IUTAM Symposium on Mechanics and Reliability of Actuating Materials: Proceedings of the IUTAM Symposium held in Beijing, China, 1-3 September, 2004 (Solid Mechanics and Its Applications)

IUTAM Symposium on Mechanics and Reliability of Actuating Materials

SOLID MECHANICS AND ITS APPLICATIONS Volume 127 Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

IUTAM Symposium on

Mechanics and Reliability of Actuating Materials Proceedings of the IUTAM Symposium held in Beijing, China, 1-3 September, 2004 Edited by

W. YANG Tsinghua University, Beijing, China

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CONTENTS ix xi xii xv

Preface Committees and Sponsors List of Participants Session Programs

1

Ferroelectrics

A switching rule for local domain wall motions and for macroscopic material response of ferroelectrics 3 H. Kessler, P. Bürmann and H. Balke The effects of sieving method and poling approach on the internal bias field in donor doped PZT ceramics D.N. Fang and F.X. Li 15 Interaction between defects and domain walls in piezoelectric materials D. Gross and R. Mueller

22

In-situ observation of electrically induced fatigue crack growth for ferroelectric single crystals F. Fang, W. Yang , F.C. Zhang and H.S. Luo 32 Crack initiation and crack propagation under cyclic electric loading in PZT I. Westram, D.C. Lupascu and J. Rödel

40

Multiaxial behavior of ferroelectric ceramic PZT53 Q. Wan, C.Q. Chen and Y.P. Shen

49

Stability analysis of 180º domains in ferroelectric thin films B. Wang, C. H. Woo and Y. Zheng

57

Stress analysis in two dimensional electrostrictive material under general loading Z.B. Kuang and Q. Jiang 68 85

Piezoelectrics

Effect of electric fields on fracture of functionally graded piezoelectric materials N. Noda and C.F. Gao 87 The charge-free zone model for conductive cracks in dielectric and piezoelectric ceramics T.Y. Zhang 96 v

vi

Electric potential drop across a crack in piezoelectrics Y.H. Chen and Z.C. Ou

107

Elastic Sv-wave scattering by an interface crack between a piezoelectric layer and an elastic substrate B. Gu, S.W. Yu and X.Q. Feng 112 A micro-macro approach to design active piezoelectric fiber composites H. Berger, S. Kari, N. Bohn, R. Rodriguez and U. Gabbert

121

FEM-techniques for thermo-electro-mechanical crack analyses in smart structures M. Kuna 131 Trefftz plane element of piezoelectric plate with p-extension capabilities Q.H. Qin

144

On piezoelectric actuator layers in plates and shells at large deflections S. Lentzen and R. Schmidt

154

Electric charge loading of a piezoelectric solid cylinder Y. Chen and R.K.N.D. Rajapakse

164

Oblique propagation of time harmonic waves in periodic piezoelectric composite layered structures M. Urago, F. Jin, Y. Mochimaru and K. Kishimoto 175 Scattering behaviour of elastic waves in 1-3 piezoelectric ceramics/polymer composites F. Jin, K. Kishimoto, Z. Qian and Z. Wang 185 Stress Analysis for an Anisotropic Solid with Variable Off-Axis of Anisotropy K. Watanabe 194 Shape Memory Alloys

205

Deformation instability and pattern formation in superelastic shape memory alloy microtubes Q. P. Sun and P. Feng 207 Theoretical consideration on the fracture of shape memory alloys W.Y. Yan and Y.W. Mai

217

vii

3D finite element simulation for shape memory alloys L.H. Han and T.J. Lu

227

Magnetostrictive Materials and Actuating Structures

237

Constitutive Models for Magnetostrictive Materials X.J. Zheng and X.E. Liu

239

Vibration analysis of a nonlinear magnetostrictive actuator Z. Zhong and Y.P. Wan

253

Test study of the feed-support system for a large radio telescope G.X. Ren, W.B. Zhu, H. Zhang, L.C. Zhu and Q.H. Lu

261

Biological Actuating Materials

271

Biofilm growth: perspectives on two-phase mixture flow and fingerings formation S. Hao, B. Moran and D. Chopp 273 Damage and fatigue of actuating heart muscles X.M. Zhang, F. Yang, N.K. Ma, Y. Zhao and W. Yang

291

Author Index

309

Preface Actuating materials hold a promise for fast-spreading applications in smart structures and active control systems, and have attracted extensive attention from scientists of both mechanics and materials sciences communities. High performance and stability of actuating materials and structures play a decisive role in their successive applications as sensors and actuators in structural control and robotics. The advances of actuating materials, however, recently encountered a severe reliability issue. For a better understanding toward this issue, scientific efforts are of paramount significance to gain a deep insight into the intricate deformation and failure behaviors of actuating materials. To examine the state of the art in this subject, the general assembly of IUTAM approved in August, 2002 at Cambridge University, UK, a proposal to hold an IUTAM symposium to summarize the relevant research findings. The main themes of the symposium are: (i) the constitutive relations of actuating materials that couple mechanical, electrical, thermal and magnetic properties, as well as incorporate phase transformation and domain switch; (ii) the physical mechanisms of deformation, damage, and fatigue crack growth of actuating materials; (iii) the development of failure-resilient approaches that base on the macro-, meso-, and micro-mechanics analyses; (iv) the investigation of microstructural evolution, stability of phase transformation, and size effects of ferroelectric ceramics, shape memory alloys, actuating polymers, and bio-actuating materials. The above problems represent an exciting challenge and form a research thrust of both materials science and solid mechanics. The IUTAM Symposium (GA.02-14) “Mechanics and Reliability of Actuating Materials” was held successfully on September 1-3, 2004 at Tsinghua University, Beijing, China. The Scientific Committee was appointed by the IUTAM Bureau. All the participants are recommended by the members of the Scientific Committee. The main aim of this symposium is to assemble top scientists working in the actuating materials to exchange their scientific results and ideas and thereby to further their collaboration in the coming years. There were total of 35 invited participants throughout the globe plus about 20 graduate students and postdocs from China to attend the symposium. The geographic distributions of the participants are: Australia (2), Canada (1), China mainland (15), Germany (7), Hong Kong, China (2), Japan (5), UK (2) and USA (1). The full list of the participants is appended in the list of participants. A website is established for the symposium. The conference venue is located at the Lecture Hall of Department of Engineering Mechanics, Tsinghua University. The full program of the symposium is also included in this proceeding. To follow the tradition of IUTAM Symposia, this symposium was conducted in only one session. Each speaker is allocated a time slot of 30 minutes (including the discussion). A total of 30 presentations were given during the symposium. Good discussions (three to five questions per each

ix

x

presentation) were conducted. The presentations addressed various parts of mechanics and reliability of actuating materials, and most of them are quite focused. Future collaborations between the participants were discussed. Part of the traveling expanses of the Germany participants is supported by the Sino-Germany research collaboration programs. Possible extension and expansion of the research collaboration in this subject were addressed. The symposium records the following scientific progresses in the topical area: 1. Various novel methods to measure domain switching zone are proposed. Interrelation between fracture and fatigue with domain switching is confirmed experimentally. 2. The importance of discharge and electric boundary condition is recognized through experiment and theory. An interesting model of charge free zone (CFZ), similar to dislocation free zone for the mechanical case, drawn large attention of the participants. 3. Interaction between domain switching and defect agglomeration is emphasized, the framework of configurational forces and microstructural evolution is under rapid development. 4. Multi-scale constitutive modeling of piezo/ferro/magneto-electric materials gains headways, as addressed through several presentations from different aspects. 5. Numerical schemes for actuating materials are near to the verge of commercial development. 6. Optimal design to maximize the performance of actuating materials in smart structures is now on the application agenda, and several ways to accomplish this goal is proposed. 7. Actuating of piezoelectric cylinders, a typical configuration in MEMS, is analyzed in detail. 8. Various approaches on the dynamics and non-destructive detection of actuating materials are explored. 9. Multi-axes testing schemes of ferroelectric ceramics and single crystals are put forward . 10. The new phenomenon of transformation spirals in nano-grained microtubes is verified by both experiments and numerical simulation. 11. Accumulated experimental data and theoretical framework for biologically actuating materials (such as hearts) and bio-films suggest a new research thrust of actuating materials.

Wei YANG Professor and Symposium Chair

xi

International Scientific Committee of the Symposium W. Yang (Chair, China) D. Gross (Germany) Y.W. Mai (Australia) R.M. McMeeking (USA) Z. Suo (USA) K. Watanabe (Japan) F. Ziegler (Austria) J. Salençon (IUTAM Representative, France) Local Organizing Committee W. Yang (Chair, China) D.N. Fang (China)

Sponsors The International Union of Theoretical and Applied Mechanics The National Science Foundation of China The Ministry of Education of China Chinesisch-Deutsches Zentrum für Wissenschaftsförderung Tsinghua University

xii

List of Participants Chang-Qing CHEN

Yi-Heng CHEN

Dai-Ning FANG

Fei FANG

Xi-Qiao FENG

U. GABBERT

Cun-Fa GAO

D. GROSS

Su HAO

Feng JIN

H. KESSLER

K. KISHIMOTO

Zhen-Bang KUANG

M. KUNA

MSSV, School of Civil Engineering and Mechanics, Xi’an Jiaotong University, Xi’an, 710049, P. R. China [email protected] School of Civil Engineering and Mechanics, Xi’an Jiaotong University, Xi’an 710049, P. R. China [email protected] Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China [email protected] Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China [email protected] Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China [email protected] Institute of Mechanics, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany, Department of Mechanical Engineering, Shizuoka University, 3-5-1, Johoku, Hamamatsu, Shizuoka 432-8561, Japan [email protected] [email protected] Institute of Mechanics, TU Darmstadt, D-64289 Darmstadt, Germany TU Darmstadt, Germany [email protected] Department of Mechanical Engineering Northwestern University, Evanston, IL 60208, U. S. A. [email protected] Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an 710049, P. R. China [email protected] Institute of Solid Mechanics, Dresden University of Technology, 01062 Dresden, Germany [email protected] Department of Mechanical and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan [email protected] Department of Engineering Mechanics, Shanghai Jiaotong University Shanghai 200240, P. R. China [email protected] Institute of Mechanics and Fluid Dynamics, Technische Universität Bergakademie Freiberg,

xiii

Lampadiusstraße 4, FREIBERG, D 09596, Germany [email protected] T.J. LU Engineering Department, University of Cambridge Cambridge CB2 1PZ, UK [email protected] N. NODA Department of Mechanical Engineering, Shizuoka University, 3-5-1, Johoku, Hamamatsu, Shizuoka 432-8561, Japan [email protected] Qing-Hua QIN Department of Engineering, The Australian National University, Canberra, ACT 0200, Australia [email protected] R.K.N.D. RAJAPAKSE Department of Mechanical Engineering, The University of British Columbia, Vancouver, Canada V6T 1Z4 [email protected] G.X. Ren Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China [email protected] Jürgen RÖDEL Department of Materials Science, Darmstadt University of Technology, 64287 Darmstadt, Germany [email protected] Rüdiger SCHMID Institute of General Mechanics, RWTH Aachen University, Germany, Germany [email protected] Qing-Ping SUN Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P. R. China [email protected] M. URAGO Department of International Development Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan [email protected] Biao WANG School of Physics and Engineering, Sun Yat-sen University, Guangzhou, P. R. China [email protected] Tie-Jun WANG School of Civil Engineering and Mechanics, Xi’an Jiaotong University, Xi’an 710049, P. R. China [email protected] K. WATANABE Department of Mechanical Engineering, Yamagata University, Yonezawa, Yamagata 992-8510 Japan [email protected] I. WESTRAM Department of Materials Science, Darmstadt University of Technology, 64287 Darmstadt, Germany [email protected] J.R. WILLIS Department of Applied Mathematics and Theoretical

xiv

Wenyi YAN

Wei YANG

Shou-Wen YU

Tong-Yi ZHANG

Xiao-Jing ZHENG

Zheng ZHONG

Physics (DAMTP), Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, UK [email protected] Computational Engineering Research Centre, Faculty of Engineering and Surveying, University of Southern Queensland, Toowoomba Qld 4350, Australia [email protected] Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China [email protected] Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China [email protected] Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P. R. China [email protected] Department of Mechanics, Lanzhou University , Gansu, 730000, P. R. China [email protected] Key Laboratory of Solid Mechanics of MOE School of Aerospace Engineering and Applied Mechanics Tongji University, Shanghai, 200092, P. R. China [email protected].

xv

Session Programs Opening Session: Opening and Welcome Address by the Chairman of the Symposium W. Yang Session 1 Chairperson: S.W Yu A switching rule for local domain wall motions and for macroscopic material response of ferroelectrics H. Kessler, P. Bürmann and H. Balke, Dresden University of Technology, Germany The effects of sieving method and poling approach on the internal bias field in donor doped PZT ceramics D.N. Fang and F.X. Li, Tsinghua University, China Interaction between defects and domain walls in piezoelectric materials D. Gross and R. Mueller, TU Darmstadt, Germany In-situ observation of electrically induced fatigue crack growth for ferroelectric single crystals F. Fang, W. Yang, F. C Zhang,, H. S. Luo, Tsinghua University, China Effect of electric fields on fracture of functionally graded piezoelectric materials N. Noda, C.F. Gao, Shizuoka University, Japan Session 2 Chairperson: D. Gross Ferroelastic toughening in PZT J. Rödel, A. B. N. Kounga and D. Lupascu, TU Darmstadt, Germany The charge-free zone model for conductive cracks in dielectric and piezoelectric ceramics T.Y. Zhang, HKUST, Hong Kong, China Electric potential drop across a crack in piezoelectrics Y. H. Chen and Z.C. Ou, Xi’an Jiaotong University, China Session 3 Chairperson: J. Rödel A micro-macro approach to design active piezoelectric fiber composites H. Berger, S. Kari, N. Bohn, R. Rodriguez and U. Gabbert, Otto-vonGuericke- Universität Magdeburg, Germany Stress analysis in two dimensional electrostrictive material under general loading Z.B. Kuang, Q. Jiang, Shanghai Jiaotong University, China Session 4 Chairperson: B. Wang Elastic Sv-wave scattering by an interface crack between a piezoelectric layer and an elastic substrate

xvi

B. Gu, S.W. Yu, X.Q. Feng, Tsinghua University, China FEM-techniques for thermo-electro-mechanical crack analyses in smart structures M. Kuna, University of Mining and Technology, Germany Trefftz plane element of piezoelectric plate with p-extension capabilities Qing-Hua Qin, The Australian National University, Australia Session 5 Chairperson: M. Kuna Electric charge loading of a piezoelectric solid cylinder Y. Chen and R.K.N.D. Rajapakse, The University of British Columbia, Canada Crack initiation and crack propagation under cyclic electric loading in PZT I. Westram, D.C. Lupascu and J. Rödel, TU Darmstadt, Germany Session 6 Chairperson: T.J. Lu On piezoelectric actuator layers in plates and shells at large deflection S. Lentzen and R. Schmidt, RWTH Aachen University, Germany Stress Analysis for an Anisotropic Solid with Variable Off-Axis of Anisotropy Kazumi Watanabe, Yamagata University, Japan Multiaxial behavior of ferroelectric ceramic PZT53 Q. Wan, C.Q. Chen and Y.P. Shen Session 7 Chairperson: K. Watanabe Oblique propagation of time harmonic waves in periodic piezoelectric composite layered structures M. Urago, F. Jin, Y. Mochimaru and K. Kishimoto, Tokyo Institute of Technology, Japan Biofilm growth: perspectives on two-phase mixture flow and fingerings formation S, Hao, B. Moran, D. Chopp, Northwestern University, USA Scattering behaviour of elastic waves in 1-3 piezoelectric ceramics/polymer composites F. Jin, K. Kishimoto, Z. Qian and Z. Wang, Xi’an Jiaotong University, China Session 8 Chairperson: T.Y. Zhang Q. P. Sun and P. Feng, HKUST, Hong Kong, China Deformation instability and pattern formation in superelastic shape memory alloy microtubes Theoretical consideration on the fracture of shape memory alloys W.Y. Yan and Y.W. Mai, University of Southern Queensland, Australia 3D finite element simulation for shape memory alloys L.H. Han and T.J. Lu, University of Cambridge, UK

xvii

Session 9 Chairperson: D.N. Fang Constitutive Models for Magnetostrictive Materials X.J. Zheng, X.E. Liu, Lanzhou University , China Vibration analysis of a nonlinear magnetostrictive actuator Z. Zhong and Y.P. Wan, Tongji University, China

Session 10 Chairperson: Q.P. Sun Test study of the feed-support system for a large radio telescope G.X. Ren, W.B. Zhu, H. ZHANG, L.C. Zhu and Q.H. Lu, Tsinghua University, China Stability analysis of 180o domains in ferroelectric thin films B. Wang, C.H. Woo, and Y. Zheng, Sun Yat-sen University, China Damage and fatigue of actuating heart muscles X.M. Zhang, F. Yang, N.K. Ma, Y. Zhao and W. Yang, Tsinghua University, China Closing Address: W. Yang

Ferroelectrics

A SWITCHING RULE FOR LOCAL DOMAIN WALL MOTIONS AND FOR MACROSCOPIC MATERIAL RESPONSE OF FERROELECTRICS H. Kessler1, P. Bürmann2 and H. Balke1 Institute of Solid Mechanics, Dresden University of Technology, 01062 Dresden, Germany 2 German Aerospace Center, Institute of Structural Mechanics, 38108 Braunschweig, Germany 1

Abstract Polarization switching is driven on the global scale by free energy reduction. The locally equivalent material force on a sharp domain wall can be expressed by Eshelby’s electromechanical energy momentum tensor. In the present paper, we deal with solids, which can be approximated by a linear response in the dissipationfree load range. In this case, the local driving force of dissipative transformations is represented by a particularly simple function of the jump of the linear material properties and of the electric field and mechanical stress on both sides of the domain wall (or before and after switching). Two generic examples for application are considered: (i) a numerical simulation of charge induced domain wall motions at a ferroelectric crystal surface; and (ii) a finite element homogenization procedure for ferroelectric or ferroelastic single domain switching in a volume element, which represents the macroscopic response of a material point. 1ˊNonequilibrium Thermodynamics of Electromechanical Processes In this section, the driving force of nonequilibrium processes is presented in terms of the Eshelby’s energy-momentum tensor [1] on the local scale and the free energy reduction on the global scale [2]. The method of derivation duplicates that of [3] with modifications required by the electrical terms. Subsequently, the general expression is reformulated in a simpler version for piezoelectric materials [4]. Consider a material volume V which is Figure 1. A volume V with a discontinuity surface S

3 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 3–14. © 2006 Springer. Printed in the Netherlands.

4

bounded by the surface wV and contains an internal discontinuity interface S (Figure 1). The symbol S may represent a domain wall. The integral entropy balance in V is given by

¨ I d3 X + ¨ [ [ I ] ] Vn d2 X + ¨ ( I s  2Is HVn ) d2 X = ¨v V

S

sV

S

n jq j 2 d X+ T

¨ T d3 X + ¨ Ts d2 X V

S

(1) where K, Ks, V and Vs denote volume density, interface density, and volume and interface production densities of the entropy, respectively. Vn and H are normal velocity parallel to nj and curvature of S in Lagrange coordinates (H < 0 in Figure 1), T the temperature and qj the heat flux density. The jump of a quantity A at the discontinuity interface is indicated by [ [ A ] ] w A2  A1 (2) where the unit normal on S (nj), points from side 2 to side 1. Gauss’ theorem allows to eliminate the surface integrals over wV. With account of discontinuities at S, eq. (1) is transformed to

¨ T d3 X + ¨ Ts d2 X = ¨ ¦¥¦¤¦ I + sX j ( T )¦¼¦»¦ d3 X + ¨ {[ [ I ] ] Vn + I s  2Is HVn  n j ¡¢ ¡¢ T °± °± } d2 X . ¦£

V

S

q j ¦²

s

V

    qj ¯ ¯

S

(3) Eq. (3) holds for arbitrary volumes V. Therefore, the volume and interface entropy balances can be localized to arbitrary locations in V or on S, respectively: qj s qj T = I + p 0 and Ts = [ [ I ] ] Vn + I s  2Is HVn  n j  ¡ ¡  ¯° ¯° p 0 (4) sX j T ¢¢ T ±± where, according to the 2nd law of thermodynamics, the entropy production cannot be negative. For isothermal processes without thermal dissipation, T = const, and eq. (4) simplifies further. n j [[ q j ]] 1 sq j T = I + p 0 and Ts = [ [ I ] ] Vn + I s  2Is HVn  p 0. (5) T sX j T

( )

Consider now the static electromechanical energy balance in V: ¨ w d3 X + ¨ [ [ w ] ] Vn d2 X + ¨ ( w s  2ws HVn ) d2 X = ¨v n jq j d2 X V

S

+¨ k i v i d X + 3

V

sV

S

2

3

2

¨v ti vi d X  ¨ qK d X  ¨v XK d X

sV

V

( vi ! velocity )

.

(6)

sV

The thermodynamic potential w = w(K,Hij,Ek) depends on the entropy, strain Hij and electric field Ek. The latter two are defined as usual by the derivatives of displacements ui and electric potential M su j ¬­ sK 1  su . (7) F ij = žž i + ­ and Ek =  sX i ®­ sX k 2 Ÿž sX j The 1st - 5th terms on the right hand side of eq. (6) account for heat flux, mechanical power of body forces ki and surface tractions ti, and for electrical power of extrinsic volume charges q and surface charges Z, respectively. Quantities ti and Z are given by

5

ti = n jT ji

and

X = n k Dk

on sV

(8)

where stress Vji and electric induction Dk satisfy the mechanical and electrical equilibrium: sT ji sDk (9) + ki = 0 and = q in V . sx j sx k Similar to the entropy balance, we may eliminate the surface terms in eq. (6) by the application of Gauss’ law. Taking into account equilibrium, assuming continuous tractions njVji and dielectric fluxes nkDk at S (valid if interface body forces and extrinsic interface charges are absent), and localizing the resulting energy balance, we arrive at sq j w = T ji F ij  Dk E k  in V and sX j . (10) w s  2w s HVn + [ [ w ] ] Vn = n jT ji [ [ vi ] ]  n k Dk [ [ K ] ] + n j [ [ q j ] ]

on S

Elimination of the heat supply in the volume entropy balance (4)1 by eq. (10)1 yields with account of T = const the Clausius-Duhem inequality for regular points of the volume V 1 T = ( T ji F ij  Dk E k  Z ) p 0 in V (11) T where the thermodynamic potential Z is defined by a Legendre-transformation Z = w  TI . (12) Similarly, we get the surface analogue of the Clausius-Duhem inequality from eqs. (4)2 and (10)2 1 Ts = ( n jT ji [ [ v i ] ]  n k Dk [ [ K ] ]  [ [ Z ] ] Vn  Z s + 2Zs HVn ) p 0 at S (13) T where the thermodynamic surface potential is defined by Z s = w s  TI s . (14) Let us calculate the velocity jump [[vi]]. The displacements at the interface S are continuous and may be written as u si ( t ) = u i ( X sj ( t ), t ) , where X sj ( t ) represents the interface position at time t. Differentiating this relation with respect to time and evaluating the jump, one obtains     su i ¯ ¯ dX sj     su ¯ ¯ ° ° Vj =  ¡ ¡ i ° ° Vn where Vj = [ [ vi ] ] =  ¡ ¡ . (15) ¡¢ ¡¢ sX j °± °± ¡¢ ¡¢ sn °± °± dt Eq. (15)2 is a consequence of the continuous displacement gradient in the tangential plane of S. The jump [ [ K ] ] can be determined in a similar way as [ [ K ] ] = [ [ Ek ] ] Vk = [ [ En ] ] Vn . (16) Substituting (15) and (16) into (13), the surface Clausius-Duhem inequality takes the form 1 (17) Ts =  ( n jn i [ [ 1 ji ] ] Vn + Z s  2Zs HVn ) p 0 at S T with Eshelby’s energy-momentum tensor

6

1 ji = ZE ji  T jl

su l + DjEi . sX i

(18)

The driving force of domain wall motions unrelated with the reduction of the surface potential Zs is [5] (19) f = n jn i 1 ji . In order to obtain the analogue of eq. (19) on the global scale, we integrate the dissipation inequality (11) in the volume V: T ¨ T d3X = ¨ ( T ji F ij  Dk E k  Z ) d3 X p 0 . (20) V

V

Let us assume prescribed displacement/traction and charge/potential boundary conditions on wV: u i ( sVu ) = u ai n jT ji ( sVt ) = tia and n k Dk ( sVX ) = Xa K ( sVK ) = Ka (21) where wV = wVu+wVt = wVZ+wVM. The right hand sides

u ai

( Xj ) ,

tai

( Xj ) , X ( Xj ) a

and K ( X j ) are given functions of the surface position. Integration of eq. (20) by parts yields with account of (21) and (9)  p0 T ¨ T d3 X = 1 (22) a

V

that is, the electromechanical processes on the global scale are driven by reduction of the potential (23) 1 = ¨ Z d3X  ¨ k i u i d3X  ¨ tia u i d2 X + ¨ qK d2 X + ¨ Xa K d2X . V

sVt

V

V

sVX

Integrating again by parts, the potential can be expressed by the free energy density u(Hij,Dk,T) [2] 1 = ¨ u d3X  ¨ k i u i d3X  ¨ tia u i d2X + ¨ n k Dk Ka d2X where u = Z + Dk Ek . V

V

sVt

sVK

(24) Let us apply these results to piezoelectric materials. The constitutive equations derived from a quadratic free energy function u(Hij,Dk,T) are linear functions of Vij and Ek, as long as the material properties do not change in the dissipation-free load range. Fij = Fijr + SijklTkl + dkijE k and Dk = Pkr + L kl El + dkijTij . (25) Consider a thermodynamic transition (domain switching) of an initial “preswitching state” 1 to a final state “post-switching state” 2. The boundary conditions (21) on wV remain fixed. During the transition, the remanent strain and polarization, Frij and Pkr , the compliance Sijkl, permittivity Nkl and piezoelectric coupling coefficients dkij change in some subregions of V, causing field changes of Hij, Vij, Dk and Ek in the whole volume V. Using equilibrium and the constitutive eqs. (9) and (25), and the corresponding free energy function u(Hij,Dk,T), the change of the potential during the transition, '3 = 32 – 31, can be expressed by the energy

7

release density 'g [4]: %1 =

¨ %g d3X

(26)

V

where 'g is nonzero only in subregions of V where the material state has changed: %g =

T1ij + T2ij T1ijT2kl T1ijE2k + T2ijE1k E1 + E2k E1 E2 %Frij + %Sijkl + k %Pkr + k l %Lkl + %dkij . 2 2 2 2 2

(27) Eqs. (26) and (27) are not valid for (nonlinear) boundary conditions more general than (21). The switching rule (27) for 'g can be also applied to local domain wall motions, if (.)2 and (.)1 are identified with fields or material properties on the two sides of the domain wall: using the continuity of ui, M, njVji and nkDk, one can prove the following relation which still requires the piezoelectric constitutive equations (25), but apart of that does not depend on external boundary conditions [4] f = n jn i 1 ji = %g . (28) 2. Charge Induced Domain Wall Motions A general framework for the description and applications of interface motion in materials is given in [5,6,7]. Domain walls are particular examples of such interfaces. Driven by external charges, they can move as has been shown by SFM imaging and manipulation of domain structures. Below, we analyze the plane configuration of an extrinsic perpendicular line charge Q near the surface intersection of a ferroelectric 90°-domain wall in a dielectric single-crystalline halfspace of isotropic Figure 2. 90°-domain wall motion, with velocity Vn < 0 permittivity N >> N’ (Figure 2). The following assumptions are used: before the appearance of Q, all polarization surface charges are neutralized by environmental charges. The domain wall motion after the extrinsic charge has been “switched on” proceeds sufficiently fast without any significant conductive currents on the surface or in the interior of the crystal until the equilibrium is reached. Therefore, new polarization charges arising at the surface where the surface locus of the domain wall moves, and at the interface where the domain wall deviates from its original 45°-slope, remain unscreened and in turn, generate additional electric fields. The kinetics and equilibrium of the domain wall is influenced by the corresponding ferroelectric anisotropy energy stored in the electric field. The ferroelastic anisotropy energy due to strain mismatch at the domain wall will be neglected in our simplified model. Other

8

energy contributions are associated with an approximately orientation independent surface energy J of the domain wall and with a “bending energy” being proportional to the square of the domain wall curvature H2 [8]. Thus, with account of (17), (27), (28), the energy dissipation rate due to domain wall motion with a normal velocity Vn consists of a ferroelectric anisotropy part, a surface energy part and a bending part g = Vn %g + 2Vn H H + 2Vn Zb H  Z b p 0

with

%g=

E1k + E2k %Pkr , 2

Zb =

1 YI ¸ ( 2H )2 . 2

(29) 2H is equal to the inverse radius of domain wall curvature, 2H = 1/R. The “bending stiffness” YI depends on Young’s modulus Y and on an effective moment of inertia I per unit width of the domain wall, which is proportional to the cube of the domain wall thickness, I ~ h3. The total dissipation rate of the domain wall follows by integration of (29) along the domain wall: s1

s1

 =¨ ¨ gds s0

s0

H¬  Vn žž %g + ­­­ ds + Ÿ R®

s1

¨( s0

)

Z Vn b  Z b ds = R

s1

H¬



¨ Vn Ÿžž %g + R ®­­­ ds  1 b p 0 . s0

(30) The right hand side of (30)2 has been written, using the total bending energy 3b: s1

1b =

¨

s0

s1

Zb ds =

¨

s0

YI ds 2R 2

V =V n

j n j ¶¶¶¶¶ l

s1

 = 1 b

¨

s1

Z b ds +

s0

¨

s1

Z b ds =

s0

¨ ( Z b  Vn s0

)

Zb ds . R

(31) Integrating (31) by parts and employing differential geometry, we obtain for arbitrary curvatures 1/R in a straightforward although tedious calculation from the energy theorem of domain wall bending s1

 b =  fb Vn ds + [ Fn Vn ]s1 + [ M8 ]s 1 s0 s0 ¨

(32)

s0

where –fb, Fn and M are intensity of distributed normal loads, transverse force and bending moment £ 2 ² d 1 YI ¦ d2 1 ¦ + 3 »¦ Fn = YI (33) fb = YI ¤¦ 2 M= ¦ ¦ R ds R R ds R ¦ ¦ ¥ ¼

( )

( )

and : represents the angular velocity of the normal nj on the domain wall. For small curvatures 1/R, (32) and (33) reduce to expressions of standard beam theory. Inserting these equations into (30) and assuming Vn = 0 and : = 0 at the endpoints s0 and s1, we arrive at the dissipation rate s1 s1 H £ 2 ² ¦ d2 1 ¦  (34) gds = ¨ ¨ Vn f ds p 0 with f = %g + R  YI ¦¤¦¥¦ ds2 R + R 3 ¦»¦¦¼ s0 s0

( )

where f represents the effective local driving force on the domain wall due to ferroelectric anisotropy energy, surface and bending energy. Let us define a kinetic relation for the normal velocity of the domain wall Vn with two free parameters, the critical driving force J and the mobility m:

9

£ ¦ m ( f  J ) sgn ( f ) if f > J . Vn = ¦ ¤ ¦ 0 if f b J ¦ ¥

(35)

J and the coercive field are related by J = EcPr. The driving force (34) and the kinetic relation (35) provide an equation for the curvature which resembles formally the elastic line in beam theory H d2 1 2 1 V + 3  = %g  J sgn ( f )  n . (36) 2 R R YI m ds R However, even at kinetic equilibrium with Vn = 0, the anisotropy force 'g(s) is not a known function of position. For our geometry, 'g can be written as

( )

{

sp r %g = ( Eex + E ey ) Pr + + ( E sp x + Ey ) P +

}

ip Eip x + Ey

Pr

where

.

w

( )1

.

+ ( . )2 . 2

(37) The electric field represents a superposition of the contributions Ee, Esp and Eip due to extrinsic, surface polarization and interface polarization charges, considered as a distribution of line charges. The generic subproblem of an isolated line charge in a bimaterial consisting of two dielectric half-planes has been solved using complex function theory [9]. Fixing the boundary between two half-planes with permittivities N >> N’ at y = 0, one can determine the electric field at (x,y) in the half-space y t 0 with permittivity N, generated by a charge Q at (xQ,yQ) £ ² ¦ 2 ¦ yQ <0 ¦ ¦ ¦ z  zQ ¦ ¦ ¦ Q ¦ ¦ E x ( z )  iE y ( z ) = ¤ » ¦ 2QL ¦¦ž 1 1 ¬­ + ­­ yQ > 0 ¦ ¦ ¦ žž ¦ ¦   z z z z Q Q® ¦Ÿ ¦ ¥ ¼

where z = x + iy, zQ = xQ + iyQ , zQ = xQ  iyQ

.

(38) The electric field in (38) is calculated at z. The symbol zQ corresponds to charge locations: (i) the extrinsic charge at (0,-a), (ii) the constant surface polarization charge density Zsp created at y = 0 between the initial and the current surface intersection of the domain wall, xo and xe, (iii) the variable interface polarization charge density Zip(s) arising along the domain wall trajectory x = x(s), y = y(s) (0 d s d sN), with s being the curve length parameter. Integration of the surface charge density contribution leads to Xsp ( x  x o )2 + y2 E sp ln x ( x, y ) = 2QL ( x  x e )2 + y2 Xsp  1 x  x e x  x o ¬­ ž cot (39) E sp  cot1 ­. y ( x, y ) = y y ®­ QL žŸ where Xsp = Pr sgn ( x o  x e ) The interface charge contribution, which cannot be integrated analytically, is given by sN sN Xip ( s ) ds Xip ( s ) ds 1 1 ip ( ) ( z )  iE E ip z = + x y ¨ ¨ ( ) 2QL z  z s 2QL z  z ( s ) . (40) 0 0 where

Xip ( s ) = Pr { sin R ( s )  cos R ( s ) }

10

In (40), T(s) denotes the angle between the x-axis and the local tangent unit vector to the domain wall at s (Figure 2). For z o z(s*) approaching the domain wall at s*, the value of the 2nd integral in (40) depends on which side of the domain wall is approached – the electric field is discontinuous at the domain wall. Its average contains a principal value integral and can be represented as s

  ip E ip x ( s )  iE y ( s )

=

N Xip ( s ) ds 1 1 iR( s ) s  s  e Xip ( s ) ln N  2QL ¨ z ( s )  z ( s ) 2QL s

0

sN

1 + 2QL ¨ 0

.

(41)

£ eiR( s )Xip ( s ) ² ¦ Xip ( s ) ¦ ¦ ¦  ¤ » ds   ( ) ¦ ¦ ( ) z s z s s s   ¦ ¦ ¥ ¼ 

The integral with a removable singularity at s* should be evaluated numerically by an open integration rule. The electric field on the domain wall shows a logarithmic singularity at the surface where s* o 0, x(s*) o xe, y(s*) o 0, as can be seen from eqs. (39) and (41). This would imply, according to eq. (29), a divergence of the local driving force, 'g(s*o 0) o f. A stability analysis of the diverging driving force 'g(s*o 0), and thus diverging velocity Vn(s*o 0), reveals two consequences: (i) domain walls with surface intersections in the range xe > xo are impossible, and the surface slope of domain walls clamped at xe = xo remains always T(s = 0) = 45°. (ii) The surface slope of domain walls moving toward xe < xo remains always T(s = 0) = 0°. Only if these conditions are met, the velocity of the wall remains finite everywhere along its trajectory. In order to simulate domain wall motions numerically, eq. (36) was implemented in a C++-program. The wall trajectory was approximated by parametric cubic Rennersplines between discrete nodes. The nodal driving forces were determined, calculating the electric field analytically or by numerical integration according to eq. (41), the bending and surface forces, evaluating the curvature and its derivatives by polynomial interpolation of the wall trajectory. The kinetic equations were integrated for all active nodes by a backward-Euler-algorithm until equilibrium was reached for all nodes. Figure 3a shows a plot of the normal displacements along the domain wall at equilibrium for the configuration sketched in Figure 3b. Surface position and slope are clamped to xo and 45°, as for Q > 0 the domain wall moves to positive x-values. Displacements un and curve length so are normalized by sc - the initial active zone of the straight domain wall characterized by a vanishing anisotropy energy. Figure 3a indicates that in the final state, the active zone has grown significantly by about 50%, although the displacements and curvatures remain small. This effect is to be attributed to the large anisotropy energy of the interface polarization charges. From eqs. (37), (38) and (41), the driving forces due to extrinsic and interface polarization charges can be estimated by order of magnitude as fe ~ QPr/(Nr) and fip ~ ZipPr/N. The quantity fe can be rewritten as fe ~ EcPr rc/r, where the initial activation radius rc is defined by fe(rc) ~ EcPr. For small deviations |'T|=|T-S/4| << 1 of the domain wall from its equilibrium slope, the interface charge density is Zip ~

11

Pr'T and fip ~ 'T(Pr)2/N. A combination of these relations yields fe LE r 1 ~ rc ¸ c ¸ where r ~ a ! rc . fip P r %R

(42)

For soft ferroelectrics, the factor NEc/Pr is small. Therefore, fip is very effective along most of the domain wall to reduce deviations relative to the equilibrium slope and presumably also the curvature. Only near the surface rc/r becomes large for small charge distances a. In this case, the curvature should increase and bending energy stabilizes the domain wall. Due to numerical problems, we were not yet able to verify this prediction.

(b) Domain wall configuration (a/sc | 0.05)

Figure 3. (a ) Normal domain wall displacements

3. Homogenization of Domain Switching Inspired by Hwang and McMeeking [10], we conducted a finite element analysis of ferroelectric/ferroelastic domain switching in a polycrystalline material point. For our purpose to illustrate and to test a new switching algorithm, the analysis was restricted to two dimensions and either an isotropic ferroelastic material with 90°switching or an isotropic ferroelectric material with 180°-switching. The material point was modeled by a mesh of 10 x 10 square 4-node elements, each representing a crystallite with a random orientation angle of the c-axis in the x-y-plane. Switching of such a crystallite-element is initiated if the associated energy release density averaged over the area of the element exceeds the material specific energy barrier J Ferroelastic :

%g =

T1ij + T2ij 2

%Frij > J,

Ferroelectric :

%g =

E1k + E2k %Pkr > J . 2

(43) The constitutive relations of a crystallite, whose c-axis is parallel to the x-direction, are given by ( 1 + O ' ) Txy Tx  O ' Ty Ty  O ' Ty 1 r Fx = + Fr , F y =  F , F xy = Y' Y' 2 Y' (44) Dx = LE x + Pr , Dy = LE y where Y’ = Y/(1-Q2) and Q’ = Q/(1-Q) are Young’s modulus and Poisson’s ratio in

12

plane strain, Hr and Pr are remanent strain and polarization of the unit cell. The boundary conditions change incrementally in small load steps. Each load increment is followed by one or more switching cycles at fixed boundary tractions or electric potential in order to determine the switching response. The iteration procedure shown in Figure 4 is based on the switching rule (27) and represents a modification of McMeeking’s analysis [10]. Obviously, the local post-switch stress V2 and electric field E2 are not known in advance, and therefore, satisfaction of the energy criterion needs confirmation after an element has been switched. If not satisfied, the element has to be switched back and remains locked in its initial state in the following switching cycles of the current load step. Once switched successfully, an element also remains locked in the switched state. In one cycle, not all possible elements attempt to switch, but only those with the highest energy release 'g. The latter is estimated using the known values V1 and E1 before the 1st cycle and the current values of the actual cycle as approximations for V2 and E2.

Figure 4. Domain switching algorithm, here for a ferroelastic material. The post-switch stress V2 has to be determined iteratively

In summary, the main features of the algorithm are: (i) All elements switched in a load step have met the exact energy criterion. (ii) Switching back and forth in infinite loops is excluded. (iii) Only a limited number and by far not all switching sequences have to be checked, which is essential for models with a larger number of elements. (iv) As a drawback, some switching sequences with a higher total energy release or a larger switching response are possibly not found by the algorithm.

13

Figure 5. Hysteresis for ferroelastic switching. One point represents a load step with 1,…,33 switch cycles (FE-runs). Hs – remanent saturation strain

Energy barrier J and remanent strain Hr or polarization Pr define the ferroelastic coercive stress Vc=J/Hr or the ferroelectric coercive field Ec=J/Pr, respectively. By using Vc and Ec, the ferroelastic eqs. (43)1, (44)1 and the ferroelectric eqs. (43)2, (44) 2, can be characterized each by a single normalized switching barrier, Vc/'V and Ec/'E, respectively, where 'V = YHr and 'E = Pr/N give the magnitude of stress fluctions and electric field fluctuations due to the mismatch of remanent strain and remanent polarization between the randomly oriented crystals. “Soft” ferroelastic or ferroelectric materials with a small normalized switching barrier show large normalized fluctuations of stress or electric field. Figure 5 shows a simulation of ferroelastic “poling” and the 1st hysteresis cycle for a material with Vc/'V = 0.1, loaded by an applied stress in x-direction at zero transversal stress. The simulation was realized in Mathematica. Due to the stress fluctuations, the equilibration step at zero load leads already to a non-zero remanent strain, and strain reversal starts at positive stresses in the 1st quarter of the hysteresis curve. The results for ferroelectric 180°-switching appear quite similar and are not shown here due to the lack of space. Acknowledgement: The present work is sponsored by the program “Structural Gradients in Crystals” of the German Research Foundation. References 1. 2.

J.D. Eshelby, 1975, The elastic energy-momentum tensor, J. Elasticity, 5, 321-335. R.M. McMeeking, S.C. Hwang, 1997, On the potential energy of a piezoelectric inclusion and the criterion for ferroelectric switching, Ferroelectrics, 200, 151-173.

14 3.

R. Abeyaratne, J.K. Knowles, 1990, On the driving traction on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids, 38, 345-360. 4. H. Kessler, H. Balke, 2001, On the local and average energy release in polarization switching phenomena, J. Mech. Phys. Solids, 49, 953-978. 5. R.E. Loge, Z. Suo, 1996, Nonequilibrium thermodynamics of ferroelectric domain evolution, Acta mater., 44, 3429-3438. 6. Z. Suo, 1997, Motions of Microscopic Surfaces in Materials, Adv. Appl. Mech., 33, 193-294. 7. B. Sun, Z. Suo, W. Yang, 1997, A finite element method for simulating interface motion-I. Migration of phase and grain boundaries, Acta mater., 45, 1907-1915. 8. E.K.H. Salje, Y. Ishibashi, 1996, Mesoscopic structures in ferroelastic crystals: needle twins and rightangled domains, J. Phys. Condens. Matter, 8, 8477-8495. 9. Z. Suo, 1990, Singularities, interfaces and cracks in dissimilar anisotropic media, Proc. Royal Soc., A427, 331-358. 10. S.C. Hwang, R.M. McMeeking, 1999, A finite element model of ferroelastic polycrystals, Int. J. Solids Structures, 36, 1541-1556.

THE EFFECTS OF SIEVING METHOD AND POLING APPROACH ON THE INTERNAL BIAS FIELD IN DONOR DOPED PZT CERAMICS D.N. Fang and F.X. Li FML, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China Abstract In this investigation, the internal bias field in donor doped PZT ceramics was investigated through measuring both the electric hysteresis loops and the butterfly loops. The effects of both poling approaches and sieving methods on the internal bias field were examined. It was found that a sparse sieving technique, which leads to more defects and high porosity in PZT ceramics, may induce a larger internal bias field than a dense sieving one. Meanwhile, for the sparsely sieved PZT ceramics, a sample poled by an impact electric loading at room temperature has fairly good piezoelectricity and a negligible internal bias field, while a sample poled with field application above the Curie point or at 120oC has a considerably large internal bias field. Space charge concentration near the grain boundary defects and pores after poling is thought to be the cause of the distinct internal bias field. 1. Introduction Due to their excellent piezoelectricity, Lead Titanate Zirconate (PZT) ceramics had been widely used as actuators, sensors, transducers, etc [1]. Their electromechanical properties can be optimized by means of doping with impurities for particular applications. It is thought that some of the doping effects on the electromechanical properties are due to an internal bias field controlled by impurity doping [2-7]. In 1962, Okazaki and Sakata observed an aymmetric D-E hysteresis loop in poled BaTiO3 and proposed the existence of an internal bias field, presumably arising during the aging process [8]. Subsequently, Takahashi found the same phenomena in Pb(Zr,Ti)O3 ceramics [3]. The effects of single impurity and multiple impurities on the internal bias field as well as the small-signal properties had been systematically studied by Uchida and Ikeda, and by Takahashi et al. [2-7]. It had been concluded from experimental results that acceptor impurities may result in an internal bias field in PZT ceramics, while donor impurities cannot. During the aging process after poling, internal bias field will build up in ferroelectric ceramics, following a usual aging law [7,9]. In an aged ceramic, the 15 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 15–21. © 2006 Springer. Printed in the Netherlands.

16

internal bias field will always orient itself along the polarization direction, thus stabilizing domain structures. On repeated cycling of hysteresis loops of an aged ceramic sample, the internal bias field will gradually decrease with the increase in cycle numbers, which is called hysteresis relaxation [7,9,10]. Several studies have been made on the origin of the internal bias field, by Jonker and Lambeck, Carl and Hardtl [9,11-13]. The possible mechanisms that might be responsible for the occurrence of an internal bias field in ferroelectric ceramics can mainly be summarized into three effects, i.e., volume effects, domain effects and grain-boundary effects [9]. In some particular cases, Carl and Hardtl, Jonker and Lambeck investigated the possible one or two mechanisms responsible for the internal bias field in ferroelectric ceramics. While for other cases or general cases, the questions of which mechanism is dominant still remain open. In this paper, unlike the predecessors who dealt with the effects of impurities, we focus our investigation on the effects of sieving methods and poling approaches on the internal bias field. This research is aroused from the observed asymmetric butterfly loops of a (La, Nb) doped PZT ceramic, which contradicts the traditional notions of the effects of donor impurities on the internal bias field. The internal bias fields in donor doped PZT ceramics prepared by a sparse sieving method and a dense sieving method are investigated. For the sparse sieved ceramics, three poling approaches, i.e., field application above the Curie point, at 120 oC and an impact electric field application at room temperature, are evaluated. The mechanism responsible for the internal bias field in donor doped PZT ceramics is analyzed. 2. Experiment The material used in this investigation is a soft PZT ceramic, doped with La and Nb. The dimension of the specimen is 10 u 10 u 10 mm3. The initial specimens were received as poled. The poling process is conducted with field application at 320 oC above the Curie point. The testing was conducted four days after poling. With the traditional Sawyer-Tower circuit, the electric hysteresis loops, as well as the butterfly loops, of the specimen were measured, as shown in Figure 1. The loading period for the testing is five minutes. It can be seen from Figure 1 that both the electric hysteresis loops and the butterfly loops are asymmetric, and the latter looks more evident. The asymmetry in these curves was caused by an internal bias field in the poled specimen. Since the symmetry of butterfly loops is rather sensitive to the internal bias field, we studied the internal bias field through measuring both the electric hysteresis loops and the butterfly loops instead of measuring the I-E curves as other researchers did. Let E1 and E2 denote the electric field corresponding the two tails in the butterfly loops, the value of the internal bias field can be given by Ei = (E1+E2)/2. Thus the internal bias field measured from Figure 1(b) is about 100V/mm. Compared with measuring the I-E curves, measuring the butterfly loops can provide more information of domain switching.

17

While for measuring strain with strain gauge, a bulk ceramic sample must be used and the magnitude of the applied electric field is limited. Since for donor doped PZT ceramics, internal bias field is negligible according to the early literatures, it is necessary to find possible mechanisms responsible for the internal bias field measured in Figure 1. Space charge effect may be a possible mechanism for the existence of internal bias field because the space charge effect is related to the poling approach [14]. Therefore, two other poling approaches, i.e., poling with field application at 120 oC and at room temperature, were used to study the internal bias field in donor doped PZT ceramics. To eliminate the effect of aging process, all specimens were aged in air, and the testing was conducted four days after poling.

Figure 1. (a) Electric hysteresis loops and (b) butterfly loops of a soft PZT ceramic poled with field application above the Curie point

Figure 2 shows the electric hysteresis loops and butterfly loops of the above mentioned soft PZT ceramics poled with field application at 120oC. The magnitude of the poling field is 2kV/mm, the duration time is 15 min. Although the negative polarization cannot saturate because of being limited by the magnitude of applied electric field, it can still be estimated from Figure 2(b) that the magnitude of the internal bias field in such a poled specimen is no less than 250V/mm, significantly larger than that in a ceramic sample poled with field application above the Curie point. Due to the existence of internal bias field, the above PZT ceramic sample is hard to pole at room temperature even under a 2.5kV/mm DC field. While a compact electric field with the magnitude of only 1.2kV/mm, about 1.4 times coercive field, is found sufficient to pole the ceramic sample at room temperature [15]. Thus, in this study, an impact electric field with a magnitude of 2kV/mm is used to pole the ceramic sample at room temperature. The rising time for the impact field is less than 1ms, the duration time is 15 min.

18

Figure 2. (a) Electric hysteresis loops and (b) butterfly loops of a soft PZT ceramic sample poled with field application at 120 oC

Figure 3 shows the electric hysteresis loops and butterfly loops of this soft PZT ceramic sample tested four days after the poling with an impact field at room temperature. It can be seen from Figure 3 that the internal bias field in ceramic sample poled with this method can hardly be noticeable.

Figure 3. (a) Electric hysteresis loops and (b) butterfly loops of a soft PZT ceramic sample poled with an impact field at room temperature

From Figure 1 to Figure 3, it is concluded that the existence of the internal bias field can only be attributed to the poling approach, especially the temperature at which the poling process is conducted. Since the internal bias field in a ceramic sample poled by an impact field at room temperature is negligible, lattice defects are excluded for the cause of internal bias field in this soft PZT ceramic sample. Space charge generated by the pyroelectric effect after poling is thought to be the cause of the internal bias field. Yet these space charges are accumulated, near the domain walls, at the grain boundary or near the pores in the ceramic, cannot be known from the above experimental results.

19

It must be noted that all the specimens used in tests mentioned above were prepared by use of a conventional industrial technique. The mesh size for sieving is 0.6 mm, larger than the mesh size typically used in laboratories. Such a sparse sieving method is likely to induce more defects in the ceramics. Therefore, a dense sieving method with the mesh size of 0.13mm was used to prepare PZT ceramics with the same ingredients as the above mentioned one. Figure 4 shows the SEM photographs of PZT ceramics prepared by using the above two sieving methods. From Figure 4, it can be seen that the grain size of the ceramic sample prepared with the dense sieving method is a little smaller and much more uniform than that prepared with the sparse sieving method. Higher porosity can be seen from the latter than the former. To study internal bias filed in the PZT ceramic sample prepared with the dense sieving method, electric hysteresis loops of an unpoled ceramic sample and the one poled at 120 oC were measured and compared with those of a ceramic sample prepared with the sparse sieving method, as shown in Figure 5. It can be seen from Figure 5 that the PZT ceramics prepared with the dense sieving method can be more easily to pole at room temperature than that prepared with the sparse sieving method. The internal bias field in the former ceramic sample is very small and henceforth negligible, as seen in Figure 5 (b).

(a)

20

(b) Figure 4. SEM photographs of a soft PZT ceramic sample prepared with two sieving methods. (a) sparse sieving; (b) dense sieving

Figure 5. Electric hysteresis loops of a soft PZT ceramic prepared with two sieving method: (a) unpoled specimen; (b) poled specimen

3. Discussions and Conclusions From Figure 4 and Figure 5, the grain boundary defects and the porosity are the cause of internal bias field in donor doped PZT ceramics, since only the grain size

21

and porosity are increased by the sparse sieving method compared with the dense sieving one. After poling at a high temperature, the released space charges due to the pyroelectricity accumulate near the grain boundary defects or pores, building up the internal bias field in the ceramic sample. The internal bias field in donor doped PZT ceramics built up in this way is usually smaller than that in acceptor doped ceramics. Not being caused the defects inside the lattice or domains, the internal bias field in donor doped PZT ceramics is actually not “intrinsic” and can be avoided by improving the preparation techniques, i.e., minimizing the grain boundary defects and porosity. Furthermore, it can easily be eliminated by an impact electric field at room temperature. Acknowledgement: The authors are grateful to Ms. X.X. Yi of the Acoustic Institute of the Chinese Academy of Siences for supplying the PZT specimens. Support by the National Natural Science Foundation of China under Grant No. 10025209, 10132010, 90208002. The support by the Key Grant Project of Ministry of Education of China (0306) is also acknowledged. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Y. Xu, 1991, Ferroelectric Materials and Their Applications. Amsterdam: North-Holland Press. N. Uchida, T. Ikeda, 1967, Studies on Pb(Zr-Ti)O3 ceramics with addition of Cr2O3 . Jpn. J. Appl. Phys., 6,1292-1299. M. Takahashi, 1970, Space charge effect in lead zirconate titanate ceramics caused by the addition of impurities. Jpn. J. Appl. Phys., 9, 1236-1246. M. Takahashi, 1971, Electrical resistivity of lead zirconate titanate ceramics containing impurities. Jpn. J. Appl. Phys., 10, 643-651. S. Takahashi, M. Takahashi, 1972, Effects of impurities on the mechanical quality factor of lead zirconate titanate ceramics . Jpn. J. Appl. Phys., 11, 31-35. S. Takahashi, 1981, Internal bias field effects in lead zirconate-titanate ceramics doped with multiple impurities. Jpn. J. Appl. Phys., 20, 95-101. S. Takahashi, 1982, Effects of impurity doping in lead zirconate-titanate ceramics. Ferroelectrics. 41, 143-156. K. Okazaki, K. Sakata, 1962, Space charge polarization and aging of barium titanate ceramics. Electrotech. J. Jpn., 7, 13-18. K. Carl, K.H. Hardtl, 1978, Electrical after-effects in Pb(Ti,Zr)O/sub 3/ ceramics. Ferroelectrics. 17, 473-486. K.H. Hardtl, 1976, Physics of ferroelectric ceramics used in electronic devices. Ferroelectrics. 12, 9-19. G.H. Jonker, 1972, Nature of aging in ferroelectric ceramics. J. Am. Ceram. Soc., 55, 57-58. G.H. Jonker, P.V. Lambeck, 1977, On the origin of the electrooptical effect in pyroelectric crystals. Ferroelectrics. 21, 641-643. P.V. Lambeck, G.H. Jonker, 1977, Ferroelectric domain stabilization in BaTiO3 by bulk ordering of defects. Ferroelectrics. 22, 729-731. K. Okazaki, H. Maiwa, 1992, Space charge effects on ferroelectric ceramic particle surfaces. Jpn. J. Appl. Phys., 31, 3113-3116. F.X. Li, D.N. Fang, A.K. Soh, 2004, An analytical axisymmetric model for the poling-history dependent behavior of ferroelectric ceramics. Smart Mater. Struct., 13, 668-675.

INTERACTION BETWEEN DEFECTS AND DOMAIN WALLS IN PIEZOELECTRIC MATERIALS

D. Gross and R. Mueller Institute of Mechanics, TU Darmstadt, D-64289 Darmstadt, Germany

Abstract In order to get some insight in processes leading to electric fatigue in piezoelectric materials, the interaction of point defects with domain walls is studied. The fundamental equations and quantities, relevant for the domain wall movement, are described within the framework of configurational forces. Using the Finite Element Method, numerical simulations have been accomplished for a number of typical defect domain wall configurations. The results are useful for understanding key features of the interaction. They indicate that a blocking of domain walls by agglomerated point defects actually seems to be possible and that this might be a dominant fatigue mechanism. 1. Introduction Ferroelectric materials under cyclic loading at sufficiently high amplitude exhibit a so-called electric fatigue effect. Macroscopically, electric fatigue is characterized among others by a gradual decrease of the mechanical output for a fixed electric excitation which finally may lead to total failure [1]. An overview about the related phenomena can be found in [2] and the literature cited therein. The origins of electric fatigue on the microlevel, although not yet fully clear, can be assumed in various mechanisms. Recent experimental observations support the hypothesis that a pinning of domain walls by point defects or their agglomerates is a dominant micromechanism [2]. These point defects - suspected are oxygen vacancies interact with domain walls and by this may block them or at least constrain their movement. Since an experimental verification of this hypothesis is very difficult, it is the intention of this paper to provide numerical simulations which may help to understand the key features of the point defect - domain wall interaction. In order to model the interaction scenario, driving forces, acting on the defect and vice versa on the domain wall, are identified. In this investigation this is done within the framework of configurational forces; for a general introduction into the theory of configurational forces see e.g. [3], [4], [5]. Once the coupled field equations are solved by Finite Elements, the configurational forces are determined 22 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 22–31. © 2006 Springer. Printed in the Netherlands.

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to investigate possible motions of the domain wall and the defects. For a discussion of material forces in the context of Finite Element discretizations see for example [6]. Numerical simulations for a number of typical configurations will demonstrate the effect of point defect position and concentration on the driving force acting on the domain wall. The results indicate that point defects are actually capable of blocking a domain wall. 2. Basic Equations The quasi-static local field equations of a piezoelectric solid are given by the mechanical equilibrium and the electrostatic condition divV  f 0  div D  q 0 (1) the kinematic and electric relations

1§ T· ¨ ’u  (’u) ¸  © ¹ 2

H

E

’M

(2)

and the coupled constitutive relations

V

^(H  H 0)  b T E

(3)

D b(H  H 0)  A E  P 0 where V is the stress tensor, f is the volume force, D is the electric displacement vector, q is the volume charge, H is the linearized strain tensor, u is the displacement vector, E is the electric field, M is the electric potential and ^ , b and A are the stiffness tensor, piezoelectric tensor and dielectric tensor, respectively. Possible irreversible strains and polarizations are taken into account by H 0 and P 0 . In conjunction with appropriate boundary conditions, equations (1) to (3) completely describe an electromechanical boundary value problem. Configurational forces may be introduced by starting with the electric enthalpy

H (H e E  x )

1 e 1 T H  ª¬^H e º¼  H e  (b E )  E ˜ ( AE )  P 0 ˜ E 2 2

(4)

with H e H  H 0 , which serves as a potential for the stresses and electric displacements,

V

wH  wH e

D



wH  wE

(5)

We now consider its gradient

’H

wH wH wH  ’H e  ˜’E  e wH wE wx

which can be written in index notation as

(6) expl 

24

H k

wH wH wH E j k  (H ij  k  H ij0k )  e wH ij wE j wxk



(7)

expl 

Using (1), (2) and (5), the simple rearrangements

H k

V ij ui kj  V ij H ij0k  D j Mkj 

wH wxk

 expl 

( H G kj ) j (V ij ui k ) j  V ij  j ui k  V ijH ij0k  ( D jM k ) j  D j  jMk  ( H G kj  V ij ui  k  M k D j ) j  ui k fi  M k q 

wH wxk

wH wxk

 V ij H ij0 k

expl 

0

expl 

can be done. Introducing through

6

H 1  (’u)T V  ’M … D

g

(’u)T f  ’M q 

wH wx

(8)

 V  ’H 0 expl 

the Eshelby-stress tensor 6 and the configurational body force g , the last equation represents the equilibrium condition for configurational forces div6  g 0  (9) Without going in details it should be mentioned that this equation can also be deduced from the variational principle G3 0 where

3

³

B

H dV  ³ ( f ˜ u  q M ) dV  ³ t ˜ u dA  ³ Q M dA wB

B

wB

(10)

is the total electromechanical potential of the system and the variation is taken with respect to the position x . Within this context, the configurational traction 6n and body force g are interpreted as forces leading to a change of the system’s energy when displaced. Furthermore, from (8) it can be seen that g vanishes in an homogeneous material without mechanical volume forces homogeneous irreversible strains

f , charges q and

0

H .

Since the elasticity tensor ^ , the piezo-electric tensor b and the dielectric tensor A can vary from domain to domain, a mobile domain wall represents a material inhomogeneity. The configurational traction acting in normal direction from now on called the driving force - is given by the jump of 6 , (11) W n n ˜ (a6 bn)  Thus the overall driving force on the domain wall with an area S is

25

Tn

³W S

n

dA 

(12)

Point defects like foreign atoms or vacancies disturb the deformation and charge state of the otherwise homogeneous material. Mechanically a point defect is associated with an inelastic eigenstrain of a center of dilatation that is localized at the defect position x D ,

H 0 o D 1G ( x  x D) 

(13) The parameter D represents the mechanical strength of the defect, allowing with D ! 0 or D  0 to model a foreign atom or a vacancy that is too large or too small for the surrounding crystal lattice. For simplicity the inelastic strain is assumed to be isotropic. It enters the problem through the constitutive equations (3). Electrically the point defect is modeled as a localized charge q o EG ( x  x D) (14) with the strength E . This part will be considered on the level of the electrostatic balance equation (1). Introducing (13), (14) into (8) leads to a single driving force acting on the defect,

G

 ³ g dV B

’(D trV  E M ) 

(15)

With the driving forces (12) and (15), the energy change of a system with a movable domain wall and n movable point defects is given by n

G3 Tn G wS  ¦ Gi G wi

(16)

i 1

where

G wS is the displacement of the domain wall in normal direction and G wi

are the point defect displacements in direction of the respective driving forces G i . It indicates that a domain wall and point defects will move in direction of the driving force and by this reduce the total potential. It should be emphasized that kinetic laws, relating the domain wall and defect velocities to the acting driving forces, cannot be deduced from energy considerations. Such laws are the subjects of experimental investigations and e.g. atomistic considerations. 3. Examples 3.1. NUMERICS AND MATERIAL DATA Restricting the attention to 2D problems, the field equations are solved using the finite elements. Without going in details it shall be mentioned that the mechanical part of the point defects is modelled by a quadrupole force system. Furthermore, the defects enter the discretized system of equations as ’external loads’ on the right hand side. Once the field quantities are calculated, the driving forces can be determined in a second step. As model systems rectangular regions without or with a domain wall are considered.

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Plane strain conditions are assumed and the following material parameters are taken to mimic the behavior of PZT that is poled in x2 -direction,

^

ª126 53 00 º « 53 117 00 »  « » «¬ 00 00 353»¼

b A

ª 00 00 170 º « 65 233 00 »  ¼ ¬

>151

(17)

0000 13@  6

4

Here the constants ^ , b and A are given in 10 N mm 2 , 10 C mm 2 and

1011 CVmm , respectively. The poling is also taken into account by setting C T T 0 (18)  H 0 > 00039 00076 00@  P > 00 02@ ˜106 2 mm As point defects positively charged oxygen vacancies in PZT have been assumed for which the approximate data

D

1011 

E 1011

C mm

2

(19)

have been chosen. 3.2. POINT DEFECT AND DOMAIN WALL First a single point defect in a sufficiently large (infinite) region without any additional external load is considered, see Figure 1.

Figure 1. A single point defect

Some components of its eigen-field are shown in Figure 2. The electric displacement D reflects the polarization in x2 -direction by the non-symmetry of the fields. In contrast, the mechanical fields which are represented by the stresses V 11 , V 22 show an almost isotropic distribution. Since no external field (load) is present, no driving force is acting on the defect, cf. (15).

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Figure 2. Field of a point defect D

We next consider a 180 -domain wall in a rectangular region, loaded solely by a potential difference of 600 Vmm , as depicted in Figure 3. The domain wall represents an inhomogeneity in the material since the remanent polarization and the material tensor b differ in the the two sub-domains. The different polarizations in the two domains produce inhomogeneous mechanical and electrical fields. Under the action of the external field, which points downwards, the left domain will shrink while the right domain will expand in vertical direction. As a consequence of the compatibility at the domain wall (continuous displacements), stresses in its vicinity appear. In this region the left domain is under tension in vertical direction, whereas the right domain is compressed. The resulting stress distribution for V 22 is plotted in Figure 4. Similarly, as can be seen from Figure 4, the electric displacement D2 exhibits a jump at the domain wall. Since the right domain is polarized downward, it favors the external electric field. The system tries to reduce

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the total energy by producing larger regions with downwards polarization. This can only be achieved by moving the domain wall to the left, in the direction of the driving forces W n which are uniformly distributed in this case, see Figure 4.

D

Figure 3. A 180 -domain wall

³22

´n

D2

D

Figure 4. Fields and driving forces at a 180 -domain wall

29

3.3. INTERACTION BETWEEN POINT DEFECTS AND DOMAIN WALL We consider again the previous system, but now with an additional point defect which is introduced at a distance d from the domain wall, see Figure 4. As a characteristic result of the simulations, typical fields and the resulting driving forces on the domain wall are plotted in Figure 5 for d 50 nm . Note the different scaling of the quantities when comparing them with the defect-free results in Figure 4.

³22

´n D2

Figure 5. Point defect and domain wall, d 50 nm

As can be seen from Figure 5, the distribution of the driving force

W n differs

essentially from the defect-free case and it changes its sign in the upper half of the domain wall. As a consequence, the overall driving force Tn decreases compared with the situation where no defect is present. This effect increases with decreasing distance d . Figure 6 shows the dependence of Tn , normalized with the defect-free 0

force Tn , from the distance d . It can be seen that the driving force is reduced by approximately 30% if the defect is shifted from d a | 09 to d a | 025 . A further reduction can be expected when the defect approaches closer to the domain wall or if more defects are present. Consequently, since a nonzero force is

30

necessary to move the domain wall, point defects have the capability to block a domain wall by reducing the driving force below a certain threshold value. To overcome the defect barrier, a higher external field is needed which increases the driving force again. Tn / Tn0

d/a Figure 6. Resulting driving force on domain wall

´n

Figure 7. Driving force for 10 randomly distributed defects

This result is underlined when simulating a more realistic situation with up to 10 randomly distributed point defects in the left domain. To avoid too strong interactions, the defects are located at least 20 nm away from the domain wall and boundaries. Figure 7 depicts the driving force for such a situation. The effect of the number of defects on the resulting driving force is shown in Figure 8. One or two defects will still result in a driving force that has the same direction as in the defectfree situation in Figure 4. Increasing the number of defects to 3-7 will result in a driving force which tries to push the domain wall in the opposite direction, i.e. to the right. A higher number of defects (8-10) will cause strong interaction between the defects themselves and thus might partly reduce the shielding effect.

31

Tn / Tn0

Number of defects Figure 8. Driving force on domain wall as a function of the number of defects

4. Conclusion The simulations support the hypothesis that point defects might block a domain wall. Nevertheless, more detailed investigations are necessary to answer remaining questions and to receive reliable answers. This refers to the used approximate material data which have to be refined. Other questions are the interaction among point defects themselves, their migration, their possible agglomeration and the accompanied shielding or amplification effect. References 1. 2. 3. 4. 5. 6.

