obtain

hx =12{hx(x, y, h, t) + hx(x, y,−h, t)} + z{f,x + 𝜎(𝜓 + 𝜅0H0x𝛹z,t)}

−𝜎𝜅0H0zz2 − h2

2𝛹x,t

hy =12{hy(x, y, h, t) + hy(x, y,−h, t)} + z{f,y − 𝜎(𝜑 − 𝜅0H0y𝛹z,t)}

−𝜎𝜅0H0zz2 − h2

2𝛹y,t

ez =12{ez(x, y, h, t) + ez(x, y,−h, t)} − z(𝜑,x + 𝜓,y)

(2.58)

Therefore, all of the electromagnetic field components are represented by means of the sixdesired functions 𝛹x, 𝛹y, 𝛹z, f , 𝜑, 𝜓 . Integrating the first and second of Eqs. (2.33) and thesixth of Eqs. (2.22) with respect to z from −h to h, we have

f,x + 𝜎(𝜓 + 𝜅0H0x𝛹z,t) =12h

{hx(x, y, h, t) − hx(x, y,−h, t)}

f,y − 𝜎(𝜑 − 𝜅0H0y𝛹z,t) =1

2h{hy(x, y, h, t) − hy(x, y,−h, t)}

𝜓,x − 𝜑,y = −𝜅0f,t

(2.59)

14 Electromagneto-Mechanics of Material Systems and Structures

The bending and twisting moments per unit length (Mxx,Myy,Mxy = Myx) and the vertical shearforces per unit length (Qx,Qy) can be expressed in terms of 𝛹x, 𝛹y, and 𝛹z as

Mxx = ∫h

−h𝜎xxz dz = D(𝛹x,x + 𝜈𝛹y,y)

Myy = ∫h

−h𝜎yyz dz = D(𝛹y,y + 𝜈𝛹x,x)

Mxy = Myx = ∫h

−h𝜎xyz dz = 1 − 𝜈

2D(𝛹y,x + 𝛹x,y)

(2.60)

Qx = ∫h

−h𝜎zx dz = 𝜋2

6𝜇h(𝛹z,x + 𝛹x)

Qy = ∫h

−h𝜎zy dz = 𝜋2

6𝜇h(𝛹z,y + 𝛹y)

(2.61)

Now, substituting Eqs. (2.19) into Eqs. (2.20), multiplying the first and second of Eqs. (2.20)by z dz, and integrating from −h to h, with the boundary condition (2.40), we find

Mxx,x + Myx,y − Qx =23𝜌h3𝛹x,tt − mM

xx

Mxy,x + Myy,y − Qy =23𝜌h3𝛹y,tt − mM

yy

(2.62)

where

mMxx =

23𝜎𝜅0h3[𝜅0(H2

0y + H20z)𝛹x,t − 𝜅0H0xH0y𝛹y,t − H0y𝜑,x − H0y𝜓,y]

mMyy =

23𝜎𝜅0h3[𝜅0H0xH0y𝛹x,t − 𝜅0(H2

0x + H20z)𝛹y,t − H0x𝜑,x − H0x𝜓,y]

(2.63)

Multiplying the third of Eqs. (2.20) by dz and integrating from −h to h, with the boundarycondition (2.40), we obtain

Qx,x + Qy,y = 2h𝜌𝛹z,tt − qM (2.64)

whereqM = 2h𝜎𝜅0{H0y𝜑 − H0x𝜓 − 𝜅0(H2

0x + H20y)𝛹z,t} (2.65)

Substituting Eqs. (2.60) and (2.61) into Eqs. (2.62) and Eq. (2.64), we have

S2{(1 − 𝜈)(𝛹x,xx + 𝛹x,yy) + (1 + 𝜈)𝛷,x} − 𝛹x − 𝛹z,x =

4h2𝜌

𝜋2𝜇𝛹x,tt −

6𝜋2𝜇h

mMxx

S2{(1 − 𝜈)(𝛹y,xx + 𝛹y,yy) + (1 + 𝜈)𝛷,y} − 𝛹y − 𝛹z,y =

4h2𝜌

𝜋2𝜇𝛹y,tt −

6𝜋2𝜇h

mMyy

𝛹z,xx + 𝛹z,yy +𝛷 = 4h2𝜌

𝜋2𝜇

1R𝛹z,tt −

6𝜋2𝜇h

qM

(2.66)

where𝛷 = 𝛹x,x + 𝛹y,y (2.67)

R = h2

3, S = 6D

𝜋2𝜇h(2.68)