J. Nuffer, D.C. Lupascu, J. Roedel, 2000, Damage evolution in ferroelectric PZT induced by bipolar electric cycling, Acta Mater., 48, 3783-3794. D.C. Lupascu, 2004, Fatigue of Ferroelectric Ceramics and Related Issues, Springer, Heidelberg. G. Maugin, 1993, Material Inhomogeneities in Elasticity, Chapman & Hall, London. M.E. Gurtin, 2000, Configurational Forces as Basic Concept of Continuum Physics, Springer, Berlin. D. Gross, S. Kolling, R. Mueller, I. Schmidt, 2003, Configurational forces and their application in solid mechanics, European Journal of Mechanics A/Solids, 22, 669-692. R. Mueller, G. Maugin, 2002, On material forces and finite element discretizations, Comput. Mech., 29, 52-60.

IN-SITU OBSERVATION OF ELECTRICALLY INDUCED FATIGUE CRACK GROWTH FOR FERROELECTRIC SINGLE CRYSTALS 1

F. Fang, 1W. Yang , 1F. C Zhang, 2H. S. Luo Failure Mechanics Laboratory, Department of Engineering Mechanics, Tsinghua University, Beijing, China 2 The State Key Laboratory of High Performance Ceramics and Superfine Microstructure, Shanghai Institute of Ceramics, Chinese Academy of Sciences, China 1

Abstract In-situ observation of electrically induced fatigue crack growth was carried out for ferroelectric single crystals under alternative electric field. Electrically-induced fatigue crack growth was observed both below and above the coercive field. The crack propagation behavior is a repeated process of a continuous increment followed by a sudden increase in the crack length. This jumped crack growth behavior was attributed to the variation of the crack boundary conditions under electric field cycling. 1. Introduction The development of microelectronic technology and miniaturization of devices places stringent demands on functional materials which can couple effectively one form of signal or energy to other forms of signals or energies. Ferroelectric materials are one important class of the functional materials. Due to the large capacity of mechatronic transformation and the controllability for different properties required, ferroelectric materials have been increasingly used in actuators and sensors in smart structures. However, the high electric driving field required to induce large displacement may cause mechanical and electrical degradations. The electrical degradation of ferroelectrics, which appears in the hysteresis loop as a decrease of remanant polarization and an increase of the coercive field, is a serious concern in applications [1]. The degradation in the mechanical sense is also found under alternate electric loading, which is termed “electrically-induced fatigue cracking” [2-6]. It was found that the actuators can fracture around the edges of their internal electrodes when subjected to high electric field [7]. Cao and Evans [2] first attacked the issue of electrically-induced fatigue crack growth in PLZT and PZT ceramics. They reported that tiny amount of fatigue crack growth, about 50Pm, occurred under an applied field less than that of 90% of the coercive field, and the 32 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 32–39. © 2006 Springer. Printed in the Netherlands.

33

crack arrested regardless the further field cycling. When the applied field was larger than 110% of the coercive field, the crack would grow in each loading cycle, and reach a steady state independent of the crack length. Lynch et al. [3] reported stable cyclic crack growth for either conducting or insulating cracks in 8/65/35 lanthanum lead zirconate titanate ceramics. They found that the impermeable fatigue crack has a wavy and bifurcated front that grows parallel to the electrode; while the conducting fatigue cracks form a tree-like structure. Zhu et al. [8] observed the apparent fatigue crack growth in PZT ceramics below coercive field. While all of the above mentioned works dealt with the propagation of cracks that were initiated by Vicker’s indentation, Nuffer et al. reported the electrically-induced fatigue cracking for PZT ceramics without pre-existing cracks [9]. It was observed that there was no fatigue when the cycling electric field was maintained at the amplitude below or equal to the coercive field. On the other hand, when cycling at 1.4Ec, edge cracks started at the electrode/ceramics boundary at cycle numbers greater than 2.5u105 and propagated toward the inside of the ferroelectric ceramics. Severe delamination was found when cycling at 1.96Ec. The crack appeared underneath the electrodes and propagated predominantly parallel to the electrodes [9]. The microstructure of ferroelectrics is characterized by domains with different orientations. In a tetragonal ferroelectric material, there are two types of domain boundaries: 90 o and 180o. Domain switching appears under appropriate mechanical or electrical load. 180o domain switching causes little strain, while 90o domain switching delivers a sizable strain, contraction along the previous polarization direction, and elongation along the current one. Considering the nonlinear electrostriction effect, Yang and Suo [10] derived the stress intensity factor on the flaws around the electrode edge under electric loading. Gao et al. [11] and Fulton and Gao [12] proposed a strip saturation model to investigate the effect of electric yielding. Yang and Zhu [13] explored the effect of crack tip domain switch to the fracture of ferroelectric ceramics. Zhu and Yang [14] predicted the fatigue crack growth under alternate electric field. All the above theoretical work dealt with the importance of 90o domain switching. However, the direct experimental evidence of the relationship between the electrically induced fatigue cracking and 90o domain switching seems still rare. Some experimental efforts were made to examine the crack propagation and 90o domain switching near the crack tip under an electric field [15-16]. In this work, ferroelectric single crystals with the thickness less than 90 Pm are used so that we can observe simultaneously the fatigue crack growth and 90o domain switching under a transmitted polarized optical microscope. The use of the single crystals also has the advantage of precluding the complexity caused by the grain boundaries [17]. 2. Experimental Procedures BaTiO3 Single crystals were grown from high temperature solution by slow cooling technique. Optically, BaTiO3 single crystals were lighted yellow in color. The crystals were edge oriented parallel to the principal crystallographic orientations

34

using an X-ray diffractometer. In order to minimize the superimposition of domains, the crystals were ground to a thickness of less than 90Pm, and polished with diamond paste (down to 0.25Pm) until a near-mirror finish was obtained on both surfaces. The samples were gold-sputtered, leaving a 600Pm gap on the surface. Silver leads were attached to the electrodes with air dry silver paste, as shown schematically in Figure 1. A high voltage power supply consisting of a function generator and a voltage amplifier was used to apply both the DC fields and the bipolar, sinusoidal electric fields to the specimen. In order to protect against electric breakdowns, specimen was immersed in a silicon oil tub made of transparent and insulating plexiglass. The height of the oil surface was just above the specimen surface in order to prevent the distortion on optical observations. An Olympus optical microscope was used to observe and to record the crack growth and the 90o domain switching instantaneously by means of a video imaging system.

sputtered Au

sputtered Au 600Pm

single crystal sample

90Pm

glass slide silver wire

E field Figure 1. Schematics of the sample setup

3. Experimental Results Figures 2(a)-(d) show the transmitted polarized optical micrographs of electricallyinduced crack propagation patterns for BaTiO3 single crystal under a cycling field of 160V/mm, which is 0.88 of the coercive field Ec. The applied electric field was sinusoidal with a frequency of 1 Hz. The upper darker area was evaporated with gold electrode. Shown in Figure 2a is the initial crack pattern for poled BaTiO3 single crystal, ABCD, which was produced during the sample preparation process. It was observed that at about cycle number 47, a very shallow crack EF which is approximately 3Pm in length appeared ahead of the crack tip E, as shown in Figure 2b. This newly formed shallow crack gradually darkened and continuously propagated as the cycle number increased. At about cycle number 5000, again a very shallow crack from G to H (about 7.7Pm in length) formed instantaneously ahead of the crack tip G (Figure 2c). Figure 2d is the crack pattern after 6000 cycles. What should be pointed out is that during cycling, parallel lines of 90o domain boundaries appeared and disappeared alternatively around the crack. The micrographs in Figure 2 did not show the domain boundary structure because what we wanted to illustrate here was the crack propagation path.

35

The crack propagation patterns for the same BaTiO3 single crystal cycled at a field of 360V/mm (2Ec) were shown in Figure 3. It was observed that the main crack (ABCDEFGHI) did not grow much during the first 695 cycles. However, it propagated quite a distance (42 Pm) all suddenly at about 700 cycle number, see Figure 3c. At the same time, several new cracks was formed just bellow the main crack. The newly formed cracks propagate quite a distance during the following 150 cycles (from cycle number 700 to 850), see Figure 3d. Also notice that the newly formed cracks touched the main crack at cycle number 850. The crack tip moved forward very little during cycle number 850-1700, despite the substantial broadening of the crack. Figure 3e shows the crack pattern at 1150 cycles. While all suddenly the crack propagated quite a distance at about 1720 cycle, and it did not show very much propagation afterwards. Shown in Figure 3f is the crack pattern at cycle number 2000. Notice that there is again a crack ahead of the main crack appeared. At about 2090 cycles, these two cracks become connected.

B

A

C

D

10Pm

C

10Pm

(a)

H

G F E D

A

B

F E D

(b)

B C

I

10Pm

H

G F E D

B C

10Pm

(c) (d) Figure 2. The transmission polarized optical micrographs of crack patterns for BaTiO3 single crystal electrically cycled at a field of 160V/mm (0.88Ec) for (a) 0, (b) 47, (c) 5000, and (d) 6000 cycles. Note that the shallow cracks of “EF” and “GH” were suddenly formed

Shown in Figure 4 are the plots for crack length versus cycle numbers. The abrupt changes in slope are associated with the sudden propagations of the main crack (at cycle number 47 and 5000 for electric field alternation at 160V/mm, and at cycle

36

numbers 700, 1720 at 360V/mm), or the linkage with a newly formed crack (at cycle number 850 at 360V/mm).

J I H G F ED I H G F ED

CB

A

CB

A

20Pm

20Pm

(a)

(b)

K

L K

I J I H

I J I H

20Pm

20Pm (c)

(d)

K L

K

I J I H

N

I

L

J

M

M

20Pm

20Pm

(e) (f) Figure 3. The transmission polarized optical micrographs of crack patterns for BaTiO3 single crystal electrically cycled at a field of 360V/mm (2Ec) for (a) 50, (b) 500, (c) 700, (d) 850, (e) 1150, and (f) 2000 cycles

During the electric cycling, one finds that the crack opened and closed alternatively, accompanied with the appearance and disappearance of the parallel lines of the 90o domain boundaries, as depicted in Figure 5. Apart from the main crack, the inclined crack which forms an angle of about 30o with the main crack formed during the

37

cycling. It was observed that the growth of this crack is accompanied by the forward movement of the 90o domain boundaries, see Figure 3a and Figure 3b.

180

160V/mm 360 V/mm

160

Crack length (Pm)

140 120 100 80 60 40 20 0 -1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Number of cycles Figure 4. Plot of the crack length versus the number of cycles for BaTiO3 single crystal electrically cycled at a field of 160V/mm, and 360V/mm

20Pm

20Pm

Figure 5. Two crack patterns recorded at cycle number 820 for BaTiO3 single crystal, showing the crack open (a) and closure (b) at the same period

4. Discussions The stress and electric fields around the flaw may lead to domain reorientation. The switched domain induces incompatible strain under the constraint of un-switched material and consequently alters the stress distribution near the flaw. If the resulting stress intensity factor is larger than the measured mechanical fracture toughness, the crack will propagate.

38

A model for fatigue crack growth under cyclic electric loading was proposed by Zhu and Yang [14]. The model predicted continuous crack growth under the assumption of well insulated crack surface. Though our single crystal specimen is submerged in the silicon bath, the rough crack surface and the surface tension of the silicon oil film would leave a partial filled gap between the crack faces. Therefore, the electric boundary condition along the crack face, from its mouth to its tip, would be a variation from impermeable to permeable state. A solution for a narrow ellipse dielectric cavity was provided by Zhu and Yang [18], which described a large variation of the electric and stress concentration near the crack tip as the effective dielectric ratio of the elliptical cavity changed. The initial field concentration near the crack tip, due to the local permeable boundary condition, would be insufficient to cause crack growth. Under repeated field cycling, the rubbing of the contacting surface, as well as the ratcheted squeezing-in of the silicon oil, make the crack surface more and more impermeable near the tip. The broadening of the main crack and the gradual approach of the domain switching zone verify this scenario. Gradually we recover the situation of a fully impermeable crack as formulated by Zhu and Yang [14]. Then the crack starts to grow after an incubation period of field cyclings. The combined silicon oil filling and fatigue cracking process can repeat itself to give a jumped crack growth behavior as in our experiment. 5. Conclusions In-situ observations of the crack propagation and 90o domain switching behavior were carried out for ferroelectric single crystals under alternative electric field. Electrically-induced fatigue crack growth was observed both below and above the coercive field. The crack propagation behavior is a repeated process of continuous increments followed by a sudden increase. This jumped crack growth behavior was attributed to the variation of the crack boundary conditions under electric field cycling. Acknowledgement The financial support by the Sino-Germany collaboration Foundation is greatly acknowledged. References 1ˊQ. Jiang, W. Cao, L.E. Cross, 1994, Electric fatigue in lead zirconate titanate ceramics, J. Am. Ceram. Soc. 77, 211-215. 2. H.C. Cao, A.G. Evans, 1994, Electric-field-induced fatigue crack growth in piezoelectrics, J. Am. Ceram. Soc., 77, 1783-1786. 3. C.S. Lynch, W. Yang, L. Collier, Z. Suo, R.M. McMeeking, 1995, Electric field induced cracking in ferroelectric ceramics, Ferroelectrics, 166, 11-30. 4. M.D. Hill, G.S. White, C.S. Hwang, et al, 1996, Cyclic damage in lead zirconate titanate. J. Am. Ceram Soc., 79, 1915-1920.

39 5. S.H. Kim, W.P. Tai, 1996, Relationship between cyclic loading and degradation of piezoelectric properties in Pb(Zr, Ti)O3 ceramics, Mater Sci Eng., B38, 182-185. 6. W. Yang, 2002, Mechatronic Reliability, THU-Springer-Verlag, Berlin. 7. A. Furuta, K. Uchino, 1993, Dynamic observation of crack propagation in piezoelectric multilayer actuators, J. Am. Ceram. Soc., 76, 1615-1617. 8. T. Zhu, F. Fang, W. Yang, 1999, Fatigue crack growth in ferroelectric ceramics below the coercive field, J. Mater. Sci. Lett, 18, 1025-1027. 9. J. Nuffer, D.C. Lupascu, A. Glazounov, H. Kleebe, J. Roedel, 2002, Microstructural modification of ferroelectric lead zirconate titanate ceramics due to bipolar electric fatigue, J. Europ. Ceram. Soc., 22, 2133-2142. 10. W. Yang, Z. Suo, 1994, Cracking in ceramic actuators xaused by electrostriction, J. Mech. Phys. Solods, 42, 649-663. 11. H. Gao, T.Y. Zhang, P. Tong, 1997, Local and global energy release rates for an electrically yielded crack in piezoelectric ceramics, J. Mech. Phys. Solids, 45, 491-510. 12. C.C. Fulton, H. Gao, 1997, Electrical nonlinearity in fracture of piezoelectric ceramics, Appl. Mech. Rev, 50, s56-s63. 13. W. Yang, T. Zhu, 1998, Switch-toughening of ferroelectrics subjected to electric fields, J. Mech. Phys. Solids, 46, 291-311. 14. T. Zhu, W. Yang, 1999, Fatigue crack growth in ferroelectrics under alternating electric loading, J. Mech. Phys. Solids, 47, 81-97. 15. F. Fang, W. Yang, T. Zhu, 1999, Crack tip 90o switching in tetragonal lanthanum-modified lead zirconia titanate under an electric field, Journal of Materials Research, 14, 2940-2944. 16. X. Tan, Z. Xu, J.K. Shang, 2000, Direct observation of electric field-induced domain boundary cracking in <001> oriented piezoelectric Pb(Mg1/3Nb2/3)O3-PbTiO3 single crystal, App. Phys. Lett., 77, 1529-1531. 17. F. Fang, W. Yang, 2002, Indentation-induced cracking and 90o domian switching pattern in barium titanate ferroelectric single crystals under different poling, Mater. Lett., 57, 198-202. 18. T. Zhu, W. Yang, 1997, Toughness variation of ferroelectrics by polarization switch under nonuniform electric field, Acta Materialia, 41, 4695-4702.

CRACK INITIATION AND CRACK PROPAGATION UNDER CYCLIC ELECTRIC LOADING IN PZT

I. Westram, D.C. Lupascu and J. Rödel Department of Materials Science, Darmstadt University of Technology 64287 Darmstadt, Germany

Abstract Crack initiation and crack propagation behavior was studied in a commercial soft PZT material under a cyclic electric field. A set of pertinent parameters that influence the crack propagation behavior was identified and their effect on the crack propagation behavior studied: the geometry of the notch, the viscosity of the surrounding liquid, the number of cycles and the electric field. To better understand the mechanism of crack initiation, linear piezoelectric FEM calculations were performed. 1. Introduction Due to their ferroelectric properties, PZT materials are nowadays widely used in industrial applications such as sensors, transducers or piezoelectric actuators [1]. However, material degradation has been observed in these devices under large electromechanical cyclic loads in the form of fatigue, microcracking, dielectric breakdown or aging. Crack initiation occurring at internal defects of the material was studied by Wang et al. [2]. They used finite element modelling to show that local polarization switching causes large stress/strain concentrations around these defects. Crack initiation from a notch was studied in PZT-5H by Chaplya and Carman [3]. In experiments and with finite element modelling they proved that polarization switching occurs near the notch in a dissimilar domain structure even though the applied electric field is lower than the coercive field strength of the material, Ec. Nuffer et al. [4-5] considered micro- and macrocracking as partial cause for fatigue behavior under bipolar cyclic electric loading in both PZT and PLZT. Crack initiation may also occur as a result of strain incompatibilities as was shown by Lucato et al. [6]. Crack propagation in ferroelectrics under a cyclic electric field has been studied since the early 1990s. First experiments performed by Cao and Evans [7] showed that cracks propagate from a Vickers indent under a cyclic electric field if the field strength is larger than 1.1Ec. The direction of crack growth is perpendicular to the 40 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 40–48. © 2006 Springer. Printed in the Netherlands.

41

applied electric field as was confirmed by further experiments [8-12]. However, mostly indented specimens with surface cracks were used for these experiments. In that case the residual stresses in the indented region were also shown to affect the crack propagation behavior [10]. It was found that different regimes of crack growth exist depending on the electric field strength. Zhu et al. [11] observed crack growth in poled specimens for an applied cyclic field of 0.8 Ec. Liu et al. [12] studied through-thickness cracks in DCB specimens and found an empirical relation between the rate of crack growth and the electric field strength. This yielded a cracking threshold of 0.797Ec below which no crack growth occurred. The crack propagation behavior in ferroelectric ceramics can furthermore be influenced by the grain size [13] and by the geometry of the specimen [14]. In the current study the effect of the medium inside the crack was first examined. Next, the effect of a static electric field on crack propagation under a static mechanical load was studied by R-curve measurements. Similar experiments were performed by Förderreuther et al. [15] in BaTiO3 ceramics. Crack initiation from a notch was also observed and a dependence of the pop-in length on the notch length was found. Also different regimes of crack propagation were observed depending on the number of applied cycles. 2. Experimental procedure The material used in the experiments is a commercially available, soft-doped PZT material (PIC 151) manufactured by PI Ceramic (Lederhose, Germany). The specimens were delivered as unpoled plates, either 40 × 5 × 1.5mm³ or 40 × 5 × 2mm³ in dimension. They were first polished down to a 1μm finish on one of their 40 × 5mm² surfaces. Silver paint was applied to both 40 × 1.5 (2) mm² faces to serve as electrodes. In order to pole the samples, an electric field of twice the coercive field strength of the material, Ec (here Ec=1kV/mm), was applied in the 5mm-direction of the samples. The 1.5mm-thick samples were used for crack propagation experiments under a cyclic electric field. The specimens were first notched with a wire saw in the middle of their 5-mm-side and then placed into a container filled with silicon oil for protection against electric breakdown. Two metal springs were used as electrical contacts. A function generator connected to a high-voltage amplifier provided either a d.c. voltage with an amplitude of up to 20kV or an a.c. sinusoidal voltage of 1 Hz and an amplitude of ±10kV. An optical microscope (Leica) with 200× magnification was used to observe the propagating crack. The container with the sample was mounted on an x/y-stage underneath the microscope, and the crack lengths could be measured with an accuracy of ±10μm. A sketch of the experimental set-up is shown in Figure 1a.

42

high voltage amplifier

V~ ±10kV

function generator

metal contacts

container with silicon oil PZT sample

Notch

Figure 1. (a) Experimental set-up for the crack propagation experiments under a cyclic electric field and (b) loading configuration for the R-curve measurements

The 2-mm-thick samples were used for R-curve measurements under a constant electric field. At one end they can be loaded with a wire connected to a load cell and are also notched with the wire saw. Due to the very low a/W-ratio of the DCB geometry, a purely mechanical mode-I loading will always result in curved cracks and therefore immediate failure of the specimen [16]. To avoid this problem, the samples were provided with a backface groove, 0.7mm in depth and approximately 0.2mm in width. A tensile load was applied and measured by a 30N-load cell. This configuration is depicted in Figure 1b. 3. Results and discussion 3.1 PARAMETER FIELD In order to examine the effect of the medium inside the crack on the crack propagation behavior, four different liquids were compared: two silicon oils (AK35 and AK1000, Wacker-Chemie GmbH, Germany) and two Flourinerts™ (FC-70 and FC-77, 3M Specialty materials, U.S.A.). The relevant properties of these liquids are summarized in Table 1. It is important that the liquid is highly insulating to avoid arcing since the electric fields applied are of the order of a few kV. Samples were cycled in each of the liquids up to 500 cycles at the field strength of 1.5 kV/mm in order to see whether any significant differences in the crack propagation behavior would occur. The crack length was recorded as a function of the cycle number and the results are plotted in Figure 2. For AK1000 the results were inconsistent while crack propagation in AK35 always occurred in a similar manner. Cycling in both Flourinerts generally resulted in a lower crack growth rate. Furthermore the wetting behavior of the four liquids was compared qualitatively since the liquid needs to be well able to penetrate the crack. The silicon oil with the highest viscosity, AK1000, showed the best wetting behavior. Both Flourinert liquids showed a rather weak

43

wetting behavior compared to the silicon oils. Since they also showed worse optical properties than the silicon oils, experiments were continued with the AK35 because of the most consistent results. Table 1: Viscosity, qualitative wetting behavior (1 = best, 3 = worst wetting behavior) and permittivity İr of four different insulating liquids. Type

Viscosity [mPas]

AK35 AK1000 FC70 FC77

35 1000 14 6.9

Wetting (qualitatively) 2 1 3 3

behavior İr 2.69 2.74 1.98 1.86

4.5

Z27 - AK35

Crack length [mm]

4

Z28 - AK35 Z29 - AK1000

3.5

Z25 - AK1000

3

Z26 - FC77

2.5

Z18- FC70

2 1.5 1 0.5 0 0

100

200

300

400

500

Cycle number Figure 2. Crack length vs. cycle number for cycling in different liquids

To examine the influence of a static electric field on crack propagation, R-curves were measured with and without the applied electric field. The procedure was similar to that used by Förderreuther et al. [15], therefore it will not be described here in detail. The stress intensity factor for the DCB geometry was calculated according to Kanninen [17]. The resulting R-curves are displayed in Figure 3. 3.2 CRACK INITIATION AND PROPAGATION When an electric field is applied to a notched sample, the field lines curve around the notch due to the different permittivities of the PZT material and the medium inside the notch [Figure 4 a]. If the electric field exceeds the coercive field strength of the material, different switching behavior of the domains around the notch will result. This in turn causes a strain incompatibility and therefore mechanical stresses.

44

In order to estimate these stresses, numerical calculations were performed with a commercial FEM software (ANSYS). The behavior of the material was modelled as linear piezoelectric in two dimensions. In Figure 4 b) Vy is plotted along a path through the middle of the notch in the 40mm-direction of the sample. As can be seen in the picture, a large tensile stress results in front of the notch if a field of 1.5Ec is applied perpendicular to the notch. Switching behavior of the material would lead to even higher tensile stresses therefore causing crack initiation from the notch.

Fracture toughness [MPa m1/2]

1.4

1.2

1.0

0.8 E=0 E = 0.5kV/mm

0.6

E = 0 (2nd meas.) E = 0.5kV/mm (2nd meas.)

0.4 0.0

0.5

1.0

1.5

2.0

Crack extension [mm] Figure 3. R-curves without and with an applied electric field of 0.5kV/mm in the polarization direction

(a)

45

(b) Figure 4. (a) Field distribution in a notched sample with Hnotch = 2 and HPZT = 2000. (b) ıy through the middle of the notch. The tip of the notch is at position 1.5mm

In the poled samples, a crack was always observed to pop into the material from the notch - perpendicular to the applied electric field - during the first half-cycle antiparallel to the poling direction if the applied electric field was larger than 1.1 kV/mm (which corresponds to 1.1Ec). In the unpoled samples a shorter pop-in was also observed during the first half-cycle. Two pictures of typical pop-ins are shown in Figure 5.

Figure 5. (a) 0.255-mm-long pop-in from a 0.29-mm-long notch; and (b) 0.57-mm-long pop-in from a 1.44mm-long notch. The pop-ins are marked by the white arrows

Pop-in-lengths were measured for different notch lengths ranging from 0.2 to 2mm in poled and unpoled samples. It was found that the length of the pop-in increases with increasing notch length up to a notch length of |0.8mm and then remains constant if the notch length is further increased. The results are plotted in Figure 6. The fracture surface of the area where the crack had popped in was also examined using a scanning electron microscope and is displayed in Figure 7. Fracture occurred mainly trans- and not inter-granular. During further cycling up to a few thousands cycles, different regimes of crack growth were observed with increasing cycle number: steady-state crack growth of one major crack usually occurred in the first 10-50 cycles although the cycle

46

number varied with the strength of the applied electric field and from sample to sample. For low electric fields (< 1.4Ec) steady-state crack growth was generally observed up to a higher number of cycles. For larger fields, the crack started bifurcating after 10-50 cycles and highly irregular crack growth occurred. This caused the crack growth rate to decrease until the crack eventually arrested. However, final crack lengths can vary by several millimeters even if the field strength is constant since the internal defects affect the crack growth behavior. Photographs of cracks of both regimes are presented in Figure 8. A possible reason for the observed behavior can be the formation of microcracks with increasing cycle number and increasing electric field strength. 0.7

Pop-in length [mm]

0.6 0.5 0.4 0.3 0.2 poled specimens

0.1

unpoled specimens

0 0

0.5

1

1.5

2

2.5

Notch length [mm] Figure 6. Pop-in-length vs. notch length after one half-cycle of an applied field of 1.5 kV/mm

3

47

Figure 7. Fracture surface of the pop-in area

(a)

(b) Figure 8. Optical pictures of cracks growth under a cyclic electric field, 200 x magnification. (a) Propagated, 1.845mm-long crack after cycling for 250 cycles with a field strength of 1.1Ec . (b) Large damage zone with strongly bifurcated cracks and many smaller cracks in between after cycling for only 12 cycles at a field strength of 1.7Ec

48

4. Conclusion Several parameters influencing crack propagation behavior in a ferroelectric material under cyclic electric loading were identified and quantified. While the viscosity of the medium inside the crack only plays a minor role, crack propagation behavior changes strongly with electric field strength. The geometry of the notch was found to affect only the crack initiation but not further crack growth.

Acknowledgement: Technical support by Emil Aulbach and financial support by the Deutsche Forschungsgemeinschaft (DFG) under contract No. RO 954-17/1 is gratefully acknowledged.

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Y. Xu, 1990, Ferroelectric Materials and their Applications, Elsevier Science Publishers B.V., Amsterdam. D. Wang, Y. Fotinich, G.P. Carman, 1998, Influence of temperature on the electromechanical and fatigue behavior of piezoelectric ceramics, J. Appl. Phys. 83, 5342-5350. P.M. Chaplya, G.P. Carman, 1998, Relation of the cracking phenomena in piezoelectrics to domain wall motion, Proceedings of SPIE, 3324, 154-160. J. Nuffer, D.C. Lupascu, J. Rödel, 2001, Microcrack clouds in fatigued electrostrictive 9.5/65/35 PLZT, J. Eur. Ceram. Soc. 21, 1421-1423. J. Nuffer, D.C. Lupascu, A. Glazounov, H.-J. Kleebe, J. Rödel, 2002, Microstructural modifications of ferroelectric lead zirconate titanate ceramics due to bipolar electric fatigue, J. Eur. Ceram. Soc. 22, 2133-2142. S.L. dos Santos e Lucato, D.C. Lupascu, M. Kamlah, J. Rödel, C.S. Lynch, 2001, Constraint-induced crack initiation at electrode edges in piezoelectric ceramics, Acta Mater. 49, 2751-2759. H. Cao, A.G. Evans, 1994, Electric-field-induced fatigue crack growth in piezoelectrics, J. Am. Ceram. Soc. 77, 1783-86. C.S. Lynch, W. Yang, L. Collier, Z. Suo, R.M. McMeeking, 1995, Electric field induced cracking in ferroelectric ceramics, Ferroelectrics 166, 11-30. H. Weitzing, G.A. Schneider, J. Steffens, M. Hammer, M. J. Hoffmann, 1999, Cyclic fatigue due to electric loading in ferroelectric ceramics, J. Eur. Ceram. Soc. 19, 1333-37. L.Z. Jiang, C.T. Sun, 1999, Crack growth behavior in piezoceramics under cyclic loads, Ferroelectrics 233, 211-223. T. Zhu, F. Fang, W. Yang, 1999, Fatigue crack growth in ferroelectric ceramics below the coercive field, J. Mat. Sci. Let. 18, 1025-1027. B. Liu, D. Fang, K.-C. Hwang, 2002, Electric-field-induced fatigue crack growth in ferroelectric ceramics, Materials Letters, 54, 442-446. F. Meschke, A. Kolleck, G.A. Schneider, 1997, R-curve behaviour of BaTiO3 due to stress-induced ferroelastic domain switching, J. Eur. Ceram. Soc. 17, 1143-1149. A. B. Kounga Njiwa, 2004, private communications. A. Förderreuther, G. Thurn, A. Zimmermann, F. Aldinger, 2002, R-curve effect, influence of electric field and process zone in BaTiO3 ceramics, J. Eur. Ceram. Soc., 22, 2023-2031. A.G. Atkins, Y.-W. Mai, 1988, Elastic and Plastic Fracture, Ellis Horwood Limited, Chichester, England. M.F. Kanninen, 1973, An augmented double cantilever beam model for studying crack propagation and arrest, Int. J. Fract. 9, 83-91.

MULTIAXIAL BEHAVIOR OF FERROELECTRIC CERAMIC PZT53

Q. Wan, C.Q. Chen and Y.P. Shen MSSV, School of Civil Engineering and Mechanics, Xi’an Jiaotong University, Xi’an, 710049, P. R. China

Abstract The coupled electromechanical behavior of a soft ferroelectric PZT53 under uniaxial, tension-torsion or compression-torsion loading has been investigated. By using plate type specimens and thin wall tube specimens, effect of uniaxial prestress in the poling direction and in the lateral direction (a direction perpendicular to the poling direction) on the electric and butterfly hysteresis response loops of PZT53 has been exploited. It is found that although both electric and butterfly hysteresis loops of PZT53 are sensitive to the pre-stress in the poling direction, the butterfly loops have additional sensitivity to the pre-stress in the lateral direction. Thin wall tube specimens are used to study the response of un-poled and poled PZT53 to proportional tension-torsion and compression-torsion loading. Initial ‘yield’ surfaces for un-poled and poled PZT53 are obtained. 1. Introduction Ferroelectric ceramics with promising electromechanical properties are widely used as sensors, transducers and actuators. Many advanced applications (e.g., actuators) require either large actuation displacement and/or large force, usually causing permanent deformation of the ferroelectric solids and the attached metallic electrodes. A thorough understanding of the nonlinear constitutive behavior of ferroelectric materials subject to severe mechanical and electric loading is essential for their application design. Most available studies are on the uniaxial electromechanical behavior of ferroelectric materials subject to simultaneously applied compression and electric loading (see, for examples [1-4]), with both mechanical and electric loading parallel with the poling direction. These studies reveal that the behavior of ferroelectric materials is highly electromechanically coupled, showing clearly the rate effects, the memory effects, and the hysteresis loops. Observed electric and butterfly hysteresis loops are also found to be very sensitive to the uniaxial pre-stress applied along the poling direction. In addition, tension-compression non-symmetry was found to be one of the salient features of ferroelectric ceramics [5,6]. Effect of 49 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 49–56. © 2006 Springer. Printed in the Netherlands.

50

uniaxial pre-stress applied in the direction perpendicular to the poling direction on the electromechanical behavior of ferroelectric ceramics has yet to be exploited. Moreover, only limited experimental results are available on the multiaxial behavior of ferroelectric ceramics. Using pressurized thin wall tube specimens, Chen and Lynch [7] studied the biaxial response of a ferroelectric ceramic PLZT and found that the stress induced switching of PLZT observes the maximum shear stress criterion, i.e., the Tresca criterion. It should be noted that, in principle, the Tresca criteria is only applicable to isotropic solids whilst poled PLZT is transversely isotropic. Fett et al. [8] investigated the multiaxial ‘yield’ of unpoled PZT and found that the onset of initial ‘yield’ of PZT can be described by the Drucker-Prager criterion. By applying electric field in various directions different from the original poling direction, Fleck and coworkers [9,10] were able to exploit the multiaxial electric behavior of several ferroelectric ceramics (i.e., PZT-5Hǃ BaTiO3 and PZT-4D). In this paper, the uniaxial and multiaxial behavior of a soft ferroelectric ceramic PZT53 was investigated. Effect of uniaxial constant pre-stress either in the poling direction or in the direction perpendicular to the poling direction on the electromechanical response of PZT53 was systematically exploited. For the multiaxial behavior, the stress-strain responses of both un-poled and poled PZT53 to combined tension-torsion and compression-torsion loading were measured. Initial yield surface in the normal and shear stress space were determined from the obtained stress-strain curves. The paper is organized as follows. In section 2, the materials and experimental procedure used in this study are summarized. Obtained experimental results are presented and discussed in section 3, followed by a few conclusions drawn in section 4. 2. Materials and Experiment methodology The material considered in this study is a soft ferroelectric ceramic PZT53, obtained in poled state from a commercial company. Un-poled state can be obtained by heating the poled material to 300-310qC, holding ten minutes, and then naturally cooling to room temperature. Two types of specimens were used in this study. Plate-type specimens with dimensions of 10 u 10 u 5mm3 were applied to investigate the effect of uniaxial prestress in the poling direction on the electromechanical behavior of PZT53. Thin wall tube specimens with inner diameter d 17.5 mm, outer diameter D 20.5 mm, and length H 22.5mm were developed to study the multi-axial behavior of PZT53 under combined axial and torsion loading. The electrodes are the inner and outer surfaces of the tubes. As a result, electric field (if any) and poling can only be in the radial direction. It should be noted that when only axial loading was applied tube specimens were also employed to study the effect of lateral stress upon the electromechanical coupling behavior of PZT53. Figure 1 shows sketches of both plate-type and thin wall tube specimens, with appropriate dimensions and electric

51

and mechanical loading marked. For plate-type specimens, the upper and bottom major surfaces were electroded with sputtered Ag. For tube specimens, the electrodes are coated on the inner and outer surfaces. Electric field was applied to the electrodes of specimens through a high voltage amplifier (Trek 30/20). We recorded readings from a capacitor connected between electrode and ground to calculate the electric displacement during deformation. The mechanical loading was imposed by a servo-hydraulic test frame (MTS 858 mini Bionixll), which is capable of simultaneously applying axial and torsion loading.

Poling

t

H

H

H

H



E

Poling L

D

H

L

H

Proportional loading

Stress

(a) (b) Figure 1. Sketches of specimens used for testing PZT53: (a) plate type specimen for uni-axial test, and (b) thin wall tube specimen for uniaxial and combined axial-torsion test.

Stain gauges and strain rosettes are used to monitor the deformation of PZT53 upon loading. Following the conventions in piezoelectricity, the poling direction is denoted by ‘3’, and lateral directions perpendicular to the poling direction are denoted by ‘1’ and ‘2’, respectively. For example, V 33 and H 33 for the plate type specimens refer to the normal stress and strain in the poling direction. The respective axial normal strain and torsion strain for the tube specimens are denoted by H 11 and J 12 and can be calculated by

H11 H 90 J 12 H 45  H135 0

0

0

(1) (2)

where H 45 H 90 , and H 135 are strains measured from strain rosettes shown in Figure 1b. The corresponding axial normal stress V 11 and shear stress W 12 are related to the applied axial force F and torque M t by 0

0

0

V 11 W 12

2F S t( D  d ) 8M t S t ( D  d )2

(3) (4)

52

where t

( D  d ) 2 is the wall thickness of tube specimens.

Note that the aims of this study are on the effect of constant uniaxial pre-stress on the electromechanical behavior of PZT53 and on its response to combined tensiontorsion and compression-torsion loading. To accomplish the first goal, the prestress (either V 33 for the plate type specimens or V 11 for the tube specimens) was increased at a rate of 10MPa/min to a specific stress level and then kept constant. A cyclic triangular electric field of frequency 0.02Hz and magnitude 1.5kVolt/mm was applied to the specimen for several complete cycles until the electric and butterfly hysteresis loops are stabilized. The pre-stress was then increased further to another stress level, and kept constant again for a different round of cyclic electric loading. Such loading procedure was repeated for all levels of pre-stress of present interest. By doing this, effect of uniaxial constant pre-stress on the electromechanical behavior of PZT53 can be elucidated. It should be noted that the electric field with very low frequency used in this study is to mimic quasi-static electric loading.

W F

E

G

D C B A

O

V

Figure 2. Illustration of the proportional axial-torsion loading paths in the

V 11 - W 12

space for the testing of

un-poled and poled PZT53 thin wall tube specimens. Paths OA to OG corresponding to

V 11 : W 12 = 1:0, 2.3:1,

1:1, 2 3 :10, 0:1, - 2 3 :10, -1:1.

For the combined tension-torsion and compression-torsion test of un-poled and poled PZT53 tube specimens, in addition to uniaxial tension and pure torsion loading, five proportional loading were considered, as indicated by paths OA, OB, …, OG in Figure 2. The loading rate is 105N/s in axial force for uniaxial loading and 1 N.m/s in torque for combined loading. For each loading path, the load was increased until specimen broke. It was found during testing that the uniaxial tensile failure strength is about 14.7MPa for un-poled PZT53 and 27.7MPa for poled PZT53 whilst their corresponding uniaxial compressive failure strength is far greater than the tensile ones. In fact, their uniaxial compressive failure stress is beyond the loading capacity of the test machine used in this study, the maximum axial force of which is 20kN (equivalent to 223.4 MPa axial stress for the tube specimens). Therefore the loading paths considered here are mainly in the tension dominated region (see Figure 2).

53

Figure 4. Electromechanical hysteresis response loops of PZT53 at various uniaxial constant pre-stress applied in the direction perpendicular to the poling direction. (a) Electric hysteresis loops, and (b) butterfly loops

3. Experimental results and discussions 3.1 EFFECT OF UNIAXIAL PRE-STRESS ON ELECTROMECHANICAL BEHAVIOR Stabilized electric and butterfly hysteresis response loops of PZT53 at various values of constant pre-stress applied in the poling direction are measured and shown in Figure 3. The results were obtained using plate type specimens, as depicted in the inset of Figure 3. Figure 4 shows the effect of the constant pre-stress applied in the direction perpendicular to the poling direction, with results obtained from thin wall tube specimens. It can be seen from Figure 3 that the electric and butterfly loops are very sensitive to the uniaxial pre-stress in the poling direction: the remanent polarization (defined as the polarization at zero electric field), coercive field (defined as the electrical field at zero polarization) and strain variation magnitude within a close loop decrease with increasing the magnitude of

54

the pre-stress. On the other hand, although the stabilized butterfly loops are also sensitive to the constant pre-stress in the lateral direction it is found that the lateral pre-stress has negligible effect on the electric hysteresis loops of PZT53 (Figure 4). Also shown in Figure 4 is the effect of tensile pre-stress on the electromechanical responses of PZT53. A comparison of Figures 3b and 4b reveals that the pre-stress in the poling direction and that in the lateral direction have a similar effect on the butterfly loops, with the compressive pre-stress decreasing the strain variation magnitude. The phenomena shown in Figures 3 and 4 can be explained by the mechanisms proposed by Wan et al. [11]. 3.2 MULTI-AXIAL BEHAVIOR OF UN-POLED AND POLED PZT53 As mentioned before, the multiaxial behavior of PZT53 was investigated using thin wall tube specimens, with their inner and outer electrode surfaces grounded. It should be noted that un-poled PZT53 is isotropic whilst poled PZT53 is transversely isotropic with 1-2 being the isotropic plane. From the measured stressstrain curves, the Young’s modulus E1 =52.3GPa and shear modulus G12 =13.4GPa in the 1-2 isotropic plane of poled PZT53 can be calculated, which are found to be not far away from those of un-poled PZT53 (i.e., E1 =55.6GPa and G12 =14.4GPa). It is found that all stress-strain curves clearly show a linear region at low stress levels and a non-linear region at high stress levels, indicating elastic-‘plastic’ deformation (the stress-strain curves are not shown here for the sake of brevity). However, transition from elastic to ‘plastic’ state can not be readily identified from the smooth stress-strain curves. As a result, it is difficult to determine uniquely the initial ‘yield’ points from the stress-strain curves and subsequently the initial yield surfaces. Chen and Lu [12] developed a theory to simulate proportional multiaxial elastoplastic behavior of solids showing smooth transition between elastic and plastic deformation. The methodology developed by Chen and Lu is employed here to model the multiaxial behavior of un-poled and poled PZT53. Following the procedure by Chen and Lu [12], the following equivalent stress V and strain H for PZT53 under combined axial-torsion loading are defined E V 2 V 112  1 W 122 G12 . (5) G H 2 H112  12 J 122 E1 It can be verified that V and H are work conjugates. Using Eq. 5, the measured stress-strain curves of PZT53 under various combined loading can be expressed in the form of V as functions of H , as shown in Figure 5a for un-poled PZT53 and Figure 5b for poled PZT53. It is seen from Figure 5 that the V - H curves for various loading paths collapse into a single master straight line in the elastic region, which is consistent

55

with the theory by Chen and Lu [12]. The obtained V - H curves can then be used to simulate the corresponding proportional behavior of un-poled and poled PZT53, within the framework laid down by Chen and Lu. From the V - H curves the initial ‘yield’ surface and its evolution can also be constructed. Figure 6 shows the corresponding initial ‘yield’ surfaces of un-poled and poled PZT53, defined at the equivalent strain of 200PH.

Figure 6. The initial yield surfaces of un-poled and poled PZT53 in the axial and shear stress space. Symbols denote experimental results, lines refer to the proposed yield surface (6)

Note that the obtained yield surfaces in shape are similar to the Drucker-Prager yield surface. Recall that the compressive failure stress of PZT53 is far greater than its tensile failure stress (at least an order of magnitude greater). It is thus reasonable to assume that in uniaxial compression PZT53 does not yield. By taking this assumption into account, the following yield criterion can be proposed for both unpoled and poled PZT53,

V 11 V 0 where

2

 A W 12 V 0  1

(6)

V 0 is the initial yield stress in uniaxial tension, and the dimensionless

56

material constant A can be obtained by curve fitting to the experimental results, giving V 0 =20.4MPa, and A =2.52 for un-poled PZT53 and V 0 =19.6MPa, and

A =1.78 for poled PZT53. The yield criterion (50) has also been included in Figure 6 for the purpose of comparison. It is seen that the proposed yield criterion can reasonably model the initial yield of both un-poled and poled PZT53 under combined axial and torsion loading. 4. Conclusion The uniaxial and multiaxial electromechanical behavior of ferroelectric PZT53 has been investigated. It is found that although both the electric and butterfly hysteresis loops of PZT53 are sensitive to the uniaxial pre-stress in the poling direction, the butterfly loops are further sensitive to the pre-stress in the lateral direction. Based upon the experimental results obtained from the multiaxial testing, ‘yield’ surfaces of un-poled and poled PZT53 are obtained. A yield criterion has been proposed. Acknowledgements: The authors are grateful for the financial support from the National Natural Science Foundation of China (No. 10302024; No. 10472088), the Foundation for National Distinguished PhD Thesis Award (No. 200129) and the Sino-German Research Center in Beijing (No. GZ 050/3). References 1 2 3 4 5 6 7 8 9 10 11 12

H. Cao, A.G. Evans, 1993, Nonlinear Deformation of Ferroelectric Ceramics. J. Am. Ceram. Soc., 76, 890-895. C.S. Lynch, 1996, The effects of uniaxial stress on the electro-mechanical response of 8/65/35 PLZT. Acta Materiala, 44, 4137-4148. D.N. Fang, C.Q. Li, 1999, Nonlinear electric-mechanical behavior of a soft PZT-51 ferroelectric ceramic. J. Mater. Sci., 34, 4001-4010. D.Y. Zhou, M. Kamlah, D. Munz, 2004, Uniaxial compressive stress dependence of the high-field dielectric and piezoelectric performance of soft PZT piezocermics. J. Mater. Res., 19, 834-841. T. Fett, D. Munz, G. Thun, 1998, Nonsymmetric deformation behavior of Lead Zirconate Titanate determined in bending Tests. J. Am. Cerm. Soc., 81, 260-272. T. Fett, D. Munz, G. Thun, 1998, Nonsymmetry in the deformation behavior of PZT, J. Mater. Sc. Lett., 17, 261-265. W. Chen, C.S. Lynch, 2001, Multiaxial constitutive behavior of ferroelectric materials. J. Eng. Mater. Tech., 123, 169-175. T. Fett, D. Munz, G., Thun J., 2003, Multiaxial deformation behavior of PZT from torsion tests. J. Am. Ceram. Soc., 86, 1427-1429. J.E. Huber, N.A. Fleck, 2001, Multiaxial electrical switching of a ferroelectric: theory versus experiment. J. Mech. Phys. Solids, 49, 785-811. J. Shieh, J.E. Huber, N.A. Fleck, 2003, An evaluation of switching criteria for ferroelectrics under stress and electric field. Acta. Mater., 51, 6123-6137. Q. Wan, C. Chen, Y.P. Shen, 2004, Effect of constant uniaxial pre-stress on the electromechanical behavior of PZT53, Acta Mech. Sinica, (In Chinese). C. Chen, T.J. Lu, 2000, A phenomenological framework of constitutive modelling for incompressible and compressible elasto-plastic solids. Int. J. Solids Struct., 37, 7769-7786.

STABILITY ANALYSIS OF 180º DOMAINS IN FERROELECTRIC THIN FILMS Biao Wang1, C. H. Woo2, and Yue Zheng1 School of Physics and Engineering, Sun Yat-sen University, Guangzhou, China 2 Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong, China 1

Abstract Ferroelectric domain switching under low voltage or short pulses is of interest to the development of high-density random access memory (FRAM) devices. Being necessarily very small in size, instability and back switching often occurs when the external voltage is removed, and creates serious problems. In this investigation, a general approach to determine the minimum size of ferroelectric domain to avoid back switching was developed. As examples, two cases were considered in detail: one is a 180o domain in a ferroelectric thin film covered by the upper and lower electrodes, the other is a 180o domain in a ferroelectric thin film induced by AFM without the top electrode. We note that our approach is generally applicable to many other fields, including phase transformation, nucleation and expansion of dislocation loops in thin films. 1. Introduction Ferroelectric materials are polar dielectrics that spontaneously polarize in the presence of an external electric field. Their usefulness is derived from the fact that different ferroelectric domains have different piezoelectric, pyroelectric, electrooptic and nonlinear optic properties. The interest in ferroelectrics has increased drastically in the last ten years, due to their wide applications in various electronic and optoelectronic devices, and as a result of recent progress in the processing of ferroelectric thin films. A ferroelectric film can maintain its polarization even in the absence of an external voltage, making it useful as a non-volatile memory device. Indeed, ferroelectric random access memory (FRAM) devices are attractive for their ideal properties such as non-volatility, high speed, and low power consumption [1,2]. The technology of switching domains of a very small size under low voltage or short pulse is required for the fabrication of high-density FRAM. Small 57 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 57–67. © 2006 Springer. Printed in the Netherlands.

58

ferroelectric domains are imprinted (or “written”) onto the surface of ferroelectric materials [3] using the atomic-force microscopes (AFMs), using the high electric field of the AFM tip. Since the macroscopic properties of ferroelectric samples can be directly influenced by the domain structure, many fabrication techniques are based on the optimization of the domain structure. Stable ferroelectric domains of dimension below 50 nm on high-quality epitaxial Pb(Zr0.2Ti0.8)O3 thin film have been successfully manipulated by using AFM [4]. It seems that smaller domains can be realized by choosing a finer AFM tip. Polarization in AFM writing is effected by the high electric field near the AFM tip; but unlike usual poling, the process does not require a top electrode, the AFM tip serves as a moving electrode. Once the AFM tip is removed, depolarization energy induced by the polarization charges, and the domain-wall energy may drive the domain to shrink to zero. The stability of written domains is crucial for nonvolatile ferroelectric memory applications. Based on the static analysis of the system free energy, Li et al. [5] found that the minimum aspect ratio of a stable cylindrical domain is of the order of 100 for PZT thin film. This is unrealistically too big. In our present paper, we will show that their analysis only gives an upper limit for the aspect ratio of a stable domain. A domain evolves under the combined action of the external applied electric field, the depolarization field, and the surface tension of the domain wall [6]. A 180o domain nucleates and expands under the action of an external electric field, but the depolarization field and the surface tension tend to destabilize the domain when the external field is removed. Wang and Woo [7] previously used the numerical calculation on phase diagram to determine the stable region of a ferroelectric domain in an infinite body. In this paper, based on a Liapunov function, a general approach was established to determine the minimum sizes of ferroelectric domain without needing the complicated numerical calculation on phase diagram. As examples, two cases were considered in detail: one is a 180o domain in a ferroelectric thin film, covered by top and bottom electrodes; the other is a 180o domain in a ferroelectric thin film induced by AFM without the top electrodes. The minimum sizes of the ferroelectric domain for the second case to avoid the back switching were found to be larger than the first case since there is no bound charge to tie the polarization charges without the top electrode, thus a higher depolarization field was induced in the domain of the ferroelectric film without the top electrode. 2. Energy expressions 2.1 A 180o DOMAIN IN A FERROELECTRIC FILM COVERED WITH TOP AND BOTTOM ELECTRODES The speed of domain evolution is proportional to the thermodynamic driving force derived from the total free energy reduction due to the change of the domain size.

59

In this part, we are interested in a ferroelectric thin film covered with top and bottom electrodes. The ferroelectric domain under consideration is assumed to have the shape of a prolate semi-spheroid that intersects the upper electrode in a circle of radius a, and have its tip at a distance c from the electrode (Figure 1). The other electrode is assumed to be far away, so that its effects on the depolarization energy can be neglected. Since the electrode on which the domain originates is an equipotential surface, the depolarization field created by the prolate semi-spheroidal domain is, based on the image approach, the same as that created by a complete prolate spheroidal domain in an infinite dielectric matrix. Such a depolarization field inside the domain has been obtained by Landauer [8], and generalized by Wang and Xiao [9]:

180o

electrode

Ferroelectric film

Figure 1. Schematic of the half of prolate spheroidal domain

E3



Ps I 3 , E1 2SH

E2

0

(1)

where for the ellipsoidal domain, I3 is given by

I3

­4S ca 2 {log[c / a  (c / a ) 2  1]  c 2  a 2 / c}/(c 2  a 2 )3/ 2 prolate spheroid c ! a °° 4S / 3 sphere ® ° 2 2 2 2 3/ 2 oblate speroid, c  a °¯ 4S a c[ (a / c)  1  arccos(c / a )] /(a  c ) (2)

where the energy expression holds for all three cases. Even if the initial shape of the domain is assumed to be a prolate semi-spheroid, during the evolution, it may become a semisphere or an oblate semi-spheroid. When a sufficiently large electrical field is applied, a 180o domain will nucleate and expand. The evolution of the domain is driven by the free energy reduction in the process. Since no strain is involved in the 180o domain, only the electrostatic energy and domain-wall energy c enter into our analysis. If the coercive field E of the ferroelectric material plays the role of the external electric field, the Gibbs free energy change can be written as [7]

'G

G  G0



1 Ei Pi *dv  ³³³ EiC` Pi *dv : 2 ³³³:

4S 2 Ps2 ca ( I 3  2 E c Ps ), 3 2SH

(3)

60

K*

where P

K 2 Ps k . Equation (3) only gives the electrostatic energy change due to

the introduction of an 180o domain. The surface energy of the wall has to be added:

'T

4S 2 Ps2 ca [ I 3  2 E c Ps ]  *(a, c)V , 3 2SH

(4)

where V designates the surface energy per unit area of the wall, and * is the surface area of an ellipsoidal domain, given by c ­ arcsin( 1  a 2 / c 2 )] prolate spheroid, a  c °2S a[a  2 2 1 a / c  °° . 4S a 2 sphere a c * ( a, c ) ® ° 2 2 ° 2S a[a  c / (a / c)  1 log( a / c  ( a / c)  1)] oblate spheroid, a ! c °¯ (5) In reality, the initial polarization charges prior to the induction of 180o domain can K K* be considered as discharged, thus one should take P Ps k in the above energy

expressions. 2.2 A 180o DOMAIN IN FERROELECTRIC FILM INDUCED BY A AFM TIP WITHOUT THE TOP ELECTRODE

Bottom Figure 2. Schematic of a 180o domain created by AFM tip

In general, AFM writing requires the preparation of the FE surface by an initial uniform polarization in a certain direction. We assume that on the bottom electrode, there is a ferroelectric thin film with thickness h. For simplicity, the initial polarization direction is assumed to be normal to the surface along the (–z) axis, which is also the easy axis of the underlying ferroelectric material. Subsequently, the polarity of the tip is reversed, and a small, cylindrical region of radius R and

61

height (or depth) Z is produced by polarization in the z direction (Figure 2). After the surface preparation and prior to writing, there is already sufficient time for the space charge to segregate, which neutralizes the ferroelectric polarization charges on the prepared surface and in the interior. Therefore, uncompensated ferroelectric polarization charges are associated with freshly written domain, as illustrated in Figure 2. Several investigators have noted that the surface charges of the AFMwritten domains are much less than the ferroelectric charges. This means that the ferroelectric charges on the surface may be instantaneously neutralized by a charge transfer from the AFM tip during writing. This constitutes the main explanation for the prediction of Li et al. [4]. If the spontaneous polarization of the ferroelectric thin film is denoted as Ps, the polarization charges on the ends of the cylindrical domain can also be assumed to be r 2 Ps since a charge transfer from AFM tip can be avoided easily during writing. A factor of 2 is introduced to reflect that the uniform polarization of the initial state is along the (-z) direction. For a cylindrical domain on the bottom electrode, the depolarization energy Udp created by the polarization charges has recently been obtained in closed form by Wang and Woo [10]. It is given by the following integral, Ps2 2S R 2S R g 0 (r , T ; r ', T ') rr ' dr ' dT ' drdT  2SH1 ³0 ³0 ³0 ³0

U dp Ps2 2SH 0

2 S R 2S R

³ ³ ³ ³ [D g (r ,T ; r ',T ')  D g 1 1

2

2

(r , T ; r ',T ')  D 3 g3 ( r ,T ; r ',T ')  D 4 g 4 (r ,T ; r ',T ')]rr ' dr ' dT ' drdT

0 0 0 0

(6) where, g 0 (r , T ; r ', T ')

2[

1

U



1 ], ( Z  U 2 )1/ 2 2

f

g1 ( r , T ; r ', T ')

2

¦{[(2h  2nh)

2

n 0 f

g 2 (r , T ; r ', T ')

1

¦{[(2Z  2nh)

2

n 0 f

g3 (r ,T ; r ',T ')

f

g 4 (r , T ; r ', T ')

2

 U 2 ]1/ 2

(1  H1 / H 0 )(H 0 / H1 ) ,D 2 (1  H1 / H 0 )



1 1 }E n  [(2nh  2h  Z )2  U 2 ]1/ 2 [(2nh  2h  Z ) 2  U 2 ]1/ 2

 

2 1  }E n [(2nh  Z ) 2  U 2 ]1/ 2 [(2nh) 2  U 2 ]1/ 2

1 1  }E n [(2nh  2h  Z ) 2  U 2 ]1/ 2 [(2nh  2h  Z ) 2  U 2 ]1/ 2

1

2 1   }E n  U 2 ]1/ 2 [(2nh  2h  Z ) 2  U 2 ]1/ 2 [(2nh  2h) 2  U 2 ]1/ 2 (1  H 0 / H1 ) (1  H 0 / H1 ) 1  H1 / H 0 ,D3  , D 4 H 0 / H1 , E , (1  H1 / H 0 ) (1  H1 / H 0 ) 1  H1 / H 0

¦{[(2h  2nh  2Z ) n 0

D1

 U 2 ]1/ 2

2

¦{[(2h  2nh) n 0

 U 2 ]1/ 2

2

U [r 2  r '2  2rr 'cos(T  T ')]1/ 2 ,

(7) and H 1 , H 0 are the dielectric constants of the ferroelectric thin film and vacuum, respectively. It is easy to see that the depolarization energy of the ferroelectric domain depends on the materials dielectric constant, spontaneous polarization, the domain size parameters R and Z, and the film thickness h. In this investigation, we

62

use expression (6) for the depolarization energy, which increases with increasing size (Z, R) of the domain. It produces the main force driving the freshly written domain to shrink, supplemented by the minimization of the domain-wall energy. For the cylindrical domain shown in Figure 2, the total domain-wall energy is, U dw 2SRZJ 1  SR 2J 2 , (8) where, J 1 , J 2 are the domain-wall energy per unit area on the cylindrical surface and the bottom surface, respectively. The energy of the upper surface does not change when the polarization reverses. Therefore it is not included in the sum in equation (8). The interaction between the AFM tip field and the spontaneous polarization field provides the main force driving the fresh creation of the written domain. When the tip is removed, there appears to be no driving force to stabilize the domain. However, the reversal of the polarization field has to be accompanied with the motion of the domain wall, which is a dissipative process characterized by a coercive field Ec. In such a case, one can treat Ec as an effective electric field in favor of the new domains, and the interaction energy between Ec and the spontaneous polarization can be written as, U int 2SR 2 ZE c Ps , (9) where the negative sign means that it is against the shrinkage of the domain. The total energy is given by taking the summation of equations (6), (8) and (9):

'T

U dp  U dw  U int

(10)

Under the assumption of a cylindrical domain, the two parameters R and Z completely describe its geometry. If the initial polarization charges are considered to be discharged before introducing the 180o domain, the spontaneous polarization should take 1/2 of its value. We assume that the process is isothermal, so that effects due to the entropy change can be neglected. Generally speaking, the spontaneous polarization is a decreasing function of temperature, approaching zero when the temperature approaches the Curie temperature of the material. Thus the effect of temperature can be taken into account by substituting the spontaneous polarization as a function of temperature. From equations (4) and (10) it can be seen that the depolarization energy and the domain wall energy drives the 180o domain to shrink, resisted by the coercive field. 3. Evolution equations and stability analysis

For a spheroidal or cylindrical domain, two parameters a and c completely describe the geometry of the domain. The rate of evolution of the domain then depends on the driving forces which are the free energy reduction rates accompanying the change of a and c, i.e.

63

fa fc

w ('T ) wa w ('T )  wc



(11)

As discussed by Loge and Suo [11], the evolution rate can be reasonably assumed to be proportional to the corresponding driving force,

dc dt da dt

Mf c Mf a

w ('T ) , wc w ('T ) M wa

M

(12)

where M is a material constant. As explained earlier, the back switching process, i.e., the shrinking of the switched domains after the removal of the applied electric field, is driven by the reduction of the depolarization energy and the domain-wall energy, whereas the coercive field provides the resistance. Equation (12) is a system of nonlinear differential equations, which we will not attempt to solve. Instead, our interest is in the conditions of stability of the solution. Knowing that ( a, c) (0, 0) is a stationary point, we can determine the condition under which the back switching can be avoided if we can find its corresponding area of attraction. All solutions of equation (12) starting in this area must end up at this point. It is very easy to understand that if f a ! 0; f c ! 0 at some point (a, c), the back switching will not happen, and if f a  0; f c  0 , definitely the domain will shrink. But the expansion and shrinkage boundary always sits in the range f a ! 0; f c  0 or f a  0; f c ! 0 . That is why it is much more difficult to determine the domain of attractions for a system of differential equations with more than one variable. The condition f a ! 0 and f c ! 0 will give the upper limit of the stable domain, whereas f a  0 and f c  0 will give the lower limit of the size of the domain. Li et al. [4] derived an upper limit of the sizes of stable domain. For a spheroidal 180o domain in an infinite ferroelectric material, Wang and Woo [10] used the numerical approach to compute the phase diagram of (a, c), and determined the region of attraction. In fact, we can establish a Liapunov function V (a, c) a 2  c 2 ! 0, for a ! 0; c ! 0 to determine the domain of attraction around a = 0; c = 0. According to the theorem 4.20 of Ref. [12], the domain of attraction, or the strict stability domain, in which

dV  0 , can be determined by dt

64

the solution of the following equation:

dV dt

wV wa wV wc ˜  ˜ wa wt wc wt

0

(13)

The physical meaning of the domain of attraction is that for any given initial state in this domain, the final state of the system will be a 0; c 0 , which determines the smallest sizes of the ferroelectric domain to avoid the back switching after the external electric field is removed. In the following, we will determine the smallest sizes of the 180o domain in the ferroelectric film covered and not covered by electrodes, respectively. Taking PZT material as an example [1], we assume a coercive field of 60 kV/cm, 2 spontaneous polarization of Ps 30 μC / cm , relative dielectric constant 2

of H r 1000 , and the domain wall energy of V 4 mJ / m . We assume that the initial polarization charges were discharged before introducing the 180o domain in the following calculation. 3.1 A 180o DOMAIN IN A FERROELECTRIC FILM COVERED WITH TOP AND BOTTOM ELECTRODES

Figure 3. Unstable area of 180o domain in ferroelectric thin film covered by electrodes

In such case, the evolution equation of a and c can be established explicitly by substituting equation (4) into equation (12). Then substituting into equation (13) yields the domain of attraction (Figure 3). In the same figure, the areas in

65

which f a ! 0; f c ! 0 , and f a  0; f c  0 are also shown. It can be seen that, in this case, the minimum stable size of the 180o domain is about a | 2 nm, c | 10 nm , which is the same as a 180o domain in the infinite ferroelectrics obtained by the numerical approach [10]. Any solution that starts from this region will eventually end up in the attractor at the origin, namely shrink to zero. Experimentally, the aspect ratio c/a of the stable 180o domains in a bulk ferroelectric body is found to be generally bigger than a certain critical value. For example, Gopalan and Mitchell et al. [5] found that the aspect ratio of dagger shape domains for TGS single crystal is 13.6 r 2.2 . Woo et al [13] found an aspect ratio of 5-14 for the dagger shape domain in PZT thin film. 3.2 A 180o DOMAIN IN FERROELECTRIC FILM INDUCED BY A AFM TIP WITHOUT THE TOP ELECTRODE

Figure 4. Unstable area of 180o domain in ferroelectric thin film without top electrode

In such case, the evolution equation of a and c can be established by substituting equation (10) into equation (12). Then substituting into equation (13) yields the domain of attraction (Figure 4) under such condition. The areas in which f a ! 0; f c ! 0 , and f a  0; f c  0 are also shown. In this case, the minimum stable size of the 180o domain is about a | 10 nm, c | 20 nm . The stable domain should be larger than the one with electrode, since there is no bound charge to tie the polarization charges without the top electrode. One should also

66

bear in mind that a is the radius of the 180o domain, and it is also the minimum stable size of the domain. Up to now, we have not found any experimental result on the critical size of the 180o domain smaller than this size. Using AFM, Paruth et al. [4] have manipulated individual stable ferroelectric domains as small as 40 nm in diameter, and Alex et al. [14] estimated the thickness of the pinned domain layers to be about 15 r 8.9 nm and 68.9 r 7.4 nm at the ferroelectric-electrode interface and the lateral free surface of the PZT thin film, respectively. In fact, these results are not contradicted with our prediction since our results are referred to the minimum stable sizes and dictated by the material constants we are taken. 4. Concluding Remarks

In this paper, a general approach was established to determine the minimum sizes of stable domain to avoid back switching in ferroelectric thin film by stability analysis on the evolution equations. As an example, two cases were considered in detail: one is the 180o domain in a ferroelectric thin film covered by the upper and the lower electrodes; the other is the 180o domain in a ferroelectric film without the top electrode as induced by AFM. It has been found that the minimum sizes of the 180o domain for the second case were larger than that for the first one since there is no bound charge to tie the polarization charges without the top electrode. The predictions on the minimum size of 180o domain in PZT thin films did not violate the present experimental results. We note that our approach is generally applicable to many other fields, including phase transformation, nucleation and expansion of dislocation loops in thin films. Acknowledgment

This project was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region (PolyU 5173/01E, 5309/03E, 5312/03E). Support from the National Natural Science Foundation of China (50232030, 10172030), and the Natural Science Foundation of Heilongjiang Province is also gratefully acknowledged. References 1. 2.

3.

4. 5. 6.

J. Scott, C. Paz de Araujo, 1989, Ferroelectric memories, Science, 246, 1400-1405. Y.J. Song, B.J.Koo, J.K. Lee, C.J. Kim, N.W. Jang, H.H. Kim, D.J. Jung, S.Y. Lee, K. Kim, 2002, Electrical properties of highly reliable plug buffer layer for high-density ferroelectric memory, Appl. Phys. Lett., 80, 2377-2379. R. Luthi, H. Haefke, K.-P. Meyer, E. Meyer, L. Howald, H.-J. Guntherodt, 1993, Surface and domain structures of ferroelectric GASH crystals studied by scanning force microscopy, Surface Science, 285, L498-L502. P. Paruch, T. Tybell, J.-M. Triscone, 2001, Nanoscale control of ferroelectric polarization and domain size in epitaxial Pb(Zr0.2Ti0.8)O3 thin films. Appl. Phys. Lett., 79, 530-532. X. Li, A. Mamchik, I.-W. Chen, 2001, Stability of electrodeless ferroelectric domains near a ferroelectric/dielectric interface. Appl. Phys. Lett., 79, 809-811. V. Gopalan, T.E. Mitchell, 1998, Wall velocities, switching times, and the stabilization mechanism of 180° domains in congruent LiTaO3 crystals. J. Appl. Phys., 83, 941-954.

67 7. 8. 9. 10. 11. 12. 13.

14.

B. Wang, C.H. Woo, 2003, Stability of 180° domain in ferroelectric thin films. J. Appl. Phys., 94, 610617. R. Landauer, 1957, Electrostatic considerations in BaTiO3 domain formation during polarization reversal. J. Appl. Phys., 28, 227-234. B. Wang, Z. Xiao, 2000, On the dynamic growth of a 180° domain in a ferroelectric material. J. Appl. Phys., 88, 1464-1472. B. Wang, C.H. Woo, 2003, Atomic force microscopy-induced electric field in ferroelectric thin films. J. Appl. Phys., 94, 4053-4059. R.E. Loge, Z. Suo, 1996, Nonequilibrium thermodynamics of ferroelectric domain evolution. Acta Mater., 44, 429-3438. L. Gruyitch, J.-P. Richard, P. Borne, J.-C. Gentina, 2004, Stability Domains, Chapman & Hall/CRC, 137. J. Woo, S. Hong, N. Setter, H. Shin, J.U. Jeon, Y.E. Pak, K. No, 2001, Quantitative analysis of the bit size dependence on the pulse width and pulse voltage in ferroelectric memory devices using atomic force microscopy. J. Vac. Sci. Technol. B, 19, 818-824. M. Alex, C. Harnagea, D. Hesse, U. Gosele, 2001, Polarization imprint and size effects in mesoscopic ferroelectric structures. Appl. Phys. Lett., 79, 242-244.

STRESS ANALYSIS IN TWO DIMENSIONAL ELECTROSTRICTIVE MATERIAL UNDER GENERAL LOADING

Zhen-Bang Kuang* , Quan Jiang Department of Engineering Mechanics, Shanghai Jiaotong University Shanghai 200240ˈ P.R.China

Abstract

A simple derivation of the body force produced by the applied electric field is given. The governing equations and boundary conditions on the problem of electrostriction with the correct constitutive equations and considering the pondermotive body force and boundary traction are obtained. Given also are the solution of first boundary problem for an infinite plate with an elliptic defect, and the asymptotic expansion of stress near the end of an elliptic defect. The stresses 1 near the end of the major axis have the form of r , but the principal part of the 1 / 2 electric displacements have the form of r if the origin of the local coordinate system is set at the focus point of the ellipse. 1. Introduction

Electrostriction is a phenomenon that the dielectric crystal deforms under an applied electric field. Contrasting with piezoelectric material, these materials may be isotropic on both elastic and electric behaviors, and the stresses are proportional to the square of the electric field. The nonlinearity plays an important role in the stress analysis. The electrostrictive effect in Pb(Mg1/3Nb2/3)O3 (PMN) and its solid solution with PbTiO3 (PMN-PT) and in some polyurethane elastomers is large and has many applications. The body force in the dielectric material subjected to an electric field has been researched by several authors. Stratton [1] gave a formula for the body force caused by an electric field in the dielectric medium. The body force vector appears as the divergence of a dielectric electrostrictive stress tensor. Some ambiguity retains in his discussion. Landau and Lifshitz [2] discussed the body force produced by an electric field. Pao [3] and Kuang [4] also discussed the body forces due to the electric fields in detail. In those papers, the relations between the electrostrictive stresses in the constitutive equations and the pondermotive force are not expressed *Corresponding author. Tel: 86-021-54743067; fax: 86-021-54743044. E-mail: [email protected]

68 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 68–84. © 2006 Springer. Printed in the Netherlands.

69

in a clear form. Simple general derivation of the body force produced by an electric field is given in this paper. High electric field brings a large strain and it may cause fracture in dielectric apparatus. The stress analysis of two dimensional electrostriction was developed in terms of the complex variable by Knops [5]. Smith and Warren [6] derived solutions of an infinite electrostrictive plane with an elliptical hole and a rigid insert. The same issue was also addressed by McMeeking [7,8]. Shkel and Klingenberg [9] discussed the measurment of the electrostrictive coefficients. The above papers gave much development in this area, but there are still suffered by two drawbacks, namely the constitutive equations can not be derived from a thermodynamic potential and the electric body force is not considered. In this paper, the constitutive equation and the pondermotive body force are given in a clear form and the two dimensional problem with defect are discussed. In the cases discussed, the stress singularity near the end of the major 1 axis of an elliptical defect has the form of r , and the main part of electric 1 / 2 if the local origin is set at the focus point of displacements have the form of r the ellipse. 2. Governing Equations and the Pondermotive Body Force

2.1 GIBBS FREE ENERGY AND CONSTITUTIVE EQUATION The constitutive equations of a dielectric solid can be derived by thermodynamics through prescribing the Gibbs free energy g eij , E i . For an electrostrictive







material without piezoelectric effect, the Gibbs free energy g eij , E i

can

be

written as [2,4]

1 1 C ijkl eij ekl  H~ij Ei E j 2 2 (1) under the isothermal condition, where eeij , Di is the inner energy density, and H~ H  l e g eij , Ei eeij , Di  Di Ei

ij

ij

ijkl kl

. The constitutive equations can be derived from Eqs. (1) and (2) by

V ij Di where V

ij

wg weij

(2)

1 C ijkl ekl  lijkl E k El 2

(3) wg ~ H ij E j H ij  lijkl E j wEi , D i , e ij , E i are the stresses, the electric displacements, the strains



and the electric fields. C ijkl is the stiffness coefficients under the constant electric field and isothermal conditions. The symbols H ik denote the permittivities under

70

the constant stress and isothermal conditions, whilst lijkl is the electrostrictive coefficients under the isothermal condition. 2.2 BODY FORCE PRODUCED BY AN ELECTRIC FIELD The body force can be produced by an electric field in a polarization dielectric medium. The body force f ' i produced by the applied electric field can be derived from the virtual electric Gibbs free energy principle [1,2,4], i.e. giving the virtual displacements Gu i under a constant electric potential on the boundaries, the decrease of the virtual electric Gibbs free energy in the dielectric medium is equal to the increase of the virtual work done by the body force f 'i . Without the loss of generality, it is assumed that the electric field in an infinite dielectric medium is produced by fixed finite electrodes in a finite region. Let virtual displacements Gu i be zero at the electrodes and infinity, one then has

1 1 G ³ Cijkl eij ekl dv  G ³ H~ij Ei E j dv . (4) v v 2 v 2 v ~ It should be noted that H ij and the electric potential M may change due to the virtual displacement Gu i . Because the values of M and H~ij at ri after virtual displacements take the values at ri  Gu i before virtual displacements, one has GM M , i  Gu i and GH~ij H~ij , j  Gu m . Noting Gu i 0 on the boundary, one  ³ f 'i Gui

G ³ g eij , Ei dv

can reduce Eq. (5) to

G ³ gdv

³ [C

v

v

e Gu k ,l  H~ij Ei GE j  Ei E j GH~ij ]dv

ijkl ij

1 1 e  l ijkl Ei E j )Gu k ,l  D j GM , j  Ei E j GH~ij ]dv . 2 2 1 1  ³ [(C ijkl eij  lijkl Ei E j ) ,l  D j , j E k  Ei E j H~ij ,k ]Gu k dv v 2 2 Using Ei E j H~ij ,k Di Ei , k  2 Di E k , j and E i , j E j ,i , we get

³ [(C v

f 'k

ijkl ij

1 § · ¨V kl  Dl E k  Dm E mG kl ¸ 2 © ¹ ,l

(5)

1 1 § · ¨ Cijkl eij  l ijkl Ei E j  Dl E k  E m E mG kl ¸ . 2 2 © ¹ ,l (6)

If f ' k

0 , the body is in an equilibrium state. If the external mechanical body m

force f i also exists, Eq. (6) dictates the following equilibrium condition

V ij , j  f i e  f i m where

0

(7)

71

V ijM, j , V ijM

f ie

D j Ei 

1 Dm E mG ij . 2

(8) e

M

We note that V ij is the Maxwell stress in dielectric medium and f i is called the pondermotive force by Landau and Lifshitz. It is also noted that the total body force f i total in an electrostrictive material induced by pure electric field is

1 · § 1 (9) ¨  l klij Ek El  D j Ei  Em EmG ij ¸ 2 ¹, j © 2 Here part of this, as caused by Geij in the variational equation, is enclosed in the f i total

e

constitutive equation, only the f i part of the total body force is caused by the variation of the particle position in the variational equation, and should be considered independently. For an isotropic dielectric medium, one has

1 a1 G ik G jl  G il G jk  a 2G ij G kl , H ij 2 Oij Okl  G G ik G jl  G il G jk .

lijkl C ijkl

H mG ij , (10)

The constitutive equation for electrostrictive material in the isothermal and isotropic case can be written as

V

ij

Di

2 Ge

 O e kk G

ij

(H m G

ij

 a 1 e ij

1 ( a 1 E i E j  a 2 E k E k G ij ) 2  a 2 e kk G ij ) E j

ij



(11)

where a 1 , a 2 are two independent electrostrictive coefficients of isotropic

H m is the permittivity at the state without strain, G ij is the Kronecker delta, O and G are Lame constants, and O can be expressed by Young’s modulus E and Poisson ratio v as O Ev /[(1  v)(1  2v)] . In this paper, we use Eq. (11)

materials,

as the constitutive equation. It is also noted that if the value of a1 and a 2 are far greater than that of

H m , the term f i e can be neglected.

With the help of Eqs. (7) and (8), the equilibrium equation becomes

V~ij , j where V~ij

V ij  V

M ij

V ij , j  V ijM, j

0

(12)

is called the pseudo total stress. Using Eqs. (4) and last

expression of (8), one may express (12) as

V~ij

a 1 ª º 2Geij  Oekk G ij  «(H m  1 ) E i E j  (H m  a 2 ) E k E k G ij » . (13) 2 2 ¬ ¼

72

2.3 BOUNDARY CONDITIONS Because the electric field also exists in the environment surrounding the body under consideration, it is obviously from Eq. (8) that the boundary condition can be phrased as

V  V n  V V~  V~ n X M ij

ij

or or where X

m i

ij

V~ij n j

en ij

j

en ij

j

m' ij

X , X

m' ij



 V ijMen n j m i

m

Xi

on S V

on S V

X

m ij

(14a) (14b)

 V~ijen n j

on S V en

(14c)

Men

is the given mechanical boundary traction; V ij , V ij

en and V~ij are the

stress, Maxwell stress and pseudo total stress in the environment. Usually if the en 0 . The displacement conditions on the environment is filled by air, we get V in boundary is

ui

ui

on S u

(15)

where u i is the prescribed boundary displacement, S u and S V are the boundary surfaces of given displacements and stresses, respectively. The total boundary consists of S S V * S u .

Two simplified electric boundary conditions will be discussed in this paper. The first one is M M , Ei M , i on S M (16a)

where

V ijMen

M is the electric potential, and S M is the conducting boundary. In this case 0 . The second one is

Dn i.e. S D is an insulating boundary, and S

0

on S D

(16b)

S M  S D is the total boundary. If the Men

permittivity of environment is small, the Maxwell stress V ij

can also be

neglected. 3. Complex Variable Solution of Electrostrictive Stress Problem

It is convenient to solve the two dimensional problem with complex variable. Because the stress V ij is proportional to the square of electric field E i E j and the electric displacement Di is proportional to eij E j , the contribution of the part of

Ei produced by eij to the stress is of the second order of eij . Therefore, when eij

73

H m Ei to solve the static electric problem and neglect

is not large one can use Di

the effect of eij . After Ei is solved, we may proceed to solve the mechanical problem [5,6,8]. This method makes the solving process easy. The complex electric field is [5,6]

Z c z

E 1  iE 2 where Z

z



(17)

is a complex potential of the electric field and Z '  z

dZ / dz .

Z  z is an analytic function of z . The real part of Z  z is the electric potential which is single-valued in any regions. The super-imposed bar indicates the complex conjugate, and this convention will be observed for other functions and variables. The solution of electrostatic problem is reduced to search an analytic function satisfying the boundary conditions. Using the pseudo total stress function U , we put

w 2U ~ , V 22 wx 22

V~

11

w 2U ~ , V 11 wx12

w 2U  . wx1wx 2

(18)

Accordingly, Eq. (5) is automatically satisfied. Using the method of Myskhelishvili [10] and Knops [5], we get the compatibility relation represented by the stress function

’ 4U

w 4U

16

2

wz w z

4N

2

w4 2

wz w z

2

>Z ' z Z ' z @.

(19)

The general solution of U can be expressed as

N

>

@

1 zI z  zI  z  F z  F z 4 2 where I  z , F  z are two arbitrary analytic functions of z . U x1 , x 2

Z z Z z 

(20)

From Eqs. (18) and (20), the pseudo total stresses in electrostrictive material are (21a) V~ 22  V~ 11 NZ c  z Z c  z  2 I '  z  I '  z

>

V~ 22  V~ 11  2 iV~ 12

>

@

NZ ' '  z Z  z  2 zI ' '  z  \ '  z

@

(21b)

where

\ z F ' z , N



1  2v a1  2a 2 . 21  v

(22)

According to Eq.(8), we have

V 22M  V11M

0 , V 22M  V 11M  2iV 12M

H m : '  z

(23)

where

:'  z

>Z ' z @2 .

Substituting Eq. (23) into Eq. (21), the mechanical stresses become

(24)

74

>

2 R1 Z c  z Z c  z  2 I '  z  I '  z

V 22  V 11

>

@

@

(25a)

NZ ' ' z Z z  2 zI ' '  z  \ '  z  2 R2 : ' z

V 22  V 11  2iV 12

(25b)

where

N

R1

2

Hm

, R2

2

.

(26)

For the plane strain case, by using the relation between the strain and the displacement e ij u i , j  u j ,i 2 , the complex displacement is



2G u1  iu 2



3  4v I z  zI ' z  \ z  N Z z Z '  z  2H m  a1 :z . 2

4

(27) In the plane stress problems, we should replace Q , E , a 1 , a 2 as

Q 1 Q

E 1  2Q

,

1  Q

2

1  2Q

, a1 ,

1 Q

a2

(28)

respectively. On S V , the stress boundary condition is

N

m'

m'

zI' (z)  I(z) \ (z)  Z(z)Z' (z) i³ ( X 1  i X 2 )ds . 2

(29)

If there are no free charge and resultant force in the material and no surface charge and traction on the boundary, the complex potential for an infinite region can be expressed by [5,10] f

Z z *3 z  ¦ J n z n

(30)

n 1

I z \ z

f

*1 z 

¦

bn z n

(31a)

z n

(31b)

n 1 f

*2 z 

¦c

n

n 1

where J n , bn , c n , n 1,2. . . are complex constants. *3 , * 1 , * 2 are determined by the electric field and tractions at infinity. Equations (17) and (21) yield

 E f , E f E1f  iE 2f (32a) 1 ~f ~f *1 V 22  V 11  NE kf E kf (32b) 4 1 ~f ~f *2  V 22  V 11  2iV~12f (32c) 2 f f where E i and V~ij ( i, j 1, 2 ) are the electric fields and the pseudo stresses at *3



the infinity.



75

4. Solution in an Infinite Plane with an Elliptical Hole

Attention is now focused on a two dimensional infinite electrostrictive plane with permittivity H m containing an elliptical hole, as shown in Figure 1. The major and minor axes of the ellipse are 2a and 2b respectively. The remote electric fields are E

f

stresses are

E  E f 2 1

E 0f e iE ( E0f f 11

f 12

V ,V ,V

f 2 2

, tan E

E 2 / E1 ) and the remote

f 22 .

The solution of problem can be explored through the conformal transformation. The function mapping the region exterior to the ellipse onto the region exterior to a unit circle is

z where R

g ]

z  z 2  4mR 2 2R

R]  m] 1 , ]

(33)

a  b / 2 , m a  b / a  b . x2

r

b

T a

x1

U

c

Figure 1. Ellipse hole

According to Myskhelishvili [10], in

] -plane Eqs. (30) and (31) are reduced to

Z ] *3 R]  Z 0 ] ,

Z 0 ]

f

¦J '

n

] n

(34)

n 1

I ] \ ]



*1 R ]  I 0 ] , I 0  z



*2 R ]  \

0

] ,

\

0

z

f

¦

b 'n ]

n

n 1 f

¦

c 'n ]

n

(35a) (35b)

n 1

where J ' n , b' n and c' n are new complex constants. According to Smith and Warren [6], Eq. (27) can be written as

Z ]  R E f ]  D ] 1

(36)

76

where

D E f E f e 2iE for the insulating defect. Due to V ijMen

conducting boundary and V Eq. (29) in

Men ij

0 on the

| 0 on the insulating boundary in the present case,

] -plane becomes

m m I ' (] ) N Z ' (] )  I (] )  \ (] )  Z (] ) i ³ ( X 1  i X 2 )ds . (37) 2 g ' ] g ' ] Let ] V on the unit circle in ] -plane, one has V 1 / V . When the inner

g ]

boundary is traction free, Eq. (37) is reduced to

g V

I ' (V )  I (V )  \ (V )  N Z (V ) Z ' (V ) 2 g ' V g ' V

0 .

(38)

Substituting Eqs. (34), (36) into Eq. (38), one has

g V

I0 ' (V )  I0 (V )  \ 0 (V )  f g ' V

0,



(39)





R*2 N R E f E f 1D V 2 D V 2 f R*1  R*1V   . (40) V 2 V 1 mV 2 V 1 mV 2 After multiplying Eq. (39) and its conjugate by 1 /[ 2Si (V  ] )] , and using Cauchy

V 2 m









integral formulae, we get

1 fd V , ³ 2Si V  ]

(41a)

f dV 1  m] 2 1 ]  I 0 ' ] . 2Si ³ V  ] ] 2 m

(41b)

I 0 ] \ 0 ]

Substituting Eq. (40) into Eq. (41), we have

\ 0 ]

R*1

I 0 ] 

mR*1



R*2 1 m] 2



2 1  m2 ]

] 2 m



]

]

R* 2



]



2





m]

N D R Ef Ef , 2]

 NR E

f

>

(42a)

@



E f 1DD  D D m ]

] 2 m

2

. (42b)

According to Eq. (36), the functions I ] and \ ] are given by

I ] R*1]  \ ] R*2]  R*1



mR*1

]



2 1 m2 ]

] 2 m



R* 2



]





R*2 1 m] 2

]

2



m]

N D R Ef Ef , 2]

 NRE

f

2

>

(43a)



@

E f 1DD  D D m ]

] 2 m

.

(43b)

77

In

] -plane, Eq. (27) is reduced to

V 22  V 11

2 R1

Z c] Z c] ª I ' ] I ' ] º  2«  » g ' ] g ' ] ¬ g ' ] g ' ] ¼

(44a)

' 2 ª§ I ' ] · ' g ] \ ' ] º ªZ ' ] º § Z ' ] · Z ] ¸¸ ¸¸ V 22  V 11  2iV 12 N ¨¨   2«¨¨ »  2 R2 « » «¬© g ' ] ¹ g ' ] g ' ] »¼ ¬ g ' ] ¼ © g ' ] ¹ g ' ] (44b) and Eq. (17) is reduced to

Z ' ] / g ' ] .

E1  iE 2

(44c)

With the help of Eqs. (33), (34) and (43), the stress fields are

ªDțE f E f  2*1 ȗ 2  m  2* 2 º ] 2 D ] D  2 R1 E E 2 Re « » ȗ2 m ] 2 m ] 2 m ¬« ¼» 2

V 22  V 11

f

f

2

2 NEf Ef D  m ] 3 ] D

V 22 V11  2iV12

] 



2

] m

>

*2 m]

]





4



ª  D NEf Ef  2*2  4m* ] 3 ]  m 1 2 « 3 2 ] « ] m ¬



>1DD  D D m@NE E ] ] f

*2] 2 2



m

3

f 2

2





m

  m  3 ]  m@º 2R E ] 1 » ]  m ]  m »¼ 2

2 ] m

2

(45a) 2

3

f 2

2

2





 ]  m



2*1 1 m2 ] 2 ] 2  m 3

2

2

2

3

2

2

2

(45b) and the electric field is

E1  iE 2





Ef ] 2 D 2

] m

.

(45c) f

From Eqs. (43) and (45), one finds that the effects of V ij and E

f

on I ] ,

\ ] and V ij are mutually independent, or there is no coupling effect between V ijf and E f . Consequently, the V ij produced by V ijf and E f can be f

f

superimposed. On the other hand, the effects of E1 and E 2 are coupled. For the special case where the mechanical traction vanishes at infinity, from Eq. (32) one gets

*1



N 4

E f E f , *2

0.

(46)

It is a straightforward matter to write down the stress field in z -plane with Eqs. (45)

78

and (46). Of particular interest is an explicit expression of V 22 , though it is rather lengthy to write it down here. 5. Asymptotic Stress Feld near the End of an Elliptical Hole

Using the local coordinates z1 and ] 1

z 0  z1 , ]

z and

]

] 0 corresponds to z

(47)

R ] 0  m] 01 . ] 0 is a branching point of

z0

g ' ] 0 0 . From Eq. (33a) we get r ] 0

g ] or

r m

and

r a 2  b 2 , i.e. z 0 is the right focus of the ellipse. Let ] is a

r z0

neighboring point of

z1

] 0  ]1

]0 (

U ei - and

m ), with the local polar coordinates ] 1

r ei K , one yields

] ] where

m  U ei -

m ]1 m ] 1

mUe

(48a)

i -

(48b)

U is a small distance from ] to ] 0 .

Substituting Eq. (48) into Eq. (45), one arrives at the following asymptotic expressions

V 22  V 11





R1 m D m D E f E f 2mU



m D e

 u e

ue

2i2i-

i-

e e

2i-

ı22 ı11 2iı12

Ef Ef 4m

3/ 2

^e U

i-

> 



 e i- R1 3m2 DD  2Nm2

i-

i-

1

i-

Ef Ef

1

2i-

2

2i-

9

f 3i-

țE E e

2

1m mD mD

@

4R2 E fe2i-mD 



Efe5i- mD

@

3mD 



1

@ ` O U

4 mȡ3

f 4i-

(49a)

1

1

4 R2 E e

@

2N  3R  mR D e D e ` 8m ^R DD 1  mD  D >e  e  3 R  2N @ m > 4N  R

D e

2i-

f

2



8 m3/ 2ȡ Efe6i2

16m

>N E @

E f e4i-mD 8m ȡ2

>N E mD 13m f

>N E mD 23e f

f

2i-

me4iT



mD mD 46e2i- e6i-

2R2 E f e6i- 11m2 2 mD 3D2 OU

(49b)

79

and



E f e i- m  D

E1  iE 2

 E 3m  D . f

(49c)

4m 2 mU The Taylor expansion of z at z 0 (] 0 ) in Eq. (33) is 1 2m 1 6m 2 3 4 z R]0  m]01  R1 m]02 ] ]0  R 3 ] ]0  R 4 ] ]0  O ] ]0 2 ! ]0 3! ]0

>

Ignoring the terms of orders higher than

]

3 1

@

(50a) , at the small region near

] 0 ( z1 o 0, ] 1 o 0 ), we get z1

§ m R¨¨1  2 © ]0

· m m ¸] 1  R 3 ] 12  R 4 ] 13 ¸ ]0 ]0 ¹

R m

] 12 

R 3 ]1 . m

(50b)

Writing in polar coordinates, we have

R

r ei K

It is obviously that K of higher order than

U 2 e 2i - 

R

U 3 e 3i - .

(51)

m m 0 when - 0 . Since U is small, one may ignor the terms

U 2 to get U

m

1 4

r /R .

(52)

Denote 2c as the focal length and U 0 as the radius of curvature at the elliptic end, we have

U 0  c 2  U 0 2 , U 0 b 2 2a . (53) When b is small, it is seen from Eq. (53) that U 0 is small value of higher order c2  b2

a

compared with b . With the help of Eqs. (49), (52) and (53), the asymptotic expression of stresses along ox1 ( K 0 , - 0 ) in z -plane are

V 22  V11

>



@



R1Ef Ef 1  D 1 D c ­° 2Ef Ef  3  D  D  DD R1   2  D  D N ® r ° 4 4 ¯ 2R1Ef Ef 2DD D D U0 ½° c ­° Ef Ef 3DD  D  D 11 R1  2 D  D  2 N  ¾ ® r ° r ° 2 8 ¿ ¯ U0 U0 ½°  Ef Ef  2DD D D R1  D  D N  Ef Ef 4DD D D R1 ¾  Or r r °¿ (54a)



>

>





@









@

80

>

 @



­° Ef 1D 2R Ef 1D NEf 1D NEf Ef 1D 1D 2  8 4 °¯

V22 V11  2iV12 ®

>



­° 2 Ef  8R EfD1D NEf  3 D D DD 2 ® 8 °¯

>

@

 @

U0 ½° c ¾ r ¿° r

U0 r





2 Ef 1D 2R2 Ef 3 D NEf 1D 2NEf Ef 1D D  3DD U0 °½ c  ¾ 8 4 r °¿ r



>

@



­° Ef 8R EfD1 2D  NEf  3 D D DD U 2 0 ® 4 r °¯

>



@



Ef 8R2 EfD1D  NEf 1D D  3DD 4



E 2R2 E 11 2D  3D  NE 1D 1D 16 f

>

f



2



U0 r

 @½°¾  Or

f

°¿

(54b) and



2E f 1  D

E 1  iE 2



§3D  Ef¨ D ¨ 4 r ©

c

4

1 (or E f

In the case for D

U 0 ·¸ r ¸¹

 O r .

(54c)

iE 2f ), we can get the stress V 22 and E2 along

ox1 from the asymptotic expansion of Eq. (54) as



ª 2 R  2 R  N E f 2 1 « 4 « ¬

V 22



2



2 4 R1  4 R2  N E

f 2

4

E1  iE 2

N E f

2



U0 º c



ª 2 2 R  2 R  3N E f 2 1 » « r »r « 2 4 ¼ ¬ f 2 2N E U0 U0 º »  O1  r 2 r » ¼

2

(55a)



2E f

c r

2

 O 1

The expression of V 22 obtained by McMeeking [8] has a

(55b)

1 / r singularity. That

conclusion is derived under the condition of r !! b / 2 . Applying the similar condition to the present solution, one has

c ª U0 º 1  »  O1 (56) 2r «¬ r ¼ 0 . This expression more or less by expanding Eq. (23) of McMeeking [8] for H f

Vy

recovers Eq. (26) of [8].



4S E y0

2

81

In the other case for D

1( E f

E1f ), the stress is

2R1  2R2  N E f 2 §¨

V 22

2 § E f ¨1  ¨ ©

E1  iE 2

U 0 ·¸

2

1  Or ¨ ¸ r © ¹ U 0 ·¸  O r r ¸¹

(55c)

(55d)

and in this case V 22 has no stress concentration. 6. Numerical Examples

A series of numerical examples of PMN material are given below with the constants taken to be

a 10 3 m, b

Q Hm

5 u 10 3 a, E (elastic modulas)

0.26, E f

5 u 10 4 e i E Vm 1

6.64 u 10 8 Fm 1 , a1

Figure 2. Stress

S / 4 ( E1f

2.704 u 10 5 Fm 1 , a 2

V 22

along

ox1

at the end of ellipse ( E

4.899 u 10 6 Fm 1

S /2)

S / 2 (only E 2f is applied) , D i or E 2f is applied) and D 1 or E 0 (only E1f is applied) will

Three examples of of D

E

281GPa,

1 or E

82

be discussed. The distributions of stress component V 22 ahead of the ellipse for three cases are shown in Figure 2, Figure 3 and Figure 4 respectively.

Figure 3. Stress

V 22 along ox1 at the end of ellipse ( E

Figure 4. The stress

V 22

along

ox1 at the end of ellipse ( E

S /4)

0)

83

The stress V 22 concentrates significantly at the end of the elliptic defect for

D 1 (Figure 2). On the contrary, the stress is very small for D 1 (Figure 4). These figures tell us that the stresses near the end of a hole have singular behaviors f f when E 2 exists and do not have singular behavior when E 2 is absent. These phenomena are obviously observed from the asymptotic expansions of Eqs. (54a) and (54b). Figure 5 gives the variations of V 22 with E at the points of x1 / a 1 , x1 / a 1  U 0 / 2a , x1 / a 1  U 0 / a and x1 / a 1  20 U 0 / a .

Figure 5. The variation of

7.

V 22

with respect to

E

at some special points

Discussions

This paper derives an accurate set of governing equations for the stress analysis in an electrostrictive problem, along with some numerical examples. The solution method recommended in this paper requires that the strain is not too large everywhere. When the minor axis b o 0 (in this case the ellipse becomes crack) f

or E is large enough, the electric fields near the end of an elliptic hole will be saturate and solutions obtained above can not be used for the region very near the end of the defect. But the solution still gives the correct stress fields in the region some distance away from the end of the defect, as in the small scale yielding case for elastoplastic crack problem. If the strain is high in a rather large region, we should solve nonlinear equation

84

system that consists of the equilibrium equations and Gauss equation, a challenge that is not discussed here. Acknowledgement

This work is supported by the National Natural Scientific Foundation of China under Grant No.10132010. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

J.A. Stratton, 1941, Electromagnetic Theory, McGraw-Hill. L.D. Landau, E.M. Lifshitz, 1960, Electodynamics of Continous Media.Pergamon Press, Oxford. Y.H. Pao, 1978, Electromagnetic force in deformable continua, S. Nemat-Nasser, Mechanics Today, Printed in Great Britain by Pitman.Bath. Z.-B. Kuang, 2002, Nonlinear Mechanics of Continuous Media Shanghai Jiaotong University Press. R.J. Knops, 1963, Two-dimensional electrostriction, Qt. J. Mech. Appl. Math. 16, 377-388. T.E. Smith, W.E. Warren, 1966, 1968, Some problems in two-dimensional electrostriction, J. Math. Phys. 45, 45-51; (corrigenda) 47, 109-110. R.M. McMeeking, 1987, On mechanical stresses at cracks in dielectrics with application to dielectric breakdown, Journal of Applied Physics, 62, 3116-3122. R.M. McMeeking, 1989, Electrostrictive stresses near crack-like flaws, Journal of Applied Mathematics and Physics (ZAMP), 40, 615-627. Y.M. Shkel, D.J. Klingenberg, 1990, Material parameters for electrostriction, J. Appl.Phys., 80, 45664571. N.I. Myskhelishvili, 1954, Some Basic Problems in the Mathematical Theory of Elasticity, Nordhoff, Netherlads. Z.-B. Kuang, F.-S. Ma, 2002, Crack Tip Fields, Xi’an Jiaotong University Publishing House.

Piezoelectrics

EFFECT OF ELECTRIC FIELDS ON FRACTURE OF FUNCTIONALLY GRADED PIEZOELECTRIC MATERIALS

Naotake Noda, Cun-Fa Gao Department of Mechanical Engineering, Shizuoka University, 3-5-1, Johoku, Hamamatsu, Shizuoka 432-8561, JAPAN

Abstract

In this paper we study the effect of electric fields on fracture of functionally graded piezoelectric materials (FGPMs) via a perturbation-based complex variable method. To illustrate the new mathematical algorithm for the fracture analysis of FGPMs, we start with an anti-plane deformation of a cracked piezoelectric body with exponentially varying elastic properties in the direction parallel to the crack plane. First, we establish a perturbation-based complex variable method, which allows us to extend the available solutions for a homogeneous body to those for a nonhomogeneous body. Using the newly established method, we derive explicit expressions of field intensity factors for an impermeable crack and a permeable crack, respectively. Finally, we discuss the effect of electric fields on the fracture of FGPMs by applying the field intensity factors as a failure criterion. It is shown that the effect of electric fields on crack propagation in the FGPMs is qualitatively the same as that in a homogeneous piezoelectric material, i.e., the gradual variation of material does not change the propagation tendencies of cracks under an electric field. 1. Introduction

Piezoelectric materials have found wide application in a variety of electronic and mechatronic devices due to their pronounced piezoelectric, dielectric and pyroelectric properties. Compared with traditional homogeneous piezoelectric materials, functionally graded piezoelectric materials (FGPMs) with gradual variation in properties have become preferred materials since they can reduce residual and thermal stresses, and improve bonding strength and toughness in the coating-substrate smart systems. Thus, fracture studies of functionally graded materials (FGMs) have received considerable interest in recent two decades, and a wealth of research results have been accumulated, e.g., one can cite the works in [114]. Recently, some of these works have been extended to the cases of the FGPMs [15-18]. Mathematically, however, the introduction of gradually changed material properties raises the difficulties of theoretical analyses of FGPMs, so that most of 87 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 87–95. © 2006 Springer. Printed in the Netherlands.

88

previous studies were based on Fourier transform technology or numerical approaches. Although a series of important results have been obtained by using these methods, it is difficult to give explicit and closed-form expressions of field variables. It is also not easy to extend the available results to the cases of FGMs or FGPMs with holes or inclusions by means of the available calculation algorithms. This may be the reason that almost all previous analyses were restricted to the cases of straight geometry boundaries. In the present work we illustrate a perturbation-based complex variable method for FGPMs, and then use it to study the effect of applied electric fields on the fracture of FGPMs with an impermeable or permeable crack. To illustrate the new mathematical algorithm for the fracture analysis of FGPMs, we start with an antiplane deformation of a cracked piezoelectric body with exponentially varying elastic properties in the direction parallel to the crack line. In theory, the newly established approaches can be extended to the general 2D deformation of FGMs or FGPMs with cracks or holes. This makes it possible to extend all known results for the cases of homogeneous materials to those for the cases of FGMs or FGPMs under the same loading and geometry conditions. 2. Basic Equations

The anti-plane solution is not only relatively simple, but also provides clear physical insights into the considered problem. Hence, we start with an anti-plane deformation of a FGPM, such that u x u y 0 , u z u z x, y , M M x, y , (1) where u x , u y and u z are the components of displacement, and M stands for the electric potential.

y

D yf

V zyf V zxf D xf

Material Gradient

x

2a

Figure 1. A mode III crack in a FGPM

The material properties are assumed to vary in an exponential manner in the

89

direction parallel to the crack plane, as shown in Figure 1: c 44 , e15 , N 11 c 440 , e150 , N 110 e Ex ,

(2)

0 11

where c , e and N are the shear modulus, the piezoelectric constant and the dielectric constants of a homogeneous material, respectively. E is a small positive number. 0 44

0 15

In this case, the constitutive equations are wu wu wM wM V zx c 44 x z  e15 x , V zy c 44 x z  e15 x , (3) wx wx wy wy wu wu wM wM , D y e15 x z  N 11 x D x e15 x z  N 11 x , (4) wy wy wx wx where V zx , V zy , D x and D y represent the components of stress and electric displacement, respectively. Equations of generalized equilibrium are wV zx wV zy wD x wD y  0, 0.  wx wy wy wx Substituting (2) into (3) and (4) leads to wu z § wM · Ex wM · Ex § wu z ¸¸e , V zx ¨ c 440  e150  e150 ¸e , V zy ¨¨ c 440 Dx

wx ¹

wx

©

©

wy

wy ¹

§ 0 wu z wM · Ex ¸e . ¨¨ e15  N 110 y wy ¸¹ w ©

wM · Ex § 0 wu z  N 110 ¨ e15 ¸e , D y x w wx ¹ ©

(5)

(6)

(7)

Then, the substitution of (6) and (7) into (5) results in wu · wM · § § c 440 ¨ ’ 2 u z  E z ¸  e150 ¨ ’ 2M  E ¸ 0, wx ¹ wx ¹ © © wu · wM · § § e150 ¨ ’ 2 u z  E z ¸  N 110 ¨ ’ 2M  E ¸ 0, wx ¹ wx ¹ © ©

where ’ 2

(8) 

(9)

w 2 / wx 2  w 2 / wy 2 .

Let u

>u ,M @

T

z

, B0

ªc 440 « 0 ¬ e15

e150 º »,  N 110 ¼

(10)

one may rewrite (8) and (9) as w 2 uˆ w 2 uˆ w uˆ  2 E 2 wx wy wx

0,

where uˆ B 0 u . For the later use we introduce the expressions of generalized stresses as

(11)

90

V

ı2

, Dy , ı1

V

T

zy

, Dx . T

zx

From (6) and (7) we have ı2

e Ex B 0

wu wy

e Ex

w uˆ , ı1 wy

e Ex B 0

wu wx

w uˆ . wx

e Ex

(12)

3. Perturbation-based Complex Variable Method

In general, it is difficult to give the exact solution of (11). Thus, we introduce a perturbation method to obtain an approximate solution of (11). To do this, we express uˆ in the following form: f

uˆ 0  ¦ E n uˆ n ,

uˆ

(13)

n 1

where uˆ 0 is the solution for the homogeneous material, and it can be considered as known. Substituting (13) into (11), one arrives at w 2 uˆ 0 w 2 uˆ 0 w 2 uˆ n w 2 uˆ n w uˆ n 1  0 ,   2 2 2 2 wx

wy

wx

wy

0 , n t 1 .

wx

(14)

Omitting some mathematical details, one can show that the general solution of (14) can be expressed as

uˆ n

­ ° g0  z  g0  z , for n 0 ° 1 ° ®g1 z, z  g1 z, z , g1 z, z f1  z  z g 0  z , for n 1 4 ° 1 ° °¯ g n z, z  g n z, z , g n z, z f n  z  4 ( I z  I z ) gn1 ( z, z ), for n t 2











(15)



 ³ g z, z dz .

where I z [g n 1 z , z ]

n 1

According to the given boundary conditions, u n ( n 0,1,2,... ) can be determined. Then, the generalized stress fields can be obtained, by inserting (13) into (12), as ı2

ª e Ex «ıˆ 20  ¬

f

¦E n 1

n

º ıˆ 2n » , ı 1 ¼

ª e Ex «ıˆ 10  ¬

f

¦E n 1

n

º ıˆ 1n » , ¼

(16)

where ıˆ 2n

w uˆ n , ıˆ 1n wy

w uˆ n , n wx

0,1,2,3... .



(17)

For n

0 , using (15) and (17) results in º ª d ıˆ 20 2 Re «i g 0 z » , ıˆ 10 ¼ ¬ dz For n t 1 , note:

º ªd 2 Re « g 0 z » . ¬ dz ¼



(18)

91

ıˆ 2n ıˆ 1n

w uˆ n wy w uˆ n wx





w uˆ n w uˆ n w w w w  ) i[ g n z , z  g n z , z  g n z, z  g n z, z ] , wz wz wz wz wz wz w uˆ w uˆ n w w w w ( n  ) [ g n z, z  g n z, z  g n z, z  g n z, z ] . wz wz wz wz wz wz i(



Then, one has from (17) and (15) that ıˆ 2n

ıˆ 1 n

^ ^











 `  `

w g n z, z  g n z, z ] , n t 1 , wz w 2 Re[ g n z , z  g n z , z ] , n t 1 . wz 2 Re[i

(19)

(20)

The intensity factor of the generalized stresses at the right crack tip x1 expressed as T K k V , k D lim 2S x  a ı 2 x . xoa

a can be

(21)

The substitution of (16.1) into (21) gives f § · K e Ea ¨ K 0  ¦ E n K n ¸ , (22) n 1 © ¹ where K 0 is the intensity factor for a homogeneous material, and K n is defined by Kn

lim 2S x  a ıˆ 2n x , n xoa

1,2,... .

 

(23)

4. Applications

4.1. FIELD INTENSITY FACTORS OF AN IMPERMEABLE CRACK For an impermeable crack located in > a, a @ along the x axis, as shown in Figure 1, it is assumed that the material is loaded only at infinity by the uniform mechanical-electric loading. In this case, the generalized stress boundary condition is ı 2 0 , a d x d a .  (24) For the case of a homogeneous material, (24) reads ıˆ 20 0 ,  a d x d a .  (25) In this case, it can be shown from (25) and (18.1) that the corresponding complex potential is [19] z i g c0 z ı f20 ,           (26) 2 2 2 z a where g c0 z d g 0 z / dz , and ı f20 From (26) we obtain i g 0 z

V

f zy

, D yf is the applied loading at infinity. T

1 2 z  a 2 ı f20 ,     2

  

 

(27)

where a constant term without any effect on the stresses and the electric fields is neglected.

92

Furthermore, the field intensity factor can be obtained as K0 Sa ı f20 .   For n 1 , (24) and (19) read ıˆ 21 0 ,  a d x d a ,  ıˆ 21

Using (15) we have

^

2 Re[i

 `

w g1 z, z  g1 z, z wz

^

(28) (29)

 `

w g1 z, z  g1 z, z ] .   wz

f 1cz 

(30)

1 1 g 0 z  z g c0 z . 4 4

(31)

Inserting (31) into (30) leads to 1 1 ª º 2 Re «i f1cz  i g 0 z  iz g c0 z » . 4 4 ¬ ¼

ıˆ 21



 

(32)

Inserting (26) and (27) into (32), and then using (29), one obtains 2 Re>i f1cx @ 0 ,  a d x d a . We have from (33) that [20] i f1cz 0 , and then f1 z 0 , where a constant without any effect on the stress fields is neglected.

(33) (34)

Inserting (26), (27) and (34) into (15) and (32), we obtain



1 1 z z 2  a 2 ı f20 , 4 2i º ª1 z 1 2 z  a 2 » ı f20 .  Re « z  2 2 4 4 z a ¼» ¬«

g1 z, z ıˆ 21



(35) 

(36)

Furthermore, inserting (36) into (23) produces 1  a Sa ı f20 . 4

K1

 

(37)

Similarly, for n t 2 , it can be shown that

 1

n

Kn

1 1 n a Sa ı f20 . 4 n n!

(38)

Inserting (28), (37) and (38) into (22) we obtain the final expression of the field intensity factor as § K a e Ea ¨1  ¨ ©

f

¦  1

n

n 1

1 § Ea · ¨ ¸ n! © 4 ¹

n

· ¸ Sa ı f20 . ¸ ¹



(39)

Note 1 § Ea · 1  ¦  1 ¨ ¸ n !© 4 ¹ n 1 f

n

n

e  Ea / 4 .

Then, (39) becomes K a e 3 Ea / 4 Sa ı f20 .

Similarly, at the left tip of the crack, the field intensity factor is

(40)

93

K  a e 3 Ea / 4 Sa ı f20 .

(41)

4.2. FIELD INTENSITY FACTORS OF A PERMEABLE CRACK For the case of a permeable crack in a FGPM, the boundary conditions on the crack faces can be expressed as  (42) V zy 0 ,  a d x d a , D y

D y , E x

E x ,  a d x d a . 

 

(43) In this case, the solution for a homogeneous material can be also given. We directly cite the results in [18] as z 1 i g c0 z 6 f20  i 2 D20 ,       (44) 2 2 2 2 z a where 6 f20

>V

f zy

@

T

, D yf  D y0 , i 2

(0,1) T , D y0

D yf 

e150 f V zy . c 440

The corresponding field intensity factor is kV

SaV zyf , k D

e150 kV . c 440

(45)

Parallel to the previous procedure, one can give the solution for the permeable case. Omitting some mathematical details we directly write out the final results of the stress intensity factor for the right crack tip as k V a e 3 Ea / 4 SaV zyf . (46) It can be shown that (45.2) always holds. Thus, the final intensity factor of electric displacement is k D a

e150 3 Ea / 4 e SaV zyf . 0 c 44

   

(47)

For the left crack tip, the corresponding results are e150 3 Ea / 4 e SaV zyf .    (48) c 440 It can be found from (40) and (41) that for an impermeable crack, the stress field factors in a cracked FGPM are only related to the applied mechanical loading, while the intensity factor of electric displacement is only related to the applied electric loading. If the field intensity factor is used as a failure criterion, one can conclude that the positive electric field may enhance the crack propagation, while the negative electric field may impede the crack propagation. That is, the effect of electric fields on crack propagation in the FGPMs is qualitatively the same as that in a homogeneous piezoelectric material. For a permeable crack, it is shown from (46)-(48) that both the singularities of stress and electric displacement are independent of the applied electric loading, i.e., the electric field has no influence on the crack propagation. This conclusion is also the same as that for a homogeneous piezoelectric material. Thus, the gradual variation of material kV  a e 3 Ea / 4 SaV zyf , k D  a

94

property only changes the magnitude of field singularities, but not the tendencies of crack propagation under electric fields. 5. Concluding Remarks

We propose an analytic method for fracture problems of functionally graded piezoelectric materials with a mode III crack. In the method a homogeneous piezoelectric material is chosen as the reference so that the solution for the nonhomogeneous piezoelectric material is treated as being perturbed from the reference solution. Based on the method, the complex potentials for the non-homogeneous piezoelectric material are derived from the corresponding solution for a homogeneous piezoelectric material, and explicit expressions of field intensity factors are presented for the cases of an impermeable crack and a permeable crack, respectively. Finally, we discuss the effect of electric fields on the fracture of FGPMs by applying the field intensity factors as a failure criterion. It is found that the gradual variation on the material property does not change the propagation tendencies of cracks under electric fields, that is, the effect of electric fields on crack propagation in the FGPMs is qualitatively the same as that in a homogeneous piezoelectric material. Acknowledgments: The authors would like to express their gratitude for the support of the Japan Society for the Promotion of Science (JSPS). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11.

12.

F. Delale, F. Erdogan, 1983, The crack problem for a nonhomogeneous plane, Journal of Applied Mechanics, 50, 609–614. F. Erdogan, 1985, The crack problem for bonded nonhomogeneous materials under antiplane shear loading, ASME Journal of Applied Mechanics 52, 823–828. H.vGao, 1991, Fracture analysis of non-homogeneous materials via a moduli perturbation approach, International Journal of Solids and Structures, 27, 1663-1682. Y. Ootao, Y. Tanigawa, 2004, Transient thermoelastic problem of functionally graded thick strip due to nonuniform heat supply, Composite Structures, 63,139-146. N. Noda, Z.H. Jin, 1993, Steady thermal stresses in an infinite nonhomogeneous elastic solid containing a crack, Journal of Thermal Stresses, 16, 181–196. Z.H. Jin, N. Noda, 1994, Crack tip singular fields in nonhomogeneous materials, ASME Journal of Applied Mechanics, 61, 738–740. N. Noda, M. Ishihara, N. Yamamoto, T. Fujimoto, 2003, Two-cracks propagation problem in a functionally graded material plate under thermal loads, Materials Science Forum, 423-425, 607-612. W. Yang, C.F. Shih, 1994, Fracture along an interlayer, International Journal of Solids and Structures, 31, 985–1002. H.A. Bahr, H. Balke, Fett, T. et al., 2003, Cracks in functionally graded materials, Materials Science and Engineering, A362, 2-16. J.H. Kim, H.P. Glaucio, 2003, The interaction integral for fracture of orthotropic functionally graded materials: evaluation of stress intensity factors, International Journal of Solids and Structures, 40, 39674001. S. El-Borgi, F. Erdogan, L. Hidri, 2004, A partially insulated embedded crack in an infinite functionally graded medium under thermo-mechanical loading, International Journal of Engineering Science, 42, 371-393. H.P. Glaucio, J.H. Kim, 2004, A new approach to compute T-stress in functionally graded materials by means of the interaction integral method, Engineering Fracture Mechanics, 71, 1907-1950.

95 13. 14. 15. 16. 17. 18.

19. 20.

S.V. Senthil, R.C. Batra, 2004, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, Journal of Sound and Vibration, 272, 703-730. Z.H. Jin, R.H.Jr. Dodds, 2004, Crack growth resistance behavior of a functionally graded material: computational studies, Engineering Fracture Mechanics, 71, 1651-1672. C. Li, G.J. Weng, 2002, Antiplane crack problem in functionally graded piezoelectric materials, ASME Journal of Applied Mechanics, 69, 481-488. S. Ueda, Y. Shindo, 2002, Crack kingking in functionally graded materials due to an initial strain resulting from stress relation, Journal of Thermal Stress, 23, 285-290. B.L. Wang, Y.W. Mai, 2004, Surface fracture of a semi-infinite piezoelectric medium under transient thermal loading (poling axis parallel to the edge of the medium), Mechanics of Materials, 36, 215-223. W.Q. Chen, Z.G. Bian, C.F. Lv, H.J. Ding, 2004, 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid, International Journal of Solids and Structures, 41, 947-964. T.Y. Zhang, C.F. Gao, 2004, Fracture behaviors of piezoelectric materials, Theor. Appl. Fract. Mech., 41, 339-379. N.I. Muskhelishvili, 1975, Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen.

THE CHARGE-FREE ZONE MODEL FOR CONDUCTIVE CRACKS IN DIELECTRIC AND PIEZOELECTRIC CERAMICS

Tong-Yi Zhang Department of Mechanical Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, China

Abstract

In the charge-free zone (CFZ) model, dielectric and piezoelectric ceramics are treated to be mechanically brittle and electrically ductile. For an electrically conductive crack under electrical and/or mechanical loading, various charge emission mechanisms may function jointly at the tip due to the high electric field concentration. Charge emission and charge trapping consume more work and thus lead to a high value of the electric toughness. The failure criterions derived from the CFZ model were verified by experimental results on poled and thermally depoled lead zirconate titanate ceramics. 1. Introduction

Internal electrodes have widely been adopted in electronic and electromechanical devices made of dielectric and piezoelectric ceramics. These embedded electrodes may naturally function as pre-conductive cracks or notches, which may lead to the failure of such devices under electric and/or mechanical loads. When a conductive crack is loaded by an electrical field parallel to the crack, electric charges in the conductive crack surfaces must rearrange themselves to produce an induced field that has the same magnitude as the applied one but with the opposite sign such that the electric field inside the conductive crack remains zero. As a result, the charges in the upper and lower crack surfaces near the crack tip have the same sign. The charges with the same sign repel each other and then have a tendency to propagate the crack. The contour-independent J-integral used in fracture mechanics can also be applied to conductive cracks [1-3] and the J-integral result for a conductive crack under purely electrical loading is very similar to that for a conventional crack under purely mechanical loading, thereby indicating that the concepts of fracture mechanics can be utilized in the study of the failure behavior of conductive cracks in dielectric and piezoelectric ceramics. Recently, Wang and Zhang [4], Fu et al. [5] and Zhang et al. [6-8] investigated the 96 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 96–106. © 2006 Springer. Printed in the Netherlands.

97

failure behavior and the failure criterion of conductive cracks in thermally depoled and poled lead zirconate titanate ceramics and proposed a charge-free zone (CFZ) model to understand and predict the failure behavior of conductive cracks in dielectric and piezoelectric ceramics under electrical and/or mechanical loading. The CFZ model treats dielectric and piezoelectric ceramics as mechanically brittle and electrically ductile. Charge emission and charge trapping consume more energy and thus lead to a high value of the electric toughness. In the CFZ model, the local electric intensity factor has a non-zero value and consequently there is a non-zero local electric energy release rate, which contributes to the driving force to propagate the conductive crack. The merit of the CFZ model lies in the ability to apply the Griffith criterion directly to link the local energy release rate to the fracture toughness in a completely brittle manner. As a result, an explicit failure criterion results from the CFZ model to predict the failure behavior of conductive cracks in dielectric and piezoelectric ceramics under electrical and/or mechanical loading and the theoretical predictions agree perfectly with the experimental observations [6-8]. The CFZ model is based on the field limiting space charge (FLSC) model [9] and analogy with the dislocation-free zone (DFZ) model [10] in the elastic-plastic fracture mechanics. In the FLSC model [9], the charge mobility had only two extremes. If the electric field, E, is lower than a critical value of Ec , the value of the charge mobility is assumed to be zero in the dielectric material, whereas the charge mobility has a finite value when E ! Ec . Based on the assumption used in the FLSC model, the level of the electric field remains the critical value in the space charge region, thereby allowing one to calculate the space charge distribution. The electric field at the tip of an electrically conductive crack is extremely high and theoretically approaches infinity. That is why an intensity factor of electric field strength, which is called the electric intensity factor in the present work for simplicity, is adopted to gauge the tip field. When the electric intensity factor reaches a critical value, charges could be emitted from the tip. Various emission mechanisms, such as the Schottky emission and the Fowler-Nordheim emission, may function jointly at the tip. The emitted charges may form a charge cloud around the tip and thus shield the tip from the applied electric field. At the onset of the failure of an electrically conductive crack in a dielectric or piezoelectric body, the charge cloud should reach a critical level. In the proposed charge-free zone model, we shall investigate this critical level of the charge cloud based on the concepts of fracture mechanics. To simplify the analysis, Zhang et al. [6-8] treated charges as line charges and the charge cloud as a charge strip. To clearly demonstrate the CFZ model, an overview with the simplified piezoelectric approach [11] is given in the present work. When the piezoelectric constant is zero, the failure criterion will reduce to that for dielectric ceramics. The anisotropic approach based on the Stroh formalism was described in [7]. Interested readers may refer [7] for details.

98

2. Simplified Piezoelectric Approach

In the simplified piezoelectric formulation [11], the number of the independent material constants is reduced to a minimum. When the poling direction is along the positive x3-direction, the simplified constitutive equations read

­V 11 ½ °V ° ° 22 ° °V 33 ° ® ¾ °V 23 ° °V 13 ° ° ° ¯V 12 ¿

ª1 « « « M« «0 «0 « ¬0

0  1º 0  1» » ­ E1 ½ 0 1 »° ° » ®E2 ¾ , 1 0 »° ° ¯ E3 ¿ 0 0» » 0 0¼

(1)

­ D1 ½ ° ° ® D2 ¾ °D ° ¯ 3¿

­ H 11 ½ °H ° 0 0 0 1 0º ° 22 ° ª0 ª1 0 0º ­ E1 ½ ° H 33 ° ° ° « » e 0 0 0 1 0 0 ®  N « 0 1 0» ® E 2 ¾ , « » 2H 23 ¾ « » ° «¬  1  1 1 0 0 0»¼ ° «¬0 0 1»¼ °¯ E 3 °¿ ° 2H 13 ° ° ° ¯ 2H 12 ¿

(2)

where * means that the corresponding constant will not appear in the model,

V ij ,

0 0 0º ­ H 11 ½ ª0 1 0 0 0» ° H 22 ° «0 ° « »°

1 0 0 0» ° H 33 ° «0 ¾  e« »® 0 0 1 0 0» °2H 23 ° «0 0 0 0 1 0» ° 2H 13 ° «1 ° « »° 0 0 0 0 ¼ ¯ 2H 12 ¿ ¬0

H ij , Di and Ei denote stress tensor, strain tensor, electric displacement vector and electric field vector, respectively, and only three independent material constants M, e and N are used to represent, on a qualitative basis, the elastic, piezoelectric and dielectric properties of the material. The equilibrium and kinematic equations are respectively given by (3) V ij , j 0, Di ,i 0 ,

H ij where ui and

1 2

u

i, j

 u j ,i ,

Ei

M ,i ,

(4)

M denote the elastic displacements and electric potential, respectively.

2.1. A SINGLE LINE CHARGE NEAR AN ELECTRICAL CONDUCTIVE CRACK Consider a semi-infinite conductive crack parallel to the poling direction. The (x, y) coordinate system is set up such that the crack tip is at its origin, the crack is located on the minus x-axis and the poling direction is along the positive x-direction.

99

Constraining the elastic displacement along the x-direction, one may denote the non-vanishing displacement component in the y-direction as u ( x, y ) . Rearranging the constitutive equations gives

V yx

Mu, x  eM , y , V yy

Mu, y  eM , x ,

Dx

 eu, y  NM , x ,

eu, x  NM , y .

Dy

(5)

Since both u and M are harmonic functions, the equilibrium conditions in Eq. (3) are satisfied automatically if u and M are expressed by the imaginary parts of analytic functions, respectively, u Im[U ( z )], M Im[) ( z )] , (6) where z x  iy . The stress, strain, electric field strength, and electric displacement intensity factors are usually defined by KV lim 2SzV yy , K H lim 2SzH yy , K E lim 2Sz E x , K D lim 2Sz Dx , z o0

z o0

z o0

z o0

(7) which also satisfy the constitutive relationships given by Eq. (5). From the definitions of the intensity factors and using the J-integral,

J

³ hn

1

*

 V ij n j ui ,1  Di E1ni d*,

(8)

1 1 V ij H ij  Di Ei , 2 2

(9)

where

h

we obtain the energy release rate

1 KV K H  K D K E . (10) 2 When a single line charge is located at z d near a semi-infinite conductive crack, G

J





the conductive crack requires the boundary conditions of for x < 0, V yy 0, E x 0,

(11)

along the crack faces. The complex potential satisfying the boundary conditions has the following form

U

0,

)





Qˆ ln z  z d  Qˆ ln§¨ z  z d ·¸, © ¹

(12)

where the overbar denotes the conjugate of a complex variable, and

Qˆ

iQ and Q

q 2SN

,

(13)

where q denotes the line charge per unit length. This line charge produces a stress field and an electric field, which are given by

100

E x  iE y

Q 2

zd  zd z ª z  z §¨ z d  z d ·¸  z d z d º «¬ »¼ © ¹ V yy  iV yx eE x  iE y ,

,

N E x  iE y .

D x  iD y

(14a)

(14b) (14c)

It is interesting to note that the line electric charge does not produce any strain field. When the electric charge is located on the x-axis, Eq. (14a) is reduced to

E x  iE y

q

xd

2SN

z >z  x d @

.

(15)

Substituting Eqs. (15) and (14b) and (14c) into the intensity factor definition of Eq. (7) leads to

KE



q , KV N 2S x d

eK E ,

KD

NK E .

(16)

These are the stress, electric and electric displacement intensity factors produced by the electric charge in front of a semi-infinity conductive crack, which are very useful in the charge-free zone model proposed in the next section. For the line charge near an electrical conductive crack, the applied tip field and the image field exert forces on the line charge. The image field is given by

E x( i )



q 2SN

x



1 x  xd

and the image force per unit length calculated from f i form:

fi



q2 1 . 2SN 2 x d

(17)

qE x( i ) ( x

xd ) takes the (18)

This image force always has the tendency to push the charge back towards the crack. On the other hand, the applied tip field exerts a driving force, f a , per unit length on the line charge, which is given by

fa

KEq 2Sx d

.

(19)

The sign of f a must be positive to emit a charge from the crack, thereby indicating that a positive (or negative) value of K E will have the tendency to emit a positive (or negative) charge. Furthermore, the driving force must be larger than the image force in order to emit a charge from the tip. For a given value of applied K E ,

101

however, there exists a critical distance from the crack tip, as shown by x0 in Figure 1. When the distance from the tip is smaller than x0 , the image force dominates, while the driving force dominates if the distance is larger than x0 . With the FLSC model, a charge moves forward in the region bounded by x1 and x2 , as shown in Figure 1, because the total electric field is higher than Ec in the region. Thus, when a charge is emitted from the tip, it must be emitted to a distance larger than x1 . Then, the charge moves forward until it reaches x2 , beyond which the total field is lower than the critical level of Ec and the charge mobility becomes zero. The analysis indicates that, microscopically, a charge-free zone is formed adjacent to the tip in the charge emission process.

f fa~1/x0.5 fc=qEc

fc fa+fi

Conductive crack

x2

x1

0

x

x0

fi~-1/(2x)

Figure 1. A single line charge in front of an electrically conductive crack subjected to the image force and the tip driving force

2.2. THE CHARGE-FREE ZONE MODEL In addition to the image force and the driving force, the interaction force between charges must be taken into account for many line charges. When more and more charges are emitted from the crack tip, these charges will entrap in the region of ba, as shown in Figure 2, where ob denotes the CFZ size. If we define f (x ' ) to be the line charge number distribution function, the charge number located at x ' in the interval dx ' is f ( x ' )dx ' . The equilibrium condition that the electrical field, Ex, equals the critical value, Ec , in the charge trap zone is described by

K Ea 2Sx

a

 Q³ b

f ( x' ) x' x ( x  x' )

dx'

Ec , b d x d a .

(20)

102

The first term on the left hand side in Eq. (20) represents the applied electric field, while the second term stands for the electric field induced by the electric charges. The uniqueness for a distribution f (x ' ) , which has zero value at b and a, requires

K Ea

2 2Sa Ec E ( S2 , k ) / S ,

(21)

S

where E ( 2 , k ) is the complete elliptic integral of the second kind and

k

1 b / a . Thus, the solution to Eq. (20) is given as 

f ( x' )

a  x' 3 ( S2 , ax('(xa'bb)) , k ) , x' ( x'b)

2 Ec b 2

S Q a

(22)

2

where 3 ( S2 , n , k ) is the complete elliptic integral of the third kind.

E

Ec

Conductive crack 0

a

b

x

Figure 2. The field distribution in front of a conductive crack, wherein ob is the size of the charge-free zone and ba denotes the charge zone

The electrical charges produce an electric intensity factor, which is calculated by a

K

i E

 2S Q ³ b

f ( x' ) x'

dx'

2

2

S

Ec

>

@

a E ( S2 , k )  b F ( S2 , k ) ,

(23)

where the superscript "i" denotes the charges, and F ( S2 , k ) is the complete elliptic integral of the first kind. Considering Eq. (21), one may write Eq. (23) as K Ei (:  1) K Ea , (24) where

:

b F a E

, , S

2

S

2

1  ab 1  ab

.

(25)

103

Using Eqs. (16) and (24), one arrives at the following expressions for the induced stress intensity factor and the electric displacement intensity factor, (26) KVi e( :  1) K Ea , K Di N ( :  1) K Ea . A local intensity factor is the sum of the applied intensity factor plus its l corresponding intensity factor induced by the charges, i.e., K K a  K i , and thus, one has

KVl

KVa  e( :  1) K Ea ,

K El

:K Ea ,

K Dl

K Hl

K Ha ,

K Da  N ( :  1) K Ea .

(27)

The applied strain intensity factor and the applied electric displacement intensity factor are usually expressed in terms of the applied stress and electric intensity factors:

K Ha

K Va  eK Ea , K Da M

§

N ¨¨1  ©

e2 MN

· a e a ¸¸ K E  KV . M ¹

(28)

Substituting Eq. (28) into Eq. (27) and then into Eq. (10), one obtains the local energy release rate as

2G l

1 KVa  eK Ea M





2





2

 N :K Ea .

(29)

Under purely mechanical loading, applying the Griffith criterion to Eq. (29) and a erasing K E yield

2*

2 1 KVo ,C , M





(30)

o

where K V ,C is the fracture toughness in terms of the critical stress intensity factor under purely mechanical loading and the subscript “C” denotes fracture or failure. Thus, the value of * can be evaluated from the experimental results under purely mechanical loading. Under purely electrical loading, the application of the Griffith a criterion to Eq. (29) and erasing K V give

§ 2 e2 · o ¨¨ N:  ¸ K E ,C M ¸¹ ©





2

2* ,

(31)

o

where K E ,C is the electric fracture toughness in terms of the electric intensity under purely electrical loading. Since the electric intensity factor may be positive or negative, we have

K Eo ,C

r

2*M . NM : 2  e 2

The parameter, : , can be evaluated from

(32)

104

ª§ K o «¨ Vo,C «¨© eK E ,C ¬

e2 NM

:

2 º · ¸  1» . ¸ » ¹ ¼

(33)

Once the value of : is known, we can calculate the ratio of b/a from Eq. (25). If the ratio of b/a is assumed as a constant, the parameter : is a constant. Thus, the application of the Griffith criterion to Eq. (29) establishes the failure criterion for conductive cracks in piezoelectric ceramics under combined electrical and mechanical loading

1 KVa ,C  eK Ea ,C M





2



 N :K Ea ,C



2

2* .

(34)

Using Eqs. (30) and (32), we can rewrite Eq. (34) in a dimensionless form:

§ KVa ,C ¨ o ¨K © V ,C

2

· 2e ¸ B ¸ 2 e  NM : 2 ¹





1/ 2

§ KVa ,C ¨ o ¨K © V ,C

· § K Ea ,C ¸¨ o ¸¨K ¹ © E ,C

· § K Ea ,C ¸¨ o ¸ ¨K ¹ © E ,C

· ¸ ¸ ¹

2

1.

(35)

Note that the negative sign is for positive electric loading and the positive sign is for negative electric loading. Equation (35) establishes the failure criterion for conductive cracks in piezoelectric ceramics under electrical and/or mechanical 2 2 loading. Mathematically, Eq. (35) has the form of x  Kxy  y 1 with

x

KVa ,C / KVo ,C

y

KVa ,C / KVo ,C ), and K

be

expressed

(or

in

x

K Ea ,C / K Eo ,C



)

B2 e / e 2  NM : 2 the

standard

and



1/ 2

K Ea ,C / K Eo ,C

y

(or

. The mathematic equation can

form

of

an

ellipse,

xˆ 2 /[2 /( 2  K )]  yˆ 2 /[2 /( 2  K )] 1 , where the ( xˆ , yˆ ) coordinator system is established by rotating 45q from the horizontal axis of the ( x , y ) coordinate



2



2 1/ 2

system. The absolute value of K is less than two due to 1  NM: / e !1 and thus Eq. (35) indeed describes an ellipse in terms of the normalized applied intensity factors. In the case that the poling direction is along the positive xdirection, e is positive. Thus, if the applied electric fields are parallel to the poling direction, i.e., under positive electrical loading, K  0 and the major semi-axis is located on the xˆ -axis, while K ! 0 and the minor semi-axis is located on the xˆ axis when applied electric fields are anti-parallel to the poling direction, i.e., under negative electrical loading. On the other hand, e is negative when the poling direction is along the negative x-direction. In this case, K  0 and the major semiaxis is located on the xˆ -axis under negative electrical loading, i.e., when applied electric fields are parallel to the poling direction, whereas K ! 0 and the minor semi-axis is located on the xˆ -axis under positive electrical loading, i.e., when the applied electric fields are anti-parallel to the poling direction. For clarification and

105

simplification, we may conclude that K  0 and the major semi-axis is located on the xˆ -axis if the applied electric fields are parallel to the poling direction and K ! 0 and the minor semi-axis is located on the xˆ -axis if applied electric fields are anti-parallel to the poling direction. For dielectric materials, the piezoelectric constant, e, is zero and thus, the interaction term, i.e., the second term on the left hand-side of Eq. (35) disappears, thereby reducing Eq. (35) to the failure criterion for conductive cracks in dielectric materials [6]:

§ KVa ,C ¨ o ¨K © V ,C

2

· § K Ea ,C ¸ ¨ o ¸ ¨K ¹ © E ,C

· ¸ ¸ ¹

2

1.

(36)

3. Concluding Remarks

The CFZ model is developed to predict the failure behavior of conductive cracks in dielectric and piezoelectric ceramics under electrical and/or mechanical loading. In the CFZ model, dielectric and piezoelectric ceramics are treated mechanically brittle and electrically ductile such that charge emission and charge trapping are assumed to occur at the conductive crack tip. The trapped charges partially shield the crack tip from the applied electrical field and the local electric intensity factor has a non-zero value. Consequently, a non-zero local electric energy release rate contributes to the driving force to propagate the conductive crack. The merit of the CFZ model, similar to the DFZ model, lies in the ability to apply the Griffith criterion directly to link the local energy release rate to the fracture toughness in a completely brittle manner. The CFZ model yields an explicit failure criterion to predict the failure behavior of conductive cracks in dielectric and piezoelectric ceramics under electrical and/or mechanical loading. Mathematically, the failure 2 2 formula takes form of x  Kxy  y 1 with x KVa ,C / KVo ,C (or

x

K Ea ,C / K Eo ,C ) and y

equation

can

be

K Ea ,C / K Eo ,C (or y

expressed

in

the

KVa ,C / KVo ,C ). The mathematic

standard

form

of

an

ellipse,

xˆ 2 /[2 /( 2  K )]  yˆ 2 /[2 /( 2  K )] 1 , where the ( xˆ , yˆ ) coordinator system is established by rotating 45q from the horizontal axis of the ( x , y ) coordinate system. If applied electric fields are parallel to the poling direction, K  0 and the major semi-axis is located on the xˆ -axis, while K ! 0 and the minor semi-axis is located on the xˆ -axis when applied electric fields are anti-parallel to the poling direction. For dielectric materials, K { 0 and the failure criterion is reduced to the failure criterion for conductive cracks in dielectric materials [6]. The failure criterions developed from the CFZ model have been verified by experimental results for thermally depoled (dielectric) [6] and poled [7] lead zirconate titanate ceramics.

106

Acknowledgements - This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China. The author would like to thank Prof. Minghao Zhao, Dr. Ran Fu, Dr. Tianhong Wang, Dr. Yi Wang and Mr. Guoning Liu for their contribution to the presented work. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

T.Y. Zhang, M.H. Zhao, P. Tong, 2002, Fracture of piezoelectric ceramics, Adv. Appl. Mech. 38, 147289. G.P. Cherepanov, 1979, Mechanics of Brittle Fracture. McGraw-Hill, New York, 317. E.J. Garboczi, 1988, Linear dielectric-breakdown electrostatics, Phys. Rev. B, 38, 9005-9010. T.H. Wang, T.Y. Zhang, 2001, Electrical fracture toughness for electrically conductive deep notches driven by electric fields in depoled lead zirconate titanate ceramics, Appl. Phys. Lett., 79, 4198-4200. R. Fu, C.F. Qian, T.Y. Zhang, 2000, Electrical fracture toughness for conductive cracks driven by electric fields in piezoelectric materials, Appl. Phys. Lett., 76, 126-128. T.Y. Zhang, T.H. Wang, M.H. Zhao, 2003, Failure behavior and failure criterion of conductive cracks (deep notches) in thermally depoled PZT-4 ceramics, Acta Mater., 51, 4881-4895. T.Y. Zhang, M.H. Zhao, G.N. Liu, 2004, Failure behavior and failure criterion of conductive cracks (deep notches) in piezoelectric ceramics I – The charge-free zone model, Acta Mate. 52, 2013-2024. T.Y. Zhang, G.N. Liu, Y. Wang, 2004, Failure behavior and failure criterion of conductive cracks (deep notches) in piezoelectric ceramics II – Experimental verification, Acta Mater. 52, 2025-2035. H.R. Zeller, W.R. Schneider, 1984, Electrofracture mechanics of dielectric aging, J. Appl. Phys., 56, 455-459. B.S. Majumdar, S.J. Burns, 1983, Griffith crack shielded by a dislocation pile-up, Int. J. Fracture 21, 229-240. H.J. Gao, T.Y. Zhang, P. Tong, 1997, Local and global energy release rates for an electrically yielded crack in piezoelectric ceramics, J. Mech. Phys. Solids, 45, 491-510.

ELECTRICAL POTENTIAL DROP ACROSS A CRACK IN PIEZOELECTRICS

Y. H. Chen and Z. C. Ou School of Civil Engineering and Mechanics, Xi’an Jiaotong University, Xi’an 710049, P.R. China

Abstract

This paper deals with the electrical potential drop across a central crack in an infinite plane of piezoelectric material. Three typical electric boundary conditions along the crack surfaces, i.e., the permeable crack model, the impermeable crack model and the so-called exact electric boundary condition accounting for the permittivity of medium inside crack gap, are considered for comparison. The influences of different media on the electrical potential drop are studied. It is found that, even though under moderate mechanical loadings the crack opening displacement jump is small, the electrical potential drop is still very high under highly applied electric field, which may lead to electrical discharge. It is also found that silicon oil inside the crack gap yields much larger values of this drop than those induced from air or vacuum. 1. Introduction

Fracture mechanics of piezoelectric ceramics received much attention in the past ten years. However, controversial results between experimental observations and theoretical estimations are remarkable [1-10]. The main obstacles in understanding this fracture problem under combined mechanical and electric loadings are: (i) the selection of the electric boundary condition; and (ii) the near-tip domain switching. Up to the date, there is no universal fracture criterion in the literature that could be used to explain all existing experimental data in piezoelectric materials. Within the framework of linear piezoelectric fracture mechanics, this paper deals with the electrical potential drop across a central crack in an infinite plane of piezoelectric material. Three typical electric boundary conditions along the crack surfaces, i.e., the permeable crack model [4], the impermeable crack model [1,2] and the socalled exact electric boundary condition accounting for the permittivity inside crack gap [3], are considered and compared. Particularly, the influences of different media inside crack gap on the electrical potential drop are studied in some detail. It is found that, even though the mechanical loading is moderate (20MPa) and the crack opening displacement jump is very small (around 10 P m ), the electrical 107 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 107–111. © 2006 Springer. Printed in the Netherlands.

108

potential drop is still very high (around several 100Volts) when the applied electric field is high ( r1MV / m ). It is expected that such high drop across the crack gap may lead to electrical discharge [7]. It is also found that silicon oil inside crack gap [1,2] yields much larger values of this drop than those induced from air or vacuum. Therefore, experiments performed in silicon oil and in air or vacuum should be quite different. 2. Electrical Potential Drop across a Central Crack

Consider a central crack with length 2a in an infinite plane of piezoelectric ceramics under combined mechanical and electric loadings (see Figure 1). The poling direction is along the x2-axis, whereas the crack is along the x1-axis. The f f remote mechanical and electric loadings are referred as V 22 and E2 , respectively. Here, the mechanical loading at infinity

V 22f is always fixed to be 20MPa, whereas

f

the electric loading at infinity E2 varies from –1MV/m to 1MV/m.

Poling

x2

2a

x1

Figure 1. A central crack in an infinite plane of piezoelectric ceramics

Assume that the electric boundary conditions on the crack surfaces are of three types: permeable [4], impermeable [1,2] and filled with some insulate medium [3], respectively. These three typical situations can be reached by selecting different values of the permittivity of medium inside the crack gap. The permittivity denoted by H a is extremely large for the permeable crack model (e.g., about 20000 times the permittivity of air or vacuum denoted by H 0 8.85 u1012 C / Vm ). It is a finite value for a specially selected medium inside the crack gap (e.g., silicon oil is 2.5 times H 0 ), and extremely small for the impermeable crack model (10-8 times H 0 ). Three different piezoelectric ceramics: PZT-4, PZT-5H and BaTiO3 are chosen in the present calculations for comparison.

109 Voltage difference at the center of the crack (MV)

0.0010

-8

Ha=10 H0

0.0008

Ha=H0

0.0006

Ha=0.5H0 Ha=20000H0

0.0004 0.0002 0.0000 -0.0002 -0.0004

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4 f

0.6



-0.0006 -1.0

PZT-4 a = 10 mm f V= 20MPa 0.8

1.0

Remote electrical loading E2 (MV/m Figure 2. Electrical potential drop across a central crack with 20mm in PZT-4 ceramics

By using the Stroh theorem and solving the corresponding Riemann-Hilbert problem, all mechanical quantities such as stresses and displacements and electric quantities such as electric field components and electric potential are obtained. Without going into the details, numerical results for a central crack with 20mm length are plotted in Figures 2, 3 and 4 for PZT-4, PZT-5H and BaTiO3, respectively. It is found that even though piezoelectric ceramics have inherent mechanical-electric coupling features, the electric potential drop across a central crack gap is approximately linear with respect to the applied electric field that varies from –1MV/m to 1MV/m. It is also found that the impermeable crack yields extremely large values of the electrical potential drop. For example, when the remote electric field is –0.2MV/m (see Figure 2), the induced potential drop across the gap is 0.001MV, i.e., 1000Volts, whereas when the remote electric field is 0.6MV/m, the induced potential drop is –0.0004MV, i.e., –400Volts. Even under purely mechanical loading with vanishing electric loading ( E2f 0 ), the impermeable crack still yields 600Volts drop across the crack. This supports the previous conclusion [5,6,7] that the impermeable crack model (or charge-free model) is physically unrealistic since it very much overestimates the influence induced from the existence of the crack on the electric field. As expected, the permeable crack does not yield any electrical potential drop across the crack gap, which means that this crack model underestimates the influence of electric field on fracture. Moreover, it is found that a crack in PZT-4 ceramic yields larger electrical potential drop than those induced from cracks in PZT-5H and BaTiO3 ceramics. Of great interest is that this drop in PZT-4 is very sensitive to the permittivity of

110

medium inside the crack gap, whereas this drop is not so sensitive to the permittivity in PZT-5H and BaTiO3 although the permittivity does yield some discrepancies. This implies that fracture behaviors may be quite different when using different piezoelectric ceramics with cracks full with different media inside the crack gaps.

Voltage difference at the center of the crack (MV)

0.0014 0.0012

-8

Ha=10 H0

0.0010 0.0008

Ha=H0

0.0006

Ha=0.5H0

0.0004

Ha=20000H0

0.0002 0.0000 -0.0002 -0.0004 -0.0006 -0.0008 -0.0010 -0.0012

PZT-5H a = 10 mm f V= 20MPa -0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4 f

0.6



-0.0014 -1.0

0.8

1.0

Remote electrical loading E2 (MV/m

0.0010

-8

0.0008

Ha=10 H0

0.0006

Ha=H0

0.0004

Ha=0.5H0 Ha=20000H0

0.0002 0.0000 -0.0002 -0.0004

BaTiO3

-0.0006

a = 10 mm f V= 20MPa

-0.0008 -0.0010 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4 f

0.6



Voltage difference at the center of the crack (MV)

Figure 3. Electrical potential drop across a central crack with 20mm in PZT-5H ceramics

0.8

1.0

Remote electrical loading E2 (MV/m Figure 4. Electrical potential drop across a central crack with 20mm in BaTiO3 ceramics

111

3. Conclusions (1) Under moderate mechanical loadings, the electrical potential drops across cracks in the three piezoelectric ceramics are remarkable when high positive or negative electric field is applied. As the crack gap (measured as being 10 P m of the normal opening displacement at the central point of the crack) is small, electrical discharge might occur especially in the near-tip region. (2) The permittivity of medium inside crack gap significantly influences the electrical potential drop. For example, to most contents, the drop induced from a central crack full with silicon oil is about twice the drop induced from the same crack full with air or vacuum since the permittivity of silicon oil is 2.5 times that of air or vacuum. (3) Even under purely mechanical loading this electrical potential drop is still large and can not be neglected. (4) The electrical potential drop for cracked PZT-4 is more sensitive to the permittivity of medium inside the crack gap than that of PZT-5H and BaTiO3. (5) The present investigations reveal that previous experimental studies on piezoelectric fracture behaviors were carried out in quite different situations, e.g., using different piezoelectric ceramics (hard or soft) and different media inside crack gaps (air or vacuum, silicon oil, NaCI solution [4]). Therefore, it is not surprising that their results were quite different or even controversial. Further investigations under similar experimental situations are needed. Only in this way, could a universal fracture criterion be proposed to explain all the existing experimental observations. Acknowledgement: The present work is sponsored by the National Natural Science Foundation of China. References 1. S.B. Park, C.T. Sun, 1995, Fracture criteria for piezoelectric ceramics. J. Am. Ceramic Soc., 78, 14751480. 2. S.B. Park, C.T. Sun, 1995, Effect of electric field on fracture of piezoelectric ceramics. Int. J. Fract., 70, 203-216. 3. T.H. Hao, Z.Y. Shen, 1994, A new electric boundary condition of electric fracture mechanics and its application. Engng. Fract. Mech., 47, 793-802. 4. V. Heyer, G.A. Schneider, H. Balke, J. Drescher, H.A. Bahr, 1998, A fracture criterion for conducting cracks in homogeneously poled piezoelectric PZT-PIC151 ceramics, Acta Mater., 46, 6615-6622. 5. R.M. McMeeking, 1999, Crack tip energy release rate for a piezoelectric compact tension specimen. Engng. Fract. Mech., 64, 217-244. 6. R.M. McMeeking, 2001, Towards a fracture mechanics for brittle piezoelectric and dielectric materials. Int. J. Fract., 108, 25-41. 7. R.M. McMeeking, 2004, The energy release rate for a Griffith crack in a piezoelectric material, Engng. Fract. Mech., 71, 1169-1183. 8. Y. Shindo, H. Murakami, K. Heriguchi, F. Narita, 2002, Evaluation of electric fracture properties of piezoelectric ceramics using the finite element and single-edge precracked-beam methods, J. Am. Ceramic Soc., 85, 1243-1248. 9. G.A. Schneider, F. Felten, R.M. McMeeking, 2003, The electric potential difference across cracks in PZT measuered by Kelvin Probe Microscopy and the implications for fracture, Acta Mater., 51, 2235-2241. 10. G.A. Schneider, V. Heyer, 1999, Influence of the electric field on Vickers indentation crack growth in BaTiO3, J. European Ceramic Soc., 19, 1299-1306.

ELASTIC Sv-WAVE SCATTERING BY AN INTERFACE CRACK BETWEEN A PIEZOELECTRIC LAYER AND AN ELASTIC SUBSTRATE

Bin Gu, Shou-Wen Yu*, Xi-Qiao Feng Key Lab. of Failure Mechanics, the Ministry of Education of China, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China

Abstract

The scattering of plane elastic waves by an interface crack between a piezoelectric layer and a homogeneous half-space is analyzed by means of the integral transform and the Cauchy singular integral equation methods. The effects of the crack configuration, the direction of the incident wave and the material combination of the piezoelectric layer and the elastic substrate on the dynamic stress intensity factors are examined. It is found from the numerical calculation that the crack configuration, denoted by the ratio of the crack length to the layer width, and the incident angle of the wave have significant influences on the dynamic response. In addition, the growth of the interface crack may be retarded or accelerated by specifying an appropriate combination of the materials. Keywords: Sv-wave scattering, interface crack, piezoelectric material, integral transform, singular integral equation. 1. Introduction

By virtue of the intrinsic coupling effects between the electrical and the mechanical fields, piezoelectric materials have found extensive applications in smart devices such as electromechanical sensors, actuators and transducers. Inherent weakness of piezoelectric ceramics is their brittleness in mechanical behavior. In most of their applications, piezoelectric crystals, ceramics and composites are exposed to severe mechanical and electrical loading conditions, which may result in structural fracture or failure. Meanwhile, piezoelectric materials usually have initial defects such as micro-cracks, micro-voids and inclusions, which may also bring about the failure of structures. Accordingly, the fracture of piezoelectric materials has attracted a * Corresponding author. Tel. & Fax: +86-10-6277 2926. E-mail address: [email protected]

112 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 112–120. © 2006 Springer. Printed in the Netherlands.

113

considerable interest in the past two decades [1-5]. Due to the complexity and difficulty in solution of mechanical-electrical coupled fields, the dynamic fracture mechanics of piezoelectric materials has not made considerable progress until recently. Chen and Yu [6,7] solved the anti-plane impact problems of a Griffith crack and a semi-infinite crack subjected to electromechanical loading, respectively. Chen and Karihaloo [8] deduced the solution of a mode-III crack subjected to arbitrary electro-mechanical impacts. Using the integral transform and the singular integral equation methods, Wang and Yu [10] investigated the transient response of a cracked strip subjected to plane electro-mechanical impacts. Their results illustrated that the dynamic stress intensity factor (DSIF) and the dynamic energy release rate (DERR) depend significantly on parameters such as the loading combination parameter and the crack configuration parameter. It was also found that in contrast with the mode-III crack as aforementioned, the dynamic electric displacement intensity factor (DEDIF) of a mode-I crack exhibits a considerable dynamic response. Chen, Karihaloo and Yu [9] solved the problem of an anti-plane Griffith crack moving along an interface of dissimilar piezoelectric materials by using the integral transform techniques. Treating the crack boundaries as electrically conducting electrodes and a vacuum zone, respectively, Li and Mataga [11,12] studied the dynamic crack propagation by means of the Wiener-Hopf and Cagniard-de Hoop techniques. As far as the scattering of waves is concerned, Narita and Shindo [13] investigated the scattering of Love waves by a surfacebreaking crack in a piezoelectric laminated medium by means of the path-integral technique. Moreover, Narita and Shindo [14] considered the scattering of horizontally polarized shear waves by a finite crack in a composite laminate containing a piezoelectric layer and obtained the numerical solution of the dynamic stress intensity factor and the dynamic energy release rate of the first symmetric mode. To date, however, investigation on plane problems of elastic wave scattering of piezoelectric materials is very limited. Using the integral transform and the Cauchy singular integral equation methods, the scattering problem of plane waves by an interface crack between a piezoelectric layer and a semi-infinite elastic substrate is investigated by Gu et al. [17] for the scattering of P wave. The present paper investigates the scattering of Sv-wave. The dependence of the dynamic response on the crack configuration, the propagating direction of the incident wave and the material combination will be discussed. 2. Formulation of the Problem

Consider a piezoelectric layer of infinite length and of thickness h, perfectly bonded to a half-space of elastic substrate, as shown in Figure 1. Assume that a crack of length 2c lies on the interface. Refer to a Cartesian coordinate system ( x , y , z ) located at the center of the crack, in which the z-axis is the poling axis of the piezoelectric medium. Assume that a wave propagates from the elastic substrate

114

with an incidence angle

T with respect to the z-axis (Figure 1). Z Piezoelectric layer

h

X

2c

© Elastic medium

Incident wave Figure 1. Crack configuration.

In the case of plane strain, the linear constitutive relations of a piezoelectric material of transverse isotropy with respect to the z-axis are expressed as

V xx c11u, x  c13 w, z  e13I, z , V zz c13u, x  c33 w, z  e33I, z , V xz c44u, z  c44 w, x  e15I, x , Dx

e15u, z  e15 w, x  N 11I, x Dz

e u  e w N I

13 , x 33 , z 33 , z , (1) where I denotes the electric potential, u and w denote the displacements in the x

and z directions, respectively,

V xx , V zz and V xz the stresses, Dx and D z the

electric displacements, c11 , c13 , c33 and c44 the elastic moduli, e13 , e33 and e15 the piezoelectric coefficients, and N 11 and N 33 the dielectric coefficients. The constitutive relations of an isotropic medium are written as V xxe (O  2G )u,ex  Ow,ez , V zze Ou ,ex  (O  2G ) w,ez , V xze Gu,ez  Gw,ex , (2) where the superscript e stands for quantities of the elastic substrate, constant, and G the shear modulus.

O the Lamé

The governing equations of the piezoelectric material read (see Qin [4])

c11u, xx  c44 u, zz  ( c13  c44 ) w, xz  ( e13  e15 )I, xz

U

w 2u wt 2 ,

w2w ( c13  c44 )u, xz  c44 w, xx  c33 w, zz  e15I, xx  e33I, zz U 2 wt , (e13  e15 )u, xz  e15 w, xx  e33 w, zz  N 11I, xx  N 33I, zz 0 .

(3)

115

By introducing the displacement potential functions M and \ , the equilibrium equations in the elastic substrate are simplified as 2

’M with cp

1 w 2M , ’ 2\ 2 2 cp wt

( O  2G ) / U e and cs

1 w 2\ , 2 2 cs wt

(4)

G / U e , where U and U e are the mass

densities of the piezoelectric medium and the elastic material, respectively. 3. Solution of the Problem

As is well known, the above problem of wave scattering can be treated as the superposition of the scattering problem of the incident wave in the same structure having no crack and the problem where the incident wave is applied to the crack surface but there is no incident wave in the far field. The solution method is the same as in Gu et al. [17]. The dynamic stress intensity factors of mode-I and modeII at the left and the right crack tips can be deduced as [17]:

ª K IIL º « L» ¬« K I ¼»

ª K IIR º « R» ¬« K I ¼»

ª  (1  J 12 )1 / 2 º ( 2)D1 Pk(D1 ,E1 ) ( 1) Ak » N « 2 », c BR ¦ « 2 1/ 2 ( 1 J   » « D 2 (D 2 , E 2 ) k 1 2 ) ( 2) Pk ( 1) Bk » «¬ 2 ¼ º ª  (1  J 12 )1 / 2 E1 (D1 ,E1 ) 2 Pk (1) Ak » N « 2 » c BR ¦ « 2 1/ 2 ( 1 J   » k 1« E 2 (D 2 , E 2 ) 2 ) 2 Pk (1) Bk » « 2 ¼ ¬

(5)

To illustrate the basic features of the solutions, numerical calculations have been carried out for two different material pairs, PZT-5H/Al and BaTiO3/Al. Their material properties are adopted from Dunn and Taya [2]. Assume here that only a wave of a single mode, either a P- or Sv-wave, is applied in the far field, as shown in Figure 1. For comparison, the non-dimensional parameters K I / K I 0 and

K II / K II 0 are introduced, in which K I 0

K II 0

V 0 (Z , k ) c in the case of an

W 0 (Z , k ) c in the case of an incident Sv-wave. Furthermore, the normalized circular frequency Zh / cs is used. incident P-wave or K I 0

K II0

Figures 2-7 indicate the effect of the incidence angle T of the wave on the DSIFs of the left and the right crack tips for two different material combinations, BaTiO3/Al and PZT-5H/Al, and different incident waves. It is emphasized that the incident angle T approximately approaches but not completely equals to ʌ / 2 or

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T cr . The phenomena happen to the mode-II DSIF of two crack tips when a Svwave is applied (Figures 2-3 and Figures 6-7 ). It can also be seen that the mode-II DSIFs under an incident Sv-wave vibrate more dramatically than the mode-I DSIFs under an incident P-wave when T approaches to the critical value ʌ / 2 or T cr .

c/h=2.0 2

|K,, /K,,0|

c/h=1.5 c/h=1.0

R

c/h=0.5 c/h=0.1 1

0.0

Figure 2. K IIR / K II0 versus

0.5

1.0

Zh/cs

1.5

2.0

2.5

Zh / cs for different values of c / h in the BaTiO3/Al system under an incident Sv-wave ( T

T cr / 2 )

In addition, comparing the results of BaTiO3/Al and PZT-5H/Al, it can be found that the maximal values of the DSIF are evidently different though their DSIFs exhibit similar variations. This means that the DSIFs may be impeded or accelerated by specifying different material combinations.

117

2

c/h=2.0

R

|K,, /K,,0|

c/h=1.5 c/h=1.0

c/h=0.5 c/h=0.1

1

0.0

Figure 3. K IIR / K II0 versus

0.5

Zh / cs

1.0

1.5

Zh/cs

for different values of Sv-wave ( T

2.0

2.5

c / h in the PZT-5H/Al system under an incident

T cr / 2 )

3

T =0

L

|K, /K,0|

2

T =S/6 1

T =S/4 T =S/3 T =S/2

0 0.0

Figure 4.

0.5

1.0

Zh/cs

1.5

2.0

2.5

K IL / K I0 versus Zh / cs for different values of T in the BaTiO3/Al system under an incident Pwave ( c / h 1.0 )

118

3

T =S/6

|K, /K,0|

2

T =S/4

R

T =0

T =S/3 T =S/2 1

0 0.0

0.5

1.0

1.5

2.0

2.5

Zh/cs

Figure 5. K IR / K I0 versus

Zh / cs

for different values of T in the BaTiO3/Al system under an incident Pwave ( c / h

1.0 )

1.5

T =Tcr/4

T =0

|K,, /K,,0|

1.0

L

T =Tcr/2

T =Tcr

0.5

0.0 0.0

Figure 6. K IIL / K II0 versus

0.5

1.0

Zh/cs

1.5

2.0

2.5

Zh / cs for different values of T in the BaTiO3/Al system under an incident Svwave ( c / h 1.0 )

119

2.5

T =Tcr

1.5

R

|K,, /K,,0|

2.0

1.0

T =0

T =Tcr/4 T =Tcr/2

0.5

0.0 0.0

Figure 7. K IIR / K II0 versus

0.5

1.0

Zh/cs

1.5

2.0

2.5

Zh / cs for different values of T in the BaTiO3/Al system under an incident Svwave ( c / h

1.0 )

4. Conclusions

The scattering of plane elastic waves by an interface crack between a piezoelectric layer and a semi-infinite elastic substrate is investigated by using the integral transform and the Cauchy singular integral equation methods. It is concluded from the numerical calculation that the ratio of the crack length to the layer width, c / h , has a significant influence on the DSIFs. On the other hand, the dynamic response of an interface crack depends, to a significant extent, on the direction of the incident wave T and the type of the incident wave. Furthermore, the dynamic stress intensity factors may be retarded or promoted by specifying different material combinations.

Acknowledgements

This project is supported by the National Natural Science Foundation of China (90205022, 10172050) and the Sino-German-Center GZ050/2 and Key Grant of project -MoE-0306, XQF wishes to thank the Education Ministry of China for financial support.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Y. Pak, 1990, Crack extension force in a piezoelectric material. Journal of Applied Mechanics, 57, 647653. M.L. Dunn, M. Taya, 1994, Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids. Journal of Applied Mechanics, 61, 474475. H. Gao, T. Zhang, P. Tong, 1997, Local and global energy rates for an electrically yielded crack in piezoelectric ceramics. Journal of the Mechanics and Physics of Solids, 45, 491510. Q.H. Qin, 2000, Fracture Mechanics of Piezoelectric Materials, WIT Press, Southampton. Q.H. Qin, X. Zhang, 2000, Crack deflection at an interface between dissimilar piezoelectric materials. International Journal of Fracture, 102, 355370. Z.T. Chen, S.W. Yu, 1997, Antiplane dynamic fracture mechanics of piezoelectric materials. International Journal of Fracture, 85, L3L12. Z.T. Chen, S.W. Yu, 1998, Semi-infinite crack under anti-plane mechanical impact in piezoelectric materials. International Journal of Fracture, 88, L53L56. Z.T. Chen, B.L. Karihaloo, 1999, Dynamic response of a cracked piezoelectric ceramic under arbitrary electro-mechanical impact. International Journal of Solids and Structures, 36, 51255133. Z.T. Chen, B.L. Karihaloo, S.W. Yu, 1998, A Griffith crack moving along the interface of two dissimilar piezoelectric materials. International Journal of Fracture, 91, 197203. X.Y. Wang, S.W. Yu, 2000, Transient response of a crack in piezoelectric strip subjected to mechanical and electrical impacts: mode-I problem. Mechanics of Materials, 33, 1120. S.F. Li, P.A. Mataga, 1996, Dynamic crack propagation in piezoelectric materials-Part I: Electrode solution. Journal of the Mechanics and Physics of Solids, 44, 17991830. S.F. Li, P.A. Mataga, 1996, Dynamic crack propagation in piezoelectric materials-Part II: Vacuum solution. Journal of the Mechanics and Physics of Solids, 44, 18311866. F. Narita, Y. Shindo, 1998, Scattering of Love waves by a surface-breaking crack in a piezoelectric layered medium. JSME Internationla Journal, 41, 4048. F. Narita, Y.Shindo, 1999, Scattering of antiplane shear waves by a finite crack in piezoelectric laminates. Acta Mechanica 134, 2743. S.P. Shen, Z.B. Kuang, 1998, Wave scattering from an interface crack in laminated anisotropic media. Mechanics Research Communication, 25, 509517. H.J. Yang, D.B. Bogy, 1985, Elastic wave scattering from an interface crack in a layered half-space. Journal of Applied Mechanics, 52, 4250. B. Gu, S.W. Yu, X.Q. Feng, 2002, Elastic wave scattering by an interface crack between a piezoelectric layer and an elastic substrate, Int. J. Fracture, 116, L29-34.

A MICRO-MACRO APPROACH TO DESIGN ACTIVE PIEZOELECTRIC FIBER COMPOSITES

H. Berger,1 S. Kari,1 N. Bohn,1 R. Rodriguez,2 U. Gabbert1 Institute of Mechanics, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany 2 Facultad de Matemática y Computación, Universidad de la Habana San Lázaro y L, CP 10400, Vedado, Habana 4, Cuba 1

Abstract

The present work deals with the modeling of 1-3 periodic active fiber composites made of piezoceramic (PZT) fibers embedded in a soft non-piezoelectric matrix. We especially focus on predicting the effective coefficients of periodic transversely isotropic piezoelectric fiber composites using representative volume element method (unit cell method). Two ways for calculating the effective coefficients, an analytical and a numerical approach, can be used. In this paper the focus is on a numerical approach based on the finite element method (FEM). The asymptotic homogenization method (AHM) is used to verify the numerical solutions. Special attention is given on definition of appropriate boundary conditions for the unit cell to ensure periodicity. With the method presented the effective coefficients were calculated for different fiber volume fractions and the results are compared and discussed. 1. Introduction

Piezoelectric ceramics, such as lead zirconate-titanate (PZT), are widely used for transducers in sonar, underwater communications, underwater or medical imaging applications, robotics and micro-machines, as well as for shape, vibration and noise control applications in adaptronics or structronic systems.[1] Even if their properties make them interesting, they are often of limited use, first due to their weight, that can be a clear disadvantage for shape control, and, as a consequence, on reason of their high specific acoustic impedance, which reduces their acoustic matching with the external fluid domain. Piezoelectric wafers have been developed by combining piezoceramics with passive nonpiezoelectric polymers. Superior properties have been achieved by these composites by taking advantage of the most profitable properties of each of the constituents and a great variety of structures have been designed. Many models are available to describe these piezocomposites [2-4]. Recently, due to the design of piezocomposites on a greatly reduced scale and 121 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 121–130. © 2006 Springer. Printed in the Netherlands.

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the use of PZT fibers instead of piezoelectric bars, new applications toward electromechanical sensors and actuators have become possible [5]. Since they are now much smaller than the wavelength of interest, however, homogenization techniques are necessary to describe the behavior of these piezocomposites. On the other hand, numerical models seem to be well-suited to describe the behavior of these materials, because there is no restrictions on the geometry, on the material properties, on the number of phases in the piezocomposite, and on the size. Therefore, the finite element method with the help of FEM system COSAR has been developed to solve problems concerning smart composite lightweight structures [6]. In the paper a methodology is proposed which consists of a two-step micro-macro mechanical approach. In the first step, a numerical homogenization procedure is applied on the micro level to calculate the overall material properties of the material system based on the geometry and the properties of the constituents, the volume fraction, the fiber distributions and orientations [7]. With a representative volume element the problem can be reduced to an investigation of a small periodic part of the structure. Based on these homogenized properties the overall behavior of a composite structure is analyzed in a second step. Finally, a test example is presented to demonstrate the dependency of the homogenized material parameters on the properties of the constituents, the volume fraction, and the thickness. 2. Mathematical Modeling of Piezoelectric Composites

Coupled piezoelectric problems are those in which an electric potential gradient causes deformation (converse piezoelectric effect), while mechanical strains cause an electric potential gradient in the material (direct piezoelectric effect). The coupling between mechanical and electric fields is characterized by piezoelectric coefficients. Those materials respond linearly to changes in the electric field, the electrical displacements, or mechanical stresses and strains. These assumptions are compatible with the piezoelectric ceramics, polymers, and composites in current use. Therefore, the behavior of the piezoelectric medium is described by the following constitutive equations, which correlate stresses ( Tij ), strains ( S kl ), electric fields ( E k ), electrical displacements ( Di ) as follows

Tij

Cijkl S kl  ekij Ek

Di

eikl S kl  İ ik E k ,

(1)

where C ijkl is the fourth-order elasticity tensor under short circuit boundary conditions, İ ik is the second-order free body electric tensor, and ekij is the thirdorder piezoelectric strain tensor. Due to the symmetry of the tensors Tij , S ij , C ijkl ,

123

and İ ij , Eq. (1) can be written in a vector/matrix notation by using the Voigt’s notation as

ªT º «D» ¬ ¼

ªC « ¬e

- e T º ªS º » « » İ ¼ ¬E ¼

(2)

where superscript T denotes a transposed matrix. For a transversely isotropic piezoelectric solid, the stiffness matrix, the piezoelectric matrix and the dielectric matrix are simplified so that only 11 independent coefficients remain. In the case of aligned fibers made of a transversely isotropic piezoelectric solid (PZT), embedded in an isotropic polymer matrix, the resulting composite is also a transversely isotropic piezoelectric material. Consequently, the constitutive equation (2) can be written as  e13eff º ªS11 º 0 0 0 0 0 ªT11 º ªC11eff C12eff C13eff » « » « » « eff eff eff e eff13 » «S 22 » 0 0 0 0 0 «T22 » «C12 C11 C13 » «T » «C eff C eff C eff eff » «  e33 0 0 0 0 0 13 33 » «S 33 » « 33 » « 13 eff «T » «0 0 0 0 0 0  e15eff 0 » «S 23 » C44 » « » « 23 » « eff «T31 » «0 0 0 0 C44 0  e15eff 0 0 » «S 31 » » « » . (3) « » « eff T 0 0 0 0 0 0 0 0 C « » «S12 » « 12 » 66 » « » «D » «0 eff eff 0 0 0 0 0 0 » «E1 » e15 İ11 « 1» « «D2 » «0 0 0 0 0 0 0 » «E2 » e15eff İ11eff » « » « » « eff eff » « E3 » «¬D3 »¼ «¬e13eff e13eff 0 0 0 0 0 e33 İ33 ¼ ¬ ¼ In this matrix the general variables of the coupled electromechanical problem were eff replaced by the appropriate values for the homogenized structure. Namely, C ij ,

eijeff , İ ijeff denote the effective material coefficients and S ij , Ei , Tij , Di denote average values. These relations represent the basis for the further considerations based on a unit cell. 3. Representative Volume Element (Unit Cell) Method

In general the object under consideration is regarded as a large-scale/macroscopic structure. The common approach to model the macroscopic properties of 3D piezoelectric fiber composites is to create a representative volume element (RVE) or a unit cell that captures the major features of the underlying microstructure. The mechanical and physical properties of the constituent material are always regarded as those of a small-scale/micro structure. One of the most powerful tools to speed up the modeling process, both the composite discretization and the computer simulation of composites in real conditions, is the homogenization method. The

124

main idea of the method is to find a globally homogeneous medium equivalent to the original composite, where the strain energy stored in both systems is approximately the same. Different analytical homogenization techniques have been developed in order to predict the effective properties of piezoelectric composites. Uni-directional (1-3 periodic) piezoelectric fiber composites can be analyzed by using asymptotic homogenization method. Inclusion problems (i.e. 0-3 periodic) can be analyzed by self-consistent methods. Moreover, numerical methods (e.g. FEM techniques) have been developed to evaluate the effective coefficients of composites. In this paper we limit ourselves to a quasistatic analysis of periodic (13) structures with perfectly bonded continuous fibers which are aligned and poled along the x 3 axis as shown in Figure 1. With the help of symmetry, such a regular piezoelectric fiber composite may be analyzed by using a representative volume element or unit cell. A unit cell is the smallest part that contains sufficient information on the above-mentioned geometrical and material parameters at the microscopic level to deduce the effective properties of the composite. Figure 1 shows the unit cell that is picked from the periodic piezoelectric fiber composite. It has infinite repetitions in all three directions. In this paper we consider a composite with a square fiber arrangement. It is assumed that the material properties are the same in the first two directions (i.e. along x1 and x2 axis). All the fibers are assumed to be straight and poled in the third direction (i.e. along axis x3). The Figure 1(b) shows the schematic diagram of the unit cell picked from the considered composite. surface B+ surface A-

x2

surface C-

x1 x3

RVE or unit cell surface A+ matrix fiber

+

surface C

surface B-

(a) (b) Figure 1. Schematic diagram of a periodic 1-3 composite: (a) and unit cell, (b) picked from the original composite

4. Numerical Solution Using Finite Element Method

4.1 PERIODIC BOUNDARY CONDITIONS TO RVE Composite materials can be represented as a periodical array of RVEs. Therefore, periodic boundary conditions must be applied to the RVE models. This implies that each RVE in the composite has the same deformation mode and there is no separation or overlap between the neighboring RVEs. These periodic boundary conditions on the boundary RVE are given by u i S ij x j  v i (4)

125

In the above Eq. (4) S ij are the average strains, v i is the periodic part of the displacement components (local fluctuation) on the boundary surfaces, which is generally unknown and is dependent on the applied global loads. The indices i and j denote the global three-dimensional coordinate directions in the range from 1 to 3. A more explicit form of periodic boundary conditions, suitable for square RVE models can be derived from the above general expression. For the RVE as shown in Figure 1(b), the displacements on a pair of opposite boundary surfaces (with their normal along the x j axis) are _

u iK



S ij x Kj  v iK

u iK



S ij x Kj  v iK

_









(5) (6)





where index ‘ K ‘ means along the positive x j direction and ‘ K ‘ means along 





the negative x j direction on the corresponding surfaces A / A , B / B 



and



C  / C  (see Figure 1(b)). The local fluctuations viK and viK around the average macroscopic value are identical on two opposing faces due to the periodic conditions of RVE. So, the difference between the above two equations is the applied macroscopic strain condition 

u iK  u iK



_





S ij ( x Kj  x Kj ) .

(7)

Similarly the periodic boundary condition for electric potential is given by the applied macroscopic electric field condition and is 

ĭK  ĭK



_





E i ( x Kj  x Kj ) .

(8)

It is assumed that the average mechanical and electrical properties of a RVE are equal to the average properties of the particular composite. The average stresses and strains in a RVE are defined by

S ij

1 S ij dV V V³

(9)

T ij

1 Tij dV V V³

(10)

where V is the volume of the periodic representative volume element. Analogously the average electric fields and electrical displacements are defined by

Ei

1 E i dV V V³

(11)

126

Di

1 Di dV . V V³

(12)

4.2 FINITE ELEMENT MODELING All finite element calculations were made with the FE package ANSYS. For modeling the RVE three dimensional multi-field 8 node brick elements with displacement degrees of freedom (DOF) and additional electric potential (voltage) degree of freedom were used. These allows for fully coupled electromechanical analyses. To obtain the homogenized effective properties we apply the macroscopic boundary conditions (Eqs. (7) and (8)) to the RVE by coupling opposite nodes on opposite boundaries. In order to apply these periodic boundary conditions in the FE analysis, the mesh on the opposite boundary surfaces must be the same. For each pair of displacement components at two corresponding nodes with identical inplane coordinates on the two opposite boundary surfaces a constraint condition (periodic boundary condition (7) or (8)) is imposed. A FORTRAN software was developed to generate all required constraint conditions automatically. First the finite element mesh is created from the ANSYS preprocessor. Then based on the generated nodal coordinates the appropriate nodal pairs are selected by the FORTRAN program and a partial ANSYS input file is created containing the constraint conditions. Using this input file ANSYS continues with assigning the constraint equations and finally with solving the problem. As an example Figure 2 shows the constraint equations for a pair of nodes on opposite surfaces A  and A  . +

-

u1A = u1A + S11 ' x1 + u2A = u2A + S21 ' x1 + u3A = u3A + S31 ' x1

-

u1A u2A u3A

-

A

A+ +

-

' x1 = x1A - x1A

Figure 2. Periodic boundary conditions for a pair of nodes on opposite surfaces

A  and A 

127

For the calculation of effective coefficients we consider a piezoceramic (PZT-5) fiber embedded in a soft non-piezoelectric material (polymer). In analytical as well as in numerical modelings, we assume that the fibers and the matrix are ideally bonded and that the fibers are straight and parallel to the x3 axis. The fiber section is circular and the unit cell has a square cross-section. The piezoelectric fibers are uniformly poled along the x3 direction. The material properties of polymer and PZT-5 are listed in Table 1, where the elastic properties, the piezoelectric constants and the permittivities are given in N/m2, C/m2 and F/m, respectively. Table 1 Material properties of composite constituent fiber (PZT-5) and matrix (Polymer) C11 C12 C13 C33 C44 C66 e15 (1010) (1010) (1010) (1010) (1010) (1010) PZT-5 12.1 Polymer 0.386

7.54 0.257

7.52 0.257

11.1 2.11 0. 386 0.064

2.28 0.064

12.3 -

e13

e33

-5.4 15.8 -

k11 (10- k33 (109 9 ) ) 8.11 7.35 0.07965 0.07965

To find the effective coefficients special load cases with different boundary conditions must be constructed in such a way that for a particular load case only one value in the strain/electric field vector (see Eq. (3) is non-zero and all others vanish. Then from one row in Eq. (3) the corresponding effective coefficient can be evaluated using the calculated average non-zero value in the strain/electric field vector and the calculated average values in the stress/electrical displacement vector. In the next section the special models for calculating different effective coefficients will be explained in detail. Every load case is calculated for six different fiber volume fractions in the range from 0.111 to 0.666 with steps of 0.111. Figure 3 visualizes the relation of fiber and matrix for the different investigated fractions. Since only the volume fiber fraction has an influence on the results the size of the RVE was chosen as a unit length.

0.111

0.222

0.333

0.444

0.555

Figure 3. Investigated ratios of fiber volume fraction

0.666

128

5. Numerical Calculation of Different Effective Coefficients eff

eff

As an example, the calculation of the effective coefficients C13 and C 33 is presented in the following. For the calculation of these effective coefficients the boundary conditions have to be applied to the RVE in such a way that, except the strain in the x 3 direction ( S 33 ), all other mechanical strains and gradients of electric potential ( Ei ) become zero. This can be achieved by constraining the 

normal displacements at all surfaces to zero except of surface C (see Figure 1(b)). 



At surface C the periodic boundary condition corresponding to surface C must be applied. Due to applied zero displacements to surface C C



in x3 direction



0 ) the periodic boundary condition in this direction according to Eq. (7) ( u3 simplifies to u 3K



_





S 33 ( x 3K  x 3K ) . C

(13)



Because this equation is independent of u 3 , instead of using constraint equations, 

an arbitrary constant prescribed displacement can be applied on surface C to produce a strain in x 3 direction. To make gradients of the electric potential in all directions zero the voltage degree of freedom on all surfaces is set to zero. Figure 4 shows the finite element mesh. In Figures 5 and 6 the distribution of the strain S 33 and the stress T33 is shown in the deformed model, respectively. For the calculation of the total average values S 33 , T11 and T33 according to Eqs. (9) and (10) the integral was replaced by a sum over averaged element values multiplied by the respective element volume. Using these total average values the eff

coefficients C13 and

eff C33

can be calculated from the matrix Eq. (3). Due to zero

strains and electric fields except S 33 , the first row becomes T11

C13eff S 33 . Then

C13eff can be calculated as the ratio of T11 S 33 . Similarly C33eff can be evaluated as the ratio of T33 S 33 from the third row of matrix Eq. (3). Figure 7 shows the variations of these effective coefficients for different volume fractions in comparison to the calculated values by AHM. The other coefficients are calculated in an analog manner (for details see [8]).

129

1.0E+10 8.0E+09

Figure 5. Strain distribution

FEM 2

C33 [N/m ]

AHM

6.0E+09 4.0E+09

eff

C13eff [N/m2]

Figure 4. Finite element mesh

2.0E+09 0.0E+00 0.0

0.2

0.4

0.6

Fiber volum e fraction

Figure 7. Variations of AHM

eff C13

0.8

S 33

Figure 6. Stress distribution

5.0E+10 4.0E+10 3.0E+10 2.0E+10 1.0E+10 0.0E+00

T33

FEM AHM

0.0

0.2 0.4 0.6 0.8 Fiber volume fraction

eff and C33 as functions of the volume fraction and comparison between FEM and

6. Conclusions

An analytical as well as a numerical approach (RVE) for predicting the homogenized properties of piezoelectric fiber composites has been presented. The numerical approach is based on the finite element method. Longitudinal and transversal elastic and piezoelectric effective coefficients have been calculated with the finite element numerical model and compared with the analytical solutions based on the asymptotic homogenization method. This permits us to estimate the range of validity of each approach and to quantify the influence of micro structural parameters, such as the volume fraction, to the effective coefficients. In the case of 1-3 periodic composites with a non-piezoelectric polymer matrix and piezoelectric ceramic (PZT) fibers, the estimation highly depends on the fiber volume fraction. The unit cell method (RVE) based on the FE and the analytical methods both have their advantages and disadvantages. The analytical approach is able to model statistic distributions and consumes less computing time than the FE analysis. FE analysis on the other hand is appropriate to estimate the effective properties of

130

composites with a given periodic fiber distribution. The FE analysis also allows the inclusion of more complex boundary conditions. A generalized approach has been developed to calculate all effective coefficients for all volume fractions by interfacing the finite element package ANSYS with a corresponding FORTRAN routines. It reduces the manual work and time and can be used as a template to determine the effective coefficients of piezoelectric fiber composites with particular arrangement of fibers such as rectangular, hexagonal or random arrangements. Acknowledgement: This work was partially supported by the German Research Foundation (DFG) in the frame of the Graduate College 828 - Micro-macro interaction of structured media and particle systems. This support is gratefully acknowledged. References 1. 2. 3. 4. 5.

6. 7.

8.

K. Watanabe, F. Ziegler, (Eds.), 2003, IUTAM Symposium on Dynamics of Advanced Materials and Structures, Kluwer Academic Publishers, Dordrecht, Boston, London. G. Hayward, J.A. Hossack, 1990, Unidimensional modeling of 1-3 composite transducers. J. Acoust. Soc. Am., 88, 599-608. W.A. Smith, 1993, Modeling 1-3 composite piezoelectrics: hydrostatic response. IEEE Trans. Ultrason., Ferroelec., Freq. Contr., 40, 41-49 A.A. Bent, N.W. Hagood, 1997, Piezoelectric fiber composites with interdigitated electrodes. J. of Intelligent. Material Systems and Structures, 8, 903-919. J. Helbig, W. Glaubitz, H. Spaniol, P. Vierhaus, R. Hansch, W. Watzka, D. Sporn, 2003, Development and technology of doped sol-gel derived lead zirconate titanate fibers, Smart Materials and Structures, 12, 987-992. H. Berger, U. Gabbert, H. Köppe, F. Seeger, 2000, Finite element analysis and design of piezoelectric controlled smart structures. J. of Theoretical and Applied Mechanics, 3, 38, 475-498. H. Berger, U. Gabbert, H. Köppe, R. Rodriguez-Ramos, J. Bravo-Castillero, R. Guinovart-Diaz, J.A. Otero, G.A. Maugin, 2003, Finite element and asymptotic homogenization methods applied to smart composite materials, Journal of Computational Mechanics, 33, 61-67. H. Berger, S. Kari, U. Gabbert, R.R.. Rodriguez, R. Guinovart, J.A. Otero, J.C. Bravo, An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites, Int. J. of Solids and Structures (accepted).

FEM-TECHNIQUES FOR THERMO-ELECTRO-MECHANICAL CRACK ANALYSES IN SMART STRUCTURES

M. Kuna Institute of Mechanics and Fluid Dynamics, Technische Universität Bergakademie Freiberg, Lampadiusstraße 4, FREIBERG, D 09596, GERMANY

Abstract A review is given about FEM-techniques to compute the coupled thermoelectromechanical boundary value problem of cracks in three-dimensional thermopiezoelectric structures. The thermoelastic and pyroelectric effects of inhomogeneous temperature fields are considered, assuming that the heat conduction problem can be treated first. In order to calculate the relevant fracture parameters precisely and efficiently, the following specialized techniques are presented in detail: i) special singular crack tip elements; ii) modified crack closure integral; and iii) definition and computation of the thermoelectromechanical Jintegral. Special emphasis is devoted to semi-permeable crack face boundary conditions using the iterative capacitor analogy. The accuracy, efficiency and applicability of these techniques are exemplified by various problems. 1. Introduction

Piezoelectric and pyroelectric materials have widespread applications in modern developments of smart structures, where they serve as sensors, actuators or transducers. In composite smart material systems they are directly embedded in a matrix material (e.g. as piezoelectric layers or fibers) or attached as coatings. As a consequence of heterogeneity, misfit of the thermoelectroelastic properties, geometrical imperfections (electrodes, edges and flaws) or field coupling effects, smart composite structures may be exposed to extraordinary high mechanical and/or electrical field concentrations resulting from manufacturing or from inservice loading. For the assessment of strength and reliability under combined electrical, mechanical and thermal loading, crack-like defects play an important role. Fracture mechanics of piezoelectric materials has been established quite well in the last decade, see the review papers [1,2,3] including the literatures cited there, whereas the coupling with thermal fields is still in the beginning, see [2]. For cracks in thermopiezoelectric solids first analytical solutions were presented in [4] for the two-dimensional case, and by [5] for a penny-shaped crack in three dimensions (3D). However, because of the mathematical complexity only a few exact solutions 131 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 131–143. © 2006 Springer. Printed in the Netherlands.

132

are available for crack problems, especially in 3D finite domains. Most of these solutions imply simplified electrical boundary conditions at the crack faces, assuming either an isolating or fully electrically permeable behavior, whereas real cracks exhibit limited permeability (dielectric medium inside). Therefore, numerical methods as finite elements (FEM) are required to analyze realistic 3D crack configurations and loading situations in smart materials and components. The aim of the present paper is to summarize the recent progress in FEM to solve efficiently 3D thermoelectromechanical crack problems and to give recommendations for applications. 2. Fundamental Equations of Thermoelectromechanical Field Coupling

The basic field variables and governing equations for a coupled thermoelectromechanical boundary value problem are summarized in Table 1. For the stationary case, CAUCHY’s equilibrium equation, GAUSS’ law of electrostatics and the stationary heat conduction equation provide the fundamental system of differential equations for the field variables ui ,M and T . Table 1: Basic relations of thermoelectromechanics Mechanics primary variables kinematical relations

displacement ui

Electrostatics electric potential ǂ

strain

electric 1 (u 2 i, j

H ij

dual stress V ij variables balance V ij, j  bi 0 equations in volume forces bi domain V balance V ij n j Ti equations at boundary S tractions Ti (normal vector ni ) constitutive law (1)

V ij

wg / wH ij

field temperature gradient

M,k

 u j,i ) Ek

Thermodynamics temperature ǂ

gk

T ,k

el. displacement Di

heat flux hi

Di,i  Zv

hi,i

0

Zs

surface charges

0

Zv

volume charges

Dj nj

 kij T ,ij

hjn j

Zs

 hs

heat transfer hs

K  wg / wT cijkl H kl  ekij Ek  Oij enkl H kl  N nm Em  pn Oij H ij  pi Ei  D T

Here cijkl , ekij ,

Dn

 wg / wEn

Oij , N ij , pi are the elastic constants, piezoelectric modulus,

133

temperature stress coefficients, dielectric constants, pyroelectric constants, respectively. 4 denotes the absolute temperature, T 4  4 0 is the temperature change from a stress free reference temperature 4 0 and kij are the heat conduction coefficients. K refers to the (volume-) specific entropy and D is the specific heat capacity per absolute temperature. As thermodynamic potential the “free electric enthalpy” or “electric GIBBS energy” is introduced with the state variables H ij , Ei and T .

g (H ij , Ei ,T )

1C H H 2 ijkl kl ij

 1 N ij Ei E j  enij EnH ij  OijT H ij  pi Ei T  1 D T 2 . (2) 2

2

For most materials the inverse thermoelastic and pyroelectric effects are very weak, i.e. the heat generationKfrom (1) can be neglected. Therefore, the studies are restricted to a piezoelectric solid under additional thermal loading, whereas the reaction of mechanical and electrical fields to thermodynamics can be decoupled. In a finite element context, the system of balance equations is solved by a weak formulation, applying the principles of virtual displacements, potential and temperature [4,5,6] together with corresponding kinematical and natural boundary conditions.

G Wu G WM G WT









³ G ui V ji, j  bi dV  ³ G ui Ti  V ji n j dS

V

St

³ GM ( Di,i  Zv ) dV  ³ GM Zs  Di ni dS

V

.

(3)

SZ

³ GT  hi,i  TK dV  ³ GT  hs  hi ni dS

V

0

0

0

S qs

This results in the following algebraic system for the corresponding nodal variables. ª K uu « «KM u « ¬ 0

K uM KMM 0

K uș º ­ u ½ »° ° KM ș » ®M ¾ »° ° K șș ¼ ¯ ș ¿

­ fu ½ ª K uu ° ° ®fM ¾ Ÿ « «¬ KM u ° ° ¯ fș ¿

K uM º ­ u ½ »® ¾ KMM »¼ ¯M ¿

°­ fu  K uș ș °½ ®f  K ș ¾ Mș ¿ ° ¯° M

­ '½ ° fu ° . ® '¾ °¯fM °¿

(4) From (4) it becomes clear that the stationary heat conduction problem can be separately solved and the thermal loading is shifted to the right hand side [5]. 3. Thermopiezoelectric Fracture Mechanics

The foundations of fracture mechanics for piezoelectric and partly thermopiezoelectric materials can be found in [1,2,3]. At each point along the crack front the asymptotic electrical and mechanical fields are controlled by the three mechanical stress-intensity factors KI, KII, KIII and the fourth “electric displacement

134

intensity factor” KIV, whereby (r,E,x3)= (x1,x2,x3) is the crack tip coordinate system, see Figure 1. The generalized tractions on the ligament ahead of the crack (E=0) have the singularity

t

1

^V 12 ,V 22 ,V 32 , D 2 `T

r

f K

K {K II , K I , K III , K IV }T ,

whereas the discontinuities of displacements ui and electric potential crack faces behave like

'u

u( r, E

S )  u( r, E

S )

^'u1, 'u2, 'u3, 'M`T

8r

S

(5)

M across the YK .

(6)

Figure 1. Crack front coordinate system and singular field quantities

The energy release rate G is defined as the difference of total electric and mechanical potential energy ǻȆ in a structure, if the crack grows by an area ǻA

1 T (7) K YK . 2 'Ao 0 Gm and Ge denote the mechanical and electric terms of G . The relation to the G

 lim ( '3 'A)

Gm  Ge

135

intensity factors is given by the generalized IRWIN-matrix Y, which depends on the elastic, piezoelectric and dielectric material constants, and on crack orientation to materials axes. tem

-integral-vector is a generalization of The thermoelectromechanical J k CHEREPANOV-RICE theory [9] to cracks in thermoelectromechanical materials. It em represents the extension of the known J k -integrals for 2D and 3D piezoelectric materials (see [1-3,10]) to additional thermal loading. The integration contour *completely encloses the crack tip. The area integral over A accounts for volume forces and charges as well as sources caused by thermal loading. Possible   mechanical Ti and electrical Zs loadings on the crack faces * and * are allowed for by the last integral term. Q jk

ƣ G jk  V ijui k D j M,k means the

generalized energy-momentum tensor Q jk .

Jktem





³ Q jk n j dS  ³ bi ui,k  Zv M,k Oij Hij T ,k  pi Ei T ,k dA

*



A

³

* *

 gˆ nk  Ti ui,k Zs Ek ds

g ƣ

(8)

 1 DT 2 2

tem The x1-component J1 G has the physical interpretation of the energy release rate. In the absence of body loads, thermal loading and material inhomogeneities, tem

the area integral vanishes and the remaining contour integral J k becomes pathindependent. Without any electrical terms, (8) simplifies to the classical thermoelastic J-integral expressions derived in [11,12]. 4. Finite Element Techniques for 3D Crack Analysis

Finite element analysis of piezoelectric structures under combined mechanical and electric loadings is meanwhile available in several commercial codes. However, the treatment of cracks requires some special techniques. Many of the numerical algorithms, developed for 3D crack analyses in pure elastomechanics (see e. g. [13]), can be generalized or adapted to the coupled electromechanical problem. The achievements are also based on experience with various FEM techniques for twodimensional electromechanical crack analysis, pioneered by [14,15,16].

136

Figure 2. Crack front segments discretized with 3D quarter-point elements

4.1 SINGULAR ELECTROMECHANICAL CRACK TIP ELEMENTS (CTE) In order to calculate accurately the intensity factors, a proper representation of the crack tip singularity (5), (6) in the finite element approach is necessary. For this purpose special crack tip elements with built-in singular shape functions are created by a particular distortion of regular hexahedron (20 noded) or pentahedron (15 noded) elements with quadratic shape functions into so-called quarter-point elements [13,16,18]. Then, the crack front is discretized by groups of concentric pentahedron elements as illustrated in Figure 2 for the segment AG . The intensity factors K are obtained by comparing the analytical crack tip solution to the FEM solution in the singular CTE elements. Thereby, the discontinuities of the nodal variables (6) across the faces ( 'u B indicates node pair B and B’, etc.) of the crack tip elements are interpreted [18]. The parameter

]  ^1, 1` is the local

element coordinate along the crack front. L denotes the length of the element edge

AC or GF . K (] )



] 2

S 8L

Y 1 {2'u B  'uC  2'u E  'u F  'u D

( 4 'u B  ' u C  4 ' u E  ' u F ) 

] 2

.

2

(9)

( 'u F  'uC  2'u D )}

4.2 MODIFIED CRACK CLOSURE INTEGRAL (MCCI) Modified crack closure integrals are well suited to compute the electromechanical energy release rate G for 2D [14,16] and 3D [17,20] crack problems. Assuming an arbitrary shaped virtual advance of the crack front around the actual position w, G equals to the electromechanical work per area 'A during a crack closure process.

137

G

1 2 'A

T

c * ³ ¬ªt ( r, 0, x3 )  t ( r ,  S , x3 )¼º 'u( r ,  S , x3 ) dA 'A

GmI  GmII  GmIII  GeIV . (10)

*

* * * Here, the tractions t {V 12 ,V 22 ,V 32 , D2* }T correspond to the situation at the T ligament before crack growth A A0 , whereas 'u {'u1, 'u2 , 'u3 , 'M }

denotes the jump across the crack faces after crack growth A

A0  'A at the

position r 'a  r . In the finite element realization of (10), G is approximated by multiplying the nodal forces Fkj and charges Z j in front of the crack tip with the corresponding nodal displacements 'ukj and electric potential 'M j behind the crack tip, see Figure 2. The first index (k) denotes the coordinate direction and the second one (j) the corresponding pair of nodes. This procedure requires only one FEM calculation and can be utilized with regular [20] or singular [17] elements. There exist various recommendations to weight the contribution of every node j. Equation (10) holds also for thermally loaded cracks as well as the crack face c loading t {T1,T2 ,T3, D2c }T , which may be due to bridging effects or semipermeability. 4.3 THERMOELECTROMECHANICAL J-INTEGRAL AS EQUIVALENT DOMAIN INTEGRAL tem

, it is advantageous to transform (8) into an For the numerical computation of J equivalent integral over the domain V as illustrated in Figure 3. For the 3D case, a vectorial weighting function qk is introduced, which equals to the virtual crack extension lk at the crack front Lc , but falls down to zero at the outer surface S and Send . Finally, the local crack front value is found by G  w see [10]. J tem

 ³ Qkj qk , j dV  ³ Qk , j qk dV V

V



³

S S

 ³ ª¬ gˆG kj  V ij ui,k  D j M,k º¼ qk ,i dV V

J

tem

/ 'A ,

Qkj n j qk dA



³

S S

ª¬ g nk  Ti ui,k  Zs M k º¼ qk dA

 ³ ¬ªbi ui,k  Zv M,k Oij H ijT ,k  pn EnT ,k ¼º qk dV

(11)

V

All FEM-techniques have been verified by various 2D and 3D testing examples with known analytical solutions, see [10,16,18,20].

138

Figure 3. Virtual crack extension along the crack front and J-integration domain

5. Verification and Application

5.1 SEMI-ELLIPTICAL SURFACE CRACK PROBLEM

Figure 4. Finite element mesh for the semi-elliptical surface crack (1/4 model shown)

A typical defect configuration in smart structures is a semi-elliptical surface crack f subjected to uniform normal traction V 22 1MPa and electric surface charge

139

D2f 0.001Cm2 . The crack front position is described by the parametric angle I of the ellipse with its major and minor semi-axes c and a=0.2c, see Figure 4. The piezoelectric material PZT-5H (12) is used with poling perpendicular to the crack [20]. Figure 4 shows a representative FEM mesh, consisting of 976 elements and 4729 nodes. Figure 5 depicts the intensity factors KI, KIV computed by CTE (9) in f dimensionless form K I g I ( a, c,I ) V 22 S a , K IV g IV ( a, c,I ) D2f S a . The distribution of G achieved by the different FEM-techniques CTE, J-integral and MCCI is given in Figure 6.

Figure 5. Semi-elliptical surface crack: Geometry functions for KI- and KIV-factors

Figure 6. Semi-elliptical surface crack: Energy release rate G for various FEM-techniques

140

5.2 PENNY-SHAPED CRACK PROBLEM UNDER THERMAL LOADING A penny-shaped crack embedded in an infinite thermopiezoelectric body has been numerically analyzed under thermal loading [19]. The temperature field has a prescribed value of T 0 100 ºC at the crack faces and fades away to zero at infinity. There exists a closed form solution [5] for the inhomogeneous steady state temperature field and the intensity factors KI and KIV. The finite element model consists of 20-node brick elements, see Figure 7. The elastic, dielectric and piezoelectric material parameters of PZT-5H are used together with selected thermal and pyroelectric properties: c11 126.0, c12 55.0, c13 53.0, c33 117.0, c44 35.3 GPa 6.5,

e31

23.3,

e33 8

N11 1.51 u 10 ,

e15

17.0 C/m 2 ,

N 33 1.30 u 10 6

O11 1.97382 u 10 ,

8

C

2 6

cT 2

Nm ,

k11 2

O33 1.4165 u 10 N Km ,

50, p3

62.244 GPa, k33

.

75 W Km ,

5.4831 u 106 C Km 2 (12)

T 

x2

x3

U T

I

x T  Figure 7. Embedded penny-shaped crack with cooled crack surfaces

At first, the stationary heat conduction analysis is performed. The calculated temperature field agrees better than 0.5% with the analytical solution [5]. The normalized stress intensity factors g I K I ( O33T0 S a ),

g IV G0

K IV ( p3T0 S a ) and the energy release rates G G0 with (O332 T 02S a ) cT are listed in Table 2. The finite element techniques involved

are: displacement extrapolation from crack tip elements CTE and regular standard elements RSE, direct extraction from CTE with (9) and the crack closure integral MCCI. One can see that the numerical results using different finite element techniques are rather close to the analytical solution [5]. Only the estimation of G by MCCI gives unacceptable accuracy, which has to be investigated further by tem means of J -integral. In all other cases, G was obtained from the IRWIN relationship (7).

141

Table 2: Dimensionless K-factors and G for thermally loaded penny-shaped crack

Exact solution CTE direct extraction, error (%) CTE displacement extrapolation, error (%) RSE displacement extrapolation, error (%) MCCI crack closure integral, error (%) 0.03

gI

g IV

G G0

0.199 +0.82 +2.65 -1.16

10.830 -4.83 -5.82 -7.89

2.394E-2 +1.2 +0.1 -2.8 -17.8

0.0025

KIV [Cm-3/2]

KIV [Cm-3/2]

0.0020

0.02

0.0015

0.01 0.0010

0.0005

0 0

0.5

0

1

40

60

80

100

120

σ22∞ [Nmm-2]

D2∞ [Cm-2]

Figure 8. KIV as function of loads

20

f and f for a semi-permeable penny-shaped crack D2 V 22

5.3 SEMI-PERMEABLE PENNY-SHAPED CRACK In order to take into account the effect of a dielectric medium inside the crack, the capacitor analogy was suggested by [21]. In this model, two opposite segments of the crack faces are considered as parallel plates of a capacitor. The whole crack interior is represented by a set of such capacitors. The electric field vector E2 depends on the potential difference 'M

and the local crack opening

displacement 'u2

E2c ( x1, x3 )



'M ( x1, x3 ) 'u2 ( x1, x3 )

,

D2c ( x1, x3 ) N c E2 ( x1, x3 )  Zs .

(13)

c The electric displacement D2 is expressed by electric permittivity N c of the medium inside the crack. It can be regarded as intrinsic crack face charge Zs . To

achieve the consistency (13) between the fields 'u2 , 'M and D2c , an iterative procedure (ICA) has to be carried out. This means, the displacements and potentials c computed by FEM in each iteration step are used to compute the new values of D2 with (13). They are transformed into adequate nodal charges to be applied to the

142

crack faces in the next iterative FEM step until convergence is reached. This technique was developed in [22] to treat 2D semi-permeable cracks and generalized to 3D problems in [23]. In the following, results for a semi-permeable pennyshaped crack in barium titanate (poled perpendicular to the crack) are shown. The f and f . In Figure 8 (left) the electric crack is exposed to remote loading V 22 D2 f =20 MPa is fixed, displacement intensity factor KIV is plotted versus D2f while V 22 f is varied at constant which shows the expected linear behavior. If contrary, V 22

D2f =0.02 C/m2 (right), the KIV-factor is weakened since the permeability is decreasing with the crack opening. 6. Conclusions

The developed FEM techniques provide a powerful tool to analyze 3D crack problems in thermopiezoelectric materials, covering a wide range of general crack configurations and loading situations encountered in smart materials and structures. Special emphasis is devoted to extend the techniques to thermal loading by stationary temperature fields. Also shown is the treatment of limited crack permeability. All techniques can be easily implemented as post-processors for standard FEM codes, whereby the J-integral requires the greatest effort and skill. The singular piezoelectric crack tip elements CTE yield most accurate predictions of electric and mechanical intensity factors compared to both other techniques. The implemented J-domain integral delivers the total energy release rate with the highest precision and should be combined with CTE. Its path independence allows for inherent error control. Good accuracy for determining the energy release rate can be obtained by the modified crack closure integral technique with standard finite elements. Compared with the J-domain integral method, the MCCI-technique has the advantage that the energy release rate can be separated into the different mechanical opening modes (I, II, III) and the electric mode (IV). Acknowledgement

The paper summarizes results of recent collaborative work with M. Abendroth, A. Ricoeur, M. Scherzer, K. Wippler and guest scientist F. Shang, which is gratefully acknowledged. References 1. 2. 3.

Y.E. Pak,1992, Linear electro-elastic fracture mechanics of piezoelectric materials. Int. J. Fracture, 54, 79-100. Q.H. Qin, 2001, Fracture Mechanics of Piezoelectric Materials. WIT Press Southampton, Boston. T.Y. Zhang, M.H. Zhao, P. Tong, 2002, Fracture of piezoelectric ceramics. Advances in Appl. Mechanics, 38, 147-289.

143 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

S.W. Yu, and Q.H. Qin, 1996, Damage analysis of thermopiezoelectric properties: Part I – crack tip singularities. Theoretical and Applied Fracture Mechanics, 25, 263-277. F. Shang, Z.K. Wang, Z.H. Li, 1996, Thermal stresses analysis of a three-dimensional crack in a thermopiezoelectric solid. Engineering Fracture Mechanics, 55, 737-750. H.S. Tzou, R. Ye, 1994, Piezothermoelasticity and precision control of piezoelectric systems: theory and finite element analysis. Journal of Vibration and Acoustic, 116, 489-495. A. Görnandt, U. Gabbert, 2002, Finite element analysis of thermopiezoelectric smart structures. Acta Mechanica, 154, 129-140. F. Shang, M. Kuna, M.Scherzer, 2002, A finite element procedure for three-dimensional analyses of thermopiezoelectric structures in static application. Technische Mechanik, 22, 235-243. G.P. Cherepanov, 1977, Invariant -integrals and their application in mechanics (in Russian), Prikladnaja Matematika i Mechanika, 41, 399-412. M. Abendroth, U. Groh, M. Kuna, A. Ricoeur, 2002, Finite element computation of the electromechanical J-Integral for 2D and 3D crack analysis. Int. J. of Fracture, 114, 359-378. W.K. Wilson, I.W. Yu, 1979, The use of the J-integral in thermal stress crack problems, Int. J. of Fracture, 15, 377-387. C.F. Shih, B. Moran, T. Nakamura, 1986, Enegy relase rate along a three-dimensional crack front in a thermally stressed body, Int Journal of Fracture, 30, 79-102. M.H. Aliabadi, D.P. Rooke, 1991, Numerical Fracture Mechanics. Computational Mechanics Publications, Kluwer Academic Publisher, Southampton/Dordrecht/Boston/London. S. Park, C.T. Sun, 1995, Fracture criteria for piezoelectric ceramics. J American Ceramics Soc., 78, 14751480. S. Kumar, R.N. Singh, 1996, Crack propagation in piezoelectric materials under combined mechanical and electrical loadings. Acta Materialica, 44, 173-200. M. Kuna, 1998, Finite element analyses of crack problems in piezoelectric structures. Computational Material Science, 13, 67-80. G. Kemmer, 2000, Berechnung von elektromechanischen Intensitätsparametern bei Rissen in Piezokeramiken, Thesis TU Dresden, Fortschritt-Berichte, VDI Verlag Düsseldorf. M. Kuna, A. Ricoeur, 2000, Theoretical investigation on the cracking of ferroelectric ceramics. In: C.S. Lynch editor. Smart Structures and Materials, SPIE, Vol. 3992, 185-196. F. Shang, M. Kuna, M. Scherzer, 2002, Analytical solutions for two penny-shaped crack problems in thermopiezoelectric materials and their finite element comparison. Int. J. Fracture, 117, 113-128. F. Shang, M. Kuna, M.Abendroth, 2003, Finite element analyses of three-dimensional crack problems in piezoelectric structures. Engineering Fracture Mechanics, 70, 143-160. T. Hao, Z. Shen, 1994, A new electric boundary condition of electric fracture mechanics and its applications. Engineering Fracture Mechanics, 47, 793–802. H. Balke, G. Kemmer, J. Drescher, 1997, Some remarks on fracture mechanics of piezoelectric solids. In: Proceedings of Micromaterials. DVM Berlin, 398–401. K. Wippler, A. Ricoeur, M. Kuna, 2004, Towards the computation of electrically permeable cracks in piezoelectrics. Engineering Fracture Mechanics, 71, 2567-2587.

TREFFTZ PLANE ELEMENT OF PIEZOELECTRIC PLATE WITH p-EXTENSION CAPABILITIES

Qing-Hua Qin Department of Engineering, The Australian National University, Canberra, ACT 0200, Australia

Abstract

This paper is concerned with the development of Trefftz p-element for twodimentional piezoelectric materials. Solutions in Stroh formalism for transversely isotropic piezoelectric materials are used for the intra-element displacement and electric potential (DEP) fields together with an independent DEP frame function field along element boundaries. The formulation is based on a modified variational functional in which electric field and strain are taken as basic variables. The final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom (DOF) at corner nodes and an optional number of hierarchic DOF associated with the mid-side nodes. Some numerical examples are considered to show the application of the proposed formulation. 1. Introduction

During the past decades the hybrid-Trefftz finite element (FE) model, originating in 1977 [1], has been considerably improved and has now become a highly efficient computational tool for the solution of complex boundary value problems. Up to now, T-elements have been successfully applied to problems of elasticity, Kirchhoff plates, moderately thick Reissner-Mindlin plates, thick plates, general 3D solid mechanics, axisymmetric solid mechanics, potential problems, shells, elastodynamic problems, transient heat conduction analysis, geometrically nonlinear plates, materially nonlinear elasticity, and piezoelectric materials. Most of these developments can be found in [2,3]. Furthermore, the concept of special purpose functions has been found to be of great efficiency in dealing with various geometry or load-dependent singularities and local effects (e.g., obtuse or reentrant corners, cracks, circular or elliptic holes, and concentrated loads) [2]. This work aims at developing a Trefftz finite element, p-element, for modeling twodimensional piezoelectric material. Solutions in Stroh formalism for transversely isotropic piezoelectric materials are used for the intra-element DEP. The modified variational functional used was based on a free energy density with strain and 144 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 144–153. © 2006 Springer. Printed in the Netherlands.

145

electric fields as independent variables. The stationary conditions of the variational functional and the theorem on the existence of extremum are discussed. Numerical results are presented to show the applicability of the proposed formulation. 2. Governing Equations and Their Trefftz Functions

2.1 BASIC FUNCTIONS In this section we recall briefly the governing equations of linear piezoelectricity. The summation convention is invoked over repeated indices. For convenience, matrices are represented by bold face letters and a comma after a variable implies differentiation with respect to Cartesian coordinates. Then, for a linear piezoelectric material, the differential governing equations of linear piezoelectric material in the Cartesian coordinates xi (i=1, 2, 3) are given by V ij , j  bi 0, Di,i  qb 0 in : (1) where Vij is the stress tensor, Di is the electric displacement vector, bi is the body force vector, qb is the electric charge density, : is the solution domain. For an anisotropic piezoelectric material, the constitutive relation is [3]

Vij

wH (İ, E) wHij

where 2 H (İ, E)

E cijkl H kl  ekij Ek , Di



wH (İ, E) wEi

eikl H kl  NikH Ek

(2)

E E is the stiffness cijkl H ij H kl  N ijH Ei E j  2ekij H ij E k , cijkl H

coefficient tensor for E=0, N ij the permittivity constant matrix for H=0, H ij and Ei are, respectively, the elastic strain tensor and the electric field intensity vector, ekij is piezoelectric stress constants. The relation between the strain tensor and the displacement, ui, is given by

2H ij

(u i , j  u j ,i )

and the electric field components are related to the electric potential I by E i  I ,i .

(3) (4)

The boundary conditions of the boundary value problem (1)-(4) are given by: u i u i on *u, ti Vij n j ti on *t, I I on *I, Dn Di ni q n Dn on *D (5) where u i , t i , q n and I are, respectively, prescribed boundary displacement, traction vector, surface charge and electric potential. An overhead bar denotes prescribed value, *=*u+ *t =*D +*I is the boundary of the solution domain :. Moreover, in the Trefftz finite element formulation, equations (1)-(5) should be completed by the following inter-element continuity requirements:

146

u ie t ie  t if

u if , I e

I f , (on *e ˆ *f , conformity),

0 , Dne  Dnf

0 (on *e ˆ * f , reciprocity)

(6) (7)

where ‘e’ and ‘f ’ stand for any two neighboring elements. Equations (1)-(7) are taken as the basis to establish the modified variational principle for Trefftz finite element analysis of piezoelectric materials. 2.2 TREFFTZ FUNCTIONS The Trefftz function plays an important role in the derivation of Trefftz finite element formulation. A complete system of homogeneous solutions to equation (1) can be generated in a systematic way from the Stroh formalism technique [4] u 2 Re{A f ( zD ) c} (8) where ‘Re’ stands for the real part of a complex number, A is the material eigenvector matrix which was well defined in the literature [4], f ( z D ) diag[ f ( z1 ) f ( z 2 ) f ( z 3 ) f ( z 4 )] is a diagonal 4u4 matrix, while

f ( z i ) is an arbitrary function with argument z i

x1  pi x 2 . pi (i=1-4) are the

material eigenvalues. Of particular interest is a complete set of polynomial solutions which may be generated by setting in equation (8) in turn (9) f ( z D ) z Dk , f ( z D ) iz Dk , (k 1,2,")

 1 . Thus the Trefftz functions of equation (1) can be given from the

where i following

f

u

¦ [ Re{A

z Dj }a j  Re{A iz Dj }b j ] .

(10)

j 1

3. Elemental Stiffness Formulation

3.1 ASSUMED FIELDS The Trefftz finite element model is based on assuming two distinct DEP fields: the ~ [2]. The field u fulfills identically the internal field u and the frame function u governing differential equations (1) and is assumed as

u

­ u1 ½ ° ° ®u 2 ¾ °I° ¯ ¿

 ­ u1 ½ ­ N 1 ½ ° ° ° ° ®u 2 ¾  ®N 2 ¾c ° I ° °N ° ¯ ¿ ¯ 3¿

 u  ¦ N jc j

 u  Nc

(11)

j 1

for a two-dimensional problem, where ci stands for undetermined coefficient, and    T  u (= {u1 , u 2 , I} ) and Nj are known functions. If the governing differential equation (1) is rewritten in a general form

147

ƒu ( x )  b ( x )

(x  : e )

0,

(12)

where ƒ stands for the differential operator matrix for equation (1), x for the T

position vector, b (={ {b1 , b2 , qb }

for the known right-hand side term, the

overhead bar indicates the imposed quantities and : e stands for the eth element



sub-domain, then u that

 u(x) and N  ƒu  b

N(x) in equation (11) have to be chosen so

0 and ƒN

0

(13) everywhere in : e . Thus Nj in (11) can be formed by a suitably truncated complete system of (10). For example, we may set

N2 j

2 Re{A z Dj } , N 2 j 1

2 Re{A iz Dj } .

(14)

The unknown coefficient c may be calculated from the conditions on the external boundary and/or the continuity conditions on the inter-element boundary. Thus various Trefftz element models can be obtained by using different approaches to enforce these conditions. In the majority of cases a hybrid technique is used, whereby the elements are linked through an auxiliary conforming displacement frame which has the same form as in the conventional finite element method. This means that, in the Trefftz finite element approach, a conforming DEP field should be independently defined on the element boundary to enforce the field continuity between elements and also to link the coefficient c, appearing in equation (11), with the nodal DEP d (={d}). The frame is defined as

~ ( x) u

~ ­ N1 ½ °~ ° ®N 2 ¾d ~ ° °N ¯ 3¿

­ u~1 ½ °~ ° ®u 2 ¾ ° °~ ¯I¿

~ Nd,

(x  *e )

(15)

for a two-dimensional problem, where the symbol “~” is used to specify that the field is defined on the element boundary only, d = d(c) stands for the vector of the nodal displacements which are the final unknowns of the problem, *e represents the

~

boundary of element e, and N is a matrix of the corresponding shape functions which are similar to those in the conventional finite element formulation. In the development of the present p-element, the following assumptions are adopted. First of all, the problem is assumed to be plane strain of transversely isotropic piezoelectric solid referred to a Cartesian system (x1, x2) [5]. Secondly, the element may be of a general quadrilateral shape or a triangle shape with three degrees of freedom (DOF) ( u1 , u 2 , I) at each corner node (see Figure 1). Thirdly, to achieve higher order variations, an optional number of extra hierarchic modes is introduced ~ along with the hierarchic DOF, a i for u~1 , bi for u~2 , pi for I, which are conveniently associated with the mid-side node C (see Figure 1). Thus, along the

148

side A-C-B of a particular element (see Figure 1), a simple interpolation of the frame DEP field can be given in the form

­ u~1 ½ °~ ° ®u 2 ¾ °~ ° ¯I¿ M

where

d AB

~ ª N1 « «0 «0 ¬

~ 0 N2 0 0 º ­ aCi ½ M » ~ ° ° i 1 0 0 N 2 0 »d AB  ¦ J Ri ® bCi ¾ ~ ~ i 1 °p ° N 1 0 0 N 2 »¼ ¯ Ci ¿

0 ~ N1 0

is

the

order

of

the

hierarchical

(16)

DOF,

{u1 A , u 2 A , I A , u1B , u 2 B , I B }T . The shape functions are 1 [ ~ 1 [ ~ i 1 2 N1 , N2 , Ri [ (1  [ ) 2 2

and

(17)

where [ is defined in Figure 1. G nDA

D

:e

… 3 DOF

…

'

M DOF

'

… A

G n AB

G nBC

' C

x1

… B

'

*e

1

'

x2

…

'

n2

G n

n1

A … [ 1

C ' [ 0

B … [ 1

…

' …

Y

…"3 DOF (u1 , u2 , I) '"variable number of DOF(aC1 , bC1 , pC1 ,")

X O Figure 1. The Trefftz p-element

The coefficient J is equal to +1 or -1 according to the orientation of the side A-C-B (see Figure 1) in the global coordinate system (X,Y):

J

­ 1 if X B  X A d YB  Y A . ® ¯ 1 if X B  X A ! YB  Y A

(18)

The purpose of using the coefficient J is to ensure a univocal definition of the frame ~ in terms of parameters a , b and p , common to two elements sharing functions u i i i the mid-side node Z. Using the definitions in (2), (5)2 and (5)4 the generalized boundary forces and electric displacements can be given as

T

­ t1 ½ ° ° ® t2 ¾ °D ° ¯ n¿

­ V1 j n j ½ ° ° ®V 2 j n j ¾ °D n ° ¯ j j¿

 ­ t1 ½ ­Q1 ½ ° ° ° ° ® t 2 ¾  ®Q 2 ¾c ° D ° °Q ° ¯ n¿ ¯ 3¿

 T  Qc ,

149

~ ­ t1 ½ °~ ° ® t2 ¾ ~ ° °D ¯ n¿

~ T







~ ­Q1 ½ °~ ° ®Q 2 ¾c ~ ° °Q ¯ 3¿

~ Qc

(19)

where t i and Dn are derived from u . 3.2 PARTICULAR SOLUTIONS



The particular solution of u can be obtained by means of their Green’s functions. The Green’s functions of (1) are as follows [5]

u ij* ( p, q )

f ½ (1) k 1 ( k ) 1 ­ (0) a ln r [aij cos(2kT)  bij( k ) sin( 2kT)¾ (20)  ® ij ¦ pq 4S ¯ k k 1 ¿

with

( xq  x p ) 2  ( y q  y p ) 2

rpq

(21)

*

where u ij ( p, q ) designates the Green’s function of the in-plane displacement (for I = 1,2) or electric potential (for i = 3) at field point p of an infinite plane when a unit (k )

point force is applied at the source point q in the j-direction. a ij calculated as: a

(k ) ij

(k ) ij

0 if i z j; k (k )

the non-zero values of a ij

0,1,2, ! , b (k )

and bij

0 if i

j; k

(k )

and bij

are

1,2, ! , and

can be found in Table 1 of Ref. [5].

Thus the particular solutions of (1) can be expressed as

    u {u1 , u 2 , I}T

where b3

³³

:

b j {u1*j , u 2* j , u 3* j }T d:

(22)

qb . The area integration in (22) is performed by numerical quadrature

using the Gauss-Legendre rule. 3.3 MODIFIED VARIATIONAL PRINCIPLE The Trefftz finite element equation for piezoelectric materials can be established by the variational approach [2]. Since the stationary conditions of the traditional potential and the complementary variational functional cannot satisfy the interelement continuity condition which is required in the Trefftz finite element analysis, some new variational functionals need to be developed. For this purpose, we present the following modified variational functional suitable for Trefftz finite element analysis:

150

¦3

3m

me

e

¦{3  ³ e

e

*Ie

~ ( I  I) Dn ds  ³

*ue

(u i  u i )~ ti ds (23)

~  ³ ( IDn  u~i t i ds} *Ie

where

3e

³³

:e

[ H (İ, E)  bi u i  q b I]d:  ³

*te

t i u i ds  ³

*De

Dn Ids

(24)

and equation (1) is assumed to be satisfied, a priori. The terminology “modified principle” refers here, to the use of a conventional functional ( ɉe here) and some modified terms for the construction of a special variational principle to account for additional requirements such as the condition defined in equations (6) and (7). The boundary *e of a particular element consists of the following parts:

*e

*ue ‰ *te ‰ *Ie

*Ie ‰ *De ‰ *Ie

(25)

where

*ue

*u ˆ *e , *te

*t ˆ *e , *Ie

*I ˆ *e , *De

*D ˆ *e ,

(26)

and *Ie is the inter-element boundary of the element ‘e’. We now show that the stationary condition of the functional (23) leads to equations (5)-(7),

(t i

~ ti on *u ) , ( Dn

~ Dn on *I ) , and present the theorem on the existence of

extremum of the functional, which ensures that an approximate solution can converge to the exact one. For the functional (23), we have the following two statements: (a) Modified complementary principle

G3 HmE

0 Ÿ (5)  (7) , (t i

~ ti on *u ) and ( Dn

~ Dn on *I )

(27)

where G stands for the variation symbol. (b) Theorem on the existence of extremum If the expression

³³ G :

2

~ G~ ti Gu i ds  ³ GDn GIds *u *I ~  ¦ ³ (GIGDn  Gu~i Gt i )ds

H (İ, E)d:  ³

e

(28)

*Ie

is uniformly positive (or negative) in the neighborhood of u 0 , where u 0 is such a value that 3 m (u 0 )

(3 m ) 0 , and where (3 m ) 0 stands for the stationary value

of 3 m , we have

3 m t (3 m ) 0 [or 3 m d (3 m ) 0 ] ~ u ~ is identical on * ˆ * has been used. in which the relation that u e f e f

(29)

PROOF: First, we derive the stationary conditions of functional (23). To this end,

151

performing a variation of 3 m and noting that eqn (1) holds true a priori by the previous assumption, we obtain

G3 m

~

~

~

³ [(u  u )G t  (t  t )Gu ]ds  ³ [(I  I)GD  ( D  ³ (t  t )Gu ds  ³ (D  D )GIds ~ ~  ¦ ³ [t G(u~  u )  D G( I  I)  u~ Gt  IGD ]ds i

*u

*t

e

i

i

i

*Ie

i

i

i

i

i

n

*D

i

i

i

n

*I

n

~  Dn )GI]ds . (30)

n

n

i

i

n

Therefore, the Euler equations for expression (30) are equations (5)-(7),

~ ( Dn Dn on *I ) , since the quantities ~ ~ Gt i , Gu i , GI, GDn , Gu~i , G~ ti , GDn and GI may be arbitrary. The principle (27) has

(t i

~ ti on *u )

,

and

thus been proved. This indicates that the stationary condition of the functional satisfies both the required boundary and inter-element continuity equations and can thus be used for deriving Trefftz finite element formulation. As for the proof of the theorem on the existence of extremum, we may complete it by way of the so-called “second variational approach” [6]. In doing this, performing variation of Gɉm and using the constrained conditions (1), we find

G23 m

~ H (İ, E)d:  ³ G~ ti Gu i ds  ³ GDn GIds *t *D ~ .  ¦ ³ (GIGDn  Gu~i Gt i )ds expression (28)

³³ G

2

:

e

(31)

*Ie

Therefore the theorem has been proved from the sufficient condition of the existence of a local extreme of a functional [6]. This completes the proof. 3.4 ELEMENT MATRIX EQUATION The element matrix equation can be generated by setting G3 me

0 . To simplify

the derivation, we first transform all domain integrals in (23) into boundary ones. In fact, by reason of the solution properties of the intra-element trial functions, the functional 3 me can be simplified to

3 me

1 1 ~ (tiui  Dn I)ds  ³ (biui  qb I)d:  ³ ( I  I) Dn ds ³ * : * Ie 2 e 2 . (32) ~ ~ ~  ³ (ui  ui ) ti ds  ³ ( Dn I  tiui )ds  ³ tiui ds  ³ Dn Ids *ue

*Ie

*te

*De

Substituting the expressions given in eqns (11), (16), and (19) into (32) produces

3 me

1 T c Hc  cT Sd  cT r1  dT r2  terms without c or d 2

(33)

in which the matrices H, S and the vectors r1, r2 are defined by

H

³

*e

Q T Nds

(34)

152

S

³

*Ie

~ N Q3ds  ³ T 3

ª N1 º *ue « N » ¬ 2¼

T

~ ª Q1 º T~ « ~ »ds  ³ *Ie Q Nds ¬Q 2 ¼

(35)

T

r1

r2

 ª N º ­t ½ 1 1  (N T T  Q T u)ds  ³ N T bd:  ³ N T3 Dn ds  ³ « 1 » ® 1 ¾ds ³ *De *te N 2 *e 2 : ¬ 2 ¼ ¯t 2 ¿ ~ T  ª N1 º ­ t1 ½ T  ~  ª Q1 º ­ u1  u1 ½ «~ » °  ° ~T Q ( ) ds ds I  I   ® ¾  » « ~ 3 ³ *Ie ³ *Ie ¬Q 2 ¼ ¯u2  u2 ¿ ³*Ie «N~ 2 » ® t2 ¾ds . « N 3 » ° Dn ° ¬ ¼ ¯ ¿

(36)

(37)

After a series of mathematical derivations and variational calculations, equation (33) leads to Kd P (38) where K=STH-1S and P=STg-r2 are, respectively, the element stiffness matrix and the equivalent nodal flow vector. The expression (38) is the elemental stiffnessmatrix equation for Trefftz finite element analysis. 4. Numerical Examples

Since the main purpose of this paper is to outline the basic principles of the Trefftz finite element method in piezoelectric materials, the assessment will be limited to two simple examples. In order to allow for comparisons with other solutions reported in reference [7], the obtained results are limited to a piezoelectric prism subjected to simple tension. Table 1 u1 of Trefftz BEM results at A for several values of M 1 2 3 4 M 10 -0.9670 -0.9671 -0.9671 -0.9671 u u 10 (m) 1

Consider a PZT-4 ceramic prism subjected to a simple tension P=10Nm-2 in the ydirection (see Figure 2 in ref. [3]). The properties of the material are the same as those in [7]. The boundary conditions of the prism are V yy P , V xy D y 0 on edges y rb

V xx

V xy

Dx

0

on edges x

ra

where a=3m, b=10m. Owing to the symmetry about load, boundary conditions and geometry, only one quadrant of the prism is modeled by 25 (x-direction) and 40 (ydirection) elements in the TFEM analysis. Table 1 shows the displacement at point A with several values of M. It is evident that the same result is obtained when M d 2.

153

5. Conclusion

A Trefftz finite element model with p-capabilities has been presented for analysis of plane piezoelectric plate. It includes a modified variational functional which are based on a free energy density with (İ, E) as the basic independent variables. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. The stationary conditions are displacement and electric potential conditions on the boundary, surface traction and surface charge condition, and inter-element continuity condition. Based on the assumed intra-element and frame fields as well as the variational functional, an element stiffness matrix equation is obtained which is implemented into the computer programs for numerical analysis. The numerical results obtained here are in excellent agreement with the analytical ones and show that they can converge to the exact solution quickly along with an increase in the order M of the p-element. Acknowledgement

The present work is sponsored by the Australian Research Council. References 1. 2. 3. 4. 5. 6. 7.

J. Jirousek, N.Leon, 1977, A powerful finite element for plate bending, Comp. Meth. Appl. Mech. Eng., 12, 77-96. Q.H. Qin, 2000, The Trefftz Finite and Boundary Element Method, WIT Press, Southampton. Q.H. Qin, 2003, Variational formulations for TFEM of piezoelectricity, Int. J. Solids Struct., 40, 63356346. Q.H. Qin, 2001, Fracture Mechanics of Piezoelectric Materials, WIT Press, Southampton. N. Khutoryansky, H. Sosa, W.H. Zu, 1998, Approximate Green’s functions and a boundary element method for electro-elastic analysis of active materials, Compu. & Struct., 66, 289-299. H.C. Simpson, S.J. Spector, 1987, On the positive of the second variation of finite elasticity, Arch. Rational Mech. Anal., 98, 1-30. H.J. Ding, G.Q. Wang, W.Q. Chen, 1998, A boundary integral formulation and 2D fundamental solutions for piezoelectric media, Comput. Meth. Appl. Mech. Eng., 158, 65-80.

ON PIEZOELECTRIC ACTUATOR LAYERS IN PLATES AND SHELLS AT LARGE DEFLECTIONS

Sven Lentzen1 and Rüdiger Schmidt2 Institute of General Mechanics RWTH Aachen University, Germany, [email protected] 2 Institute of General Mechanics RWTH Aachen University, Germany, [email protected] 1

Abstract

Several numerical examples are presented in which geometrical nonlinearity plays a considerable role when the actuation of piezolaminated plates and shells is considered. In each numerical example the comparison between linear and geometrically nonlinear approximations is drawn. For the numerical simulation of the actuation of piezolaminated shells a geometrically nonlinear finite shell element has been developed based on the total Lagrangian formulation. The straindisplacement relations are based on the assumptions of small strains and moderate rotations of the mid-surface normals. The displacement field in transverse direction is assumed to vary linearly according to the Reissner-Mindlin hypothesis. Keywords: geometrical nonlinearity, piezoelectric shells, actuators 1. Introduction

In modern engineering a big effort is made to reduce the weight of structures. Reducing the weight bears advantages which lead to lower manufacturing and operational costs and less required raw materials. However, light-weight structures tend to be more sensitive to static as well as dynamic instabilities. A solution to this problem without drastically changing the structural weight seems to be the implementation of smart materials to sense as well as to control the instabilities. An option which has been extensively investigated in recent years is the integration of piezoelectric patches in these mostly plate- or shell-like structures. In order to better understand the behaviour of these piezo-integrated structures many finite element methods have been developed. A large amount of papers can be found on geometrically linear so-called ’smart’ beams (see e.g. Crawley and Luis [1] and Robbins and Reddy [2]). For large deflections and vibration amplitudes geometrically nonlinear beam elements are developed (see e.g. 154 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 154–163. © 2006 Springer. Printed in the Netherlands.

155

Mukherjee and Chaudhuri [3], Chroscielewski et al.[4] and Wang and Varadan [5]). Piefort [6] has pointed out that the modelling of piezointegrated beams is not accurate enough when collocated systems are considered. Piezoelectric plate and shell elements seem to be more appropriate and versatile to model thin smart structures. As a result, many linear (e.g. Piefort [6], Lammering [7], Tzou and Tseng [8], Batra et al. [9], and Lee et al. [10]) as well as geometrically nonlinear (e.g. Yi et al. [11] and Chandrashekhara and Bhatia [12]) finite elements for piezolaminated plates and shells have been developed. It has been pointed out by the present authors [13], that the geometrical nonlinearities are more profound when the sensing properties of the piezoelectric layers are considered rather then its actuation properties. Generally, within a physically realistic range of actuation voltages, for most structures the induced deformations are not large enough to create a significant difference between the results of a geometrically linear and nonlinear approach. Nevertheless, for some structures, like e.g. a hinged beam or plate, or certain combinations of boundary conditions and geometrical dimensions large estimation errors occur if only linear theories are considered [4]. The present nonlinear finite shell element is based on the moderate rotation straindisplacement relations, developed by Schmidt and Reddy [14]. The displacement field as well as the electric field is assumed to vary linearly across the thickness. The actuators are assumed to be voltage driven, which leads to a decoupled electromechanical system. The finite element has been validated several times by comparison with results found in literature, e.g. [13] and [15]. In the present work, a short introduction is given into the applied fundamental equations. Next, a numerical example discussed by Kioua and Mirza [16] is taken to present the different results obtained by linear and nonlinear plate elements. The example is followed by another one in which a parameter variation is performed on a piezoactuated simply supported cylindrical shell. This example clearly distinguishes between ranges of parameter combinations affected by nonlinear effects and ranges which are not affected. 2. Fundamental Equations

States of equilibrium of an arbitrarily deformed configuration, denoted by the left 2 superscript 2, can be found by the condition that the internal virtual work įWi 2

equals the external virtual work įWe 2

įWi  2 įWe

0

(1)

As a continuation of earlier work by Kreja and Schmidt [17] and Palmerio et al. [18], a total Lagrangian approach is adopted. In this approach all quantities to be 2 calculated in the deformed configuration C referring to the initial undeformed

156 0

configuration C . The internal and external virtual work can be expressed as 2

2

GWi

GWe

³

0

³

0

V

V

0

0

GH 0 dV

with

U 0F iGVi 0 dV  ³

0

AV

0

0

H

0

s ij 0 H ij  0 D i 0 GEi

V iGVi 0 dAV  ³

0

AQ

0

and

(2)

D i 0 niGI 0 dAQ ,

(3)

where 0 H is the electric enthalpy, 0 s is the 2nd Piola-Kirchhoff stress tensor,

0

H

denotes the Green-Lagrange strain tensor, 0 D is the electric displacement vector

E is the electric field vector, respectively, referring to the initial configuration. Furthermore, 0 U denotes the mass density, 0 F is the body force and

0

V is the prescribed external stress, 0 D i 0 ni denotes the prescribed external charge and V and I are the displacement vector and the electric potential vector,

0

for each electrode pair, respectively. 2.1 STRAIN-DISPLACEMENT RELATIONS The nonlinear Green-Lagrange strain-displacement relation

1 Vi | j  V j | i  Vk | iV k | i , (4) 2 Vi g i ( gi and g i are the co- resp. contravariant base vectors in

H ij

with V

V i gi

space), can be approximated considering the following orders of magnitude [14]: 2 2 small strains H ij O T with T  1 ; small rotations about the normal



ZDE

O T 2 ; and moderate rotations of the normal ZDE

O T .

It is further assumed that the displacement field varies linearly across the thickness. The final strain-displacement relations are described by Schmidt and Reddy [14]. 2.2 LINEAR CONSTITUTIVE EQUATIONS The piezoelectric effect can be described by two constitutive equations, namely the direct and the converse piezoelectric effect: (5) ^ 0 D` > e@^ 0 İ`  > į@^ 0 E` T

(6) ^ 0 S` >c @^ 0 İ`  >e@ ^ 0 E`  where ^ 0 S ` denotes the stress vector, ^0 H ` the strain vector, ^ 0 D` the electric displacement vector and ^ 0 E` the electric field vector. The elasticity matrix is denoted by > c @ , >G @ is the dielectric constant matrix and > e @ > d @ > c @ , where

157

> d @ is the piezoelectric constant matrix. The converse piezoelectric effect describes the effect of an applied voltage on the additionally generated stresses in the material. If voltage controlled actuators are investigated, then it is sufficient to take only this equation into consideration. When charge controlled actuators or piezoelectric sensor are considered, then it becomes necessary to include the direct piezoelectric effect. In the present work, solely voltage controlled actuators are investigated, which decouples the electric quantities from the mechanical ones. 2.3 FEM FORMULATION The presence of the geometrical nonlinearity necessitates the application of an incremental finite element formulation [19]. After introducing the shape functions to interpolate all possible quantities, the internal virtual work can be written in an incremental finite element formulation as [17]:

įWi

T § ª1 ¨«0 ©¬

^įq`

˜

·

K u »º¼  «ª¬ 10 K g »º¼ ¸¹ ˜ ^q`  ­¯® 10 Fi ½¿¾ 

where ^q` is the nodal displacement vector,

ª1 «0 ¬

(7)

K u º»¼ denotes the first term of the

tangential stiffness matrix containing the mechanical constitutive stiffness terms, ª1 «0 ¬

K g º»¼ is the geometric part of the tangential stiffness matrix and

­1 ®0 ¯

Fi ½¾¿ is the

internal force vector. The additional piezoelectrically generated mechanical stresses are included in the terms

ª1 «0 ¬

K g º»¼ and

­1 ®0 ¯

Fi ¿½¾ . These additional stresses can be

calculated a priori, because voltage controlled actuators are considered. The linearly varying electric field between an electrode pair of a piezoelectric patch can be prescribed by one variable, namely the difference between the electric potentials between the upper and the lower electrode. The external virtual work is expressed in finite element formulation as:

įWe

T

^įq` ˜ ^ Fe ` 

(8)

where ^ Fe ` is the externally applied nodal force vector. By means of a NewtonRaphson [19] iteration method, points of equilibrium are found with the condition įWi įWe . 3. Numerical Examples

3.1 CANTILEVERED PLATE The first numerical example deals with a problem discussed by Kioua and Mirza

158

[16]. A cantilevered >302  0@s T300/976 graphite/epoxy composite plate is covered with a PZT G1195 layer on the upper and lower surface. The configuration is depicted in Figure 1 and the material properties can be found in Table 1.

3

2 1 a a l Figure 1. Cantilevered composite plate covered with piezoelectric layers

Table 1 Material properties

E11 (GPa) E22 (GPa) Ȟ12 G12 (GPa) d31 ( 10

10

PZT G1195 T300/976 PVDF Epoxy piezoceramic graphite/epoxy piezo film substrate 63 150 2.9 1.9 63

9

2.9

1.9

0.3

0.3

0.33

0.3

24.2

7.1

1.09

0.731

-

0.23

-

m/V) 2.54

Each composite ply is 0.138 mm thick and both PZT layers, with opposite poling directions, are 0.254 mm thick. The length a is chosen to be 25.4 cm. The effect of the length of the PZT layers l on the transverse tip deflection w2 at position ’2’ and the twist, defined as w3  w1 , is investigated if both PZT layers are loaded with 100 V. The comparison between the results of geometrically linear and nonlinear FE-analysis and the results obtained by Kioua and Mirza [16] are displayed in Figure 2.

159

deflection (100 x w2/a), twist (100 x |w3−w1|/a)

2

1.5

1

Kioua and Mirza (2000) linear moderate rotations

twist

0.5

deflection

0

−0.5 0

0.5

0.25

0.75

1

length of the PZT patch ( l /a) Figure 2. Tip deflection and twist of a composite cantilevered plate at 100 V actuation voltage

deflection (10 x w2/a), twist (10 x |w3−w1|/a)

2

linear moderate rotations 1.5

1 twist 0.5

0

−0.5 0

deflection

0.25

0.5

0.75

1

length of the PZT patch ( l /a) Figure 3. Tip deflection and twist of a composite cantilevered plate at 1000 V actuation voltage

A reasonable agreement is noticed between the linear and the referenced results. The difference is explained by the fact that Kioua and Mirza [16] used a Ritz method. Detailed comparisons of results obtained by the cited, the present and other

160

authors can be found in [15]. Nevertheless, it is interesting to see that the linear and moderate rotation approximations differ significantly. They agree well in terms of qualitative trends, but quantitative errors of up to 25% are predicted when only linear theories are considered. As an illustration of the increasing geometrically nonlinear effect at higher actuation voltages, the results at 1000 V actuation voltage is displayed in Figure 3. 3.2 SIMPLY SUPPORTED CYLINDRICAL SHELL The next example deals with a simply supported cylindrical epoxy shell covered with a PVDF film in the middle of the upper surface as depicted in Figure 4. The choice of the materials is based on those used by Moskalik and Brei [20]. They used the same materials for their C-block actuators. The material parameters can be found in Table 1.

R

a

b

Figure 4. Simply supported cylindrical epoxy shell covered with a PVDF film

The thickness of the epoxy substrate layer is kept constant at 0.1 mm. The thickness of the PVDF film as well as the surface area of the substrate layer covered by the actuator layer are varied. The lengths a and b are chosen to be 5 cm and 2.5 cm, respectively. The effect of the parameter variation on the mid-point transverse displacement at 1000 V actuation voltage for two different radii of curvature (2.5 cm and 25 cm) is displayed in Figures 5 and 6, respectively.

161

mid−point deflection [mm]

0.4

linear moderate rotations 0.2

0

−0.2 0 0.2 PVD 0.4 0.6 F thi 0.8 ckne 1 ss [m m]

20 40 60 100 80 ] strate layer [% covered sub

Figure 5. Parameter variation of a simply supported cylindrical epoxy shell ( R

0

25 cm)

mid−point deflection [mm]

0.2

linear moderate rotations

0.1

0 0 0.2 PVD 0.4 0.6 F thi 0.8 ckne 1 ss [m m]

20 40 60 100 80 ] strate layer [% covered sub

Figure 6. Parameter variation of a simply supported cylindrical epoxy shell ( R

0

25 cm)

Comparing the results of the deep and the shallow shell, it is noticed that the geometrically nonlinear effect is more pronounced when the shell is shallow. In both cases the difference between the geometrically linear and nonlinear results is the largest when the shell is thinnest. This can be explained by the fact that for

162

thinner shells the deformations are larger. Further reduction of the actuator layer thickness to zero would result into a sudden drop of the mid-point deflections to zero as well.

Figure 7. Deformed cylindrical epoxy deep shell: 40% covered, 0.1 mm PVDF, deformations multiplied by 10

Figure 8. Deformed cylindrical epoxy deep shell: 100% covered, 0.1 mm PVDF, deformations multiplied by 25

Looking at the parameter variation of the deep shell, another interesting phenomenon can be noticed. When the actuator layer gets thin and the covering percentage of the epoxy shell reaches 100%, the transverse direction into which the mid-point deflects changes from upwards to downwards. The deformed configurations at area coverages of 40% and 100% are shown in Figures 7 and 8, respectively. The deformations are multiplied by a factor 10 in the first case and a factor 25 in the latter. 4. Conclusions

A repeatedly tested geometrically nonlinear finite shell element has been used to numerically simulate the actuation of piezolaminated plates and shells. Geometrically nonlinear effects usually are more profound when the sensor abilities of the piezoelectric layers are calculated rather than when the actuation of these layers is considered. Nevertheless, examples are discussed in this work which clearly distinguishes between geometrically linear and nonlinear results of piezoelectrically actuated plates and shells. References 1. 2. 3. 4.

E.F. Crawley, J. de Luis, 1987, Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal, 25, 1373–1385. D.H. Robbins, J.N. Reddy, 1997, Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Computers & Structures. 41, 265–279. A. Mukherjee, A.S. Chaudhuri, 2002, Piezolaminated beams with large deformations. Int.J. of Solids and Structures. 39, 1567–1582. J. Chroscielewski, P. Klosowski, R. Schmidt, 1997, Modelling and FE-Analysis of large deflection shape and vibration control of structures via piezoelectric layers. Fortschritts-Berichte VDI, 11, 53–62.

163 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Q. Wang, V.K. Varadan, 2003, Transition of the buckling load of beams by the use of piezoelectric layers. Smart Mater. Struct., 12, 696–702. V. Piefort, 2001, Finite element modelling of piezoelectric active structures. PhD thesis, Universit Libre de Bruxelles. R. Lammering, 1991, The Application of finite shell elements for composites containing piezoelectric polymers in vibration control. Computers & Structures. 41, 1101–1109. H.S. Tzou, C.I. Tseng, 1991, Distributed vibration control and identification of coupled elastic/piezoelectric systems: finite element formulation and application. Mechanical Systems and Signal Processing. 5, 215–231. R.C. Batra, X.Q. Liang, J.S. Yang, 1996, Shape control of vibrating simply supported rectangular plates. AIAA Journal, 14, 116–122. S. Lee, N.S. Goo, H.C. Park, K.J. Yoon, C. Cho, 2003, A nine-node assumed strain shell element for analysis of a coupled electro-mechanical system. Smart Mater. Struct., 12, 355–362. S. Yi, S. F. Ling, M. Ying, 2000, Large deformation finite element analyses of composite structures integrated with piezoelectric sensors and actuators. Finite Elements in Analysis and Design, 35, 1–15. K. Chandrashekhara, K. Bhatia, 1993, Active buckling control of smart composite plates – finite-element analysis. Smart Mater. Struct., 2, 31–39. S. Lentzen, R. Schmidt, 2004, Nonlinear finite element modelling of composite structures with integrated piezoelectric layers. Proc. High Performance Structures and Materials II. 67–76. R. Schmidt, J.N. Reddy. 1988, A refined small strain and moderate rotation theory of elastic anisotropic shells. Journal of Applied Mechanics. 55, 611–617. S. Lentzen, R. Schmidt, 2004, Geometrically nonlinear composite shells with integrated piezoelectric layers. Proc. Appl. Math. Mech.. 4, in print. K. Kioua, S. Mirza, 2000, Piezoelectric induced bending and twisting of laminated composite shallow shells. Smart Mater. Struct., 9, 476–484. I. Kreja, R. Schmidt, 1995, Moderate rotation shell theory in FEM application. Zeszyty Naukowe Politechniki Gdanskiej (Research Transaction of Gdansk University of Technology). 522, 229–249. A. F. Palmerio, J. N. Reddy, R. Schmidt, 1990, On a moderate rotation theory of elastic anisotropic shells, Part II: FE analysis. Int. J. Non-Linear Mech., 25, 701–714. K.J. Bathe, 2002, Finite-Elemente-Methoden., Springer, Heidelberg. A.J. Moskalik, D. Brei, 1997, Deflection-voltage model and experimental results for polymeric piezoelectric C-block actuators. AIAA Journal, 35, 1556–1558.

ELECTRIC CHARGE LOADING OF A PIEZOELECTRIC SOLID CYLINDER

Y. Chen and R.K.N.D. Rajapakse Department of Mechanical Engineering The University of British Columbia Vancouver, Canada V6T 1Z4

Abstract

Piezoelectric elements are commonly used as sensors and actuators in adaptive structures. Among the many types of piezoelectric elements, solid cylindrical elements are used in a broad range of practical applications. This paper presents a theoretical study of a piezoelectric solid cylinder under electric charge loading applied to the ends. The material is assumed to be transversely isotropic and the general solution of the governing equations is obtained in terms of a Fourier-Bessel series containing Bessel and modified Bessel functions of the first kind. A boundary-value problem corresponding to uniform electric charge loading applied to the ends of a cylinder is solved by expanding the applied load in terms of a Fourier-Bessel series. Selected numerical results are presented to portray the basic features of the electroelastic field of a cylinder for different length-radius ratios of the cylinder and piezoelectric materials. 1. Introduction

Piezoelectric materials generate electric charge when subjected to mechanical loading. Conversely, they produce deformations under an applied electric field. Piezoelectric actuators are based on the converse piezoelectric effect and are used in modern engineering applications such as smart structures, ultra precision machining, telescopic mirrors, printer heads, etc. These actuators are available in a wide range of sizes and shapes and can be distributed along a structure without greatly increasing its mass. Among the many types of piezoelectric actuator and resonator elements, the cylindrical (solid and hollow) shape is used in a broad spectrum of practical applications. The study of electroelastic field in a piezoelectric cylinder under electric loading is therefore one of the fundamental problems in modern actuator technology. Stress analysis of elastic cylinders under various boundary conditions is one of the classical problems in elasticity and has a rich history. Wei and Chau [1,2] and Chau 164 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 164–174. © 2006 Springer. Printed in the Netherlands.

165

and Wei [3] reviewed the past studies on elastic cylinders and presented a new Fourier-Bessel series solution to analyse axisymmetric response of solid cylinders. Past studies on elastic cylinders lay a strong foundation for the advancement of theoretical analysis of piezoelectric cylinders. Rajapakse and Zhou [4] used Fourier integral transforms to derive a theoretical solution for a long piezoceramic cylinder subjected to axially symmetric electromechanical loading. Their solution examined the effects of coupling between mechanical and electric fields and zones of mechanical and dielectric field concentration. The main objective of this study is to analyze the coupled field in a piezoelectric solid cylinder due to electric charge loading applied to the ends. Rajapakse et al [5] recently derived the analytical general solution for an annular piezoelectric cylinder of finite length subjected to axisymmetic electromechanical loading. Their solution is specialized here for a solid cylinder and applied to solve a boundary value problem corresponding to uniform electric charge patch load applied to the ends. Selected numerical results for different piezoelectric materials and cylinder dimensions are presented and the salient features of the coupled field in a cylinder are identified. 2. Analytical Solution

Figure 1 shows a solid cylinder of radius b and height 2h . A cylindrical polar coordinate system r,T , z is used with the z-axis along the axis of symmetry of the cylinder. The cylinder is made out of a transversely isotropic piezoelectric material or a poled ceramic with the poling direction parallel to the z-axis. Loading is assumed to be axisymmetric and applied to the top and bottom surfaces. z b0

r D0

b

h

T x h

Figure 1. Piezoelectric finite cylinder subjected to electric charge loading at the top and bottom ends

166

The constitutive equations for piezoelectric materials which are transversely isotropic or poled along the z-axis can be expressed as (Parton and Kudryavtsev [6]), (1a) V rr c11Ƥrr  c12Ƥșș  c13Ƥ zz  e31 E z (1b) V TT c12Ƥrr  c11ƤTT  c13Ƥ zz  e31 E z (1c) V zz c13Ƥrr  c13ƤTT  c33Ƥ zz  e33 E z (1d) V rz 2c44Ƥrz  e15 E r (1e) Dr 2e15Ƥrz  H 11 E r ; Dz e31Ƥrr  e31ƤTT  e33Ƥ zz  H 33 E z where V ij , Ƥij , Di and Ei i, j r,T , z denote the components of stress tensor, strain tensor, electric displacement vector and electric field vector respectively; c11 , c12 , c13 , c 33 and c 44 are the elastic constants under zero or constant electric field; e31 , e33 and e15 are the piezoelectric constants; and H 11 and H 33 are the dielectric constants under zero or constant strain. The field equations of a piezoelectric material undergoing axisymmetric deformations about the z-axis can be expressed as, wV rr wV zz V rr  V TT wV rz wV zz V rz (2a)   0;   0 wr wz r wr wz r wDr wDz Dr (2b)   0. wz wr r The relationship between the electric field Ei i r, z and the electric potential I can be expressed as, wI ; wI . (3) Er



Ez

wr



wz

It is prudent to nondimensionalize all field variables. The coordinates r and z and the displacements ur and u z are nondimensionalized by the outer radius b which is set as the nondimensional unit length parameter. The stresses and elastic constants are nondimensionalized by c44 . The electric displacements and piezoelectric constants are nondimensionalized by e31 . For convenience, the nondimensional coordinates, displacements, stresses, electric displacements, elastic constants and piezoelectric constants are denoted by the same symbols without loss of generality. In addition, the following new nondimensional quantities are introduced. e31 c44 c44 \ (4) I I; \ ; H 11 H 11 ; H 33 H 33 . 2 2 2 c44 b b e31 e31 The corresponding general solutions of displacements and electric potential of a solid cylinder are (Rajapakse et al [5]): u r r

3

­

¦ ®¯2 A

0i

i 1

f

r

¦t m 1

m

Aim J 1 t m r cosh t m z i 

f

¦ n 1

½

Oi s n Gin I 1 s n ri coss n z ¾ ¿

(5a)

167 f f ­° ½° A tm k1i ® 4 0i z  Aim J 0 t m r sinh t m z i  s n Gin I 0 s n ri sin s n z ¾ (5b) Oi Oi °¯ °¿ i 1 m 1 n 1 f f 3 ­° ½° A t I z ¦ k 2i ® 4 0i z  ¦ m Aim J 0 t m r sinh t m z i  ¦ s n Gin I 0 s n ri sin s n z ¾ (5c) Oi Oi °¯ °¿ m 1 n 1 i 1 nS where sn , z i z Oi , ri Oi r ; J 0 t m r and I 0 s n ri are the Bessel and h modified Bessel functions of the first kind respectively (Bland [7]); tm is the m-th

u z z

3

¦

¦

¦

root of J 1 tm 0 ; A0i , Aim and Gin i 1,2,3 are arbitrary functions to be determined from the boundary conditions. In addition O i i 1,2,3 and k ij i 1,2; j 1,2,3 where the subscript j identifies the corresponding root O i are obtained by solving the following equations. 1  1  c13 k1  1  e15 k 2 c33k1  e33k 2 (6a) O c11 k1  1  c13  e15 k 2 1  1  c13 k1  1  e15 k 2 e33k1  H 33 k 2 (6b) O. c11 e15 k1  1  e15  H 11k 2 The general solutions of stresses and electric displacements can be obtained from Eq. (5).

3. Electric Charge Loading of a Solid Cylinder

In this study, a cylinder subjected to electric charge loading as shown in Figure 1 is considered. The boundary conditions can be expressed as, Dr 0, V rr 0, V rz 0, at r b (7a) Dz

­ D0 0 d r d b0 , V zr ® ¯ 0 b0 d r d b

qr

0, V zz

0

at

z

rh .

(7b)

In order to solve the boundary-value problem corresponding to Eq. (7), the general solutions for relevant stresses and electric displacements are first derived from Eq. (5). Thereafter, applied loading q r is expanded into a Fourier-Bessel series of the following form. f

qr Q0  ¦ Qm J 0 t m r

(8)

m 1

where Q0

³ 

b0

0

D0 rdr b

³ rdr 0



b02 D0 b2

(9a)

168

Qm

³ 

b0

D0 rJ 0 t m r dr

0

³

b

0

rJ t m r dr 2 0



2b0 J 1 b0 t m D0 . t m b 2 J 02 t m b  J 12 t m b

>

(9b)

@

Using the general solutions, boundary conditions (7a) and (7b), and Eqs. (8) and (9), the unknown arbitrary functions A0i , Aim and Gin i 1,2,3 appearing in the general solution can be determined. 4. Numerical Results and Discussion

There are not any solutions reported in the literature for piezoelectric cylinders. The available solutions for elastic cylinders are therefore used to establish the accuracy of the piezoelectric solutions by setting the piezoelectric and dielectric constants to negligibly small values. Vendhan and Archer [8] considered a magnesium solid cylinder subjected to the following boundary conditions. * 2 (10) V rr 0, V rz 0, at r b ; V zz V (1  2r ), V zr 0, at z r h The properties of magnesium are: c11 5.64 , c33 5.86 , c44 = 1.68, c12 2.30 ,





1.81 1010 Nm 2 . Figure 2 shows a comparison of present solutions for radial and hoop stresses of a magnesium cylinder with those obtained by Vendhan and Archer [8]. The two solutions agree very close and confirm the high accuracy of the present numerical results. c13

The basic features of coupled electroelastic field of a solid piezoelectric cylinder are examined next. Three piezoelectric materials and different aspect ratios of cylinder (i.e. h b 0.5, 0.75, 1.0, 2.0, 5.0 ) are considered in the numerical study. Table 1 shows the material properties of the three piezoelectric materials, i.e. BaTiO3 , PZT-4 and PZT-5H. 

1.0 0.8

0.4



VTT/V

*

0.6

r=0 (Present) r=0 (Vendhan and Archer) r=0.5b (Present) r=0.5b (Vendhan and Archer)

0.2 0.0 -0.2 0.0

0.2

0.4

0.6 z

0.8

1.0

*

(a) Nondimensional hoop stress profiles along z-axis (h/b=1)

169  0.10

0.05

0.00



V/V

*

-0.05

*

-0.10

Vrr/V (Present) *

Vrr/V (Vendhan and Archer)

-0.15

*

VTT/V (Present) *

VTT/V (Vendhan and Archer)

-0.20

-0.25 0.0

0.2

0.4

0.6

r

0.8

1.0

*

(b) Nondimensional radial and hoop stress profiles along r-axis (z=0, h/b=0.2)

-0.004

0.004

-0.008

0.002

-0.012

0.000

-0.016

*

0.006

Ez

Vzz

*

Figure 2. Comparison of solutions for stresses of a solid magnesium cylinder

BaTiO3 PZT-4 PZT-5H

-0.020

-0.002 BaTiO3 PZT-4 PZT-5H

-0.004 -0.006 0.0

0.2

0.4

0.6 r

0.8

-0.024

1.0

*

(a) Nondimensional vertical stress profiles along the r-axis (z=0)

-0.028 0.0

0.2

0.4

0.6 r

0.8

1.0

*

(b) Nondimensional vertical electric field profiles along the r-axis (z=0)

Figure 3. Vertical stress and electric field profiles of a piezoelectric solid cylinder under vertical electrical charge loading (b0/b=0.5, h/b=1)

Figure 3 shows the variation of the nondimensional vertical stress, 2 V zz* V zz e31 c44 D0 , and electric field, E z* E z e31 c 44 D0 , along the radial axis

r

*



r b at z

value of V

* zz

0 of a cylinder with h b 1 . For PZT cylinders, the maximum

occurs at the center and the magnitude decreases with the radial

distance. Vertical stress experiences a change in sign in the vicinity of r * 0.65 for PZT cylinders and both compressive and tensile stresses exist at the middle of the

170

cylinder. In the case of a BaTiO3 cylinder, V zz* remains tensile but relatively smaller in magnitude and nearly constant. The profiles of the vertical electric field are almost parallel to each other with a minor decrease in magnitude with the radial distance and the largest nondimensional electric field is generated in PZT-4 followed by PZT-5H and BaTiO3 . 0.07

0.006

0.06

0.004

BaTiO3 PZT-4 PZT-5H

0.05 0.002 *

Uz

Vzz

*

0.04 0.000 BaTiO3 PZT-4 PZT-5H

-0.002 -0.004

0.03 0.02 0.01 0.00

-0.006 0.0

0.2

0.4

0.6 z

0.8

0.0

1.0

0.2

*

0.4

0.6 z

(a) Nondimensional vertical stress profiles along the z-axis (r=0)

0.8

1.0

*

(b) Nondimensional vertical displacement profiles along the z-axis (r=0)

Figure 4. Vertical stress and displacement profiles of a piezoelectric solid cylinder under vertical electrical charge loading (b0/b=0.5, h/b=1)

Table 1 Material properties of selected piezoceramics. PZT-4

PZT-5H

15.0

13.9

12.6

14.6

11.5

11.7

6.6

7.78

7.95

6.6

7.43

8.41

4.4

2.56

2.3

2

11.4

12.7

17.0

2

-4.35

-5.2

-6.55

2

17.5

15.1

23.3

9.87

6.45

15.38

11.15

5.62

12.76

BaTiO3

 10 Nm 10 Nm 10 Nm 10 Nm Cm Cm Cm 10 Fm 10 Fm

c11 1010 Nm 2

c33 c12 c13 c44

e15 e31 e33

H 11

H 33

10

2

10

2

10

2

10

2

9

1

9

1





Figure 4 shows the nondimensional vertical stress, V zz* , and vertical displacement, u *z



u z e31 bD0 , profiles along the z-axis z *



z b at r

0 . Vertical stress should be zero at the top end of the cylinder, but this boundary condition is not exactly

171

satisfied due to precision errors. Nondimensional vertical stress in BaTiO3 cylinders remains nearly constant with depth and much smaller in magnitude when compared to the stress generated in PZT cylinders. Maximum value of V zz* occurs in the vicinity of z * 0.5 . It is interesting to note that V zz* of a PZT-5H cylinder is compressive along the length whereas nondimensional vertical stress corresponding to other materials is tensile. Nondimensional vertical displacement increases with z * . PZT materials generate a larger overall vertical displacement (stroke) when compared to BaTiO3 . * Figure 5 shows the variation of nondimensional hoop stress, V TT V TT e31 c44 D0 , along the radial axis at z 0 of a cylinder with h b 1 . Hoop stress is tensile with its maximum value at the centre of the cylinder and then becomes compressive for r ! 0.5 . Hoop stress in a BaTiO3 cylinder is smaller than that in a PZT cylinder.

0.004 0.003

BaTiO3 PZT-4 PZT-5H

0.002 0.001

VTT

*

0.000 -0.001 -0.002 -0.003 -0.004 0.0

0.2

0.4

0.6 r

0.8

1.0

*

Figure 5. Nondimensional hoop stress profiles along the r-axis (z=0) of a piezoelectric solid cylinder under vertical electrical charge loading (b0/b=0.5, h/b=1) 0.004

-0.15

0.002

-0.20 -0.25

0.000

-0.004 -0.006 -0.008 0.0

*

h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5

0.2

0.4

0.6

0.8

Dz

Vzz

*

-0.30

-0.002

-0.35

h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5

-0.40 -0.45

1.0

*

r

(a) Nondimensional vertical stress profiles along the r-axis (z=0)

-0.50 0.0

0.2

0.4

0.6

0.8

1.0

*

r

(b) Nondimensional vertical electric displacement profiles along the r-axis (z=0)

Figure 6. Vertical stress and electric displacement of a PZT-5H solid cylinder under vertical electric charge loading for different h/b ratios (b0/b=0.5)

172

Next, the influence of cylinder geometry h b on the electroelastic field generated by electric loading is examined by considering cylinders made of PZT-5H with different h b ratios. Figure 6 shows the nondimensional vertical stress and electric displacement, D z* D z D0 , along the r -axis at z 0 (mid-plane) for different h b ratios. Nondimensional vertical stress has a large compressive value at the center of short cylinders and decreases with radial distance and becoming tensile close to the boundary of the cylinder. For a relatively long cylinder h b t 2 , V zz* is nearly zero. For very short cylinders h b 0.5 , vertical stress at the mid-plane changes its sign twice and becomes compressive near the boundary. From Figure 6(b), it can be seen that vertical electric displacement decreases rapidly as h b increases. For a long cylinder h b ! 1 , the radial distribution of D z* is nearly uniform. It is noted that vertical electric displacement is equal to -0.25 for h b 2 and 5 and this value is equal to the 1-D solution for a long cylinder. Figure 7 shows the nondimensional hoop stress profiles along the radial axis at z 0 for different h b ratios. As h b increases, it approaches zero which is consistent with the 1-D response of a long cylinder.

0.020

h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5

0.015 0.010 0.005

VTT

*

0.000 -0.005 -0.010 -0.015 -0.020 -0.025 0.0

0.2

0.4

0.6

0.8

1.0

*

r

Figure 7. Nondimensional hoop stress profiles along the r-axis (z=0) of a PZT-5H cylinder under vertical electric charge loading for different h/b ratios (b0/b=0.5)

Figures 8 and 9 show the nondimensional vertical stress, vertical electric displacement and vertical electric field profiles along the center of cylinders of different h b ratios. Maximum vertical stress occurs inside the cylinder near the loaded end except in the case of h b 0.5 and 0.75. For a long cylinder (e.g.

173

h b t 1 ), the maximum value of V zz* is much larger than the value at the middle of

the cylinder. For shorter cylinders h b  1 , V zz* decreases rapidly along the z-axis and the peak value occurs at the mid plane of the cylinder. Vertical electric displacement decays rapidly with depth from the loaded end and approaches the 1D solution for a long cylinder (e.g. h b t 2 ). For short cylinders, Dz* also decreases from the top end but the value at the mid plane is larger than the 1-D solution. The variation of vertical electric field along the cylinder axis is nearly identical to the variation of vertical electric displacement.

0.002 0.000

Vzz

*

-0.002 h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5

-0.004 -0.006 -0.008 0.0

1.0

2.0

3.0 z

4.0

5.0

*

Figure 8. Nondimensional vertical stress profiles along the z-axis (r=0) of a PZT-5H cylinder under vertical electric charge loading for different h/b ratios (b0/b=0.5)

-0.01

-0.2

-0.02

-0.4

-0.03 -0.6

-1.0

1.0

2.0

3.0 z

4.0

h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5

*

Ez

*

Dz

-0.8

-1.2 0.0

-0.04

h/b=0.5 h/b=0.75 h/b=1 h/b=2 h/b=5

-0.05 -0.06 -0.07

5.0

*

(a) Nondimensional vertical electric displacement profiles along the z-axis (r=0)

-0.08 0.0

1.0

2.0

3.0 z

4.0

5.0

*

(b) Nondimensional vertical electric field profiles along the z-axis (r=0)

Figure 9. Vertical electric displacement and electric field of a PZT-5H cylinder under vertical electric charge loading for different h/b ratios (b0/b=0.5)

174

5. Conclusion

A generalized displacement potential function method together with a FourierBessel series expansion is successfully applied to study fully coupled electroelastic field of a solid piezoelectric cylinder of finite length under electric charge loading applied to the end faces. Current solutions agree well with the existing solutions for the limiting case of an ideal elastic cylinder. It is found that 3-D analysis is needed only if the length-radius ratio of a cylinder is less than two. Vertical stress has its maximum value near the loaded end for long cylinders but it becomes the maximum in the centre for short cylinders. Electroelastic field in a solid cylinder shows complex dependence on materials properties. For example, the magnitude of electric field and vertical displacement generated in a PZT cylinder are substantially higher than those in a BaTiO3 cylinder. The two PZT materials show significantly different vertical stresses (one tensile and the other compressive) and have higher displacements (stroke) compared to BaTiO3 under electric charge loading. The analytical general solution used in this paper can be further extended to solve a more complex and practically useful mixed boundary value problem involving a cylinder with end electrodes subjected to voltage loading and a mechanical bias load. This problem requires a much more complex analysis and is currently under study (Senjuntichai et al [9]). Acknowledgment

The work presented in this paper was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

X.X. Wei, K.T. Chau, R.H.C. Wong, 1999, Analytic solution for axial point load strength test on solid circular cylinders. Journal of Engineering Mechanics, ASCE 125, 1349-1357. X.X. Wei, K.T. Chau, 2002, Analytic solution for finite transversely isotropic circular cylinders under the axial point load test. Journal of Engineering Mechanics, ASCE 128, 209-219. K.T. Chau, X.X. Wei, 2000, Finite solid circular cylinders subjected to arbitrary surface load. Part I – Analytic solution. International Journal of Solids and Structures, 37, 5707-5732. R.K.N.D. Rajapakse, Y. Zhou, 1997, Stress analysis of piezoceramic cylinders. Smart Materials and Structures, 6, 169-177. R.K.N.D. Rajapakse, Y. Chen, T. Senjuntichai, 2004, Electroelastic field of a piezoelectric annular finite cylinder. International Journal of Solids and Structures, submitted. V.Z. Parton, B.A. Kudryavtsev, 1988, Electromagnetoelasticity. Gordon and Breach, New York. D.R. Bland, 1961, Solutions of Laplace’s Equation. Routledge and Kegan Paul Ltd., London. C.P. Vendhan, R.R. Archer, 1977, Axisymmetric stresses in transversely isotropic finite cylinders. International Journal of Solid and Structures, 14, 305-318. T. Senjuntichai, W. Kaewejua, R.K.N.D. Rajajapkse, 2004, Piezoelectric cylindrical actuator under voltage and mechanical bias loading, in preparation.

OBLIQUE PROPAGATION OF TIME HARMONIC WAVES IN PERIODIC PIEZOELECTRIC COMPOSITE LAYERED STRUCTURES

M. Urago1, F. Jin2, Y. Mochimaru1 and K. Kishimoto3 1 Department of International Development Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan 2 Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an 710049, P.R. China 3 Department of Mechanical and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan Abstract

Piezoelectric material has played an important role of the modern engineering such as ultrasonic transducers, ultrasonic receivers, ultrasonic motors, SAW devices and crystal oscillators to date. An elastic wave analysis of the material is indispensable to enlarge the applicability of the material. In this paper, we investigate a plane-like SH wave obliquely propagating through a periodic piezoelectric composite layered structure and derive a dispersion relation of the wave. 1. Introduction

The piezoelectric phenomenon is a coupling phenomenon between deformation of material and electric field discovered by Pierre and Jacque Curie in 1880 [1]. Therefore, we have to solve motion of the material and electric field simultaneously to clarify engineering properties of the piezoelectric material [2]. Jin et al. [3] focused on a periodic piezoelectric composite layered structure to reinforce the brittleness of the piezoelectric material and revealed the dispersion relations of a horizontally polarized shear wave (a SH wave) propagating along the layer of the structure and perpendicular to the layer of the structure. The dispersion relation links the phase velocity c and the wave number k of the wave and gives us the transmission and the reflection effects of the layered structure to the wave. Sve [4] focused on a periodic isotropic elastic structure and derived the dispersion relation of the plane-like wave obliquely propagating through that structure. Delph et al. [5] also derived the dispersion relation of the plane-like wave obliquely propagating through that structure. In this paper, we derive the dispersion relation of a plane-like SH wave obliquely propagating through the periodic piezoelectric composite layered structure. 175 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 175–184. © 2006 Springer. Printed in the Netherlands.

176

2. Problem Description

The periodic piezoelectric composite layered structure is shown in Figure 1. The layered structure is an unbounded three-dimensional structure. That consists of transversely piezoelectric material and isotropic elastic material without piezoelectricity. These materials perfectly and alternately bond. Thickness of a layer of the isotropic elastic material is h1 and that of the transversely piezoelectric material is h2 . Total thickness of two layers is H h1  h2 and is also the smallest period of the layered structure. We seek a dispersion relation of a planelike SH wave obliquely propagating through the layered structure. The SH wave means a horizontally polarized shear wave. The angle T is defined as a propagation angle measured from the x-axis. We assume that deformation of the layered structure and electric field of that are sufficiently small to linearize the governing equations of the plane-like SH wave and that the change of electric field with respect to time is sufficiently slow to neglect the effect of time dependency of the electric field. 3. Equations of Electric Field

The equations of electric field for both materials are as follows:

G E {E x , E y , E z }T G divD 0 ,

 grad M ,

(1) (2)

G G where E is the electric field and M is the electric potential and D is the electric flux density. The constitutive equations of electric flux density are as follows. For the isotropic elastic material:

G G D H11E .

(3)

The dielectric constant of the isotropic elastic material is H 11 . For the transversely piezoelectric material, we assume that the transversely piezoelectric material is formed into layers with the constitutive equations (4) and (10).

­ Dx ½ ° ° ® Dy ¾ °D ° ¯ z¿

ª0 0 0 « «0 0 0 / / / «e31 e31 e33 ¬

0 e15/ 0

e15/ 0 0

­ Hx ½ °H ° y ° ªH11/ 0 0 º ­ Ex ½ 0º ° ° ° H »° z ° « »° ° / 0» ® ¾  « 0 H11 0 » ® E y ¾ . 2H zy ° « 0 0 H11/ » °¯ Ez °¿ 0»¼ ° ¼ °2H zx ° ¬ ° ° °¯2H xy °¿

(4)

177 /

/

/

/

The symbols e31 , e33 and e15 are the piezoelectric constants, H 11 is dielectric constant and H x , H y , H z , H zy , H zx and H xy are the Cartesian components of a strain tensor, e.g.

Hx

wu / wx .

4. Equations of Motion

The equations of motion are as follows:

U mat U mat U mat U mat where

­U / ® ¯U

wV x wW yx wW zx   , wx wy wz w 2v wW xy wV y wW zy   , wt 2 wx wy wz w 2 w wW xz wW yz wV z ,   wz wt 2 wx wy w 2u wt 2

for the transversely piezoelectric material for the isotropic elastic material

(5) (6) (7) ,

(8)

U mat is the mass density of material, u , v and w are the x component, the y

component and the z component of displacements of material respectively and V x ,V y ,V z ,W yx ,W zx ,W xy ,W zy ,W xz and W yz are the Cartesian components of Cauchy stress. The constitutive equations of stress are as follows. x For the isotropic elastic material:

0 0 º­ H x ½ ­V x ½ ª c11 c12 c12 0 °V ° « ° ° 0 0 »» ° H y ° ° y ° «c12 c11 c12 0 °°V z °° «c12 c12 c11 0 0 0 » °° H z °° »® ® ¾ « ¾. 0 0 c 44 0 0 » °2H zy ° °W zy ° « 0 °W zx ° « 0 0 0 0 c 44 0 » ° 2H zx ° »° ° ° « ° 0 0 0 0 c 44 »¼ °¯2H xy °¿ °¯W xy °¿ «¬ 0 The symbols c11 , c12 and c44 (c11  c12 ) / 2 are elastic constants.

x For the transversely piezoelectric material:

(9)

178

­V x ½ °V ° ° y° °°V z °° ® ¾ °W zy ° °W zx ° ° ° °¯W xy °¿

ª c11/ « / «c12 «c13/ « «0 «0 « «¬ 0

c12/

c13/

0

0

c11/

c13/

0

0

/ 13

/ 33

0

0

0

0

/ 44

0

0

0

0

/ c44 0

c

c

0 / 11

c

0 / 12

0 / 13

/ 33

The symbols c , c , c , c

and c

0 º­ H x ½ ª 0 »° ° « 0 »° H y ° « 0 0 » °° H z °° « 0 »® ¾« 0 » ° 2H zy ° « 0 0 » ° 2H zx ° «e15/ ° « / / »° c12 c11 2 » ¼ °¯2H xy °¿ «¬ 0 / 44

0 0 0 e15/ 0 0

are the elastic constants

/ º e31 / » e31 »­E ½ x e33/ » ° ° » ® E y ¾ . (10) 0 »° ° ¯ Ez ¿ 0» » 0 »¼ / / and e31 , e33 and

/ 15

e are the piezoelectric constants. 5. Governing Equations of a SH Wave and Continuity Conditions

We assume that the wave motion through each material can be described as follows: u 0 , v 0 , w w( x, y, t ) , (11)

M M ( x, y , t ) .

(12) This wave is called a horizontally polarized shear wave (a SH wave), and whose governing equations are listed below.

x For the transversely piezoelectric material: w2w U / 2 c44/ 'w  e15/ 'M wt / H11'M e15/ 'w ww wM  H11/ D y e15/ wy wy ww / wM W yz c44/  e15 . wy wy x For the isotropic elastic material: w2w U 2 c44 'w wt H11'M 0 wM Dy H11 wy ww W yz c44 . wy

(13) (14) (15) (16)

(17) (18) (19) (20)

179

The continuity conditions of physical quantities are as follows. The left hand sides of the continuity conditions specify the physical quantities of the transversely piezoelectric material and the right hand sides specify those of the isotropic elastic material.

x Continuities of displacements along each interface w( x, jH  0, t ) w( x, jH  0, t ) (21) w( x, jH  h2  0, t ) w( x, jH  h2  0, t ) . (22) x Continuities of electric potentials along each interface (23) M ( x, jH  0, t ) M ( x, jH  0, t ) (24) M ( x, jH  h2  0, t ) M ( x, jH  h2  0, t ) . x Continuities of normal components of electric flux density along each interface Dy ( x, jH  0, t ) Dy ( x, jH  0, t ) (25) D y ( x, jH  h2  0, t )

Dy ( x, jH  h2  0, t ) .

x Continuities of z components of traction along each interface W yz ( x, jH  0, t ) W yz ( x, jH  0, t )

W yz ( x, jH  h2  0, t ) W yz ( x, jH  h2  0, t ) .

(26) (27) (28)

The symbol j is an arbitrary integer, H is the smallest period of the layered structure. The term ‘  0 ’ means the limiting process from below and the term ‘  0 ’ means the limiting process from above. The number of continuity conditions is infinite. Other physical quantities V y , W yx and E z become zero with Eqs. (11)(12). Therefore, the quantities

V y , W yx and E z satisfy the continuity conditions

along each interface. The x component of electric field E x also satisfies the continuity conditions along each interface by Eqs. (23) and (24). 6. Solutions of a Plane-like SH Wave

We try to seek the plane-like SH wave having the following forms: ik ( nx x  n y y ct )

w( x, y, t ) F ( y )e F ( y  H ) F ( y) ,

M ( x , y , t ) G ( y )e G( y  H )

ik ( nx x  n y y ct )

G( y) ,

,

(29) (30)

,

(31) (32)

 1 , F ( y ) and G ( y ) are the periodic unknown functions with the smallest period H , nx cosT and n y sin T are the direction cosines defining the direction of propagation, k is the wave number and c is the phase velocity of where i is

the wave.

180

It is sufficient to solve the unknown functions F ( y ) and G ( y ) over the smallest period of the layered structure [ h1 , h2 ] because of these periodicities. Therefore, the following expressions are introduced:

F ( y)

­ f p ( y) ® ¯ fe ( y)

[0, h2 ] [h1 ,0]

for the transversely piezoelectric material , for the isotropic elastic material (33)

G( y)

­ g p ( y) ® ¯ ge ( y)

[0, h2 ] [h1 ,0]

for the transversely piezoelectric material . for the isotropic elastic material

(34) Substituting Eq. (33) and Eq. (34) into Eqs. (13)-(20), we obtain the ordinary differential equations as follows. For the isotropic elastic material:

­°§ c · 2 ½° f  2ikn f  k ®¨¨ ¸¸  1¾ f e °¯© csh ¹ °¿ g e//  2ikn y g e/  k 2 g e 0 // e

/ y e

csh

2

c44

U

0

(35) (36)

.

(37)

The symbol csh is the bulk phase velocity of the isotropic elastic material. The solutions of equations (35) and (36) are as follows:

fe ( y) g e ( y)

e

 ikn y y

e

^Pe ^R e

ikDy

 ikn y y

 Qe  ikDy `,

kn x y

 Se

 kn x y

(38)

`,

(39)

2

§ c · ¨¨ ¸¸  nx2 , © csh ¹

D

(40)

where P , Q , R and S are the unknown constants. For the transversely piezoelectric material:

­°§ c · 2 ½° f  2ikn y f  k ®¨¨ / ¸¸  1¾ f p 0 °¿ °¯© csh ¹ e/ 1 g //p  2ikn y g /p  k 2 g p  k 2 15/ / 2 f p H11 csh // p

/ p

2

(41)

(42)

181

1 § / e15/ 2 · ¨c  ¸ . U / ¨© 44 H11/ ¸¹

csh/

(43)

/

The symbol csh denotes the bulk phase velocity of the transversely piezoelectric material. The solutions of Eq. (41) and Eq. (42) are as follows:

g p ( y)

e15/

H

/ 11

f p ( y)

e

ikn y y

^Ae

e

 ikn y y

ikEy

^Ae

ikEy

 Be  ikEy `,

 Be ikEy ` e

ikn y y

^Ce

(44) knx y

`

 De knx y , (45)

2

§ c · ¨¨ / ¸¸  nx2 , © csh ¹

E

(46)

where A , B , C and D are unknown constants. 7. Treatment of the Continuity Conditions

The continuity conditions are applied to the functions f p ( y ) , f e ( y ) , g p ( y ) and

g e ( y ) to determine the constants A , B , C , D , P , Q , R and S . If one substitutes Eq. (29) into Eq. (21), the following equation is obtained: ik ( n x  n jH ct )

ik ( n x  n y jH ct )

F ( jH  0)e x y F ( jH  0)e x F ( jH  0) F ( jH  0) . The application of the periodicity of F ( y ) leads to: F (0) F (0) . This equation can be rewritten in term of f p ( y ) and f e ( y ) as: f p (0)

f e (0) .

,

(47) (48) (49)

(50)

Appling the above-mentioned procedures to equations (22)-(28), one obtains the following equations: f p (h2 ) f e (h1 ) (51)

g p (0) g p (h2 ) / 15

' p

g e (0)

(52)

g e (h1 ) / 11

' p

(53) ' e

e { f (0)  ikn y f p (0)}  H {g (0)  ikn y g p (0)} H11{g (0)  ikn y g e (0)} (54) e15/ { f p' (h2 )  ikn y f p (h2 )}  H11/ {g 'p ( h2 )  ikn y g p (h2 )} H11{g e' (h1 )  ikn y g e ( h1 )} (55) / c44 { f p' (0)  ikn y f p (0)}  e15/ {g 'p (0)  ikn y g p (0)} c44 { f e' (0)  ikn y f e (0)}

(56)

182 / c44 { f p' (h2 )  ikn y f p (h2 )}  e15/ {g 'p (h2 )  ikn y g p (h2 )} c44 { f e' (h1 )  ikn y f e ( h1 )} .

(57) The functions f , f , g and g mean the differentiations of f p , f q , g p and ' e

' p

' e

' p

g q with respect to y , respectively. 8. Dispersion Relation

In order to determine the unknown constants A , B , C , D , P , Q , R and S , we substitute the solutions (38), (39), (44) and (45) into the simplified continuity conditions (50), (51)-(57). The homogeneous simultaneous linear equations are obtained as follows:

G M[ G

[

MA

MB

~ and P

ª 1 « e15/ « H/ 11 « « 0 ~ « iEP « ikEh « e 2 « e15/ ikEh2 « / e « H11 « 0 ~ ikEh2 «iEP ¬ e

G [ M A M B ][

G 0,

(58) T

{ A, B, C , D, P, Q, R, S} , 1 0 e15/ 1 /

H11 0

H11/

e ikEh2

nx e15/ 0

~  iEP e15/

H

/ 11

e ikEh2

e knxh2

H11/ e kn h

0 ~ ikEh2  iEP e

1 ª « 0 « « 0 «  i c44 D « ikn y H ikDh1 « e e « 0 « « 0 « ikn y H ikDh1 e «¬ iDc44e / c44  e15/ 2 / H11/ .

e

x 2

e15/ nx e knxh2

(59)

0

º » 1 » »  H11/ » /  nx e15 » », 0 » » knx h2 e » »  H11/ e knx h2 »  e15/ nx e knxh2 »¼

1

0

0

1

0

 H11

iDc44

0

ikn y H ikDh1

iDc44e

e

0

e

0

 H11e

ikn y H ikDh1

e

0

0 ikn y H

e knxh1

ikn y H

0

(60)

e knxh1

º » 1 » » H11 » 0 » » 0 ikn y H knx h1 » e e » ikn y H knx h1 » H11e e » 0 »¼

(61)

183

We assume k is not zero and n x is not zero because of the form of the solutions (9) and (45). The third, fourth, seventh and eighth rows of M are already divided by k and the third row and the seventh row of M are already divided by n x . In order to exist a non trivial solution of Eq. (58), the determinant of M must be zero. This leads to the dispersion relation of the plane-like SH wave obliquely propagating through the layered structure. det M (c, k ) 0 . (62) 9. Numerical Results and Conclusion

The determinant of M is calculated with LU factorization as follows: ZM LU , (63) det M det L ˜ det U / det Z , (64) where L is the lower triangular matrix, U is the upper triangular matrix, Z is the permutation matrix and the determinants of L , U and Z are easily calculated. The LU factorization is also valid when M is a singular matrix [6]. Unfortunately, the determinant of M is a complex number. We have to search the common zero contour lines of two equations as follows: Re[det M (c, k )] 0 , (65)

Im[det M (c, k )]

0.

(66)

We calculate the determinants of M on the grid points of k and c and draw the zero contour lines with a graph tool. The material constants are used as follows. For the isotropic elastic material: H 11 = 0.2036 u 10-10F/m, h1 = 0.02m. c44 = 0.128 u 1010N/m2, U ̓= 1.18 u 103kg/m3, ǂ y axis 3000

]

Propagation direction

0 -h1

z axis

x axis

[

T H

y

h2

2000

1000

isotropic elastic material

0 degree

transversely piezoelectric material 0 0

200 100 Wave number k [rad/m]

Figure 1. Periodic piezoelectric composite layered structure (a )

300

184

(c)

(b)

Figure 2. (a), (b), (c): Numerical results of the dispersion relations

For the transversely piezoelectric material: / c44 2.3 u1010 N/m2, U / 7.5 u 103 kg/m3, e15/

2 17 c/m ,

277.0 u 10 10 F/m, h2 0.08 m The numerical results of the dispersion relations are shown in Figure 2. Fortunately, the determinant of the zero degree case only has a pure real part. Therefore, the dispersion relation is clearly observed. The others of Figure 2 have a lot of vertical lines. The vertical lines are not common zero contour lines of the real part and the imaginary part of the determinant. Therefore, they are not dispersion relations. The dispersion relation is deformed with respect to the propagation angle.

H11/

We investigate the plane-like SH wave obliquely propagating through the periodic piezoelectric composite layered structure and derive the dispersion relation of the wave. References 1. 2. 3. 4. 5. 6.

B. Jaffe, W.R. Cook Jr, H. Jaffe, 1971, Piezoelectric Ceramics, Academic Press, 1-5. G.J.S. Little, 1957, Piezoelectricity, Her Majesty’s Stationery Office, 33-41. Z. Qian, F. Jin, Z. Wang, K. Kishimoto, 2004, Dispersion relations for SH-wave propagation in periodic piezoelectric composite layered structures, International Journal of Engineering Science, 42, 673-689. C. Sve, 1971, Time-harmonic waves traveling obliquely in a periodically laminated medium, ASME Journal of Applied Mechanics, 38, 477-482. T.J. Delph, G. Herrmann, R.K. Kaul, 1979, Harmonic wave propagation in a periodically layered, infinite elastic body: plane strain, analytical results, ASME Journal of Applied Mechanics, 46, 133-119. M. Galassi, J. Davis, J. Theiler, B. Gough, G. Jungman, M. Booth, F. Rossi, 2002, GNU Scientific Library Reference Manual, Network Theory Limited, 173-175.

SCATTERING BEHAVIOUR OF ELASTIC WAVES IN 1-3 PIEZOELECTRIC CERAMICS/POLYMER COMPOSITES

F. Jin1, K. Kishimoto2, Z. Qian1 and Z. Wang1 Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an 710049, P.R. China; 2 Department of Mechanical and Control Engineering, Tokyo Institute of Technology 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan 1

Abstract

The scattering behavior of elastic waves in piezoelectric composites with 1-3 connectivity is taken into account. As the most utilized two-phase connectivity types, the functional piezoelectric composites with 1-3 connectivity, usually consists of piezoelectric cylinders aligned in a direction parallel to the poling direction and surrounded by polymeric medium, can be found many sensing and actuating applications. The method of wave function expansions is applied to study the scattering of a plane acoustic wave by piezoelectric cylinders surrounded by polymer medium. The dynamic stress concentration factor around the cylinders due to the incident wave is investigated and some other characters are studied. 1. Introduction

Following the advances in structural applications, composite structures are being used commonly in transducer applications to improve acoustic, mechanical and electrical performance of piezoelectric devices. Functional composite transducers for sensors and actuators generally consist of ceramics and polymers, the disadvantage of the strength and brittleness nature of the piezoelectric ceramics can be overcome to a certain extent and the structures good for sensing and actuating can be allowed for building up. As the most studied, understood and utilized two-phase connectivity types, the functional piezoelectric 1-3 connectivity composites, usually consisting of piezoelectric cylinders aligned in a direction parallel to the poling direction and surrounded by polymeric medium, can find many transducer applications [1-7]. For example, in pulse-echo medical ultrasonic imaging due to their superior performance in comparison with the monolithic ceramics and polymers, these researches reveal the fine prospective applications of the piezoelectric cylinder composites with 1-3 connectivity in the fields of acoustics, medical appliance, and 185 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 185–193. © 2006 Springer. Printed in the Netherlands.

186

nondestructive evaluation. Piezoelectric 1-3 composites provide many advantages including high electromechanical coupling coefficient, low acoustic impedance and adjustable relative permittivity [8]. Of particular importance is the performance and reliability of piezoelectric composites devices. Recently, Honavar et al [9], Shindo et al [10, 11] and Liu et al [12] have studied the wave scattering behavior by cylinders or interface cavities. The dynamic stress and electric field concentration due to the presence of inclusion are the key points for the design of piezoelectric composites and nondestructive evaluation, so the electro-elastic wave scattering behavior received considerable attention. While in the case of 1-3 piezoelectric composites, prediction of scattered fields from inclusions is more complicated due to the intrinsic coupling of electrical and mechanical fields. However, the above-mentioned research works provide the confidence and motivation for the present work. 2. Statement of the Problem

Theoretical model of our present work can be described as follows. The incident wave is a harmonic time-varying plane P-wave of frequency Z / 2S at an incident angle J . It is scattered by transversely isotropic piezoelectric circular cylinders submerged in an isotropic polymeric medium. Without loss of generality, the scattering behavior of incident waves by a single cylinder will be investigated, as shown in Figure1. The piezoelectric cylinder is of infinite length in z-direction and its radius is a . A cylindrical coordinate system ( r ,T , z ) is chosen with the zdirection coinciding with the axis of the cylinder. The polling direction of the piezoelectric cylinder is along the z-direction.

Figure 1. Theoretical model of the scattering problem and coordinate system

The incident wave can be represented by the following expression:

I i

I0 e i k

 x  k z x Zt



(1)

187

k cos J , k z

k sin J , and k

Z c p is the wave number of the compressional wave in the medium outside the piezoelectric cylinder, Z the angular frequency and I 0 is the amplitude. in which k 

The mechanical displacement vector and the electrical potential function can be written in terms of four scalar potentials I1 , I2 , F and \ [13] as follows,

G

G

K u (i )

’I1(i )  ’ u ( F (i ) ez )  a’ u ’ u (\ (i ) ez )

M (2)

(2) 2

wI wz

(2)



(3)

Where i 1 , for the medium outside the cylinder; i 2 , for the piezoelectric cylinder, and a is the radius of the piezoelectric cylinder, which keeps a constant in our analysis. The wave fields in the piezoelectric cylinder will be taken into account. Equations (2) and (3) are substituted into the differential motion equations in terms of the mechanical displacement components and the electrical potential function, we can ( 2) is decoupled. see that the scalar potential F Then the normal mode expansion method is used and the solutions of I1(2) , I2(2) , F (2) and \ (2) can be found to have the following forms: f

¦[ A J

I1(2)

n

n

( s1r )  q1 Bn J n ( s2 r )  q2Cn J n ( s3 r )]cos( nT )ei ( kz z Z t )

(4)

( s1r )  Bn J n ( s2 r )  q4Cn J n ( s3 r )]cos(nT )ei ( k z z Z t )

(5)

( s1r )  q6 Bn J n ( s2 r )  Cn J n ( s3 r )]cos(nT )ei ( kz z Z t )

(6)

( s4 r ) sin( nT )ei ( k z z Z t )

(7)

n 0

f

¦ [q A J

I2(2)

3

n

n

n 0 f

\ (2)

¦ [q A J n

5

n

n 0

f

F (2)

¦D J n

n

n 0

2

2

2

where J n is the Bessel function of the first kind of order n , s1 , s2 and s3 are three roots of equation

[

c11 (c44 11 e152 ) s 6  [ s 4  ] s 2  K

0 , in which

( UZ 2  2c13k z2 )(e152  c44 11 ) 11 [c132 k z2  c11 ( UZ 2  c33 k z2 )]  k z2 [c44 (c11 33 e312 )  2e15 (c13e31  c11e33 )]

188

]

( UZ 2  c33 k z2 )[11 ( UZ 2  c44 k z2 )  c11 33 k z2  (e31  e15 ) 2 k z2 ] 2  k z2 ( UZ 2  c13 k z2 )(2e15e33  c44 33 )  k z4 [(c44  c13 )(2e31e33  c13 33 )  c11e33 ]

K

2 2 k z2 ( UZ 2  c44 k z2 )[33 ( UZ 2  c33k z2 )  e33 kz ] .



c

 c12 , and q1 , q2 , q3 , q4 , q5 and q6 can be determined explicitly (expressions are omitted here). An , Bn , C n and Dn are s 42

2 UZ 2  c 44 k Z2

11

undetermined constants. Next, the wave fields in the medium around the piezoelectric cylinders will be taken into account. As the incident wave impinge on the piezoelectric cylinders, three kinds of waves are reflected from the boundary: P-waves (compressional), SH-waves (horizontally polarized shear) and SV-waves (vertically polarized shear), r r r which can be represented by I , F and \ , respectively. The total wave can be expressed as the sum of the incident wave and the reflected wave as follows

I (1) I (i )  I ( r ) , F (1)

F (r ) ,\ (1) \ (r ) .

(8)

The outgoing scattered waves can be assumed to possess the following forms [13], f

I

(r )

¦E

n

H n(1) ( k  r ) cos( nT ) e i ( k z z Z t )

n 0 f

F (r )

¦F H n

(1) n

( k t r ) sin( nT ) e i ( k z z Z t )

(9)

n 0 f

\ (r )

¦G

n

H n(1) ( k t r ) cos( nT ) e i ( k z z Z t )

n 0

in which

kt

the cylinders,

Z 2 / cs2  k z2 H n(1)

is the shear wave number in the medium outside

is the Hankel function of the first kind of order

n , and En , Fn

and Gn are the undetermined coefficients. Meanwhile, the expression of the incident wave is expanded in terms of the cylindrical wave function as follows, f

I (i )

I 0 ¦ H n i n J n ( k  r ) cos( nT ) e i ( k

z z Z t )

(10)

n 0

where

H n is the Neumann factor ( H n = 1 for n = 0, and H n = 2 for n t 1 ).

The continuous boundary conditions along the interface of the piezoelectric cylinder and the polymeric medium can be expressed as follows,

189

u r1

u r2 , uT1

V rr1

V rr2 , V r1T

M

1

2

uT2 , u z1

V rT2 , V rz1

1

u z2

V rz2

(11)

2

M , Dr

Dr

1

where M is the electrical potential function of the medium outside the piezoelectric cylinders, which results from the reflected waves of the piezoelectric cylinder surface. It can be represented as follows with undetermined constants H n : f

¦H

M (1)

n

H n(1) ( ik z r ) co s( nT ) e i ( k z z  Z t )

(12)

n 0

and therefore, f

(1) r

D

  ik z ¦ H n H n(1)c (ik z r ) cos(nT )ei ( kz z Zt )

(13)

n 0

where  is the dielectric constant of the matrix medium. Up to this stage, P-wave scattering problem by the piezoelectric cylinders as shown in Figure 1 becomes the determination of the unknown coefficients An , Bn , Cn , Dn , En , Fn , Gn , H n under the continuity conditions (11), which means

ª a11 a12 «a « 21 a22 « a31 a32 « « a41 a42 « a51 a52 « « a61 a62 «a a72 « 71 «¬ a81 a82 Elements aij and bi

a13

a14

a15

a16

a17

a23

a24

a25

a26

a27

a33

0

a35

0

a37

a43

a44

a45

a46

a47

a53

a54

a55

a56

a57

a63 a73

a64 a74

a65 0

a66 0

a67 0

a83

0

0

0

0

0 º ª An º 0 »» «« Bn »» 0 » « Cn » »« » 0 » « Dn » 0 » « En » »« » 0 » « Fn » a78 » « Gn » »« » a88 »¼ «¬ H n »¼

ª b1 º «b » « 2» «b3 » « » «b4 » . «b5 » « » «b6 » «0» « » «¬ 0 »¼

(14)

in equation (14) can be given explicitly (detailed expressions

are omitted here for brevity). The eight undetermined constants can be solved from Eq. (14) for any given frequency and position angle T . 3. Numerical Results

Effects of the normalized frequency ka of the incident wave, the incident angle

J

and the cylinder radius a on the dynamic stress concentration around the cylinders, the mechanical displacement component in z-direction and the electrical potential distribution in the cylinders will be considered, and PZT-5H ceramic cylinders surrounded by polythene material combination will be chosen to show the results.

190

S /2:

(a) t

0 , (b)

J , T S /2:

(a) t

0 , (b)

Figure 2. Variations of hoop stress versus wave number for different values of a , T t T / 4 , and (c) absolute values of hoop stress

Figure 3. Variations of hoop stress versus wave number for different values of

t

T / 4 , and (c) absolute values of hoop stress

191

Figure 4. Distributions of mechanical displacement u z for different wave numbers, (b) t

T

S ,J

5D : (a) t

0,

T / 4 , and (c) absolute values of u z

The hoop stress around the boundary of the piezoelectric cylinder is: V TT V TT V 0 r a , where V 0 (O  2 P )k 2I0 denotes the stress intensity of the incident wave in the direction of propagation.

V T T is a dimensionless

parameter and can be considered as the Dynamic Stress Concentration Factor for the P-waves scattered by piezoelectric cylinders problem. Variations of hoop stress are shown in Figure 2 and Figure 3. Distributions of

2 mechanical displacement component u z ( u z u z ikI 0 ) in the piezoelectric cylinder and the electrical potential

M (M

M 2 M 02 ) in the piezoelectric

cylinder due to the incident wave are shown in Figure 4 and Figure 5, respectively.

192

Figure 5. Distributions of electrical potential (b) t

M

for different position angles T , k

200, J

5D : (a) t

0,

T / 4 , and (c) absolute values of M

4. Conclusions

The following conclusion can be drawn. (1) When the incident angle is specified, the effect of cylinder radius on dynamic stress concentration factor is negligible. The dynamic stress concentration around the cylinders depends upon the properties of the incident wave; on condition that the cylinder radius is specified, the effect of the incident angle on the dynamic stress concentration is obvious for the low frequency wave, while the effect is not obvious for the high frequency wave. (2) Wave numbers have important influence on the distributions of mechanical displacement u z in the piezoelectric cylinders; position angles have important effects on the distribution of electrical potential

M

in the piezoelectric cylinders.

These results can provide meaningful suggestions for the design of 1-3 connectivity piezoelectric composites.

Acknowledgement: The present work is sponsored by the National Natural Science Foundation of China (No. 50135030).

193

References 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13.

J.F. Tressler, S. Alkoy, A. Dogan, R.E. Newnham, 1999, Functional composites for sensors, actuators and transducers, Composites Part A, 30, 477-482. W.A. Smith, 1992, The key principles for piezoelectric ceramic/polymer composites, Proceedings of the Conference on Recent Advances in Adaptive and Sensory Materials and Their Applications, Blacksburg, VA. H. Gordon, A. John, 1985, Unidimensional modeling of 1-3 composite transducers, J. Acoust. Soc. Am. 88, 599–608. H. Lai, W. Chan, U. Joseph, 1989, Simple model for piezoelectric ceramic/polymer 1-3 composites used in ultrasonic transducer applications, IEEE Trans. U. F. F. C., 36, 434–441. W.A. Smith, et al, 1990, The application of 1-3 piezocomposite in acoustic transducers, Pro. 7th Inter Sympl Appl Ferro., 145–152. G.G. John, A.K. John, et al, 1994, Piezoelectric materials for acoustic wave applications, IEEE Trans. U. F. F. C., 41, 53–59. Q.M. Zhang, H. Wang, L.E. Cross, 1993, Piezoelectric tubes and tubular composites for actuator and sensor applications, J. Mater. Soc., 28, 3962–3968. D.T. Wang, K. Li, H.L.W. Chan, 2004, High frequency 1-3 composite transducer fabricated using solgel derived lead-free BNBT fibers, Senaors and Actuators A, in press. F. Honarvar, et al, 1996, Acoustic wave scattering from transversely isotropic cylinders, J. Acoust. Soc. Am., 100, 57–63. Y. Shindo, K. Minamida, F. Narita, 2002, Antiplane shear wave scattering from two curved interface cracks between a piezoelectric fiber and an elastic matrix, Smart Mater. Struct., 11, 534-540. Y. Shindo, H. Moribayashi, F. Narita, 2002, Scattering of antiplane shear waves by a circular piezoelectric inclusion embedded in a piezoelectric medium subjected to a steady-state electrical load, Z. Angew. Math. Mech., 82, 43-49. Y.J. Liu, Z.Y. He, H.M. Fan, 2004, Scattering of SH-waves by an interface cavity, Acta Mechanica, 170, 47-56. Y.H. Pao, C.C. Mow, 1973, Diffraction of Elastic Waves and Dynamic Stress Concentration, Crane Russak & Company Inc., New York, 217–220.

STRESS ANALYSIS FOR AN ANISOTROPIC SOLID WITH VARIABLE OFF-AXIS OF ANISOTROPY

Kazumi Watanabe Department of Mechanical Engineering, Yamagata University, Yonezawa, Yamagata 992-8510 Japan E-mail: [email protected]

Abstract

Stress analysis for anisotropic elastic solids under anti-plane shear deformation is carried out. The axis of anisotropy is varying with depth from horizontal to normal and repeating its variation in the medium. Two problems, stresses in a thick plate and stress distribution around a Griffith crack, are considered. For both problems, exact closed form solutions are obtained and numerical examples show the nature of the variable off-axis of anisotropy. 1. Introduction

Stress analysis, so far done for anisotropic solids, has been limited to the case of uniform axis of anisotropy. However, fibers in injection molded fiber reinforced plastics (FRP) show the nonuniform orientation and are changing their orientation with depth. In the FRP, an off-angled interface, so called “weld line,” is formed by counter flows and the estimation of its strength is one of crucial engineering problems [1]. The fiber near the weld line is changing its direction with the distance and the weld line constitutes an interface of off-angled axis of anisotropy. The similar nature of the off-angled interface can be found in the grain boundary in metals. Figure 1. Anisotropic solid with variable off-axis of anisotropy

Recently, Watanabe and Adachi [2] introduced a mathematical model for the anisotropic solid with variable off-axis of anisotropy. It states that the off-angle of anisotropy axis is varying with depth as shown in Figure 1. 194 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 194–204. © 2006 Springer. Printed in the Netherlands.

195

Hooke’s law for the anti-plane shear deformation is assumed as

§ V xz · ¨ ¸ © V yz ¹

H sin y · § u z , x · S (c44  c55 ) §1  H cos y ¸, ¨ ¸¨ 1  H cos y ¹ © u z , y ¹ 2h © H sin y

(1)

where cij are shear moduli in the fiber and its normal directions and dimensionless variables, x  S x / h, y  S y / h , are introduced. The reference length h is a half period of off-axis variation and the parameter H is defined as 1  H (c44  c55 ) /(c44  c55 )  1 . (2) The similar modeling for austenite welds was introduced by Abraham and Wickham[3]. Equilibrium equation in terms of the anti-plane displacement u z is given by

(1  H cos y )u z , xx  2H sin yu z , xy (1  H cos y )u z , yy H cos yu z , x H sin yu z , y

0 ,

(3) where the subscript after the comma denotes the partial differentiation with respect to the subscript variable. Equation (3) is one of Hill’s equations and its exact solution has been found by the author [4]. It is given in the form of Fourier integral, uz

1 2S

f

ª

­°

¬

¯

³ «« A([ ) exp ®°2 | [ | tan

f

1

§ 1 H y · ½° tan ¸¸ ¾ ¨¨ 2 ¹ ¿° © 1 H

(4)

­° § 1 H y · ½°º tan ¸¸ ¾» exp ª¬ i[ ^ x  log(1  H cos y )`º¼ d[  B ([ ) exp ® 2 | [ | tan 1 ¨¨ 2 ¹ ¿°» © 1 H ¯° ¼

The present paper shows two applications of the solution for discussing the stress field around the off-angled interface. One is a thick plate with the interface as depicted in Figure 2 and the other is the stress distribution around Griffith crack on the interface. 2. Plate with Off-angled Interface

Let us consider a thick plate composed of two layers as in Figure 2. The upper layer occupies the region, 2I1 d y d S , and the lower does 2I1 t y t S . Two layers constitute a weld line at their interface, y and

2I1 and y

2I2 , where I1

I2 are off-angles at the interface. A uniform shear stress Q0 with width 2a is

applied on the surface of each layer. Then, boundary conditions are given by V yz(1) | y 2I1 V yz(2) | y 2I2 , u z(1) | y 2I1 u z(2) | y 2I2 ; 0 d| x | f ,

(5)

196

V yz(1) | y

V yz(2) | y

S

S

­Q0 ; 0 d| x |d a , ® ¯ 0; a | x | f

(6)

where physical quantities in the upper and the lower layers are classified by the superscripts (1) and (2), respectively.

y S

y S

Interface off-angle

2I1

y

I1

S / 2  I1

I2

S / 2  I2

2I2

y

y (a)

    

S H !0

y

S



    (b) H  0 Figure 2. Off-angled interface in a plate

Applying the standard technique of Fourier transform [5] and integration formula [6] f

sinh(ax)

³ sinh(cx) sin(bx) 0

we have for the stress

S Q0

V yz(1)

dx x

­ §Sa · § S b ·½ tan 1 ® tan ¨ ¸ tanh ¨ ¸¾ , © 2c ¹ ¿ ¯ © 2c ¹

(7)

V yz ,

ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0  l1  l2 ) ½º tan 1 « tan ® ¾ tanh ® ¾» 2(2S  m1  m2 ) ¯ ¿¼ ¬ ¯ 2(2S  m1  m2 ) ¿ ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0  l1  l2 ) ½ º  tan 1 « tan ® ¾ tanh ® ¾» 2(2S  m1  m2 ) ¯ ¿¼ ¬ ¯ 2(2S  m1  m2 ) ¿ ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0 ) ½º  tan 1 «cot ® ¾ tanh ® ¾» ¯ 2(2S  m1  m2 ) ¿¼ ¬ ¯ 2(2S  m1  m2 ) ¿ ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0 ) ½ º  tan 1 «cot ® ¾ tanh ® ¾ » ; 2I1 d y d S ¯ 2(2S  m1  m2 ) ¿ ¼ ¬ ¯ 2(2S  m1  m2 ) ¿ (8)

197

S Q0

V yz(2)

ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0  l1  l2 ) ½º tan 1 « tan ® ¾ tanh ® ¾» 2(2S  m1  m2 ) ¯ ¿¼ ¬ ¯ 2(2S  m1  m2 ) ¿ ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0  l1  l2 ) ½º  tan 1 « tan ® ¾ tanh ® ¾» 2(2S  m1  m2 ) ¯ ¿¼ ¬ ¯ 2(2S  m1  m2 ) ¿ ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0 ) ½º  tan 1 «cot ® ¾ tanh ® ¾» ¯ 2(2S  m1  m2 ) ¿¼ ¬ ¯ 2(2S  m1  m2 ) ¿ ª ­ S (S  m( y )) ½ ­ S ( x  a  l ( y )  l0 ) ½º  tan 1 «cot ® ¾ tanh ® ¾» ;  2I2 d y d S ¯ 2(2S  m1  m2 ) ¿¼ ¬ ¯ 2(2S  m1  m2 ) ¿ (9)

where

l ( y ) log(1  H cos y ), l0 m( y )

l (rS ), l1

§ 1 H y· 2 tan 1 ¨¨ tan ¸¸ , m1 2¹ © 1 H

l (2I1 ), l2

m(2I1 ), m2

l (2I2 ),

m(2I2 ) .

(10) (11)

V yz in the plate. The thickness in the upper layer is S  2I1 and the lower, S  2I2 . Total thickness of the plate is 2(S  I1  I2 ) . Thus the thickness for each sub-figure is different from other two. The ordinate in the figures normalized by ( y  2I1 ) / S for the upper layer and ( y  2I2 ) / S for the lower. The abscissa is also by x / S . It should be noticed that Figures 3 and 4 show stress contours for

the stress gradient across the off-angled interface is discontinuous, but is continuous only when zero off-angle. 3. Stress Distribution around a Crack

When a Griffith crack is present on the off-angled interface, y

2I1 , 2I2 , the

following boundary conditions are employed:

V yz(1) | y

V

(1) yz y 2I1

V

|

u z(1) | y

V

V yz(2) | y

2I1

2I1

|

|

u z(2) | y

(1) yz y of

2I2

(2) yz y 2I2 2I2

0, V yz(2) | y

; 0 d| x | f ,

Q0 ; 0 d| x | a 0; a | x | f f

0; 0 d| x | f

(12) (13) (14)

where the superscript also classifies the quantities in the upper and the lower half spaces.

198

This crack problem is very a classic one and we can solve exactly by applying Sneddon’s technique [5] and the integration formulas [7], f

³ J1 (ax) cos(bx) exp(cx)dx

I c ( a , b, c )

0

f

³ J (ax) sin(bx) exp(cx)dx

I s ( a , b, c )

1

0

b  b 2  p12 ab b  b 2  p12 ab

where a, b, c are positive constants and 1 p1 ( a  b) 2  c 2  ( a  b) 2  c 2 , p2 2

^

`

1 2

^

p22  b 2  b b 2  p12 ( a  b ) 2  c 2 ( a  b) 2  c 2



p22  b 2 b  b 2  p12



( a  b) 2  c 2 ( a  b) 2  c 2 (15)

`

( a  b) 2  c 2  ( a  b) 2  c 2 . (16)

The stress components are given by

1 H 2 sgn{x  l1  l ( y )}I s (a,| x  l1  l ( y ) |, m( y )  m1 ) 1  H cos y

1 (1) V xz Q0  1 (1) V yz Q0

H sin y I c (a,| x  l1  l ( y ) |, m( y )  m1 ), 1  H cos y

(17a)

aI c (a,| x  l1  l ( y ) |, m( y )  m1 ),

for the upper half space, 2I1 d y  f , and

1 (2) V xz Q0

 

1 (2) V yz Q0

1 H 2 sgn{x  l2  l ( y )}I s (a,| x  l2  l ( y ) |,  m( y )  m2 ) 1  H cos y

H sin y I c (a,| x  l2  l ( y ) |,  m( y )  m2 ), 1  H cos y

(17b)

aI c (a,| x  l2  l ( y ) |,  m( y )  m2 ),

for the lower half space, 2I2 t y ! f . Figures 5-7 show the contour for the maximum shear stress,

V

2 xz

 V yz2

1/ 2



/ Q0 .

The ordinate and abscissa are normalized as those in the plate and the crack length is 2a S . In Figure 5, the off-axis of anisotropy is getting normal with distance from the cracked interface and the contour is deformed to the upper/lower direction. On the other hand, in Figure 6 where the angle is getting horizontal, the contour is suppressed and deformed to the horizontal direction. The stress gradient across the

199

cracked interface is discontinuous as was shown in the case of the plate. When two off-angles at the interface are different to each other, the maximum shear stress has a jump/discrepancy across the interface, because of the discrepancy in the stress V xz .

200

201

202

203

Figure 6. Contours of the maximum shear stress

Figure 7. Contours of the maximum shear stress angle,

(V xz( j ) ) 2  (V yz( j ) ) 2 / Q0 for H

0.5

(V xz( j ) ) 2  (V yz( j ) ) 2 / Q0 in the case of the non-symmetric off-

I1 S / 3, I2 S / 6

204

References 1. 2. 3. 4. 5. 6. 7.

R. Selden, 1997, Effect of processing on weld line strength in five thermoplastics, Polym. Eng. Sci., 37, 205-218. K. Watanabe, K. Adachi, 2001, SH wave in a transition layer of anisotropy, Int. J. Solids & Struct., 38, 4825-4838. I.D. Abrahams, G.R. Wickham, 1992, The propagation of elastic waves in a certain class of inhomogeneous anisotropic materials, Proc. R. Soc. Lond. A, 436, 449-478. K. Watanabe, B. N. Abu Hanipah, 2004, Exact closed form solution for an anti-plane deformation of anisotropic media, Quart. Appl. Math., to appear. I.N. Sneddon, 1951, Fourier Transforms, McGraw-Hill. I.S. Gradshteyn, I.M. Ryzhik, 1994, Table of Integrals, Series and Products, 5th Ed., Academic Press. V.I. Fabrikant, G. Dome, 2001, Elementary evaluation of certain infinite integrals involving Bessel functions, Quart. Appl. Math., 59, 1-24.

Shape Memory Alloys

DEFORMATION INSTABILITY AND PATTERN FORMATION IN SUPERELASTIC SHAPE MEMORY ALLOY MICROTUBES Q. P. Sun and P. Feng Department of Mechanical Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, P. R. China Email: [email protected]

Abstract This paper reports briefly the observed deformation instability and domain morphology evolution during stress-induced austenite o martensite (AoM) phase transformation in a superelastic NiTi polycrystalline shape memory alloy microtube. High-speed data and image acquisition techniques were used to investigate the dynamic and quasistatic events which happened in displacement controlled quasi-static tensile loading/unloading process of the tube. These events include dynamic formation, self-merging, topology transition and front instability of a macroscopic deformation band. The reported phenomena brought up several important issues in the fundamental understanding of the instability and pattern evolution in polycrystals under mechanical force. These issues are believed to be essential in the theoretical modeling and worth further investigation in the future. 1.

Introduction

Considerable effort and advances, both theoretical and experimental, have been seen in the past 20 years on the investigation of mechanical behaviors of shape memory alloys. However, in the aspect of deformation instability of the material during phase transition, especially for polycrystals, experimental effort has been relatively weak. Much of the previous experimental researches on NiTi (see [1-3] and the references therein) were focused on isothermal and/or non-isothermal transitions in strip and wire samples. The observed instability phenomena are mainly about a straight smooth front or necking zone propagating in the specimen. It is known that phase transition in NiTi polycrystal is of the first order in nature and involves intrinsic material instability at different length scales. Thus complicated microstructure interaction and evolution at microscale are inherent in such a process and may percolate up to the macroscopic level as demonstrated through mechanical response and surface morphology. However, exact and detailed observations of pattern formation and evolution in NiTi thin walled tubes have not been seen in the literature. 207 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 207–216. © 2006 Springer. Printed in the Netherlands.

208

Experimental research on deformation instability of NiTi microtubing started only in recent years [4,5]. In addition to the facilitation of biaxial loading, the advantage of using the tube configuration is that an isolated deformation domain (martensite band) could survive in the tube after nucleation and that deformation patterns associate with the band morphology evolution during the loading process could be produced and observed. Preliminary study by the authors showed that deformation instability by phase transition under tension manifest itself not only through sharp stress jumps in the nominal stress-strain curves of the tube but also through the formation of various evolving deformation patterns as revealed by careful tube surface observations. This short paper reports some key results from the authors’ recent experimental investigation on the superelastic polycrystalline NiTi microtubes. The purpose of the research is to obtain systematic and quantitative information on the spacial and temporal evolution of the deformation patterns and eventually to provide a critical experiment for future theoretical development of this type of materials. 2.

Material and Test Procedures

The material used in the experiment is a polycrystalline NiTi tube with composition of 55.4 wt. % Ni (Nitinol Devices & Components, USA). The original outer diameter is 1.78 mm and the inner diameter is 1.47mm. The grain size is in the range of 50~100 nm and strong texture exists in the tube. Using differential scanning calorimeter (DSC) the measured austenite finish temperature (Af) of the tube is 1 q C so the material is in austenite state and will exhibit superelastic behavior at room temperature. The sample was chemically etched by hydrofluoric acid into a dog-bone shape and then mechanically polished by fine grained sand papers to reach a final roughness of less than 0.15 ǂm. An electro-mechanical-optical testing system suitable for NiTi microtubing is developed to facilitate a series of precise measurements such as high speed data acquisition and synchronized recording of load-displacement response and image of the tube surface. The testing system consists of four subsystems: loading, measurement, surface observation and automation. Detailed descriptions of each subsystem are given in references [6,7]. The systems were tested with a calibrated specimen and the reliability and accuracy of the developed system is thus assured. The austenite to martensite (AÆM) phase transition of the tube is induced by a quasistatic displacement-controlled uniaxial tension at room temperature on the manufactured testing system. In order to minimize the self-heating effect caused by the transformation latent heat, all tests were performed at a very low elongation rate of 1.2 Pm s-1with a corresponding nominal strain rate of 2.6×10-5 s-1.

209

3.

Key Results

The typical measured macroscopic nominal stress-strain (S-S) curve of the tube during loading and unloading is shown in Figure 1. During loading process (i.e., monotonic increase of the tube length), the deformation consists of the following stages where distinct physical processes can be identified from both the S-S curve and the surface morphology. They are:

Figure 1.

1. Linear elastic deformation of austenite followed by macroscopically homogeneous partial transformation of austenite (pre-burst incubation), see “Loading I” in Figure 1 and the corresponding surface morphology in Figure 2 A, B. 2. Dynamic formation (started at peak load) of transformation band and its rapid growth into a helical shaped band (by autocatalysis reaction), see Figure 1 B-C with the blow up of the S-S curve. The corresponding surface morphology of the tube is shown Figure 2 B-C and (a). The variation of front velocities at middle point and sharp tip of the front with the applied stress was measured and is shown in Figure 3. 3. Stable helical band growth followed by its self-merging at which the band topology was transformed from helix to cylinder (see Figure 1 D-F with the blow up of S-S curve during merge and the tube surface morphology in

210

Figure 2 C-E and (b)). At the same time the original smooth front of helix branched into several sharp tips, see Figure 2 E-F and (c) for detail. 4. Further growth of the cylindrical band via the convoluted propagation of the macroscopic branched front, exhibiting a nearly steady state advancing and rotating pattern in the tube, see “Loading IV” in Figure 1 and Figure 2 G-H and (c).

Figure 2.

To avoid the tube end effect, the specimen was unloaded far before the fronts reached the tube end. Unloading process involved the following deformation stages which are not simple reversals of those in loading process and the reverse transition

211

took place on much lower stress plateau due to the hysteresis of transformation. They are: 1. Elastic unloading of the austenite and martensite band regions with the tube surface pattern unchanged, see Figure 1 G-H and Figure 2 G-H. 2. At the lower stress plateau, shrinkage of the cylindrical band through the reverse convoluted motion of the branched front, see Figure 1 I-K and Figure 2 I-K. 3. Front instability happened when the band length is reduced to a critical value, the branched front became straight smooth and the band became an inclined ring. Reloading the tube to the upper stress plateau gave the opposite result, the smooth front of inclined ring switched back to the dendritic shape with several long sharp tips, see Figure 1 “Unloading-III”, Figure 2 K-L and Figure 4 for details. 4. Further unloading will change the ring topology, via the merge of the fronts at the nearest location, into an lenticular inclusion which annihilated dynamically, signalling the completion of reverse transformation process, see Figure 1 “Unloading-II” and the blow up of S-S curve, Figure 2 (d) and Figure 5 (a) (surface morphology) and (b) (3-D drawing) for details. 5. Elastic unloading of the austenite (Unloading-I in Figure 1). 4.

Discussions

4.1 PHYSICAL ORIGIN OF THE INSTABILITY IN POLYCRYSTALS

MACROSCOPIC

DEFORMATION

Before macroscopic instability, the loading curve exhibited very small amount of nonlinearity due to the formation of many microscopic martensite bands in the grains but macroscopically the deformation is still homogeneous. This macroscopic homogeneous deformation (here termed as pre-burst incubation) continues until the peak load that marks the beginning of macroscopic burst transformation. Here we must distinguish the microscale band formation (nucleation and growth) during the rising portion of the S-S curve with the macroscopic band formation during the load drop. There are so far two explanations for the possible physical origins underlying the pre-burst and burst deformations (shown as Loading-I and Loading-II in Figure 1). One speculated mechanism is the autocatalytic nucleation and coalescence of many micro-martensite bands inside the NiTi polycrystal. The autocatalytic transformation happened because the interaction among the local stress fields of the micro-bands become increasingly strong with loading and eventually these micro-events percolate up to the macroscopic level and lead to the strain softening of a macroscopic representative material element until it is fully transformed. The phenomenon of autocatalysis, wherein a region of elastic phase transformation creates stresses sufficient to drive alone further transformations, was also observed

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in transformation process of other material systems such as ceramics and metals. Another possible mechanism leading to the macroscopic band formation, without invoking the intrinsic material softening, is the geometric instability due to the finite transformation strain [8] as in the neck formation observed in some polymers whereas the material’s true stress-strain relation is still stable. Recent numerical analysis by finite element method [9] on band morphology in NiTi tube indicated that only geometry effect of finite deformation is not sufficient to quantify the experimental observation and that material instability dominates the observed helical band formation. There is a notable theoretical effort in recent years (see [10-12] and the references therein) on the modelling of the nucleation peak, propagation plateau and hysteresis phenomena, emphasizing the discreteness and nonlocality of the phase transition process. These works captured some key features of experimental observations and indeed provided some insight in understanding the nature of the phenomena. The pre-burst, burst and post-burst transformations in the present experiment consistently demonstrate that microscopic instability developed before and after the macroscopic homogenized system becomes unstable. Quantitative modeling for real polycrystals along this line of thought will still be a challenge topic.

4.2 IMPACT OF THE FRONT ENERGY ON THE BAND MORPHOLOGY AND PATTERN EVOLUTION One of the successes in the present investigation is the identification of front energy and the revelation of its essential role in the formation and evolution of deformation patterns in polycrystal. First, it is seen that the formation and growth of the helical band broke the symmetry of axial loading and tube geometry. This symmetry-breaking morphology evolution could be attributed to the energy preference of the final “equilibrium shape” of a domain in which the transformation strain is shear in nature. Though it is not appropriate to express the structures interested as the minimizers of free energy for dissipative systems like SMA, the concept of “equilibrium shape” provides a simple criterion to estimate and to understand the domain shapes and their evolution during mechanical loading. Due to the difference in the dependence on domain size of bulk energy (such as chemical/elastic strain energy) and interfacial energy, the equilibrium shape obtained by minimizing their sum is also size-dependent. In particular, the interfacial energy plays a dominant role at small sizes, whereas the bulk energy gradually becomes dominant in the domain shape at larger sizes [13]. Second, the front energy played an important role also in the band merging/vanishing process. When the two fronts came close enough the driving force from the sudden reduction in surface energy of the system makes the fronts merge very quickly producing a rapid load drop in the S-S curve. Finally, the role of front energy manifested again during the formation of the branched front and the front instability. The competition between bulk strain energy and front (surface) energy during unloading/reloading

213

may lead to branched ring l smooth ring and smooth ring l inclusion topology transitions. One way of accounting the surface energy quantitatively is to implement numerical simulations within the nonlocal and nonconvex framework, which are currently under the way. 4.3 MOBILITY OF FRONT DURING BAND GROWTH AND PATTERN EVOLUTION A significant progress in the present work is that we are able to measure the front velocity at different points of the domain boundary where the driving force changes during the pattern evolution. Thus a quantitative constitutive kinetic relationship between the macroscopic front velocity and the driving force could be obtained for the material. It must be noted that the macroscopic front propagation is actually realized through the successive transformation of numerous grains in front of it. In the sense of continuum theory, the measured kinetic relationship captures the macroscopic effect of numerous micro-events that facilitated the front motion in polycrystal and is very important for a continuum description of transformation process [14]. Generally the front velocity v is expressed as v = v(f) where f is the thermodynamic driving force at the front and f is usually a function of the externally applied force, domain configuration and temperature.

Figure 3.

214

It is known that the latent heat generated as the front propagates will reduce the chemical driving force, so the front velocity is limited by heat dissipation. Also, as the band grows rapidly during the dynamic band formation, the resulted load drop will in turn reduce the driving force of the front. This leads to a rapid decrease in front velocity as demonstrated in Figure 3. Such thermomechanical coupling in the measured kinetic relationship has been analyzed and incorporated into a one-dimensional setting [15,16]. It is shown that our measured results in tube configuration agree quite well with both the previous experimental data from wire specimen [15] and the 1-D theoretically predicted front kinetics [15,16] as shown in Figure 3. Notice that the data of tip velocity are shifted because the driving force at the tip is higher than that at the middle point which is roughly equal to the applied stress, thus causing the data shift in Figure 3.

Figure 4.

Figure 5.

Under the applied tensile stress the driving force for the front of a lamellar band observed here varies along the front and is the largest at the tips and the smallest at the middle portion of the fronts. Therefore the band will initially grow all around but with the tip velocity larger than the transverse velocity, making the band

215

spiraled in the tube configuration. In the late stage of growth, the fronts of this helical band became very close and started to attract each other, leading to the band topology transition via self-merging. In the near steady state band growth stage, the moving pattern in the tube is produced by the highly coordinated motion of the front. In all the observed phenomena, however, the controlling mechanisms that determine the front instability and tip splitting and govern the coordinated motion of the front still need to be investigated in the future. 5.

Non-concluding Remarks

Taking the advantage of tube configuration and using the synchronized load-deformation and surface morphology observation technique, we are able to observe the whole nucleation and growth process of a single stress-induced macroscopic deformation band under displacement controlled uniaxial tension. The observation revealed several interesting pictures of pattern evolution during loading, unloading and reloading. Important processes and events such as pre-burst incubation, dynamic band formation, front convolution and instability and band topology transitions were recorded and measured quantitatively. These phenomena brought up several important issues for phase transition under mechanical forces. The underlying physical mechanisms are only briefly discussed and are still not well understood. The following remarks on the experimental findings are not conclusive and just listed for open discussions: There is an incubation stage where micro-instability occurred before the macroscopic deformation of the tube became unstable. After the macroscopic band formation, the propagation of the macroscopic front is effected through the numerous successive micro-instabilities associated with the transformation of individual grains in front of the macroscopic interface. Such multi-scale scenario of deformation process reflects the discrete and nonlocal natures of the material hierarch. The macroscopic transformation front carries its own energy. The competition between this surface energy and the bulk energy may play an essential role in the band morphology and the pattern evolution. Though such interfacial energy can be estimated by fitting the experimental data, understanding and quantifying this energy term in polycrystalline material remain open for the future investigation. The kinetics of the front is measured directly in the experiment. The measured front speed and driving force relationship agrees quite well with the previous one-dimensional theoretical relationship considering heat transfer effect. It seems that the combined effort of kinetic law and energetic preference governs the observed various aspects of pattern formation and evolution, but exact conditions and detailed mechanisms on Why, When and How they happen remain to be unveiled through future systematic research.

216

Acknowledgements  The authors are grateful for the financial support from the Research Grants Council of Hong Kong SAR, China (through Project No. HKUST 6234/01E and Project No. HKUST 6245/02E). References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16.

J.A. Shaw, S. Kyriakides, 1995, Thermomechanical aspects of NiTi. J. Mech. Phys. Solids, 43, 1243-1281. J.A. Shaw, S. Kyriakides, 1997, On the nucleation and propagation of phase transformation fronts in a NiTi alloy. Acta Mater., 45, 683-700. P.H. Leo, T.W. Shield, O.P. Bruno, 1993, Transient heat transfer effects on the pseudoelastic behavior of shape-memory wires. Acta Metal. Mater., 41, 2477-2485. Z.Q. Li, Q.P. Sun, 2002, The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension. Int. J. of Plasticity, 18, 1481-1498. Q.P. Sun, Z.Q. Li, 2002, Phase transformation in superelastic NiTi polycrystalline micro-tubes under tension and torsion - From localization to homogeneous deformation. Int. J. Solids and Structure, 39, 3797-3809. P. Feng, Q.P. Sun, 2004, submitted to J. Mech. Phys. Solids. P. Feng, 2005, PhD thesis, Hong Kong University of Science and Technology. D. Favier, Y. Liu, L. Orgeas, R. Rio, 2001, Solid Mechanics and Its Applications, Vol. 101, Q.P. Sun ed., Kluwer Academic Publisher, 205-212. Z.D.Hu, Q.P. Sun, 2003, unpublished work. L. Truskinovsky, A. Vainchtein, 2004, The origin of nucleation peak in transformational plasticity. J. Mech. Phys. Solids, 52, 1421-1446. R.D. James, 1996, paper for the Symposium in honor of J. L. Ericksen, Maryland, USA. R. Abeyaratne, C. Chu, R.D. James, 1996, Kinetics of materials with wiggly energies: Theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy. Philosophical Magazine A, 73, 457-497. W.C. Johnson, J.W. Cahn, 1984, Elasticity induced shape bifurcations of inclusions. Acta Metal., 32, 1925-1933. R. Abeyaratne, J.K. Knowles, 1997, On the kinetics of an austenite->martensite phase transformation induced by impact in a Cu-Al-Ni shape-memory alloy. Acta Mater., 45, 1671-1683. O.P. Bruno, P.H. Leo, F. Reitich, 1995, Free boundary conditions at austenite-martensite interfaces. Phys. Rev. Lett., 74, 746-749. T.W. Shield, P.H. Leo, W.C.C. Grebner, 1997, Quasi-static extension of shape memory wires under constant load. Acta Mater., 45, 67-74.

THEORETICAL CONSIDERATION ON THE FRACTURE OF SHAPE MEMORY ALLOYS Wenyi Yan1 and Yiu-Wing Mai2 1 Computational Engineering Research Centre, Faculty of Engineering and Surveying, University of Southern Queensland, Toowoomba Qld 4350, Australia 2 Centre for Advanced Materials Technology, School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney NSW 2006, Australia

Abstract The application of shape memory alloys (SMA), as an advanced material, has accelerated in recent years, especially in biomechanical engineering. However, there is a lack of understanding of the fracture behavior of SMA devices. This paper presents our consideration on the theoretical study of the fracture properties of SMA. Owing to the existence of a new transformed phase near the crack-tip region, the transformation strain, including the transformation volume strain and the shear strain, the plastic deformation, and the mismatch of elastic property will alter the crack-tip stress field and hence govern the fracture behavior of the SMA material. Therefore, it is vital to clarify the influence of these factors on the fracture toughness. Following this consideration, the paper reports our recent research progress in this direction. First, a simple study is carried out to show the influence of transformation consisting of pure volume contraction. These results reveal that the phase transformation with volume contraction in SMA tends to reduce their fracture resistance and increase the brittleness. Second, a constitutive model is established to quantify the effect of stabilization of plasticity on the stress-induced martensitic trans-formation. Third, the effect of transformation strain with shear and volume components on the fracture toughness of a superelastic SMA is studied. 1. Introduction Shape memory alloys are well-known smart materials that possess two important properties: shape memory effect and superelastic deformation. These behaviors are due to the intrinsic thermoelastic martensitic transformation at different temperatures. At relatively low temperatures, with the application of an external force, the initial parent phase (austenite) is transformed to a martensitic phase, thus resulting in macroscopic deformation. After unloading, the material remains in a 217 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 217–226. © 2006 Springer. Printed in the Netherlands.

218

martensite state with a residual strain. Upon heating to a certain temperature, the martensite can transform back to the initial austenite phase with the material returning to the initial shape. This is the shape memory effect. For the same material, at some higher temperature, the transformed martensite induced by stresses can transform in reversal to austenite during unloading and recovers a large amount of deformation. This extraordinary recoverable deformation behavior is called superelasticity or pseudoelasticity [1]. Both the shape memory effect and superelasticity have been exploited to design SMA-based functional and smart structures in mechanical and biomedical engineering [2-4]. A number of commercial products are already available on the market. For instance, couplings and fasteners based on shape memory effect have been extensively developed and applied. A historical example is the large-scale application of SMA coupling to connect titanium hydraulic tubing in the aircraft F-14 in 1971 [5]. Many more potential applications of SMA have been investigated. For example, it was recently discovered that shape memory alloy NiTi had super resistance against wear due to its superelasic deformation and would be applied in tribological engineering [6]. Other important applications include SMA-based composites [7, 8]. Historically, research on SMA has developed in three stages. Early studies focused on the crystallography and thermodynamics of martensitic transformation. For example, Wechsler et al. [9] proposed a crystallographic habit plane theory on the formation of martensite, and Olson and Cohen [10] studied the martensitic nucleation process. Due to the extraordinary mechanical behaviors of SMA, constitutive laws were developed to describe the stress-strain-temperature relations during the forward and reverse trans-formation processes, which dominated the second stage of research on SMA in the past decade. Thus, Liang and Rogers [11] proposed a model based on the thermo-mechanical theory. Sun and Hwang [12] established the micromechanics constitutive model based on an internal variable theory. Crystallographic theory of martensitic transformation was also considered in the models [13, 14]. The third stage or the current trend is to study the failure mechanisms of SMA, that is, their fracture and fatigue mechanisms. With the significant increase in applications of SMA as functional and smart structural materials, characterization of their failure mechanisms is rapidly gaining momentum. A few experimental studies performed on the fatigue of SMA have been reported in a recent overview paper [15]. But as pointed out by McKelvey and Ritchie [16], “a systematic examination on the role of temperature, microstructure, and constitutive behavior on crack-growth rates in NiTi” is still lacking. Van Humbeeck argued that “little is known on … fracture mechanics, [and] fatigue” of SMA [17]. Compared to conventional metals, SMA exhibits much more complicated failure mechanisms; this is due to its intrinsic phase transformations. As shown in Figure 1, the thermal force, brought about by the change of temperature, and/or mechanical

219

force will induce the forward austenite-to-martensite transformation or the reverse martensite-to-austenite transformation. In addition, the mechanical force will cause an elastic-plastic deformation in the fracture process zone in the SMA. Some initial experimental results suggested that plastic deformation might hinder the reverse transformation [16], which should be an important part of the constitutive behavior of SMA. Also, the forward austenite-to-martensite transformation yields transformation volume contraction and the reverse martensite-to-austenite transformation yields transformation volume expansion in the SMA [18, 19]. In the absence of any rigorous analysis, it can be expected that transformation volume strain will affect transformation yield stresses, which in turn will affect the failure process because the material near the tip of a growing crack is in a state of high hydrostatic tensile stress.

Temperature variation

Transformation

Transformation strain

Material mismatch

Mechanical force

Elastic-plastic deformation

Elastic-plastic strain

Fracture of SMA

Figure 1. Influencing factors on the fracture of SMA

220

Transformation in the SMA not only results in a new phase but also induces transformation strains, including volume and shear strains. Like transformation toughness in zirconia-containing ceramics, transformation strains will certainly alter the crack growth resistance in SMA. In addition, a transformed phase normally has different elastic properties and different failure-resistance from the parent phase. For example, for binary NiTi SMA the Young’s modulus of the martensite is about one-third to one-half of the Young’s modulus of the austenite [20]. The change of Young’s modulus will change the stress field near the crack-tip, which will directly change the crack driving force. As illustrated in Figure1, all of the aforementioned factors, like transformation strain, material properties mismatch, and elastic-plastic deformation and especially their interaction will affect the failure processes of the SMA materials under either cyclic loading or static loading conditions. In our view, to understand the failure mechanisms and the fracture toughness of SMA, it is important to clarify the relative effect of all these factors on the fracture process in the SMA. In the following sections, we report some of our recent theoretical results in this direction. 2. Effect of Transformation Volume Contraction Unlike the martensitic transformation from the tetragonal to the monoclinic phase in zirconia, the transition from austenite to martensite in SMA will be accompanied by a volume contraction. Although the magnitude of the volume strain is small, about -0.4% for NiTi, recent experimental research [16] indicated that the transformation volume strain in shape memory alloy NiTi might have hindered the transformation process near a fatigue crack-tip with high triaxial stresses. Recently, we studied the effect of pure transformation volume contraction on the fracture toughness of SMA by using a simple forward-reverse transformation volume model, where the transformation shear strain is excluded [21]. Assuming a small-scale transformation zone near the crack-tip, we utilize the linear elastic fracture mechanic theory for this problem. The crack-tip stress intensity factor, Ktip, can be obtained from the applied stress intensity factor, Kapp, and the transformation-induced stress intensity factor, Ktran: K tip K app  K tran . (1) The transformation zone near an advancing crack-tip as a result of forward and reverse transformation is estimated by using the finite element method and the transformation- induced stress intensity factor Ktran is evaluated by [22]

K tran

EH vtr 6 2S (1  Q)

³ r³ A

3/ 2

cos(3T / 2)dA

(2)

221 tr

where H v is the transformation volume strain, A is the transformation area, E is the Young modulus and v is the Poisson ratio of the material. Figure 2 shows the results of the normalized transformation-induced stress intensity factor with crack growth for different transformation volume strains, where h is the half-height of the transformation zone for a stationary crack. Clearly, the numerical result is close to zero before crack advance for all three cases with different values tr of transformation volume strain, H v , which is consistent with the theoretical analysis [23]. More importantly, Figure 2 shows that the transformation-induced stress intensity factor is positive once the crack advances. Thus, the effective stress intensity factor near the crack-tip, K tip , increases. This implies that crack growth would occur at a higher driving force. That is, the apparent toughness of the SMA material would decrease. It is also clear that K tran / K app asymptotically approaches to a constant value after the crack extension exceeds about 5 times the half-height of the transformation zone. Figure 2 indicates that the steady value of the transformation-induced stress intensity factor increases with the amplitude of the transformation volume strain. The increase of the crack-tip effective stress intensity factor is due to the transformation volume contraction, which results in a tensile stress field around the crack-tip, and as a result, K tip increases. 0.20

Ktran/Kapp

0.16

0.12

0.08

Htrv = -0.3% Htrv = -0.39%

0.04

Htrv = -0.60%

0.00

0

2

4

6

8

'a/h Figure 2. Variation of the normalized transformation-induced stress intensity factor with normalized crack growth for different transformation volume strains

222

3. Constraint of Plasticity on Reverse Transformation Recently, McKelvey and Ritchie [16] observed monoclinic martensitic structure in an unloaded NiTi superelastic bar after having experienced stress-induced forward trans-formation and plastic deformation. Furthermore, they have found that the more intense the plastic deformation occurred, the less strain due to the forward transformation could be recovered. Effectively, this means that the plastic deformation would stabilize the stress-induced martensite so that no or only part reverse transformation to austenite could occur after the removal of the applied load. This influence of plastic deformation on the reverse transformation of NiTi shape memory alloy can be represented by the stabilized irrecoverable martensite volume fraction f sta . Quantitatively, f sta is assumed to be dependent on the level of prior plastic strain, H

pl

, in our theoretical study [24], i.e.,

f sta

F ( H pl ) .

(3)

pl

The function F ( H ) can be calibrated from the strain curves measured in the uniaxial tensile tests. The plastic strain in Eq. (3) can be determined by traditional plasticity theory, such as von Mises isotropic hardening theory. The forward-reverse martensitic transformation is described by a phenomenological model which takes into account the influence of the hydrostatic stress in our study [24]. For example, for the forward transformation, the potential function is: F for (ı, f ) V eq  3DV m  Y for ( f ) . (4) tr

From the normality hypothesis, the transformation strain rate H during the forward transformation process can be determined by:

wF for

3 s (5) DI) , f ! 0 . wV 2 V eq The martensite volume fraction rate f is obtained based on the consistency İ tr

Ef

Ef (

condition for transformation, that is:

f

1 3 s : s  3 DV m ) , ( H for 2 V eq

f ! 0 for forward transformation. (6)

In Eqs. (5) and (6) above, the parameters D and E represent the capacity of the macroscopic transformation strains. The maximum transformation volume strain is: H trv 3D E (7) and under uniaxial tensile loading condition, the maximum transformation strain in the tensile direction is, tr H11 E (1  D). (8) One of the advantages of the phenomenological transformation model is that the material constants can be easily calibrated from conventional macroscopic tests. For

223

example, based on the uniaxial experimental results in [16], we can derive: E 4.13% . D 3.15%

(9)

4. Effect of Transformation Strain on Fracture of SMA The transformation model described in Section 3 is utilized to analyze numerically the fracture toughness of SMA. Here, both transformation volume strain and transformation shear strain are considered while the plastic deformation is neglected. Equation (1) is still suitable for this study based on the assumption of small-scale transformation zone. Now the transformation-induced stress intensity factor is calculated from [25]

K tran

E 2 8S (1  Q

2

³³ r ) A

3 / 2

M ( EJGtr , E )dA

(10)

tr

where M ( E rG , E ) is a function of the equivalent transformation strain tensor

E rtrG and the location angle E , which are defined in [25]. Figure 3 shows the variation of the normalized transformation-induced stress intensity factor with the crack growth for different values ofҏǂfrom ҏ finite element analysis. In all cases, K app

3.15% . Figure 3 shows that

40 MPa m and a

a negative value of Ktran will be induced after the crack commences. And, a constant value of Ktran can be achieved once the crack grows over about 4 mm. Negative transformation-induced stress intensity factor implicates that the effective crack-tip stress-intensity factor, Ktip, will be less than the applied stress intensity factor, Kapp, according to Eq. (1). Therefore, transformation in SMA will increase its fracture E represents the recoverable transformation strain under toughness. In Eq. (8), ǂ D , the value of ǂE actually uniaxial loading condition. For a given value of ǂ indicates the magnitude of maximum transformation shear strain. A higher transformation shear strain will result in a larger fracture toughness. For examples, tr when E 1.0% and D 3.15% , which corresponds to H 11 0.97% , the plateau

E

 K tran / K app

15% , which implies K tip / K app

3.0% and D 3.15% ,  K tran / K app 43% , and K tip / K app

i.e.,

H 11tr

2.91% ,

85% . When the

plateau

57% .

The influence of the transformation volume contraction on the fracture toughness is indicated by the two curves shown in Figure 4. The solid line represents the results tr 0 and the dotted line the results from pure transformation shear strain with H v tr

from the transformation volume strain, H v 0.39% . Comparing these two curves, the values of Ktran/Kapp has increased 5% from -58% to -53%, which indicates that transformation volume contraction reduces the negative value of Ktran,

224

i.e., increases the effective value of Ktip or reduces the fracture toughness of the SMA. This result accords the conclusion obtained from the simple study in Section 1. 70

60

40

E = 1.0% E = 1.5% E = 3.0% E = 4.13%

30

-K

tran

/K

app

(%)

50

20

10

0

0

2

4

6

8

10

'a (mm) Figure 3. Variation of the normalized transformation-induced stress intensity factor with crack growth for different values of E 80 70

-Ktran/Kapp (%)

60 50

Htrv = 0

40

Htrv = -0.39%

30 20 10 0 0

2

4

6

8

10

'a (mm) Figure 4. Variation of the normalized transformation-induced stress intensity factor with crack growth for pure shear transformation, compared with transformation volume strain of -0.39%

225

5. Concluding Remarks Theoretical study of the fracture behavior of SMA is discussed in this paper. In order to completely understand the fracture mechanism of SMA, the influence of transformation strains, including volume strain and shear strain, plasticity and material mismatch and their interaction on the fracture process should be investigated. Our results presented in this paper indicate that the phase transformation with volume contraction in SMA tends to reduce their fracture resistance and increase the brittleness. In contrast, the trans-formation shear strain can increase the fracture toughness and this increment will be enhanced with higher value of transformation shear strain. A constitutive model is also reported to quantify the effect of plasticity on stress-induced martensite transformation. In this model, a constraint equation is introduced to quantify the phenomenon of the stabilization of plasticity on the stress-induced martensite based on experimental results. Further work, including experimental study is being carried out to explore the fracture behavior of SMA. Acknowledgement: The present work is sponsored by the Australian Research Council (ARC). Most of the calculations were carried out at the National Facility of the Australian Partnership for Advanced Computing through an award under the Merit Allocation Scheme to W Yan. Y-W Mai wishes to thank the ARC for the award of a Federation Fellowship tenable at the University of Sydney.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

J.M. Christian, 1982, Deformation by moving interfaces. Metall. Trans. A, 13A, 509-538. H. Funakubo, 1987, Shape Memory Alloys, Gordon and Bleach, New York. J. Van Humbeeck, 1999, Non-medical applications of shape memory alloys. Materials Science and Engineering A, 273-275, 134-148. K. Otsuka, C.M.Wayman, 1998, Shape Memory Materials. Cambridge University Press, Cambridge, UK. K.N. Melton, 1998, General applications of SMA’s and smart materials. In: Shape memory materials. Eds: K. Otsuka, C.M. Wayman, Cambridge University Presss, Cambridge, UK. D.Y. Li, 2000, Exploration of TiNi shape memory alloy for potential application in a new area: tribological engineering. Smart Mater. Struct. 9, 717-726. G.A. Porter, P.K. Liaw, T.N. Tiegs, K.H.Wu, 2000, Ni-Ti SMA-reinforced Al composites. JOM-J. Miner. Met. Mater. Soc. 52, 52-56. P. Sittner, R. Stalmans, 2000, Developing hybrid polymer composites with embedded shape-memory alloy wires. JOM-J. Miner. Met. Mater. Soc. 52, 15-20. M.S. Wechsler, D.S. Lieberman, T.A. Read, 1953, On the theory of the formation of martensite. AIME Trans. J. Metals 197, 1503-1515. G.B. Olson, M. Cohen, 1981, Theory of martensitic nucleation: a current assessment. In: Solid-Solid Transformations (edited by H. I. Aaronson et al), 1145-1164. Warrendale, Pennsylvania. C. Liang, C.A. Rogers, One-dimensional thermomechanical constitutive relations for shape memory materials. J. Intelligent Mat. Syst. Struct., 1, 207-234. Q.-P. Sun, K.-C. Hwang, 1993, Micromechanics modeling for the constitutive behavior of polycrystalline shape memory alloys. J. Mech. Phys. Solids 41, 1-33. E. Patoor, A. Eberhardt, M. Berveiller, 1988, Thermomechanical behavior of shape memory alloys. Arch. Mech. 40, 775-794. W. Yan, Q.-P. Sun, K.-C. Hwang, 1998, A generalized micromechanics constitutive theory of single

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3D FINITE ELEMENT SIMULATION FOR SHAPE MEMORY ALLOYS

L.H. Han and T.J. Lu Engineering Department, University of Cambridge Cambridge CB2 1PZ, UK

Abstract In this paper, a methodology based on three-dimensional (3D) finite element procedures is developed and used to simulate the superelastic behavior and the shape memory effect (SME) of shape memory alloys (SMAs). A 3D constitutive model for SMAs is implemented as a user defined subroutine for the finite element code ABAQUS. The kinetic law based on an experimentally defined stress-temperature phase diagram provides the phase fraction history for a given loading path and specified initial value of the martensite fraction. With a simple 3D model, we demonstrate the capability of the proposed methodology to simulate the shape memory effect, pseudoelasticity and hysteresis behavior. This methodology provides a computational tool for the design of 3D actuators using SMAs. 1. Introduction The shape memory effect (SME) and pseudoelasticity (or superelasticity) [1], as shown in Figure 1, are the two most significant characteristics of shape memory alloys (SMAs). Both properties are derived from the first order crystalline phase transformations (e.g. martensite œ austenite; temperature-induced martensite Ÿ stress-induced martensite). After a large inelastic deformation at low temperatures, a SMA can recover its original shape by heating over its phase transition temperature, a phenomenon commonly known as the SME. pseudoelasticity refers to the ability of the material to undergo a large inelastic deformation due to stressing at high temperatures, and recover it without permanent plastic deformation upon unloading. These unique features make the SMAs attractive in many applications, particularly in the medical and aerospace fields [2-4]. More specifically, the SMAs are increasingly used to design actuators and smart structures because of their high stiffness/strength, large recoverable strain (up to 8%) and simple actuation mechanism [2,5]. To utilize SMAs efficiently in smart structures, accurate and reliable predictions of the SME and pseudoelasticity are required. During the last two decades, a number of constitutive models have been proposed 227 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 227–236. © 2006 Springer. Printed in the Netherlands.

228

for SMAs. Most of these fall into three categories [6]: (a) Phenomenological models [7-10], (b) Micromechanical models [1,6,11-16] and (c) Microscopic thermodynamics models [17-21]. In the phenomenological models, the martensite fraction typically is used as an internal variable, and the distinction amongst these models is the formulation of transformation kinetics. In the micromechanical models, the free energy potentials due to the transformation are constructed by performing a micromecahnical analysis and then by using suitable averaging schemes. Phenomenological models are generally more suitable for engineering applications due to their simplicity. The micromechanical approach is more suited for fundamental research than for quantitative description of macroscopic behaviors. Microscopic thermodynamics models are helpful for the understanding of micro-scale behaviors, such as the nucleation and growth of a martensite plate, and the evolution of the interface between different phases. Based on the aforementioned constitutive laws, a few numerical simulations with the finite element (FE) method have been carried out. Built upon the plastic flow theory, Trochu and Qian [22] performed a numerical simulation of pseudoelasticity using nonlinear finite elements. Brinson and Lammering [23] developed an one-dimensional finite element procedure to characterize both the SME and pseudoelasticity. Under the framework of generalized plasticity theory, Auricchio [24-26] presented 1D and 3D simulations for SMAs. A similar method was used by Panoskaltsis et al. [27] for isothermal SMA modelling. A user-defined material subroutine to simulate the 3D superelasticity of Nitinol alloys is now available in the commercially available FE code ABAQUS 6.4. Govindjee and Hall [28] presented the numerical implementation of Abeyaratne’s 1D model [18]. The 3D numerical implementation of a thermomechanical constitutive model was presented by Qidwai and Lagoudas [29], but they did not consider the SME. Among the numerous numerical simulations, most concern 1D/3D pseudoelasticity or 1D/3D SME. Few deal with the 3D one-way SME as well as pseudoelasticity, i.e. the non-isothermal behavior of SMAs. It is inevitable that practical applications require complete thermomechanical loadings including non-isothermal ones. Moreover, even though the most popular SMA applications are in the form of wires or ribbons in which the SMA material behavior is primarily one-dimensional, 3D numerical simulations of SMAs are still required if SMA wires are used in a 3D structure. Since SMA devices are often operated under small thermal and/or mechanical loads that are insufficient to complete the transformations [30], hysteresis subloops (i.e. hysteresis loops due to incomplete transformations) should also be considered. The aim of this research is to provide such a computational tool for the analysis of 3D SMA devices. Although many alloys exhibit the SME and pseudoelasticity, the most popular SMA for applications is the nearly equi-atomic polycrystalline NiTi alloy due to its excellent structural properties, corrosion resistance, and superior memory capability (up to 8%). It has been shown experimentally that the different types of SMAs such as NiTi and Fe-based SMAs [31] have completely different behaviors, and therefore

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should obey different constitutive laws. Here, we limit our study on NiTi. In this work, a 3D finite element simulation procedure is developed. The distinct features of SMAs such as pseudoelasticity and SME are considered. The hysteresis subloop is also considered.

Figure 1. Schematic diagram of NiTi: (a) SME; (b) pseudoelasticity with complete and incomplete phase transformation

2. Thermodynamic Constitutive Model 2.1 CONSTITUTIVE LAW e

Decomposing the total strain H into an elastic part H and a transformation part H tr , and assuming the elastic thermal effects are negligible in comparison with the transformation strain, once can write the stress tensor T as [25]: T = De : ( H  H tr ) (1) where De is the fourth-order compliance tensor that depends on the volume fraction of martensite Y . Here, a linear form of Y is proposed so that De is expressed as: e De = DAe + Y(DM  DAe ) (2)

e e and DM are the elastic compliance tensors when the material is where DA

complete martensite and austenite, respectively. Other forms of De could also be used [32], but the one given in (2) is used here due to its simplicity. It is assumed that the transformation strain rate H tr is proportional to the rate of detwinned martensite fraction, Ys , i.e.

230

H tr = -Ys

(3)

where - is the transformation tensor which determines the transformation strain direction and is assumed to have the following form [29]:

3 S £ ¦ F , ¦ ¦ 2 LS ¦ -=¤ ¦ ¦tr ¦ FL tr , ¦ ¦ ¥ H

Ys > 0 (4)

Ys < 0

Here, FL is the maximum uniaxial transformation strain, H tr is the transformation strain tensor at the reversal of phase transformation, and

1 S = T  tr (T )1 , S = 3

3 S , H tr = 2

2 tr H 3

(5)

where 1 is the second-order identity tensor. 2.2 TRANSFORMATION KINETICS In order to describe both SME and pseudoelasticity, three phases are chosen: twinned martensite, detwinned martensite and austenite, with YT , Ys and YA corresponding to the volume fractions of each phase, respectively. The total volume fraction of martensite can be represented as Y = YS + YT = 1  YA . Austenite can form twinned martensite by cooling in the absence of stressing. When the stress is sufficiently high, both austenite and twinned martensite can convert to detwinned martensite, a process which is typically associated with a large transformation strain. By heating, twinned or detwinned marteniste can convert to austenite. Detwinned martensite converted from austenite due to stressing can be reverted to austenite by unloading. Figure 2 shows a typical tension-temperature ( T  T ) phase diagram of a NiTi alloy, with the shadow areas representing the possible transition zones. Since phase transformation depends on the loading path as well as the initial conditions, the concept of “switching points” [10] is used to represent directional changes of the loading path ( , starting from point S and ending at point E . ( can be divided into several segments by the switching points (e.g., points A, B, C), which are defined as the points where the direction or sense of the transformation changes. Phase transformation can occur in five different zones, depending on the combination of stress and temperature levels. A phenomenological cosine model of transformation kinetics has been developed [8], which could be represented as follows: (1) Zone1 (conversion to twinned martensite)

231

£ 1  YT0 ¦ ¦ [ cos (aM (T  M f ) ) + 1 ], YT = ¦ ¤ 2 ¦ ¦ Y0, ¦ ¥ T

G G U
(6)

otherwise

(2) Zone 2 (conversion to detwinned martensite)

£¦ 1  Ys0 £ ¬² 1 + Ys0 Q ¦ž cr ­ ¦ ¦ T  T cos , ( ) ­» + ¤ f ž cr cr ¦ ¦¥ 2 ®­ ¦¼ Ys = ¦¤ 2 ¦ žŸ Ts  T f ¦ ¦ ¦ Y 0, ¦ ¦ ¥ s

YT =

G G U
(7)

otherwise

1  Ys 0 Y 1  Ys0 T

(8)

(3) Zone 3 (conversion to twinned and/or detwinned martensites)

£ 1  Ys0 £ ¬² 1 + Ys0 G G Q ¦ ¦ cr ­ ¦ ž ¦ T T U 0 cos ,  ( ) ­» + ¤ f ž cr cr ¦ 2 ­ ¦ ž ¦ 2 ®¦ (9) Ys = ¤ ¦Ÿ Ts  T f ¦ ¥ ¼ ¦ 0 ¦ otherwise Y , ¦ ¦ ¥ s 1  Ys 0 YT = Y + %T Y (10) 1  Ys0 T £ 1  YT0 G G ¦ ¦¦ U 0 [ cos (aM (T  M f ) ) + 1 ], %T Y = ¤ 2 (11) ¦ ¦ 0, otherwise ¦ ¥ (4) Zone 4 (conversion to detwinned martensite) £¦ 1  Ys0 £¦ ² 1 + Ys0 Q ¦ ¦ T  Tcrf  C M (T  Ms ) )» + cos ¤ cr , ( cr ¦ ¦ 2 ¦ Ts  Tf ¦ 2 Ys = ¤ ¦ ¦ ¥ ¼ ¦ 0 ¦ ¦ ¦ Ys , ¥

YT =

1  Ys 0 Y 1  Ys0 T

(5) Zone 5 (conversion to austenite):

{

G G U
(12)

otherwise

(13)

}

£Y    T ¬¯ ­ ° + 1 , UG 0 ¦¦ 0 cos ¡ aA žžT  As  ­ ¦ C A ® °± ¡¢ Ÿ Y =¤2 (14) ¦ 0 ¦ Y , otherwise ¦ ¦ ¥ Y0 Y0 (15) Ys = s Y , YT = T Y Y0 Y0 G In the above relations, a M = Q ( M s  M f ) , a A = Q ( Af  As ) , U is the G G G vector tangent to the loading path ( , nA , nM and nMT represent the directions of transformation change, corresponding separately to austenite,

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detwinned martensite and twinned martensite phases; M s and M f

denote

martensite start and finish temperatures; As and Af denote the austenite start and finish temperatures; the superscript “0” in Y 0 , YT0 and Ys0 denotes the initial conditions; the constants C M and C A are material properties describing the relationship between the temperature and critical transformation stress. The critical stress values below M s are taken to be constant, and are denoted by Tscr and

Tcrf at the start and the finish of the transformation. For 3D problems, the stress

T in equations (6) to (15) is replaced by the equivalent stress S defined in equation (5).

Figure 2. Typical phase diagram of a NiTi SMA

3. Numerical Simulations Using a backward-Euler integration method and a return mapping (elastic predicator-transformation corrector) algorithm [25,33], the above constitutive model of SMAs is implemented in a user supplied subroutine of ABAQUS. To demonstrate the proposed approach, a simple model with one 3D solid element having six faces is used for analysis. One face of the model is fixed, and a uniaxial

233

loading is applied on its opposite face. The material properties used (Table 1) are taken from experimental measurements [7]. The maximum recoverable transformation strain FL is 6.7%. The predicted stress-strain curves at different temperatures are shown in Figure 3, with the initial detwinned martensite fraction Ys0 taken as zero for all the cases. For temperatures below As , the material is consisted of all temperature-induced martensite. The shape memory effect exhibits at temperatures less than As (Figure 3a): following the elastic unloading, a large residual strain remains due to the conversion of the detwinned martensite from loading. This residual strain is completely recovered upon heating above Af . There is good agreement between experimental measurements [7] and model predictions when the material is fully martensite (e.g., T < 9 o C ). Figure 3b o presents the pseudoelasticity and hysteresis behavior. At T = 52 C , above Af , the material transforms to detwinned martensite during loading. It exhibits a complete hysteresis loop upon complete unloading; and a hysteresis subloop upon partial unloading and reloading. Table 1. Material properties for the Nitinol alloy (Ni55-Ti) [7] Young’s moduli

Transformation temperatures

EM = 26.3 ×103 MPa Mf = 9oC o EA = 67 ×103 MPa Ms = 18.4 C As = 34.5 o C Af = 49 o C

Transformation consts.

CM = 8MPa o C-1 Tscr = 100MPa CA = 13.8MPa o C-1 Tcrf = 170MPa

Figure 3. Stress-strain curves: (a) shape memory effect; (b) pseudoelasticity (complete and incomplete transformations )

234

Figure 4. Stress response with cyclic heating ( Ys0 = 0.1 )

To demonstrate the constitutive response of the model in restrained recovery, a 3D SMA beam with an initial residual strain of 0.67% ( Ys0 = 0.1 ) is constrained at two ends, and a heating cycle is applied uniformly to the beam (Figure 4). Two-node 3D truss elements are used. When detwinned martensite transforms to austenite upon heating, the residual strain recovers, causing a large internal stress in the beam due to geometric restraint. Figure 4 shows this stress as a function of temperature, with the phase transformation diagram added to illustrate the phase transformation process. Note that when the initial residual strain is more than 0.67% , the material starts to yield before the austenite transformation is completed. 4. Conclusion A computational tool for designing actuators and smart structures with SMAs is presented. A 3D finite element procedure to simulate the behavior of SMAs is developed and implemented in ABAQUS. To assess the viability of the proposed approach, the behavior of Nitinol alloys is simulated for a wide range of temperatures. It is demonstrated that, in addition to the well-capturing of the shape memory effect and pseudoelasticity, the model can faithfully describe the hysteresis loop and hysteresis subloops. Acknowledgement The present work is partially supported by the DARPA MURI project on multifunctional materials.

235

References 1

2 3 4 5 6 7 8

9 10 11

12 13 14 15 16 17 18 19 20 21 22 23 24

25 26 27

Q.P. Sun, K.C. Hwang, 1993, Micromechanics modelling for the constitutive behavior of polycristalline shape memore alloys I: Derivation of general relations, Journal of the Mechanics and Physics of Solids, 41, 1-17. J.V. Humbeeck, 1999, Non-medical applications of shape memory alloys, Materials Science and Engineering, A273-275, 124-138. V. Birman, 1997, Review of mechanics of shape memory structures, Applied Mechanics Review, 50, 629-645. T. Duerig, A. Pelton, D. Stöckel, 1999, An overview of Nitinol medical applications, Materials Science and Engineering, A273-275, 149-160. J.E. Huber, N.A. Fleck, M.F. Ashby, 1997, The selection of mechanical actuators based on performance indices, Proceedings: Mathematical, Physical and Engineering Science, 453, 2185-2205. M. Huang, X. Gao, L.C. Brinson, 2000, A multivariant micromechanical models for smas: Part2. Polycrystal model, International Journal of Plasticity, 16, 1371-1390. C. Liang, C.A. Rogers, 1990, One dimensional thermomechanical constitutive relations for shape memory materials, Journal of Intelligent Material Systems and Structures, 1, 207-234. L.C.Brinson, 1993, One dimensional constitutive behavior of shape memory alloys: Thermo-mechanical derivation with non-constant material functions, Journal of Intelligent Material Systems and Structures, 4, 229-242. K. Tanaka, S. Kobayashi, Y. Sato, 1986, Thermomechanicals of transformation pseudoelasticity and shape memory effect in alloys, International Journal of Plasticity, 2, 59-72. A. Bekker, L.C. Brinson, 1998, Phase diagram based description of the hysteresis behavior of shape memory alloys, Acta Materialia, 46, 3649-3665. Q.P. Sun, K.C. Hwang, 1993, Micromechanics modelling for the constitutive behavior of polycristalline shape memore alloys II: Study of the indidual phenomena, Journal of the Mechanics and Physics of Solids, 41, 19-33. E. Patoor, A. Eberhardt, M. Berveiller, 1987, Potential pseudoelastique et plasticite de transformation martensitique dans les mono et polycristaux metalliques, Acta Mettall, 35, 2779-2789. E. Patoor, A. Eberhardt, M. Berveiller, 1996, Micromechanical modelling of superelasticity in shape memory alloys, Journal de Physique IV, 6, C1-277-292. M. Huang, L.C. Brinson, 1998, A multivariant model for single crystal shape memory alloy behavior, Journal of the Mechanics and Physics of Solids, 46, 1379-1400. X. Gao, M. Huang, L.C. Brinson, 2000, A multivariant micromechanical model for SMAs: Part1. Crystallographic issues for single crystal model, International Journal of Plasticity, 16, 1345-1369. Z. Bo, D.C. Lagoudas, 1999, Thermomechanical modeling of polycrystalline SMAs under cyclic loading, part i: Theoretical derivations, International Journal of Engineering Science, 37, 1089-1140. F. Falk, 1980, Model free energy, mechanics and thermodynamics of shape memory alloys, Acta Mettall, 28, 1773-1780. R. Abeyaratne, S. Kim, J.K. Knowles, 1994, A one-dimensional constinuum model for shape memory alloys, International Journal of Solids and Structures, 31, 2229-2249. Y. Huo, I. Müller, S. Seelecke eds., 1993, Quasiplasticity and pseudoelasticity in shape memory alloys. ed. Visintin, A. Lecture notes in mathematics 1584, Springer-Verlag, 87-146. S. Seelecke, I. Müller, 2004, Shape memory alloy actuators in smart structures: Modelling and simulation, Applied Mechanics Review, 57, 23-45. I. Müller, H.B. Xu, 1991, On the pseudo-elastic hysteresis, Acta Mettall. Mater., 36, 263. F. Trochu, F.F. Qian, 1997, Nonlinear finite element simulation of superelastic shape memory alloy parts, Computers and Structures, 62, 799-810. L.C. Brinson, R. Lammering, 1993, Finite element analysis of the behavior of shape memory alloys and their applications, International Journal of Solids and Structures, 30, 3261-3280. F. Auricchio, R.L. Taylor, 1997, Shape-memory alloys: Modelling and numerical simulations of the finite-strain superelastic behavior, Computer Methods in Applied Mechanics and Engineering, 143, 175-194. F. Auricchio, 1995, Shape memory alloys: Applicatios, micromechanics, macromodelling and numerical simulations, in Civil Engineering, University of California a Berkeley. F. Auricchio, 2001, Considerations on the constitutive modeling of shape-memory alloys. CMINE, Barcelona. V.P. Panoskaltsis, S. Bahuguna, D. Soldatos, 2004, On the thermomechanical modeling of shape memory alloys, International Journal of Non-Linear Mechanics, 39, 709-722.

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Magnetostrictive Materials and Actuating Structures

CONSTITUTIVE MODELS FOR MAGNETOSTRICTIVE MATERIALS

Xiao-Jing Zheng and Xin-En Liu Department of Mechanics, Lanzhou University , Gansu, Lanzhou, 730000, P. R. China e-mail: [email protected]

Abstract A new general nonlinear constitutive model is proposed for giant magnetostrictive materials. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are respectively given for an isotropic material and their applicable regions are also identified. It is found that the new model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation illustrates that, the new model, whether for magnetostrictive rods or for thin films, can effectively describe the effects of the pre-stress or residual stress on the magnetization and the magnetostrictive strain curves, which can not be captured by previous models. Keywords: giant magnetostrictive materials, constitutive model, nonlinear behavior, actuating materials 1. Introduction Giant magnetostrictive material (GMM) is a typical kind of actuating materials and is a promising candidate for future high-performance transducers or actuators. It is successfully used in various devices such as ultrasonic transducers, sonar projectors, active vibration absorbers, micro-positioners, micro-pumps and micro-valves in MEMS [1]. Many experiments have exhibited that magnetostrictive materials, such as Terfenol-D, have nonlinear and coupled constitutive relations. Their magnetization curves and magnetostrictive strain curves not only have saturation and slight hysteresis features, but also are remarkably affected by the axial pre-stress or in-plane residual stress [2,3]. In the known nonlinear models, the standard square model (the SS model) suggested by Carman and Mitrovic [4] is a well-known one. By comparison with the experimental results obtained by Moffet et al. [5], it’s found that the SS model can neither describe the saturation trend nor provide any accurate prediction for the 239 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 239–252. © 2006 Springer. Printed in the Netherlands.

240

low pre-stress levels. Wan et al. [6] presented two new 1-D models, i.e. the hyperbolic tangent model (the HT model) and a model based on density of domain switching (the DDS model). However, there are still large discrepancies between the predicted values and the experimental ones (e.g. the errors are about r 30 ~ 40% in the high field when the compressive pre-stress is 65.4MPa ). Furthermore, the HT model and the DDS model haven’t been generalized to the corresponding 2-D and 3-D models. Another kind of the models is the D-H model suggested by Hsu [7], which fails to exhibit good quantitative agreement with the experimental results, due to the inherent obstacle of failing to deflect the change of the maximum magnetostrictive strain with the pre-stress. Besides these, none of the above models can adequately capture the main coupling characteristics of magnetostrictive thin films, such as the maximum magnetostrictive strain at the saturation changing with the in-plane residual stress. Based on the physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress, a nonlinear constitutive model for GMM is suggested in this pape. Its simplified expressions for 3-D, 2-D and 1-D models are given for an isotropic material. The comparison with the experimental data confirms that the new model can accurately predict the magnetostrictive strain curves of a Terfenol-D rod in the low, moderate and high magnetic field regions for various compressive pre-stress levels. Moreover, whether for magnetostrictive rods or for thin films, the new model can effectively characterize the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves. 2. General Form of the New Model To obtain polynomial constitutive relations, the elastic Gibbs free energy function G (ı ij , M k ) is expressed as the Taylor series expansion of independent variables of

ı ij

stress

and

magnetization

Mk

about

the

(ı ij , M k )

(0,0) , i.e.

G (ı ij , M k )

w 3G 1 w 2G 1 V ij V kl  V ij V kl V mn  " 2 wV ij wV kl 3! wV ij wV kl wV mn

reference

point



. (1) w 3G w 4G 3 6 V ij M k M l  V ij V mn M k M l  " 3! wV ij wM k wM l 4! wV ij wV mn wM k wM l



w 4G 1 w 2G 1 MkMl  MkMlMiM j " 2 wM k wM l 4! wM k wM l wM i wM j

Here only the coupling terms containing M k M l , such as V ij M k M l ,

V ij V mn M k M l and so on, are reserved in Eq. (1). The expressions of strain H ij

241

and magnetic field H k can be respectively obtained as follows

H ij



w 2G 1 w 3G V kl  V kl V mn  " wV ij wV kl 2 wV ij wV kl wV mn 1 w 3G w 4G  (  V mn  ") M k M l 2 wV ij wM k wM l wV ij wV mn wM k wM l

P0 H k

,

w 2G 1 w 4G Ml  MlMiM j " wM k wM l 3! wM k wM l wM i wM j 1 w 3G w 4G ( V ij  V ij V mn  ") M l 2 wV ij wV mn wM k wM l wV ij wM k wM l

(2a)

. (2b)

According to the fact that the saturation value of the axial magnetostrictive strain of a Terfenol-D rod can be enhanced by an exerted axial compressive pre-stress before a magnetic field is applied along the rod axis [7], the elastic strain, expressed by the terms independent of magnetization M k in Eq. (2a), can be divided into two parts. The first part reflects the intrinsic elasticity and is linear with respect to stress ı ij , (s )

which can be described by the intrinsic (or saturation) flexibility tensor S ijkl (i.e. the flexibility tensor when the magnetic saturation is reached). And the second part reflects the effect of the stress anisotropy and is nonlinear with respect to stress ı ij , which can be expressed by a nonlinear tensor function O 0 ij (V mn ) . Thus, the elastic strain in Eq. (2a) can be rewritten as



w 2G 1 w 3G V kl  V kl V mn  " wV ij wV kl 2 wV ij wV kl wV mn

(s) S ijkl V kl  O0ij (V mn ) .

(3)

The magnetostrictive strain, expressed by the terms dependent on M k in Eq. (2a), is often denoted as Oij ( M k , V mn ) , which is produced by the magnetization M k under a given pre-stress (i.e. M k

V mn . It remains to be zero before the rod is magnetized

0 ) and will approach the maximum value dependent on the pre-stress

V mn when the magnetization approaches saturation (i.e. M k M k

M s2 , where

M s is the saturation magnetization). So, the change of the maximum magnetostrictive strain under a given pre-stress should generally be  O 0ij (V mn ) , which is independent of the direction of the magnetization. Thus, the magnetostrictive strain in Eq. (2a) can be expressed as

242

O0ij (V mn ) w 4G 1 w 3G V mn ")M k Ml [mijkl  G kl ]M k Ml . (4)   ( 2 wV ij wM k wMl wV ij wV mnwM k wMl M s2 Here, the constant tensor mijkl is introduced to describe the free magnetostriction without a pre-stress ( V mn independent of stress ( V mn

0 ), and G kl is the Kronecker delta. The terms

V mn in Eq. (2b) represent the free magnetization relation

0 ). Thus, these terms can be rewritten as

w 2G 1 w 4G Ml  MlMiM j  " wM k wM l 3! wM k wM l wM i wM j

P 0 f k1 ( M l ) ,

(5)

For the terms dependent on V mn in Eq. (2b), they represent the effect of the pre-stress on the magnetization process, i.e. the inverse magnetostrictive effect. These terms come from the coupling terms of Eq. (1) as same as those dependent on M k in Eq. (2a). So, they can be expressed as

(

w 3G 1 w 4G V ij  V ijV mn  ") M l wV ij wM k wM l 2 wV ij wV mn wM k wM l w  wM k

³

V ij

0

[mijkl 

Here, / 0 (V mn )

O0ij (V mn ) M s2 V mn

³

0

.

(6)

/ (V ) 2[mijklV ij  0 2mn G kl ]M l Ms

G kl ]M k M l dV ij

O0ij (V mn )dV ij is the primary function of O 0ij (V mn ) .

Substituting Eqs. (3) - (6) into Eqs. (2a) and (2b), the new constitutive model can be expressed as the following compact form

H ij Hk

(s) S ijkl V kl  O0ij (V mn )  [mijkl 

f k1 ( M l )  2P 01[mijklV ij 

O0ij (V mn ) M s2

G kl ]M k M l ,

/ 0 (V mn ) G kl ]M l . M s2

(7a) (7b)

Equation (7) is a general form of the new model derived in this paper. It should be noted that not only the magnetization relation when V mn 0 but also the stress-strain relation when M k 0 are nonlinear, and the maximum magnetostrictive strain changes nonlinearly with the given pre-stress in the new model. In addition, it is very important for applying the new model in practice to 1 determine the nonlinear tensor or vector functions O 0ij (V mn ) and f k ( M l ) . According to the experiments on Terfenol-D rods, O 0ij (V mn ) in 1-D case can be approximated by the hyperbolic tangent function tanh(x) . Thus, expression (7) in

243

1-D case can be rewritten as follows

V V Os ­ 2 °O s tanh(V )  [1  tanh(V )] M 2 M V ° s s s H ® Es ° Os 2V 1 2V O tanh( )  [1  tanh( )] s2 M 2 °¯ 2 2 Vs Vs Ms O V ­ 2{ V V s ln[cosh( )]} s 2 M ° V s P0M s 1 1 § M · ° ¸¸  ® H f ¨¨ k © M s ¹ °2{V  V s ln[cosh( 2V )]} O s M °¯ 4 Vs P 0 M s2

V t0 Vs (8a) V 0 Vs V t0 Vs (8b) V 0 Vs Here, V s O s /(1 / E 0  1 / E s ) O s E s E 0 /( E s  E 0 ) , E0 and E s are the initial and saturation Young’s modulus respectively, M s and O s are the saturation magnetization and magnetostrictive coefficient respectively, k is a relaxation factor. The nonlinear function f (x ) can be chosen as (9) f ( x) coth(x)  1 / x 3. Simplified Form of the New Model When the pre-stress

V mn is not very large, the nonlinear function O0ij (V mn ) can

always be linearized as follows (m) O0ij (V mn ) | S ijkl V kl .

Defining a new flexibility tensor S ijkl

S

(10)

(s) ijkl

S

(m) ijkl

, the new model, expressed by

Eqs. (7a) and (7b), can be simplified as

Hij S ijkl V kl  (mijkl  Hk

(m) V mn S ijmn

M s2

G kl ) M k M l ,

f k1 ( M l )  P01 (2mijkl V ij 

(m) S ijmn Vij V mn

M s2

(11a)

G kl ) M l

(11b)

which means the stress-strain relation becomes linear when M k 0 and the maximum magnetostrictive strain changes linearly with the pre-stress. For the convenience of engineering applications, the 3-D, 2-D and 1-D models for an isotropic magnetostrictive material will be given according to the simplified model expressed by Eqs. (11a) and (11b). 3.1 3-D CASE Taking account of the symmetry, SijklV kl and mijkl M k M l can be reduced to

244

Os

1 [(1 Q )V ij QV kkGij ] , m ijkl M k M l E0

SijklV kl

3 1 ( M i M j  M k M k G ij ) . 2 M 2 2 s

(12)

(m)

Similarly, SijmnV mn can be reduced to ( m) V mn Sijmn

Os 3 Os ~ 1 ( V ij  V kk G ij ) V ij . Vs 2 Vs 2

(13)

and

 P 01 (2mijkl V ij   P 01  P 01

(m) V ij V mn S ijmn

M s2

G kl ) M l

Os

3 1 3 1 [2( V kl  V mmG kl )  ( V mnV mn  V mmV nn )G kl / V s ]M l 2 2 2 2 M 2 s

Os

M ~ Here, V mn

2 s

(14)

[2V~kl  (I V2  3IIV )G kl / V s ]M l

3V mn / 2  V kk G mn / 2 . The first and second invariants of the stress tensor are denoted by V mn I V V mm and II V (V mmV nn  V mnV mn ) / 2 . 1

Having chosen isotropic vector function f k ( M l ) as a scalar function, that is

1 1 § M · ¸¸G kl M l . f ¨¨ (15) kM © Ms ¹ in which the nonlinear scalar function f (x ) is given by Eq. (9), Eqs. (11a) and f k1 ( M l )

(11b) can be simplified in the 3-D case as follows

H ij Hk

O 3 1 1 [(1Q )V ij QV kkG ij ]  s2 [ M i M j  M k M k ( G ij  V~ij / V s )] 2 E Ms 2 {

§ M 1 f 1 ¨¨ kM © Ms

· Os ¸¸G kl  [ 2V~kl  ( I V2  3IIV )G kl / V s ]}M l 2 P0M s ¹

(16a)

(16b)

3.2 2-D CASE To describe the magnetoelastic coupled behavior in magnetostrictive thin films, a 2-D constitutive model is necessary. Since all the off-plane stress components can be neglected, i.e. V z W zx W zy 0 , and the magnetic field is usually applied along a direction parallel or perpendicular to the film plane, i.e. H z

Hx

0 or

0 , the expressions of V~ij and I V2  3IIV in Eq. (16) become

Hy 1 2

1 2

1 2

V~x V x  V y , V~y V y  V x , V~z  (V x  V y ),W~xy

3 W xy , W~yz W~zx 0 , (17a) 2

245

I V2  3IIV

V x2  V y2  V xV y  3W xy2 .

(17b)

Thus, the 2-D constitutive model for an isotropic magnetostrictive material can be derived from Eqs. (23a) and (17b) in the following matrix forms (using the engineering shear strain, i.e. J xy 2H xy ). 1) In an in-plane magnetic field ­ H x ½ ª 1 / E Q / E 0 º­V x ½ ° ° O ° ° « 0 »»®V y ¾  s2 ®H y ¾ «Q / E 1 / E Ms °J ° « 0 0 1 / G»¼°¯W xy °¿ ¯ xy ¿ ¬

ª 1V~x / V s 1 / 2 V~x / V s 0º­ M x2 ½ ° , (18a) « »° ~ 1V~y / V s 0»® M y2 ¾ «1 / 2 V y / V s «  2W~xy / V s 3»¼°¯M x M y °¿  2W~xy / V s ¬ 2 º ·­ M x ½ . 2W~xy ­ H x ½ §¨ 1 1 § M · ª1 0º Os ª2V~x  (IV  3IIV ) / V s ¸¸ «  f ¨¨ » ¸¸® ¾ ® ¾ ¨ » 2 2 « ~ ~ H   2 W 2 V ( I 3 II ) / V 0 1 kM M M P « » ¹¯M y ¿ ¼ xy y s¼ V V 0 s ¬ © s ¹¬ ¯ y¿ ©

(18b) 2) In a transverse magnetic field

­H x ½ ° ° ®H y ¾ °J ° ¯ xy ¿

Hz

­1/ 2 V~x / V s ½ ª 1/ E Q / E 0 º­V x ½ ° ° O ° ° «Q / E 1/ E 0 »»®V y ¾  s2 ®1/ 2 V~y / V s ¾Mz2 , « Ms ° ~ ° «¬ 0 0 1/ G»¼°¯W xy °¿ ¯  2W xy / V s ¿ 1 1 § M z · Os [2V~z  (IV2  3IIV ) / V s ] ¸¸  Mz . f ¨¨ k P 0 M s2 © Ms ¹

In the special case of the biaxial residual stress can be simplified as ( V~ x 2

IV  3IIV

V~

y

Vx

(19a)

(19b)

V y V , Eqs. (18) - (19) V , W~xy W~yz W~zx 0 ,

V / 2 , V~ z

2

V )

1) In an in-plane magnetic field (along the x-direction)

Hx

O [1V /(2V s )] 2 1Q V s Mx , H y E M s2

1 Q 1 O (1  V / V s ) 2 V s M x , J xy E 2 M s2

0, (20a)

Hx

1 1 § M x · O sV (1  V / V s ) ¸¸  f ¨¨ Mx, k P 0 M s2 © Ms ¹

Hy

0.

(20b)

2) In a transverse magnetic field (along the z-direction)

Hx Hz

Hy

1 Q 1 O (1  V / V s ) 2 V s M z , J xy E 2 M s2

1 1 § M z · O sV (2  V / V s ) ¸¸  f ¨¨ Mz. M k P 0 M s2 © s¹

0,

(21a)

(21b)

246

3.3 1-D CASE Terfenol-D rods are often used in various magnetostrictive actuators, which need a 1-D model to describe the relations between stress V , strain H , magnetic field H and magnetization M along the rod axis. In that case, the expressions of V~ij and I V2  3IIV in Eq. (16) are

1 2

V x

V y  V ,

I V2  3IIV

V z V ,

Wxy

Wyz Wzx 0 ,

V 2.

(22a) (22b)

Thus, the 1-D magnetostrictive constitutive model can be written as

H H

V

Os

V )M 2 , Vs

(23a)

1 1 § M · O sV (2  V / V s ) ¸¸  f ¨¨ M. k P 0 M s2 © Ms ¹

(23b)

E



M

2 s

(1 

It is obvious that above model must be limited within a proper region since the simplification for nonlinear function O0ij (V mn ) is true only when the pre-stress

V mn is not very large. In the 3-D case, assuming O01 , O02 and O03 as three principal strain components and O01 d O02 d O03 , the magnetic domain analysis will give the limit  O s / 2 d O01 d O02 d O03 d O s ~ ~ ~ or  V s / 2 d V 1 d V 2 d V 3 d V s for a positive magnetostrictive material ( O s ! 0 ), where V~1 , V~2 and V~3 (and V~1 d V~2 d V~3 ) are the three principal stress components of V~ij . Due to V~1  V~2  V~3 0 , we have V~3 d 2V~1 so that a more compact expression of the applicable region can be given as follows (24a) V~min V~1 t V s / 2 ( O s ! 0 ). Similarly, for a negative magnetostrictive material ( O s  0 ), the applicable region can be expressed as V~max V~3 d V s / 2 ( O s  0 ). (24b) Equation (24) can also be written in the following uniform dimensionless form min(V 1 , V 2 , V 3 ) t 1 / 2 , (25)

V~i / V s . In the 1-D and 2-D cases, equation (32) can be rewritten as  1/ 2 d V d 1 , (26) where V V / V s . where

Vi

247

4. Verification and Discussion

The new constitutive model will be used to simulate experimental data on magnetostrictive strain for a Terfenol-D and Ni6 rod under different compressive pre-stress levels. It is obvious from Figure 1 that the predicted results by Eqs. (8) and (20) are in much better agreement with the experimental data and have wider applicability and higher precision, especially in the region of the high field, than the previous models.

Figure 1. Comparison predicted strain (solid lines) with experiment data

Various graphs in Figure 2 are the predicted magnetization, magnetostrictive strain and Young’s modulus of the Terfenol-D rod respectively under the various compressive pre-stress levels by Eq. (8). From Figure 2a, it can be found that the values of the magnetization increase with the applied field under several pre-stress levels, and will reach a constant in the region of the high field. It means that the magnetization of the rod subjected to different pre-stresses will be saturated to the same level, i.e. the saturation magnetization. For a given magnetic field, the magnetization decreases with an increasing compressive pre-stress, indicting that the axial compressive pre-stress makes the axial magnetization of the rod more difficult. For the magnetostrictive strain shown in Figure 2b, it increases with an increasing magnetic field for a given pre-stress and decreases with an increasing compressive pre-stress for a given magnetic field of low or moderate strength, similar to the magnetization shown in Figure 2a. However, the similarity only takes place in the region of low and moderate fields. When the applied magnetic field is in the high field region, or near the saturation field of the material, the magnetostrictive strain of the rod will reach different saturation values for different pre-stresses (see Figure 2b). The larger the compressive pre-stress, the larger the maximum magnetostrictive strain. In the other words, the maximum magnetostrictive strain of the rod increases with the axial compressive pre-stress, which is obviously different from the characteristics in the region of the low and moderated fields. These predicted results of the new model coincide with the observed experimental phenomena [5], but none of the previous models (e.g. the SS Model, the HT Model, the DDS Model and the D-H Model) can capture all of them

248

qualitatively or quantitatively. The calculated curves of Young’s modulus versus compressive stress for some given magnetic bias fields are plotted in Figure 2c which shows that the magnetic bias field has obvious influence on the calculated Young’s modulus of the rod and confirms that the new constitutive formula can describe the 'E effect of Terfenol-D rods. The numerical simulation indicates that the new model can effectively capture the main magnetoelastic coupling characteristics of magnetostrictive materials. For Terfenol-D thin films, experiments have found that in-plane residual stresses have remarkable influence on the magnetization and magnetostrictive strain curves and the influence is different with the direction of residual stresses. In order to confirm that the new model can capture this kind of experiment facts, the magnetization and magnetostrictive strain curves are calculated according to Eqs. (20) - (21) for an in-plane ( // ) and transverse ( A ) magnetic field. The results for three special cases, i.e. V V s / 2 (tensile), V 0 (free) and V V s (compressive), are shown in Figure 3 and Figure 4, in which material parameters are taken as Os 500ppm , P 0 M s 0.5T , F m 5 and V s 80MPa .

Figure 2. Predicted results by the new model

From Figure 3, it can be found that the magnetization curve for the residual stress V parallel to the magnetic field, i.e. the in-plane magnetization curve, is obviously different from that for the residual stress V perpendicular to the magnetic field, i.e. the transverse magnetization curve when V z 0 (anisotropic). However, two curves coincide when V 0 (isotropic). It is because, from the viewpoint of magnetic domains, that an in-plane tensile stress tends to align the magnetic domains in the film plane, and an in-plane compressive stress tends to align them along the perpendicular direction. Hence, an arbitrary in-plane direction becomes an easy direction of the magnetization and the transverse direction becomes a difficult one when the stress is tensile ( V V s / 2 ), as shown by Figure 3a. It is called in-plane anisotropy. On the contrary, when the stress is compressive ( V V s ), as shown by Figure 3b, the transverse direction becomes an easy direction of the

249

magnetization and an arbitrary in-plane direction becomes a difficult one. It is called perpendicular anisotropy. So, the new model can give coherent prediction between the domain analytical results and the experimental observations [3].

Figure 3. Predicted magnetization curves of a Terfenol-D thin film by the new model

Figure 4 displays the influence of the residual stress on the magnetostrictive strains. When the film is in a in-plane magnetic field along with the x-direction, its in-plane components of the magnetostrictive strain, i.e. O x and O y , are respectively positive and negative one shown in Figure 4a. The component the magnetic field and approaches the saturation value

O x increases with

Os when the stress is zero

0 ). However, the component O x exhibits different increment when the stress is applied ( V z 0 ). When the stress is tensile ( V V s / 2 ), the component (V

O x increases more rapidly in the region of the low magnetic field, due to the in-plane anisotropy, but reaches a lower saturation value ( 3O s / 4 ) in the region of the high field. On the contrary, when the stress is compressive ( V V s ), the component O x increases more slowly in the region of the low or moderate magnetic field, due to the perpendicular anisotropy, but approaches a higher saturation value ( 3O s / 2 ) in the region of the high field. The residual stress has

O y to O x . For example, O y O x / 2 when V 0 , O x and O y 0 respectively for V V s / 2 and V V s . When

influence on the ratio of and

Oy

the film is in a transverse magnetic field along the z-direction, the magnetostrictive strain components O x and O y , shown in Figure 4b, are always equal and negative no matter the stress is tensile or compressive. The absolute value of or

Ox

O y increase with the magnetic field and finally approaches its saturation value

250

(  Os / 2 ) when the stress is zero ( V similar to the curve of

0 ). When the stress is tensile ( V

V s / 2 ),

O x ~ H for V  0 in Figure 4 (a), the absolute value of

O x or O y increases more slowly in the region of low or moderate magnetic field, due to the in-plane anisotropy but finally approaches a higher saturation value (  3O s / 4 ). When the stress is compressive, a curve similar to the curve of

O x ~ H for V ! 0 in Figure 4a should be given due to the perpendicular anisotropy. When V V s , an ideal perpendicular anisotropy is achieved and the transverse magnetic field can not produce any magnetostrictive strain, i.e. O x O y 0 , since the domain switch motion in the the magnetization process takes place only in the direction perpendicular to the film plane. The above predictions also coincide with the domain analytical results and the experimental observations [3]. However, the prediction given by the D-H model, shown in Figure 5, is different from those given by the new model. So, the numerical results indicate that the new model can more effectively describe the effect of the in-plane residual stress on the magnetization and magnetostrictive strain curves of a thin film than the previous models, especially the change of the maximum magnetostrictive strain with the residual stress.

(a) H H x (in-plane field) (b) H H z (transverse field) Figure 4. Predicted magnetostrictive strain curves of a Terfenol-D thin film by the new model

Additionally, it is convenient for the new model to be used in engineering applications since the adopted constitutive constants are the parameters of magnetostrictive materials, such as the saturation magnetization M s , the saturation magnetostrictive strain

O s , the intrinsic (or saturation) Young’s modulus

E s and the initial Young’s modulus E0 as well as the linear magnetic susceptibility

F m , which are all easily measured in experiments.

251

(a) H H x (in-plane field) (b) H H z (transverse field) Figure 5. Predicted magnetostrictive strain curves of a Terfenol-D thin film by the D-H model

5. Conclusions

1.

2.

3.

4.

For magnetostrictive materials, the general form of a new constitutive model is presented, which can describe the nonlinear change of the maximum magnetostrictive strain with the pre-stress (dependent on the nonlinear elastic strain part arising from domain rotation or movement). The new model is simplified by means of linearizing the nonlinear elastic strain part. For an isotropic material, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are given and their applicable ranges are discussed in detail. For a Terfenol-D rod, the new model can give the calculated magnetostrictive strain curves perfectly coinciding with the experimental data for various compressive pre-stress levels and low, moderate and high field regions, which indicates that the new model has wider applicability and higher precision than the previous models. Whether for a Terfenol-D rod or for a Terfenol-D thin film, the new model can effectively characterize the effect of the axial pre-stress or in-plane residual stress on the magnetization and magnetostrictive strain curves of the rod or the thin film, while none of the previous models can capture all of them.

Acknowledgements

The support of the National Natural Science Foundation of China and the Fund of Ph.D project in Universities of the Ministry of Education of China are gratefully acknowledged.

252

References 1. 2. 3. 4. 5. 6. 7.

F. Claeyssen, N. Lhermet, R. Le Letty, P. Bouchilloux, 1997, Actuators, transducers and motors based on giant magnetostrictive materials, Journal of Alloys and Compounds, 258, 61-73. M.E.H. Benbouzid, G. Reyne, G. Meunier, 1995, Dynamic modelling of giant magnetostriction in Terfenol-D rods by the finite element method, IEEE Trans. Magn. 31, 1821-1824. F. Schatz, M. Hirscher, M. Schnell, G. Flik, H. Kronmüller, 1994, Magnetic anisotropy and giant magnetostriction of amorphous TbDyFe films, J. Appl. Phys. 76, 5380-5380. G.P. Carman, M. Mitrovic, 1995, Nonlinear constitutive relations for magnetostrictive materials with application to 1-D problems, J. Intell. Mater. Syst. Struct. 6, 673-683. M.B. Moffet, A.E. Clark, M. Wun-Fogle, J. Linberg, J.P. Teter, E.A. McLaughlin, 1991, Characterization of Terfenol-D for magnetostrictive transducers, J. Acoust. Soc. Am., 89, 1448-1455. Y.P. Wan, D.N. Fang, K.C. Hwang, 2003, Non-linear constitutive relations for magnetostrictive materials, Inter. J. Non-linear Mechanics, 38, 1053-1065. J.L. Butler, 1988, Application Manual for Design of Etrema Terfenol-D Magnetostrictive Transducers, Edge Technologies, Inc., Ames, IA.

VIBRATION ANALYSIS OF A NONLINEAR MAGNETOSTRICTIVE ACTUATOR

Z. Zhong and Y. P. Wan Key Laboratory of Solid Mechanics of MOE School of Aerospace Engineering and Applied Mechanics Tongji University, Shanghai, 200092, People’s Republic of China

Abstract

In this paper, the magnetostrictive actuator was analyzed with the nonlinear constitutive equation of magnetostriction and, for comparison, the linear piezomagnetic equation. Theoretical model shows that, for the case of relatively large excitation magnetic field, the nonlinear constitutive equation should be adopted in order to give an accurate prediction in the design of actuators. The actuator resonates not only at the natural frequencies of the vibration system, but also arises when the frequency of the excitation current in the coil equals half of any one of the natural frequencies of the system. This conclusion cannot be reached by the formulation with the linear piezomagnetic equation. It is verified that the results obtained from the nonlinear equation reduce to be that of linear piezomagnetic equation when the amplitude of the excitation magnetic field provided by the coil is very small compared to the bias magnetic field and its frequency does not induce resonance of the system. Key Words: nonlinear constitutive equation, the magnetostrictive actuator, vibration analysis 1. Introduction

Actuators, as one kind of smart devices, have wide application in modern engineering and technology. The Tb-Dy-Fe (Terbium-Dysprosium-Iron) metallic compound generally holds giant magnetostriction under external magnetic field and has become an increasingly important candidate material for actuators [1]. The Tb-Dy-Fe Giant Magnetostrictive Materials (GMMs) generally take on obvious nonlinear behavior under mechanical and magnetic loads [2]. Analysis with the linearized constitutive equations, such as the linear piezomagnetic equations, cannot reflect the total response of the GMMs [3,4]. Therefore, it is necessary to use the *E-mail: [email protected]

253 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 253–260. © 2006 Springer. Printed in the Netherlands.

254

nonlinear constitutive equation to analyze the GMMs actuators. In this paper, the standard square nonlinear constitutive relation of magnetostriction and the linear piezomagnetic equation are respectively incorporated into the mathematical model to analyze the vibration of an actuator. A Tb-Dy-Fe rod, biased by a permanent magnet and a compressive prestress, is driven to vibrate with an excitation magnetic field, which is provided by a coil carrying an alternating current. The intrinsic loss of the Tb-Dy-Fe rod and the damp of the vibrating system are not included in this analysis in order to focus on the effect of the nonlinear constitutive behavior of the GMMs on the vibration of actuators. 2. Longitudinal Vibration of a Tb-Dy-Fe Actuator

2.1

STANDARD SQUARE NONLINEAR CONSTITUTIVE RELATION OF MAGNETOSTRICTION

The GMMs exhibit nonlinear magnetostriction as the magnetic field increases. On the magnetostriction loop, there are generally three regions in terms of the response of magnetostrain [4]. Under a low magnetic field, magnetostrain is very small. In the moderate region of magnetic field, strain responds to external field quite sensitively. A small increase of magnetic field will produce much larger strain. Under an intensive magnetic field, magnetostrain tends to saturate. In addition, the response of the GMMs varies for different prestresses. Therefore, to adequately model the constitutive behavior of the GMMs, a nonlinear constitutive relation is required. Due to the small energy loss of the GMMs in the whole magnetization loop, the hysteresis can be temporarily excluded in developing theoretical model of constitutive equation [3-5]. The standard square nonlinear constitutive relation of magnetostriction for 1-dimensional problem is as follows [3,4]:

H sV  mH 2

(1) where s is the compliance of the GMMs in the absence of magnetic field, m the magnetostrictive modulus, H the external magnetic field, H the strain, and V the stress. It has been verified that the theoretical value of the standard square nonlinear constitutive relation of magnetostriction is in agreement with the experimental results in the first and second region of magnetic field, while in the third region of magnetic field, the standard square model cannot predict the saturation of magnetostrain. Thereby, this nonlinear constitutive relation is applicable in the first and second stage of magnetic field, where the GMMs actuator is generally designed to work [2]. Besides the nonlinear constitutive equation, the following linear piezomagnetic equation was used in previous works for the GMMs [3,6];

H sV  dH

(2) where d is the piezomagnetic coefficient [3,4,6], whose physical implication is the magnetostrain produced by per unit magnetic field at the bias magnetic field,

255

H0 d

2mH 0 .

(3)

2.2 ANALYSIS MODELS OF VIBRATION Figure 1 depects a simplified schematic of a magnetostrictive actuator. A compressive stress is exerted on a Tb-Dy-Fe rod to avoid tensile stress during vibration. In the meantime, a bias magnetic field, H 0 , usually generated by a permanent magnet, is used to preclude the double-frequency effect. The total magnetic field is a summation of the permanent magnetic field and the magnetic field produced by the coil, i.e. H H 0  h0 sin Zt (4) where Z is the angular frequency of the current in the coil, h0 the amplitude of the excitation magnetic field produced by the coil. As indicated by Figure 1, the displacement is output through a rigid leader. The magnetostrictive actuator is generally used in very low frequencies and sometimes in quasi-static state for accurate displacement control, such as the case for valve control in micro pumps. The Tb-Dy-Fe rod is hence mainly subjected to the compression of the spring, and the inertia of the rigid leader can be neglected. Figure 2 is the simplified analysis model of the Tb-Dy-Fe actuator. With reference to the standard square nonlinear constitutive equation of magnetostriction in Eq. (1), the formulation, including the governing equation, the associating boundary conditions and initial conditions, is outlined as follows:

w 2u 1 w 2u  wt 2 a 2 wx 2 wu u x 0 0, wx x

ut where a

0

>m( H

0

0 (mH 2  sV 0 )l

(5)

1

@

) 2  sV 0 ˜ lx,

wu wt

0 t 0

l sU , U is the density of the GMMs, l the length of the Tb-Dy-Fe

rod. V 0 the prestress offered by the spring. t the variable for time, whose unit is second. x is the coordinate along length of the rod, which is dimensionless and has the range of x  [0,1] . This problem can be solved by the method of impulse theorem, which is basically solved in terms of the physical meaning of string vibration.

256

7

8

6

4

1

1. Tb-Dy-Fe rod; 2. Coil; 3. Permanent magnets ; 4. Pure Iron ; 5. Base ; 6. Fixed cover; 7. Displacement leader; 8. Spring for prestress

3

2

4 5 Figure 1. Schematic diagram of a magnetostrictive actuator

l

V0

x

O

H(t)

Figure 2. The analysis model of the vibrating Tb-Dy-Fe actuator

The solution to the problem in Eq. (5) can be found as

>m(H )

u

2

@

 sV 0 ˜ lx  W(1) ( x, t )

(6)

in which f

W(1) ( x, t )

¦ (16acZ ) n 0

(1) n (2n  1)S sin x 3 3 2 (2n  1) S

­ (2n  1)S º ½ (2n  1)S ª (2n  1)S º ª (2n  1)S t » ° (7) Z« cos sin Zt Z sin t  cos 2Zt » ° Z « 2a ¬ 2a 2a ¼  G ¬ 2a ¼° °G 2 (2n  1)S º ª (2n  1)S º ° º ° ª (2n  1)S º ª (2n  1)S ª  Z» ¾ ˜ 2Z   Z» ˜ « ® «¬ 2a «¬2Z  2a »¼ «¬ 2a »¼ ¼ ° ¼ ¬ 2a ° ° ° (2n  1)S t ° ° G sin 2a ¿ ¯ in which G h0 H 0 denote the ratio of the amplitude of the excitation magnetic

field to the bias magnetic field, c is a combined parameter c output displacement of the rod ( x

'u x 1

u x 1 ut

2m( H 0 ) 2 l . The

1 ) is 0, x 1

.

(8)

257

Insertion of Eqs. (6) and (7) into (8) leads to 'u(1) m( H 0 ) 2 l G 2 (sinZt ) 2  2G sin Zt  W(1) (1, t )

>

@

(9)

where W(1) (1, t ) W(1) ( x, t ) . When the alternating current in the coil is very small, x 1 i.e. G h0 H 0  1 , and G 2 is a high-order small quantity and is generally negligible, Eq. (9) can be simplified as (10) 'u x 1 2m( H 0 ) 2 lG sin Zt  W(1) (1, t ) . The formulation incorporating the linear piezomagnetic equation in Eq. (2) can be derived by analogy to that in Eq. (5), and takes on the following form

w 2u 1 w 2u  wt 2 a 2 wx 2 wu u x 0 0, wx x ut

0

0 ( dH  sV 0 )l

.

(11)

1

>dH 0  sV 0 @˜ lx, wu wt

0 t 0

The solution to the problem described by Eq. (11) is

u

>dH  sV 0 @˜ lx  W(2) ( x, t )

(12)

in which f

W( 2) ( x, t )

¦ (16acZ ) n 0

(1) n (2n  1)S sin x 3 3 (2n  1) S 2

­ (2n  1)S º ª (2n  1)S °° Z « 2a sin Zt  Z sin 2a t » ¬ ¼  G sin (2n  1)S ® G 2a ª (2n  1)S º ª (2n  1)S º °  Z» ˜ « Z» « °¯ ¬ 2a ¼ ¬ 2a ¼ and there is c (x

½ °° t¾ ° °¿

(13)

dH 0 l in terms of Eq. (3). The output displacement of the rod

1 ) renders 'u( 2)

dH 0lG sin Zt  W( 2) (1, t )

(14)

where W( 2) (1, t ) W( 2) ( x, t ) . x 1 3. Discussions

It is obvious that Eqs. in (9) and (14), obtained from the nonlinear and linear constitutive equations respectively, are different. To give insight into the correlation of these two formulations and the system vibration, plots based on the solutions are presented and, as an example, the following constants are used for the Tb-Dy-Fe rod [4,5]. E 20 u 10 9 Pascal , l 0.2meter , U 9.2 u 10 3 kilogram / meter 3

258

m

9 u 10 14 meter 2 / Ampere 2 , H 0

80 u 10 3 Ampere / meter ,

V 0 32 u 10 6 Pascal With the above constants, the following parameters are obtained. a 0.13564 u 10 3 second , c l 1152 u 10 6 . The natural frequencies of the vibration system in the absence of magnetic field can be estimated by (2n  1)S (15) Z e( n ) , n 0,1,2, " . 2a (1) When the amplitude of the alternating current in the coil is very small, i.e. G h0 H 0  1 , and G 2 is a high-order small quantity, the Tb-Dy-Fe actuator is excited in the vicinity of the bias magnetic field and equation (10) can be used for theoretical prediction. Suppose that the frequency of the current in the coil does not ( n) equal half of any one of the natural frequencies, i.e. Z z Z e 2 n 0,1,2, " , Eq. (10) reduces to Eq. (14). Therefore, only in this case can the linear piezomagnetic equation be employed to analyze vibration of the Tb-Dy-Fe actuator. (2) To obtain a larger displacement output, an actuator usually works with a higher excitation magnetic field. Figure 3(a) shows the output displacement of an actuator ( x 1 ), by using the Eqs. (9) and (14), with the associating parameters chosen as f 1Hz , Z 2Sf 2S , G 0.5 , where the abscissa records time in the unit of second, and the ordinate is the dimensionless output displacement 'u l . There exists a large gap of almost 25% between the predicted values by the linear and nonlinear constitutive equation. Figure 3(b) demonstrates the case of an excited current in the coil with a relatively high frequency, f 100 Hz , and G 0.5 . The figure indicates obvious difference predicted by two formulations. This is because the GMMs exhibit obvious nonlinearity when the range of vibrating magnetic field is relatively wide, and the linear approximation is no longer acceptable. Consequently, the nonlinear constitutive relation of magnetostriction should be used in the analysis and design of the magnetostrictive actuator, if the excitation magnetic field is relatively large compared to the bias magnetic field. (3) As indicated by Figure 4, the response of the vibrating system, predicted by the solution in Eq. (7), which is obtained from the formulation in Eq. (5) with nonlinear constitutive equation, tends to be infinite as time elapses when the frequency of the excitation magnetic field coincides with one of the natural frequencies or half of them, i.e. Z Z e( k ) , k 0,1,2, " , or Z Z e( k ) 2 , k 0,1,2, " . However, this prediction cannot be given by the formulation with the linear piezomagnetic equation, which finds that the system resonates only when the frequency of the excitation magnetic field equals one of the natural frequencies, i.e. Z Z e( k ) , k 0,1,2, " .

259

-4

0.0008

8.0x10

G=0.5, Z=2S

-4

4.0x10

0.0000

'u/l

'u/l

0.0004

-0.0004

-4.0x10

0.0

G=0.5, f=100

nonlinear linear

nonlinear linear

0.0

-4

0.2

0.4

0.6

t

0.8

1.0

0.0

-3

2.0x10

-3

-3

4.0x10

6.0x10

-3

8.0x10

-2

1.0x10

t

(a) (b) Figure 3. Comparison of the output displacement with the Eqs. (9) (nonlinear) and (14) (linear). (a) f 1Hz and G 0.5 (b) f 100 Hz and G 0.5

(a) (b) Figure 4. The output displacement when Z Z e( 0) 2 and G 0.5 (a) with the nonlinear solution (Eq. 9), (b) with the linear solution (Eq. 14)

4. Conclusions

The GMMs possess large magnetostrain and high energy density, and are of increasing importance in the application of the modern actuators. Under the combined action of mechanical and magnetic loads, the constitutive behavior of the GMMs is essentially nonlinear. Both the standard square nonlinear constitutive relation of magnetostriction and the linear piezomagnetic equation are incorporated in the formulation to study the longitudinal vibration of a Tb-Dy-Fe actuator. 1. On the condition that the excitation magnetic field is relatively large compared to the bias magnetic field, such as in the design of an actuator expecting a big displacement output, the nonlinear constitutive relation of magnetostriction should be employed in the analysis. 2. Solution obtained from the standard square nonlinear constitutive relation of

260

magnetostriction indicates that the actuator system resonates not only when the frequency of the excitation current, Z , equals any one of the natural frequencies of the system Z e( k ) , k 0,1,2, " , but also occurs at half of these natural frequencies Z Z e( k ) 2 , k 0,1,2, " . This suggests that in the analysis of design, attention should not only be paid to the natural frequencies, Z e( k ) , k 0,1,2, " , but also to half of the natural frequencies, Z e( k ) 2 , k 0,1,2, " . Acknowledgements

This work was supported by the National Natural Science Foundation of China (no. 10125209) and the Teaching and Research Award Fund for Outstanding Young Teachers in High Education Institutions of MOE, P. R. China. References 1 2

3 4 5 6

A.E. Clark, 1980, Magnetostrictive rare earth-Fe2 compounds, in: Ferromagnetic materials, Edited by E. P. Wohlfarth, (Amsterdam, North Holland, Vol 1), 531-589. M.B. Moffet, A.E. Clark, M. Wun-Fogle, J. Linberg, J.P. Teter, E.A. McLaughlin, 1991, Characterization of Terfenol-D for magnetostrictive transducers. Journal of Acoustics Society of America, 89, 1448-1455. G.P. Carman, M. Mitrovic, 1996, Nonlinear constitutive relations for magnetostrictive materials with applications to 1-D problems. Journal of Intelligent Material Systems & Structures 6, 673-683. Y.P. Wan, D.N. Fang, K.-Ch. Hwang, 2003, Nonlinear constitutive relations for the magnetostrictive materials. International Journal of Non-linear Mechanics, 38, 1053-1065. Y.P. Wan, D.N. Fang, A.K. Soh, K.-Ch. Hwang, 2003, Experimental and theoretical study of the nonlinear response of a giant magnetostrictive rod. Acta Mechanica Sinica, 19, 324-329. J.H. Huang, W.S. Kuo, 1997, The analysis of piezoeletric/piezomagnetic composite materials containing ellipsoidal inclusions. Journal of Applied Physics, 81, 1378-1386.

TEST STUDY OF THE FEED-SUPPORT SYSTEM FOR A LARGE RADIO TELESCOPE

G.X. Ren1, W.B. Zhu2, H. Zhang1, L.C. Zhu 2 and Q.H. Lu1 Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P.R. China 2 National Astronomical Observatories, Chinese Academy of Science, Beijing 100012, P.R. China.

1

Abstract

This paper introduces the 1:12 model test of the feed support system of a 500-meter aperture radio telescope. The main purpose of this research is to study the controllability and performance of a cable-driven Gough-Stewart platform used as a positioning platform for the feed of a large radio telescope. The Gough-Stewart platform is used for vibration control, the upper platform of which is driven by long cables and the lower one is to be stabilized in tracing the observation trajectories. The similarity for the scaled model and the implementation are illustrated. The test study has been carried out in two steps. The first step involves the uncoupled cable-car tracking control tests and the Gough-Stewart platform vibration suppression tests, while the second step carries out the coupled test of the system. The general aspects of the control system and control law are introduced and the problems arising in the tests are also addressed. Results on the 50m×50m×12m are also presented. 1.

Introduction

A Five hundred meter Aperture Spherical Telescope (FAST) to be built in the unique Karst area of southwest China [1] has been under research for many years. This telescope has an adaptive spherical reflector that requires the receivers to trace on a focus surface, a half radius from the main reflector, referring to Figure 1 for the optics of the telescope. The telescope is pointed to the source by moving the receivers along the locus to counter the rotation of the earth, and by adjusting the reflector in the illuminated area accordingly. This paper studies the feasibility of a platformless feed support configuration [2-3] for FAST, where the receivers are carried by a cable car driven by long cables, see Figure 2. To observe 5-GHz radio waves, the receivers are required to trace trajectories to a precision of RMS 4mm. The expected wind-induced vibration of the cable car might be around 0.5m, much higher than the specification. So a Gough-Stewart platform [4] is arranged on the 261 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 261–270. © 2006 Springer. Printed in the Netherlands.

262

car to actively isolate the platform of the payload from the vibration induced by wind and other disturbances, see Figure 2.

Figure 1.

Optical geometry of FAST

Tower

1:

Suspension

Tracking

Pre-tension

Figure 2.

Platformless feed support configuration Pointing

The

Stewart Figure 2.

feed

1

2

Cable

The cable-car feed support configuration for the telescope

The cable car in the feed support configuration is suspended and driven by cables that can be fed in/out individually by the motors. The Gough-Stewart platform is suspended under the cable car, see Figure 3. The orientating mechanism, set in between, has two orthogonal orientating axes driven by servomotors and is used to

263

orient the upper platform relative to the trolley. From the viewpoint of control, the system is an integration of three subsystems, respectively, the cable drive architecture, the orientating mechanism, and the Gough-Stewart platform. The Cable car is driven by simultaneously adjusting the lengths of the ten cables. Cables 1-4, referred to as the driving cables, are anchored to the trolley, see Figure 3. Cables 5 and 6, the suspension cables, passing through both the fixed pulleys on the towers and those on the trolley, form an orthogonal flexible rail for the cable car. Cables 7-10, the downward pre-tension cables, are used to adjust the stiffness of the architecture.

Figure 3. Assembly of the Cable car, the orientating mechanism and the Stewart platform elements

Researches on the dynamics and control of the Gough-Stewart platform are intensive, however in most of them the base platform is fixed on the ground [5-7]. Literatures [8-9] have investigated the use of Gough-Stewart platform for vibration isolation, where vibration of relatively small amplitude is concerned. Although widely used in structure engineering, the use of cables in high-precision robots is rarely found. Literatures [10-11] have investigated the dynamics and feedback control of cable suspended robots with 6-DOFs. Vibration control of the platformless feed support configuration has been studied through analytical and numerical methods [12-13]. In this application the cable-driven architecture controls the position of the upper platform. The orientating mechanism controls the orientation of the upper platform. The Gough-Stewart platform is used to further suppress position and orientation error of the lower platform. The main purpose of this research is to study the feasibility of the cable-car support configuration and the Stewart platform as the feed stabilizing equipment. A set of control laws was proposed and implemented to

264

study the controllability and performance of the mechanism and were tested in the experiments. 2.

Similarity for the Model of Feed Support System

Table 1 No

Variable

1

Length Displacement

2

Time Vibration period‫ޔ‬Control period

3

Linear density

4

Mass, force, unit cable axial stiffness

5

Diameter Load cable

6

Diameter of trimming cable

7

Velocity

8 9

Similarity for time varying cable structure Similarity Model (model : Value Parameter Prototype)

Prototype parameter

50m 1mm

600m 12mm

1 12

1s

3.46s

OU

0.0646

0.356kg/m

5.51kg/m

Ol OU

0.00538

1kg 1N

185.9kg 185.9N

OU Ol

0.0734

2.5mm

3cm

OU 1  Ol

0. 243

7.3mm

0

Ol

1 / 12

1m/s

3.46m/s

Wind velocity

OU Ol

0.88

1m/s

1.1m/s

Inertial moment

O3l OU

Ol

mass

of

Ot

1 12

Ol

satisfied

To make a scaled model predictive for the performance of the prototype system, a set of similarity had to be satisfied. The feed support system is a time-varying cable structure, coupled with the cable-car, the orientation mechanism and the Gough-Stewart platform taken as the rigid body systems. The similarity to be satisfied is mainly mechanical. The set of similarity adopted in the model construction is depicted on Table 1. The key constraint for this similarity comes from that the gravity acceleration for the model and the prototype are the same. Hence the scales for time and length had to be dependent as shown in Table 1. This dependence makes it not practical to use metal cables to satisfy the inertia and the

265

stiffness requirements simultaneously. The way to circumvent this difficulty is to use two sets of cables for each cable of the prototype, namely the load cable and the trimming cable, the former is selected according to the similarity of stiffness, and the latter that of inertia. Through dimensional analysis, scales of diameters of the two set cables are related to scales of length and the cable linear mass density. The side effects of this doing is the wind velocity scale is not the same as that for car moving velocity due to that the trimming cable contributes the main part of the wind excitations. Scale requirements for the inertial moments are automatically satisfied. However, it is impossible to control the damping both in the model and the prototype. Fortunately, the scope of motion is away from resonance where damping is sensitive. 3. Vibration Control Test of Elastically Suspended Gough-Stewart Platform

The Stewart parallel mechanism in this application has two platforms, one is the platform to be stabilized, and the other is the base platform that is mounted on the cable car. When the base platform vibrates with wind-induced cable-car, the vibrations of the stabilized platform can be alleviated by adjusting the six actuators with appropriate control laws. Inevitably, the stroking of the six actuators would induce dynamic interactions between the two platforms. To study the feasibility of the Gough-Stewart platform in this application, an elastically supported Stewart platform is first set up and tested. The working diagram of the Stewart parallel mechanism with elastically restrained base platform is shown in Figure 4. The system includes the base platform, the stabilized platform, the actuators and the elastic restraints for the base platform. % ase platform

z

450

y x

A

$ ctuator ka

B

kz cx

cz ky cy

kx

C

6 tabilized platform

Control force fc

( lastic restraints Figure 4. The working diagram of the elastically restrained Stewart parallel mechanism for vibration control

266

The payloads are installed on the stabilized platform, the vibration of which can be alleviated by adjusting the six actuators of the Stewart parallel mechanism when the base platform is disturbed by winds or other unknown excitations. Seeking the appropriate control laws for adjusting the six actuators with measured position outputs of the base platform is the target of this research. 3 al nt Disturbance Base platform

xa

Actuators 6 tabilized

platform

xb

Controlled variable

u

Controller

x

­ xa ½ ® ¾ ¯ xb ¿

Figure 5. The diagram for the control system

In this problem, the target is the elastically restraint Stewart mechanism, the controlled variables of the system are displacements of the stabilized platform and the control variables of the system are the strokes of the six actuators. Assumed the displacement of the two platforms can be observed, the control system can be illustrated with the following diagram Figure 5: The control problem can be described as follows: through observing the responses x of both the base platform and the stabilized platform㧘we determine a control u(t), namely the strokes of each actuator, such that the response xb of the stabilized platform is under the given root-mean-square specification. The implemented test system is shown in Figure 6. The system is suspended with the rubber ropes from roof. The fundamental natural frequency of the Stewart platform on the ropes is adjusted according to that of the cable-car on the cables in the prototype which is computed by FEM at typical positions of the cable car. With the control law proposed in [12], a series of control test is carried out, and a typical result is shown in Figure 7, which shows that Stewart platform is effective for vibration suppression. However, the control instability predicted by [12] is also observed, to which the main affective factors includes the resonance between the actuating frequencies of the actuator and the fundamental frequency of the system on the suspension, also the time delay effects in the observer and actuator, especially the optical instruments. Based on the assumption of one step-delay for the control system from the measurement, the stability margin is predicted as shown in Figure 8, and the observed phenomenon matches qualitatively with the predicted one. The control updating periods corresponding to the maximum absolute value of

267

the eigenvalue less than 1 are stable. The controls shown in the picture are different only on the predict difference schemes used.

Figure 6. The test system of the elastically supported Stewart platform

The maximum absolute value of the eigenvalues

1.3 control 1 control 2 control 3 control 4

1.25

1.2

1.15

1.1

1.05

1

0.95

0

Figure 7. The control effectiveness

0.2

0.4

0.6 0.8 1 C ontrol updating period(in second)

1.2

1.4

Figure 8. The stability of the control system: damped case

4. Coupled Test of Feed Support System

With the understanding of the control property of the Stewart platform, a coupled model of the cable car, the orientation mechanism and the Stewart platform model was constructed based on the similarity parameters specified in Table 1. The size of the model is of 50m×50m×12m. The model is built on the campus of Tsinghua University, see Figures 9 and 10.

268

Three Laser Automatic Total Stations with the positioning precision of 10mm+2ppm and the sampling rate of 5Hz are employed to measure three points on the upper platform. Two Laser Trackers with positioning precision 10ppm and sampling rate 75Hz are employed to measure the centers of the upper platform and lower platform respectively. With rotation angles of the two orientating axes and lengths of the six actuators acquired from servo feedback, positions and attitudes of all the platforms are obtained. It is worthy to emphasize in this measurement scheme that the positions of the upper platform and the lower platform, critical to the feedback control, are directly acquired with high-performance instruments. The frame chart of the control system is shown in Figure 11. The system output, including positions and attitudes of all the platforms and tensions in all the cables, are acquired in real-time. The reference data include positions and attitudes of all the platforms as a function of time and tensions in all the cables. Control commands are given to the motors in velocity.

Figure 9. The 50m×50m×12m model

Figure 10. Cable-suspended car with Stewart platform

269

Figure 11. The control system frame chart

Figure 12. The position error history of the center of the controlled platform(m)

The experiment is carried out in the natural wind environment, a series of the tests shows that the proposed control law in [12] and the standard PID method yield comparable results, whereas the later is more robust with the available optical positioning measuring instruments. In Figure 12, a typical position error history of the center of the lower platform is shown. The position has been controlled to very high precision considering the size of the model. According to the similarity used for this model, the corresponding position error for the prototype is 12 times that of the model. Hence the accuracy of a few millimeters is achievable for the large telescope. In the experiments, the control results are mainly confined to the displacement and orientation measurement of the upper platform. With coarse measurement data from the total station, the raw data always induce instability of the control system. If averaging method is used to smooth the data, unavoidably time delay will be induced and hence still result in instability. With the laser tracker, the data is much better and faster, but uncertainty in the timing remains and affects the effectiveness of the control system.

270

5. Concluding Remarks

The test results confirm the controllability and the effectiveness of the proposed feed support configuration for the large radio telescope in high-accuracy positioning of the receiver provided the optical measurements on the displacement and orientation of the upper platform is accurate enough. In addition to the instrumentation of high quality observer and the harsher test conditions, further work should be devoted to the dynamically optimized Stewart platform design and the robust control law. Acknowledgement

The present work is supported by National Astronomy Observatory, the National Natural Science Foundation of China (Profject No. 10172049), and by the Tsinghua 985 fund for Solid Mechanics. Thanks are gratefully extended to the university and all the faculties and students involved in the field test. References 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

R. Nan, B. Peng, 2000, A Chinese concept for the 1 km 2 radio telescope, Acta Astronautica, 46, 66. G. Ren, Q. Lu, Z. Zhou, 2001, On the cable car feed support configuration for FAST, Astrophys. Aerosp. Sci., 278, 243-247. B. Duan, Y. Zhao, J. Wang, G. Xu, 1996, Study of the feed system for a large radio telescope from the viewpoint of mechanical and structural engineering. in Proc. of the LTWG-3&W-SRT, Edited by: Strom R.G., B. Peng, R. Nan, 85-102,. D. Stewart, 1965, A platform with six degrees of freedom, Proc. Inst. Mech. Eng., 180, 371-386. E. F. Fichter, 1986, Stewart platform-based manipulator: general theory and practical construction, Int. J. Robot. Res., 5, 157-182. G. Lbret, K. Liu, F.L. Lewis, 1993, Dynamic analysis and control of a Stewart platform manipulator, J. Robot. Syst., 10, 629-655. D. Li, S.E. Salcudean, 1997, Modeling, simulation, and control of a hydraulic Stewart platform, IEEE Int. Conf. Robotics and Automation, 4, 3360-3366. Z. Geng, L.S. Haynes, 1993, Six degree-of-freedom active vibration isolation using Stewart platform manipulator, J. Robot. Syst., 10, 725-744. R. Graf, R. Dillmann, 1997, Active acceleration compensation using a Stewart platform on a mobile robot, in Proc. Euromicro Workshop on Advanced Mobile Robots (EUROBOT), 59-64. T. Satoshi, M. Yoshio, H. Manfred, M. Rie, K. Hideaki, M. Toshiyuki, 2002, A motion base with 6-DOF by parallel cable drive architecture, IEEE ASME Transactions on Mechatronics, 7, 115-123. B.A. Abdullah, K.A. Sunil, 2002, Cable suspended robots: design, planning and control, Proceedings-IEEE International Conference on Robotics and Automation, 4, 4275-4280. G. Ren, Q. Lu, N. Hu, R. Nan, B. Peng, 2004, On vibration control with Stewart parallel mechanism, Mechatronics, 14, 1-13. Y. Cheng, G. Ren, S. Dai, 2003, Vibration control of Gough-Stewart platform on flexible suspension, IEEE transactions on robotics and automation, 19, 489-493.

Biological Actuating Materials

BIOFILM GROWTH: PERSPECTIVES ON TWO-PHASE MIXTURE FLOW AND FINGERINGS FORMATION

S. Hao1, B. Moran1, D. Chopp2 Department of Mechanical Engineering 2 Department of Engineering Science and Applied Mathematics Northwestern University, Evanston, IL 60208, U. S. A.

1

Abstract

The motion of biofilm in aqueous environment is modeled as two-phase mixture flow which is governed by a generalized Novier-Stokes equation. Noticed that the propagation of bacteria colonies obeys the diffusion law with the similar instability mechanism in the coarsening of metallic grains during solidification, a thermodynamic framework has been derived which reveals the two competing mechanisms in biofilm growth: the absorption of nutrition from water phase that tends to maximize the contact area to environment and a loose-surface layer formation that tends to minimizes biofilm surface. Therefore an additional condition has been obtained which governs the formation of finger-shaped biofilm. 2D numerical simulation has been performed. 1. Introduction

A biofilm is an aggregation of millions of microorganisms that consist of bacteria cells immobilized in an organic extracellular polymer matrix of microbial origin which is anchored to biological or non-biological substrate surfaces in aqueous environment. This surface association is an efficient means of lingering for bacteria colonies in a favorable microenvironment, rather than being swept away by the current. Biofilm growth is the evolution and decay of such a self-monitoring organism system through the interior and exterior cycles of energy exchange, representing the significant and incompletely understood mode of the propagation of bacteria in natural environment and during the distribution of infectious diseases which involves the interactions between ecological, biological, chemical and mechanical processes. Therefore, progresses in biofilm research will bring significant impacts to both modern industries and the daily life of humen-beings. This also provides challenges and opportunities for mechanical engineers to work with environmentalists, biologists, and applied mathematicans in biofilm research. Reviews of recent development of biofilm can be found in [1-4]. 273 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 273–290. © 2006 Springer. Printed in the Netherlands.

274

Phenomenologically, a biofilm growth is a mechanical process that couples diffusion, transportation, chemical reaction, and interaction between fluid and polymer-like soft substance (biofilm) with many underlying biological and ecological mechanisms. In the viewpoint of continuum mechanics, the motion of biofilm can be modeled as the two-phase flow where the water phase can be assumed as incompressible flow and the biofilm is governed by generalized convection-diffucsion equations, which will be discussed in details in this research. The biofilm phase contains two components: the bacteria cells that consists of one or several active species (bacterium families); and the biomass that consists of water and extracellular polymeric substances (EPS), primary biotic origin, such as the accumulation of dead cells [2,5]. The active bacteria cells continuously emit synthase signals to the environment; this process is termed “quorum sensing”. When the concentrations of the signals and active cell density are over their threshold values, the signals interact with certain transcriptional regulators which modulate the expression of quorum sensing-regulated genes and catalyze the synthesis of the bacteria cell. This process is accomplished by the energy exchange through the consumption of the nuitrition, which is termed “limiting growth substrates”, from the water phase[2,5]. Figure 1 illustrates this life cycle of bacteria in biofilm, which is the dominant mechanism in the mechanical system introduced in this analysis.

Figure 1. Life cycle of bacteria in biofilm in aqueous environment

Therefore, in the framework of continuum mechanics, the dynamics of biofilm

275

growth is characterized by the evolution of the interface between biofilm and water phase via the convection-diffusion field solution which is governed by the three key-biological variables: total living cells mass concentration mb , quorum sensing concentration m a , and the limiting growth substrate concentration mO . As compared to other fluid-solid interaction problems, an interesting and important feature of biofilm is its heterogenenuous nature that contains the cycle of deaths and creatures of bacterium through the diffusion-consumption of the limiting substrate absorbed from water phase.

Figure 2. Finger-shaped biofilm

Observations reveal that the propagation of biofilm usually presents a finger-shaped interface to the water environment (Figure 2). This morphology is elected by bacteria inhabitant to achieve optimized inhabitable condition and, thus, it governs the evolution and decay for this class of biofilms. While considerable studies have been reported in the respects of microbiology, ecology, and physiology, it seems that relatively little attention has been paid to the bio-mechanical functions and responses for such a fingering morphology and underlying physics in biofilm growth. In a two-phase mixture flow, the outer profile of the biofilm defines the boundary condition for a boundary value problem and, thus, determines the mechanical processes such as slough, detachment, bacteria cells loss and growth in the associated models. In recent years, an increasing interesting can be found in mathematic and mechanical modeling and simulation [2,6-8]. Among them, Dockery and Klapper first addressed the issue of fingering formation and obtained an one dimensional mathematic model which offers solutions ranging from linear unstable to fingering-shaped instability [6]. In this research we will focus on the physics incentive of this phenomenon and the model of multi-dimensional fingering

276

formulation. As verification, numerical simulation has also been performed. The finger-like morphology can be found in many diffusion-convection systems in nature, among which the alloys precipitate, solidification, and coarsening is a well-known process with dendritic moving boundary that has been thoroughly investigated in materials science[9,10]. Applying the diffusion instability theory in metallurgical analysis, a model of dentritic biofilm growth has been proposed in the framework of a proposed generalized Navier-Stokes equation in this research. Hereafter both the terminologies “dentritic” and “finger-like” refer to the same waived morphology of biofilm surface. This paper is organized as follows: the next section gives the derivation of the proposed generalized Naviks-Stocks equation with a brief introduction of biofilm evolution equation. Section 3 presents the proposed model of the dendritic biofilm propagation. Examples of numerical simulation using proposed model and discussions are given in the section 4. The conclusions are given in the section 5. In this paper boldface symbols denote tensors, the order of which is indicated by the context. Plain symbols denote scalars or a component of a tensor when subscript-indices are attached. Repeated indices are summed. For rank two tensors a and b: a aij , b bij : a ˜ b aik bkj , a : b aij bij , and

ab

>aij bkl @.

and ’

2

> @

> @

>

@

>

The symbol ’ represents the gradient operator: ’

@

>w w i @ ;

’˜ ’.

2. A Model of Biofilm Growth and Governing Equations

2.1 A GENERALIZED “NAVIER-STOKES EQUATION” As illustrated in Figure 3a, a biofilm growth is modeled as a two-phase mixture flow coupling diffusion and reaction where the latter represents the life cycle of bacteria cells without inquiring into its biologic basis. In the biofilm model of 2 Figure 3b, a region (:  R ) is divided into a biofilm region : b and an aqueous region : w with the interface surface *b . We use the subscript ‘w’, ‘O’, ‘a’, ‘EPS’, and ‘k’, respectively, to denote in turn the variables associated with the liquid water (H2O), limiting substrate(O2), signal of quorum sensing, extracellular polymeric substances (EPS), and the kth specie of bacterium in biofilm; when totally n species of bacteria exist, then k 1,2, ,..., n . Let ‘m’ to denote the mass concentration and ‘ U ’ to denote the density, e.g. ‘ m EPS ’ is the mass concentration of EPS and ‘ U EPS ’ is the density of EPS, then, for the boundary value problem defined in Figure 3b:

277 n

mW  mO  ma  m EPS  ¦ mk

1

(1a)

U.

(1b)

k 1

and n

U W  U O  U a  U EPS  ¦ U k k 1

Hence, the mass concentrations of each above mentioned species, together with the velocity field v >vi ,@ total density U , and the energy per unit volume ‘e’, are the n  9 governing variables that define the boundary value problem of biofilm growth. Alternatively, using the subscripts ‘b’ and ‘d’ to denote the active bacteria cells and deactived biomass, respectively; then in : b we have n

mb

¦m

and

k

md

mEPS  mw .

(2)

k 1

Figure 3. An illustration of the analyzed problem

For this boundary value problem of biofilm growth, the following approximations are employed: (1) incompressible flow; (2) isothermal process; (3) the interface surface *b is defined by mcr , a critical value for the total concentration of the active bacteria cells and deactived biomass (3) mcr mb  md ;

>

@*

b

and (4) a boundary layer enhanced to *b exists within biofilm phase; which has a constant thickness b* and is associated with an extra additional surface energy density

F  x .

278

We consider the problem in a Eularian coordinate system

^x`

and begin with the

I  x which is characterized by the flux field jI that is proportional to the gradient of I (Fick’s law): jI  BI ’I (4)

diffusion problem of a field variable

where BI is the diffusion coefficient which is assumed to be a constant. When there is no transportation, the difference between the reaction rate qI  x x and the divergence of jI

I

represents the rate of net change of

of ǂҏ ҏ at

I at this point:

qI ’ ˜ jI

(5)

In an aqueous environment characterized by the flow velocity field v, the net change of I at the spatial point x is the summation of the change of I itself and the transportation due to the flow, expressed as ’ ˜ I v :

dI  x, t dt

I I , t ’ ˜ I v i.e.

wI  ’ ˜ I v . wt

(6)

Substituting (4) and (6) into (5), the latter becomes:

’ ˜ I v  BI ’ 2I  qI

I,t

(7)

or, alternatively

I,t  ’ Iv  BI ’ 2I  qI Remarks: 1) When I

0.

(8)

U , the total density, so q U

“mass conservation law”. 2) Considering Newtonian flow so

I

0 and (8) becomes the conventional

U v and qI

U f , where f is the

body force vector, the correspondant constitutive relation is:

ı





C C P, CN : d  Ip 0 and d

>d @ , ij

d ij

1 v i , j  v j ,i 2





(9)

where ı is stress tensor, C is the stiffness matrix which is the function of shear and volumetric viscosity coefficients C P and CN , I is the rank two unit tensor and

p 0 is the initial hydrostatic pressure.

Then (8) becomes the “momentum conservation law” in conventional fluid dynamic. 3)When I Ue and qI q H , the heat source per unit mass, then (8) degenerates to the energy conservation law; where e is the total energy per unit mass in either water phase or biofilm phase:

279

1 2

M el  RC p T  vi vi

e

(10)

where M el is the elastic energy; T is the absolute temperature, Cp is the isothermal heat capacity, R is the universal gas constant; the product of RCpT represents the intrinsic energy of unit mass in the system. Alternatively, M el is also termed as an “energy potential” and the product RCiT is considered as the internal energy per unit mass. 4) The above mentioned five partial diffenrential equations (mass conservation, momentum conservation, and energy conservation) are the conventional Navier-Stokes equations. More details of fluid dynamics can be found, e.g. in [11]. For the biofilm problem, (8) can be rewritten in a general form:

ij,t  ’ ˜ v ij  ’ ˜  B˜ ’ij  Q

0

(11)

@

>

mW , mO , ma , m EPS , m1 , m2 ,..., mn , U , e, v with totally n + 9 where ij components (n + 8 components for two-dimensional case), B is a matrix determined by system stiffness and diffusion coefficients; Q 0, qO , q a , q EPS , q1 , q 2 ,..., q n ,0, q H ,0,0,0 is a reaction vector

>

@

corresponding to each term in ij ; where the source for ij , can be expressed as

Ii , the ith component of

Ni

qi

¦/

ik

jk

(12)

k 1

where Ni is the number of reactions involved for

Ii ; j k is the rate of the kth

reaction and / ik is the corresponding “mass stoichiometric coefficient”. The detail expresses of

j k and / ik will be discussed at the next subsection.

In this paper the partial differential equations (11) is termed as the “generalized Navier-Stokes equation” for the two-phase mixture flow with biofilm. 2.2 BIOFILM KINETICS – EVOLUTION EQUATIONS As an organic system, the evolution and decay of a biofilm are the life cycles of bacteria involving the following four major energy exchange and transformations processes [5]: (1) creation (new born) of bacteria population through the absorption of the nutrition, i.e. the “limiting substrate”; the heredity and variation of the bacteria are governed by synthase signals represented by the signals concentration “ma”; (2) quorum sensing: creation and emission of the synthase signals; (3) death of bacteria that creates inert biomass to form the extracellular polymeric substances(EPS);

280

(4) inorganic chemical reactions in the EPS and water, such as the biodegradation of the inert biomass. The correspond mathematic expression of the reaction rates have been studied in [2] which are applied in this research. For the case of single bacterium specie (n=1) without heat source (qH=0), so Q 0, qO , q a , q EPS , qb ,0,0,0,0,0 and the

@

>

corresponding

coefficients are

listed

in

Table

1,

KO

where

is

the

“half-maximum-rate concentration” for utilization of substrate; Yx / O , qˆ O , b , and Jˆ are the constants termed as “yield of active biomass”, “maximum specific substrate utilization rate”, “rate of endogenous decay”, and “chemical oxygen demand” for the degradation of a unit of active biomass; E 1 and E 3 are the coefficients of basal rate of signal production whereas E 2 is the coefficient for the additional quorum sensing represented by the Heaviside function

xt0 . x0

­1 H x ® ¯0

(13)

Table 1 The mass stoichiometric coefficients and reaction rates [2] qb 2

qa 3

qEPS 2

qO 2

mb

0.2mb

mb

E1 ˜ P mO , K 0

b ˜ P mO , K 0

 qˆ 0 ˜ P mO , K 0

/ i2

mb Y X / O qˆ 0 ˜ P mO , K 0 mb

J2

 b ˜ P mO , K 0

Ni

/ i1 j1





mb

mb

0.8mb

E3

YW / O qˆ 0 ˜ P mO , K 0

 bJˆ ˜ P mO , K 0

/ i3

mb

J3

E 2 H mb  md  mcr

Also in Table 1 the

P  x, K 0





P  x, K 0 denotes the Monod kinetic: x . KO  x

(14)

2.3 INITIAL CONDITION AND BOUNDARY CONDITIONS For any variable

Ii in (11) the initial condition is: t 0 : Ii Ii 0 where Ii 0 Ii  x, t 0  x : .

(15)

 CN Hii  p 0 .

(16)

at t Particularly, for the hydrodynamic pressure p: at t t 0 : p p 0 otherwise

p

281

Hence, for incompressible flow: so Hii { 0

p { p0 .

Boundary conditions on * in Figure 3 are: Ii * Ii or

n˜ ’Ii

where J i is the flux on * for

(17)

Ji

(18)

Ii .

2.4 LEVER-SET REPRESENTATION OF INTERFACE *b Let the summation:

\  x mb  x  m d  x

(19)

to be a lever-Set function, so *b , the interface between biofilm and water phase, can be represented as following:

w\  F ’\ wt

’\ ˜ u*b ’\ is the velocity vector of *b when \ x mcr .

where u *b

F

0;

(20)

3. Thermodynamic Framework and Application to Fingering Formation

In this section we revisit the mass and energy conservations in biofilm with the reactions introduced by Table 1. A thermodynamic framework will be established which determines the dynamics and kinetics of biofilm growth. 3.1 MASS CONSERVATION FOR BIOFILM REACTION By superposition of equations in (11) for ij

>mw , mO , ma , mEPS , m1 , m2 ,..., mn @ ,

we obtain the conventional mass conservation law:

U ,t ’ ˜ v U 0 which

leads to an additional constraint to the rection rates: n 9 Ni

n9

¦q ¦ ¦/ i

i 1

ik

jk

0.

(21)

i 1 k 1

This relation implies that mass can neither be created nor dismissed through chemical reactions, so there is at least one coefficient among those listed in Table 1 is predetermined by (21). 3.2 RATE OF ENTROPY PRODUCTION The second thermodynamic law is the fundamental law that governs any nature process including biofilm growth. Let s  x to be the entropy function, the local

282

form of the second thermodynamic law is: 's t 0 . According to Gibbs’ equation the rate of entropy change is expressed as:

U

ª 's º lim « U » ¬ 't ¼

T

't o 0

u

p U ’ ˜ v  ¦ K k m k T T k

(22) (23)

where p, T and K k are pressure, temperature, and the chemical potential for the kth specie, respectively; u is the internal energy per unit mass that is defined as below in this study:

u x where

e x  F  x  )  x 

1 v˜v 2

(24)

F  x is the interfacial energy per unit mass when x is within the surface

layer enhanced to *b ; ) is the summation of the potential energy for each component that defines the general force field f k :

)

¦m ) k

k

and

fk

’ ) k .

(25)

k

By substituting (24) and (25) into the conservation of energy in (11), we obtain

Uu

U F ’ ˜ ’T  ı : ’v

here u

du , F dt

dF . dt

(26)

In the derivation of (26), several additional relations are applied which is listed in Appendix I. The chemical potential for ideal solution is employed in (23): Gk H k  s k T (27) K k Gk  RT logmk and where G k , s k , and H k are the Gibbs free energy, entropy, and enthalpy for the kth component of the system, respectively. The physical meaning of enthalpy is the chemical bonding energy of molecule. The Gibbs free energy for many biological C-H-O compounds are given in [1]. By substituting into (24-27) into (23) then (22) and applying (11), we obtain the “Clausius-Duheim” equation of the biofilm system, i.e. the local form of the second thermodynamic law: N ­ ½ 1 1 p U U s ’ ˜ ’T  ı : ’v  ’ ˜ v  F  ¦ K k ®’ ˜ Bk ’mk  ¦ / kr jr ¾ t 0 T T T T T k r 1 ¯ ¿ (28) where Bk is the diffusion coefficient for m k . In (28) the first term refers to the entropy change caused by heat flow; the following terms denote in turn the changes of visco-elastic strain energy, volumetric strain energy, interfacial energy, diffusion induced entropy and reaction induced entropy.

283

4. Finger-Shaped Biofilm Formation and Diffusion Instability

Experimental observations reveal that biofilm usually has a thick coherent surface layer over the film body that consists of semi-contiguous organisms unit, e.g. particles made of EPS and bacteria cells, and water channels [12,13]. As the “frontier” to “fresh” water, the biological function of the surface layer is to extract more limiting substrate, mainly O2, to supply the nutrition for the film body. From the viewpoint of continuum mechanics, this surface layer is a nature shield that confines a favorable internal environment for the bacteria life cycle meanwhile prevents the semi-contiguous organisms to be sloughed or washed out. To form such a surface layer requires extra bonding energy to make the layer to be relatively coherent as compared to the inside part. This extra energy defines the function F  x that is introduced in (25). We propose the two competing mechanisms which may govern the evolution of interfacial surface *b . Since growth of biofilm is a bacteria population increase that results in volumetric enlargement of active biomass, this process requires the supply of nutrition from fresh water which tends to maximize its surface area, so as to absorb more limiting substrate. On the other hand, to maintain a stationary state, the energy equilibrium requires the system to minimize its surface because the latter requires extra coherent energy. The consumption of entropy, stated by the second thermodynamic law, adjusts the system under a balance between these two mechanisms and, thus, determines the surface morphology.

Figure 4. A model for analyzing finger formation

To describe quantitatively these two competing mechanisms, a biofilm surface evolution model is proposed which is illustrated in Figure 4. In this diagram three pieces of biofilm are ploted which have the same interior area but different shapes of interfacial surface boundary. The circler biofilm (b) has the smallest boundary length, whereas the elliptic (a) and the qua-poles biofilms can be considered as waved boundaries with the first and the third order frequencies, respectively. Then the problem can be addressed as: finding entropy consumption for the transformation from (b) to (a) or (b) to (c) or verse versa; the signs of the entropy

284

consumption in these transformations determine the favarable morphology of the biofilm. For simplification, we consider the case of b œ a for the biofilm with single bacterium specie (n=1) with the following approximations: A1: isothermal process, i.e. T const ; A2: diffusion process dominates, so the convection terms can be omitted, i.e. v #0; A3: no bacteria cell penetrating the biofilm boundary ( ’mb

*b

0 ) and the

creation of bacteria only resulting in the expansion of biofilm; A4: constant diffusion coefficients in (2.11); A5: continuities for all variables of ij on and inside of *b ; A6: the product of quorum sending signal is small, i.e.

>’ma  ’mO @*

>ma  mO @*

b

and

;

b

A7: axial symmetric distributions for all variables of ij in the case (b) of Figure 4.

According to the approximations A1, A2, A4, and A6, the entropy change of the biofilm, represented by the integration of (3.8) over the biofilm domain in Figure 4, becomes

ª º 2 2  B K ’ m  B K ’ m  F  K q « »d: t 0 (29) ¦ O O O b b b k k T :³b ¬ k O ,b ¼ and Bb are diffusion coefficients. Applying the following relations

's d: 't o 0 ³ 't :b lim

where BO

a’ 2 b



U

’ ˜ a’b  ’a ˜ ’b ,

³ ’ ˜ a’b d: ³ a’b ˜ nd* , b

:b

and ’mO

JO ,

*b

where n is the unit outer normal vector of *b , then (29) can be rewritten in the form as

lim

't o 0

's

³ 't d:



:b

U

B K T ³ O

O

J O ˜ n*b d*b 

*b

where the approximation A7 ( ’mb

*b

U T

³ >~s  F @d: t 0

(30)

:b

0 ) is also applied. In (30) the first term

represents the nutrition (limiting substrate) that flows into the : b under the chemical potential

~ s

K O ; whereas ~s is the summation of the following terms

¦K q k

k

 BO ’K O ’mO  Bb ’K b ’mb

(31)

k O ,b

For the transformation, e.g. difference below:

b Ÿ a , the entropy rate can be expressed as the

285

ª º « ³ 'sd:» »¼ b o a «¬ :b

ª 's º ª 's º 't « ³ d:»  't « ³ d:» . «¬ :b 't »¼ a «¬ :b 't »¼ b

(32)

Keeping in mind that no changes in area and substituting (30) into (32), so the difference for the first term of (30) is

ª º ª º  « ³ BOK O J O ˜ n*b d*b »  « ³ BOK O J O ˜ n*b d*b » «¬ *b »¼ b «¬ *b »¼  a

BOK O* J O* '*b  o'*b

(33) where K O* and J O* are the K O and J O on *b for the case (b) in Figure 4. Hence, (33) represents the increment of nutrition flow due to the length increment of surface layer when (b) shifts to (a) (or (c)). Consequently, there should be also a correspondent term for the increment of surface energy:



³ > F @ d:  ³ > F @ d: b

:b

a

:b

'F '*b b*  o'*b 't

(34)

where b* is the thickness of the boundary layer enhanced to *b . By the elementary calculation detailed in Appendix II, we know that the rest terms in (32) are in the order of o'*b . When the ǂҏin case (a) and (c) of Figure 4 is a small perturbation parameter, then '*b is also a small quantity. Finally, by leaving out the terms in the order of o'*b and taking 't to be unity, (32) becomes:

lim ³ 'sd: bo a

BOK O* J O* '*b  Fb* '*b t 0

(35)

:b

The underlying physics of (35) can be explained through two extreme cases: when the surface coherent energy density is infinitesimal or vanishes, i.e. F Ÿ 0 and the biofilm is the simply aggregation of bacteria cells and deactive biomass without surface coherence, so

lim ³ 'sd: bo a

BOK O* J O* '*b t 0

(36)

:b

which implies that a fingering shape formation, which increases '*b , is a thermodynamically favorable motion during the biofilm growth. On the other hand, when F z 0 and b* F !! BOK O* J O* , i.e., the biofilm has a well coherent surface, then (35) becomes:

lim ³ 'sd: |  Fb* '*b t 0 . bo a

:b

(37)

286

To ensure that the greater or equal sign holds in (37), '*b must vanish; i.e. the biofilm intends to keep a spherical shape. Hence, the surface morphology of biofilm is determined by the amplitude of the surface layer coherence energy density. When a biofilm is an aggregation of loosely contiguous particle with minimized surface coherence, then according (36) a finger-typed surface is a preferred morphology. Whereas for a biofilm that has a well-formed coherent surface layer, the smooth spherical surface is the preferred morphology. This phenomenon is essentially the same as the metal grain solidification during cooling process where the melted metal tends to maximized the surface area of the precipitated solid colonies so as to promote heat conduct whereas the capillary force tends to minimized the surface area so as to minimize system energy. This metallurgical process has been thoroughly investigated in [9, 10,14]. 5. Numerical Example

Figure 5. A simulation of biofilm figuring formation

Figure 6. Snap-shots of the merger of biofilm – conjunction of moving boundaries

As an extension of one-dimensional bacteria evolution model introduced in [2], a

287

simulation of a growing “Pseudomonas aeruginosa” biofilm has been performed using a MLE technique (MPFEM[15] + Level Set [16] based on E-FEM enrichment[17]). Figure 5 is a simulation of fingering formation of P. Aeruginosa biofilm where a constant mO a given at the outer boundary of the domain and an initial value of mb is given at the middle of domain. Figure 6 is an example of merger process of three pieces of biofilm because it remains as challenge in numerical analysis for simulating the moving boundaries conjunction. The third example is given in Figure 8 with a qualitative comparison with the experimental observation shown in Figure 7 [18].

Figure 7. An experimental observation of the evolution of biofilm [18]; where the arrow at upper-right corner represents the flowing direction and the small arrow new the middle indicates a high concentration of biomass

Figure 8. A simulation of biofilm evolution for the case in Figure 5; where the contours represent the volume fraction of the active biomass of the “Pseudomonas aeruginosa”; the grey net is the back-ground finite element mesh

288

6. Conclusions

1) Based on the previous theoretical frameworks in [1,2,4], a boundary value problem of biofilm growth has been established associated with a drived generalized Navier-Stokes equations (GNSE) that contains n + 8 ( n + 9 for 3D) partial difference equations where n the number of active bacteria species. 2) According to mass conservation, an additional constraint has been obtained for the biofilm evolution, which indicates that there is at least one coefficient is predetermined among those coefficients in biofilm kinetics 3) Based on Gibbs equation and generalized Navier-Stokes equation, a thermodynamic framework of biofilm growth has been established which indicates two competing mechanisms which may govern the evolution of biofilm surface. They are the absorption of nutrition that intends to enlarge the contact area to water phase and the system energy balance that minimizes the biofilm surface. The quantitative expression of the surface evolution kinetics has been obtained according to the second law of thermodynamics. 4) To verify the proposed growth kinetics, two dimensional simulation of biofilm growth has been performed and been compared with experimental observations. Acknowledgements

The first authors would like to thank Professor P. Voorhees for the illuminated discussion. Appendix I

By substituting (3.4) and (3.5) into (2.11) for ij >e@ and after some tedious derivations, the energy conservation can be split into the following three independent equations which are engaged for the derivation of (3.6):

w § Uv ˜ v · · § Uv ˜ v  ı ¸v  ı : ’v  ¦ U k f k ˜ v ¸  ’˜¨ ¨ wt © 2 ¹ ¹ © 2 k w § U) · ¸  ’ ˜  U)v  ¦ U k f k ˜ v 0 ¨ wt © 2 ¹ k wUu  ’ ˜  Uuv  ’ ˜ ’T  ı : ’v 0 wt

0

a.1) (a.2) (a.3)

Appendix II

As illustrated in Figure 4, after the transformation

b Ÿ a

the maximum

normal derivation of *b is denoted as ǂ G . By elementary calculation it can be proven that the surface layer length increament '*b and the area variation,

289 

denoted as 'S and 'S



in Figure 4, are the same order asҏǂ G , i. e.:

S  ~ S  ~ rG

'*b ~ G ,

(b.1)

where r is the radius of the sphere in Figure 4b. In the vicinity of the boundary *b , any function can be expressed as a Talyor series:

f x

df  x df  x 0 d cG d 1 (b.2) 'x  ... f 7b  cG G , dx dx is the value of f x on *b . For an integral of f x , after the

f 7b 

where f 7b

transformation

b Ÿ a , the difference is

ª df  x º ª df  x º G » d:  ³ «cG G d:  ³ « cG dx ¼  a dx »¼  a :b :b S ¬ S ¬ (b.3) Applying the mid-value principle of integration and the second relation of (b.1) to (b.3), it can be simplified as F G

³ > f  x @ d:  ³ > f  x @ d: b

a

df x 2 F G c~G r G , dx

0 d c~G d 1

(b.4)

So the difference (32) for the second term in (30) reads



³ >~s  F @ d:  ³ >~s  F @ d: b

:b

a

:b

'F '*b b*  o'*b . 't

(b.5)

References 1㧚 B.E. Rittmann, P.L. McCarty, 2001, Environmental Biotechnology. McGraw-Hill International Editions. 2㧚 D.L. Chopp, M.J. Kirisits, B. Moran, M.R. Parsek, 2003, The dependence of quorum sensing on the depth of a growth biofilm. Bulletin of Mathematical Biology, 1-34. 3㧚 P. Watnick, R. Kolter, 2000, Biofilm, city of microbes. Journal of Bacteriology, 182, 2675-2679. 4㧚 L. Hall-Stoodley, J.W. Costerton, P. Stoodley, 2004, Bacterial biofilms: From the natural environment to infectious diseases. Nature Reviews Microbiology, 2, 95-108. 5㧚 E. Bruce, P.L.M. Rittmann, 2001, Environmental Biotechnology. Biological Science Series.: McGraw-Hill. 6㧚 J. Dockery, I. Klapper, 2002, Finger formation in biofilm layers. Siam Journal on Applied Mathematics, 62, 853-869. 7㧚 S. Bordas, 2003, Extended finite element and level set methods with application to growth of cracks and biofilms, in Mech. Engr., Northwestern University: Evanston. 8㧚 N.G. Cogan, J.P. Keener, 2004, The role of the biofilm matrix in structural development. Mathematical Medicine and Biology-a Journal of the Ima. 21, 147-166. 9㧚 P.W. Voorhees, M.E. Glicksman, 1982, Analysis of multiparticle diffusion. Journal of Metals, 35, A84-A84. 10㧚 P.W. Voorhees, M.E. Glicksman, 1984, Solution to the multi-particle diffusion problem with applications to Ostwald ripening .1. Theory. Acta Metallurgica, 32, 2001-2011. 11㧚 T.J.R. Hughes, L.P. Franca, M. Mallet, 1986, A new finite-element formulation for computational fluid-dynamics .1. Symmetrical forms of the compressible Euler and Navier-Stokes equations and the 2nd Law of thermodynamics. Computer Methods in Applied Mechanics and Engineering, 54, 223-234. 12㧚 S. Okabe, T. Yasuda, Y. Watanabe, 1997, Uptake and release of inert fluorescence particles by mixed population biofilms. Biotechnology and Bioengineering, 53, 459-469.

290 13㧚 M.R. Wirthlin, G.W. Marshall, R.W. Rowland, 2003, Formation and decontamination of biofilms in dental unit waterlines. Journal of Periodontology, 74, 1595-1609. 14㧚 W.W. Mullins, R.F. Sekerka, 1963, Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Physics, 34, 323. 15㧚 S. Hao, W.K. Liu, T. Belytschko, 2003, Moving particle finite element method with global smoothness. Int. J. Numer. Meth. Engr, in press. 16㧚 J.A. Sethian, 1999, Level set methods and fast marching methods: Evolving interface in computational geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge, U. K.: Cambridge University Press. 17㧚 T. Belytschko, T. Black, 1999, Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45, 601-620. 18㧚 P. Stoodley, J.D. Boyle, D. DeBeer, H.M. Lappin-Scott, 1999, Evolving perspective of biofilm structure. Biofouling, 14, 75-90.

DAMAGE AND FATIGUE OF ACTUATING HEART MUSCLES

Xiang-Ming Zhang 1, Nian-Ke Ma 1, Fan Yang 1, Yong Zhao 2 and Wei Yang 1 1 School of Aerospace, Tsinghua University, Beijing, 100084, P.R.China 2 Medtronic Inc., Minneapolis, Minnesota, USA

Abstract

Heart is the pump to make blood circulate in the vessels. The actuating heart muscles can generate intrinsic force making the heart chambers contract and diastole rhythmically under the stimulation of electronic signals. We carried out in-vivo experiments that screwed the lead tip of a heart pacer into the designated sites of heart and applied on controlled external excitation on the lead-myocardium interface. A special design of the helix tip using strain gauges and fiber optic sensors is instrumental for the measurement of the push-pull forces of lead-myocardium interactions during the tests. We observed that the interaction was mainly caused by the heart beating and influenced by the lungs breath. The in vivo heart had an ability of self-adjusting to protect itself from outside excitations. The security of the pacer attachment to the heart muscle is estimated for the fatigue and damage tests. In-vitro heart muscle specimen tests are also carried out for the parameters that characterize the constitutive relation of heart muscles. Keywords: actuating hear muscles, in vivo tests, damage, fatigue, cardiac pacing lead 1. Introduction

More than 300,000 cardiac pacing leads are implanted into patients annually [1]. A cardiac pacemaker consists of two major structural parts: the pulse generator and the conductor lead. The pulse generator is composed of a programmer, a battery, and other parts, and is implanted under the skin in the pectorals region to generate the desired electrical pulse. The transvenous pacemaker lead connecting with the generator can be implanted at various sites inside the heart chambers and epicardial surface through the help of various lead delivery systems. A helix electrode configuration is typical for the heart pacing lead distal tip [2], and has a more secured attachment. The lead distal tip is attached or fixed to a certain critical surface of the heart. A pacing lead is implanted to a desired location of the heart

Corresponding author. Tel: 86-010-62782642. Fax: 86-010-62781824 Email: [email protected]

291 W. Yang (ed.), IUTAM Symposium on Mechanics and Reliability of Actuating Materials, 291–307. © 2006 Springer. Printed in the Netherlands.

292

with either an active fixation (screw-in helix) or a passive (tines) fixation mechanism at its distal tip. Two critical issues arise: the pertinent parameter ranges for the lead distal fixation that guarantee an easy implant of the lead into the heart and the least damage to the heart; and the cyclic damage to the heart as well as the heart/device interface under the normal heart beats and the external excitation in the post-implant clinical services of up to ten years. The pacing lead/heart interaction is the main cause for the failure of lead tip and damage of the myocardium. Investigators traditionally performed in vitro testing of medical devices to predict their in vivo mechanical performance. Other investigators developed 3-D finite elements analysis (FEA) modeling to study the heart/device interaction and analyzed the mechanical performances of lead or heart muscle [3-7]. An innovative 3-D reconstruction technique based on the image-analysis was applied to compare intracardiac pacing lead bending stresses and motion characteristics at different sites of heart [8]. A new energy convert was developed to study the long-term in vivo tests of the totally implantable heart system [9]. But the aforementioned tests and methods cannot quantify the interactions though they are crucial to the lead designers and clinicians. The interaction between the actuating heart and the heart pacer is revealed by a unique design of the lead tip that appropriately uses the tiny strain gauges and fiber optical sensors. Generally the lead/heart interactions include pulling and pressing during heart beat and torsion during the screw-in of the helix tip. We measured the push and pull forces under two statuses: normal heart beat and heart beat under controlled outside excitations including fatigue motions. Several frequencies and amplitudes of the outside excitations were selected to investigate the mechanical performances of the lead/heart interaction. The pulling force during heart beating and the force of drawing the lead out of the heart muscle are compared to estimate the security to attach the pacing lead to heart muscle. The pull-out forces in chronic tests and acute tests at two certain sites were compared. The in-vitro sample tests were carried out to study the general mechanical properties of heart muscles. The passive stress-strain relationships at various sites of the heart were established on the data of tension and relaxation tests. 2. In-vivo Damage and Fatigue Tests of Pacing Lead-Myocardium Interaction

2.1 TESTING PROCEDURES The in-vivo tests were carried out in the Animal Lab of Beijing Fuwai Angiocardiopathy Hospital. An ECG monitor was connected to monitor the heart beating rate and the rhythm. A right lateral cephalic vein was cut open and intravenous catheter was placed into the vein and reached the designated site in the heart chambers. A pacing lead with force sensors was delivered into the heart

293

chambers through the catheter and reached a certain site of heart with the help of X-ray imagery. The lead helix tip was screwed into the myocardium slowly and it moved with heart beating. The force sensors recorded the data of lead/heart interactions. 2.2 TESTING EQUIPMENT The integrated system of testing included three sub-systems (Figure 1). (1) A fiber optic sensor (Fabry Perot strain gauge) which converts the mechanical signals into the optical ones is installed. The assembly of the sensor contains a signal conditioner to convert the optical signals to the electric ones. (2) Strain gauges are connected with a dynamic strain indicator to identify and to amplify the electric signals. (3) The metal filament connecting to the helix tip of the pacing lead is excited by the outside vibrator through a controller. The test system had a relative error of 2%. The external exciter and its controller were customly designed.

Computer

Signal conditioner

Controller

Vibration exciter

Computer

Dynamic strain indicator

Metal filame

Fiber

Conduct Heart Lead tip and force sensors

Figure 1. The testing system.

2.3 RESULTS AND ANALYSES We obtained data of the lead/heart interactions under three conditions: normal heart beating, heart beating with external excitation of different frequencies and amplitudes, and under continuous long-time cyclic motions (fatigue tests). 2.3.1 Lead/heart Interaction under Normal Heart Beating The helix tip reached the designated site of right ventricle apex through veins and was screwed into the myocardium. When it began to interact with the heart under the normal heart beating, we recorded a series of push-pull forces which indicated the lead/heart interaction in temporal series. We could investigate the response and adaptation of the heart to the outer stimulation (the implanting electrode helix tip).

294

Figure 2. Push-pull force wave of the lead tip in 0, and 12 minutes after implantation at right ventricle apex

Figure 2 was a recorded wave pattern typical of the push-pull force under normal heart beating. It recorded the forces every 4 minutes after the helix tip had just been screwed in. Several features are observed. (1) The force oscillated with the heart beat at a frequency of 2.7Hz. (2) The frequency and the amplitude of the push-pull force wave were nearly invariant immediately after the helix tip had been screwed into heart muscle. (3) There was a carrier wave with a frequency of about 1/4 Hz which coincided with the breath frequency. These phenomena indicated that: (1) Heart beating was the main cause of the actuating forces. They were induced by the heart motion. The motions of lungs which pressed the heart periodically modulated the force amplitudes. (2) The force-time curves become stable quickly after the helix tip had been screwed in. It took a very short time (within seconds) for the heart to adjust itself to adapt the implanted lead. The canine heart had an excellent ability of self-adapting. The forces under normal heart beating would not damage the heart muscles.

295

Security Estimation 1400

electroede-1 electroede-2 electroede-3 electroede-4 electroede-5

1316

1300 1200

Pull/Push Force(g)

1100 1000 900

825

800

722

700 600 500 400 300

240 85

100

206

184

200

36

24

18

0

3.44

15.49

5.08

11.51

30.32

Safety Coefficient (PP/PN) Figure 3. Attaching-security estimate of five different types of helix tips at right ventricle apex

We compared the minimum pull-out forces value and the peak value of force under normal heart beating of the same type of helix tip. The ratio of these two force values was defined as the safety coefficient for one type of helix tip at the right ventricle apex (Figure 3). This coefficient represents the firmness to attach the helix tip to the heart muscle. The bigger it is, the safer the helix tip and heart muscle are attached. All the coefficients were bigger than 3, and that puts a comfortable safety margin to prevent the helix/heart interface from disengagement under normal heart beating. 2.3.2 Lead/heart Interaction under Heart Beating with External Excitation After recorded the lead/heart interaction under normal heart beating, we applied a controlled cyclic motion on the helix tip through the vein by a thin metal filament which connected the helix tip and the outside exciter. We recorded two groups of data for the push-pull forces: (1) the helix tip was continuously vibrating at a fixed frequency and amplitude for a long time (fatigue test); and (2) the helix tip was subjected to a series of vibrations of various frequencies and amplitudes. We examined the influences of the cyclic loading on the changes of push-pull forces (Figures 4 to 6) and those of frequencies of the exciter (Figure 7).

Figure 4 demonstrates the changes of push-pull forces with time under outside cyclic excitation. When the outside exciter was applied to the helix tip, the force was almost thrice of that under normal heart beating condition. The amplitudes of the force reduce as the cyclic excitation continues and gradually become stabled.

296

The stabled force amplitude was still twice of that under the normal heart beating. The self-adjustment of the in vivo heart attributes to the initial stiff resistance to the outside excitations. The cardiac muscle has the “consciousness” to protect itself from serious damages. Later on, the fatigue takes over and the damage along the heart/pacer interface occurs, leading to moderate relaxation of the push-pull forces.

297

Figure 4. A series of push-pull forces under outside cyclic excitations at frequency of 10Hz and amplitude of 3mm. From the top to the bottom, force variations under normal heart beating and those after 2400, 4800, 7200, 12000, 14400 and 16800 cycles are plotted

The force waves in Figure 4 in the time domain was converted to the frequency domain to get the power spectrum (Figure 5). The normal heart beating at a frequency of 2.54Hz was observed as the main cause of the lead/heart interaction and the outside excitation at the frequency of 10Hz influenced the interaction obviously. As the time went, the energy of interaction caused by both heart beating and outside excitation decreased, while the ratio of the outside excitation to the normal heart beating increased. These data suggest that the actuating heart has the ability of self-adjusting to protect itself. After the termination of the outside excitation, we measured the push-pull forces under normal heart beating for a long-time period. The heart could recover immediately and the forces stabilized quickly. The peak to peak value of the forces obviously decreased and the waveform became less regular (Figure 6). These data supported the conjecture that the heart muscles would be fatigued by the lead/heart interaction under outside excitation.

298

Power Spectra

Spectra Analysis of Fatigue Test

35000 32500 30000 27500 25000 22500 20000 17500 15000 12500 10000 7500 5000 2500 0

2400 cycles 4800 cycles 7200 cycles 9600 cycles 12000 cycles 14400 cycles 16800 cycles Natrual heart beat

0

2

4

6

8

10

12

Frequency(Hz)

2.4

2.7

9.6

9.9

Figure 5. Power spectrum of the push-pull forces under cyclic excitation at the frequency of 10Hz

14

299

Figure 6. The left graphs plot the push-pull forces under normal heart beating before outside excitation and the right graphs plot those under normal heart beating after the termination of outside excitation Electrode 2 No.168 canine

Peak to peak value of pull/push force(g)

750 700 650 600 550 500 450 2

3

4

5

6

7

8

9

10

11

Excitation frequency (Hz)

Figure 7. The amplitude of push-pull forces versus the exciting frequency

2.3.3 Chronic Test Chronic tests refer to the case when the screwed-in helices were left in the myocardium untested for a time period of two months. Pathological changes were developed in myocardium near the fixed electrode. At the right ventricle apex and Hisbundle, the heart tissues underwent physiological changes and encysted

300

completely the helix tip and its lead. There were tissues binding on the lead when the electrode was pulled out. At these two sites, the preliminary testing data even suggested safer attachment between heart muscle and helix tip than other sites in a long working time (Figure 8).

Pull force comparison under acute state and chronic state tests Electrode 2-right ventricle apex Electrode 5-HI S bundle

600

Pull Force (g)

500 400 300 200 100 0

PC/PA=3.31

PC/PA=9.58

Figure 8. Comparison of the pull-out forces in acute and chronic tests. PC was the force value obtained in chronic test and PA was that obtained in acute test

3

In-vitro Heart Muscle Specimen Tests

After the in-vivo tests, the hearts were sliced into specimens. The tests of specimens of regional muscle were completed and the constitutive relations of passive stress and strain were established. 3.1 TESTING EQUIPMENT Soft tissue test device (Figure 9) was designed in the specimen tests. It is composed of four parts: temperature controlling system, loading system, imagery system and force measuring system. The device is capable of carrying out uniaxial tension, biaxial tension and relaxation tests. The strain error of this machine is 2.5P H when the specimen has clear grains. When the specimen has no clear grains, the relative error of the strain measurement was less then 5.6%. The relative error of stress measurement is less than 7%.

301

1.Test specimen 2. Fixation edge 3. Solution tank 4. Step motor 5. Leading screw 6. Force sensor 7. Heating rob 8. Temperature sensor 9. CCD 10. Temperature contoller 11. Step motors driver 12. Stub card 13. A/D card 14. Image card 15. Computer

Figure 9. The soft-tissue test device. The top figure shows all the assembling parts and the system. The bottom figure is the top view for the load system

3.2 RESULTS AND ANALYSES 3.2.1

De-actuating Cycle Tests

Before each tension tests, the specimens were de-actuated (or preconditioned) by 20

302

cycles of load-unload. Figure 10 reveals that the hysteresis loops were formed while preconditioning. The external force did positive work on the specimen. The subsequent loading circles led to the suppression of the hysteresis loops. At the same time, the stress at the same strain level was significantly reduced and the upturn curvature of stress-strain was slightly increasing. Finally the tension curves of different cycles were stabilized, and the heart specimen was de-actuated. This attributes to the declination of the active forces as well as the visco-elastic effect.

Figure 10. The preconditioning loops. The left figure shows the first three cycles and the right figure shows the 10th- 12th cycles. The curves were obtained at right atrium appendix

3.2.2 Tension Tests The stress-elongation ratio curves of the heart muscle were obtained along its fiber axis at a series of strain rates (Figure 11). From Figure11, one could observe that: (1) the strain rate effect, at least in the range tested, was not a significant factor for the mechanical properties of heart muscles; and (2) the stress was slightly bigger at higher strain rates.

Figure 12 shows the stress-elongation ratio curves at different sites of the heart. From this figure, one could reach three conclusions. (1) The mechanical properties of heart muscle at different sites were quite different. (2) The “elastic” modulus of the heart muscle in the ventricle was much larger than that at atrium. Usually the heart muscle at epicardium of the left ventricle had largest stiffness; high ventricular interseptum, ventricular interseptum, right ventricular lateral wall and His-bundle had comparable stiffness; while atrial interseptum, right atrium lateral wall and right atrium appendix had smaller stiffness. (3) The maximum elongations at sites in atrium were much larger than those in ventricle. The former could reach 2.0 and the latter were about 1.4-1.7. The differences of the mechanical properties at different sites of heart muscle are derived from the internal structure of the heart. In the ventricle, the tissues were denser and had more elastic myocardium fibers while in atrium there were more connective collagenous fibers.

303

Figure 11. Stress-elongation ratio curves of different strain rates. The left figure was obtained at right ventricle apex and the right one was obtained at lateral wall of right ventricle

Figure 12. Stress-elongation curves of different test sites in same canine

3.2.3 Relaxation Tests The specimen was tensioned to a certain elongation at a series of strain rates and relaxed for a period of time. The time history of force variation was recorded. Figure 13 plotted typical normalized relaxation curves for heart muscle at different sites and showed the stress response after a step change in the elongation ratio. Several observations can be drawn from our tests. (1) The relaxation was the intrinsic properties of the heart muscle and was influenced by the degree of the freshness. The faster the test was conducted after the removal of the heart, the more pronounced the relaxation effect. (2) The larger the tension rate, the faster the relaxation, as predicted by our modified model. (3) The heart muscle of atrium had

304

deeper relaxation than that of ventricle.

Figure 13. Left: the normalized relaxation curves after different tension rates to the same elongation ratio; right: the stress response curve after a step change in the elongation ratio from 1.249 to 1.204. The data was obtained at right ventricle apex

3.2.4 Rupture Tests The rupture tests were carried out at ten sites of the heart. The heart muscle was sliced to the “I” shape samples. The ends of the sample were much larger than the gauge area to avoid the tearing at the fixation ends. Figures 14 and 15 show the stress-time curves of rupture tests at various sites. L0 was the original length of the gauge area of the sample.

Figure 14. The rupture curves of the right ventricle apex (left) and the lateral wall of right ventricle (right)

Several observations could be drawn from these data. (1) The rupture forces of the heart muscle vary at different testing sites, and a majority of them are within the range from 100kPa to 300KPa. (2) Rupture in most specimens did not occur as brittle failure, and the stress did not drop steeply. The muscle was basically torn up, especially at the right ventricle apex and the outflow tract of right ventricle. The parallel muscle fiber bundles were separated and ruptured gradually as the stress increased.

305 180

Stress (Kpa)

100

outflow tract of right ventricle

right atrium appendix

160

L0=10.282 mm

L0=6.418 mm tension rate: 0.6mm/s

140

tension rate: 0.6mm/s

Stress (Kpa)

120

80

60

120 100 80 60

40

40 20

20 0

0

0

4000 8000 12000 16000 20000 24000 28000 32000 36000 40000

10000

20000

30000

40000

50000

60000

Time (ms)

Time (ms)

Figure 15. Rupture curves of the outflow tract of right ventricle (left) and the right atrium appendix (right)

3.2.5 Modified HMT Constitutive Model of Heart Muscle In HMT model [4], the stress is given by:

T Ta (O , t , Ca 2 )  TP (O )

(1)

where Ta is the active stress, and Tp is the elastic passive stress given by the pole-zero model. We modified the HMT model by introducing the item of viscoelasticity [10]. The passive stress Tp is given instead by t

T p (t ) TPe >O (t )@  ³ T pe >O (W )@ o

wG (t  W ) dW wW

(2)

where T pe is the elastic response given by pole-zero model [4,11] and kernel G (t ) is the normalized relaxation function given by:

ce D ¦ G t ¦c

 it

i

(3)

i

We take only the first three terms of G (t ) as:

G t

1  c1e D1t  c2 e D 2t 1  c1  c2

(4)

That allows us to predict the viscoelastic response of the passive force. The parameters of Eq. (2) were determined by the tension and the relaxation tests of regional cardiac muscle specimen. In our modified model, the passive force depends on the time. The model predicts that: (1) if a specimen is stretched to a fixed elongation, the peak stress will be larger at a larger strain rate; and (2) the stress at different strain rates will converge after a long relaxation time. These predictions were verified by our tests (Figure 16).

306

Figure 16. The tensile relaxation curves at different tension rates. These curves were obtained at canine right ventricle apex

We fitted the uniaxial tests curves by the modified HMT model and determined its parameters (Figure 17).

Figure 17. The modified HMT model fitts for the test data of uniaxial tension and relaxation

4

Conclusions

We accomplished the in-vivo lead/heart interaction tests under normal heart beating and heart beating with external excitation. The in-vitro specimen tests including tension, relaxation and rupture tests were also carried out. The following conclusions are reached: (1) The interaction was mainly caused by the heart beating and the lung breath. (2) The heart has an excellent ability of self-adapting. The force-time curves become stable in 10 seconds after the helix tip had been screwed in. The forces under normal heart beating would not damage the heart muscles. (3) The in vivo heart had an ability of self-adjusting to protect itself from external excitation. The push-pull forces of lead/heart interaction quickly increased with

307

(4) (5) (6) (7)

(8)

the loading cycles and then decreased slowly to a stable value about twice of the normal push-pull forces. The pull-out forces were much larger than the transmitted forces under normal heart beating. Stronger heart/pacer adherence can be achieved in chronic tests. The actuating effect of living heart muscles diminishes as the load cycles. The mechanical properties of heart muscle at different sites were quite different. The stiffness of the heart muscle in the ventricle was much larger than that at atrium. The modified constitutive relations for passive force based on pole-zero elastic model were verified by tests. The parameters of the model were determined by correlating to our test curves.

Acknowledgement The authors thank Dr. Shu Zhang and PhD candidate Guodong Niu in Beijing Fuwai Angiocardiopathy Hospital for their clinical supports and animal preparing. The authors also acknowledge the financial support of the Medtronic Cardiac Rhythm Management, USA.

References 1㧚 K.A. Ellenbongen, G.N. Kay, B.L. Willkoff, 1995, Preface in: K.A. Ellenbongen, G.N. Kay, B.L. Willkoff (Eds), Clinical Cardiac Pacing, W.B. Saunders, Philadelphia, PA, P. Xvi 2㧚 Y. Zhao, 2001, Challenges in Modeling Implantable Medical Devices, In: Proceedings of ABAQUS User Conference, 641-654. 3㧚 D. Nobel, 1995, The development of mathematical models of the heart, Chaos, Solitons and Fractals, 5, 321-333. 4㧚 P.J. Hunter, A.D. McCullonch, H.E.D.J.ter Keurs, et al, 1998, Modeling the mechanical properties of cardiac muscle, Progress in biophysics and molecular biology, 69, 289-331. 5㧚 W.W. Baxter, A.D. McCulloch, 2001, In vivo finite element model-based image analysis of pacemaker lead mechanics, Medical Image Analysis, 5, 255-270. 6㧚 N.P. Smith, P.J.Mulquiney, M.P. Nash, C.P. Bradley, D.P. Nickerson, P.J. Hunter. 2002, Mathematical modeling of the heart: cell to organ, Chaos, Solitons and Fractals, 13, 1613-1621. 7㧚 Y. Zhao, 2003, The use of nonlinear FEA modeling to determine the in vivo stresses in cardiac pacing lead coils for fatigue evaluation”, 2003 Summer Bioengineering Conference, June 25-29, Sonesta Beach Resort in Key Biscayne, Florida. 8㧚 J.Y. Zhang, S.J.Chen, D.J.Cooke, et al, 2003, Analysis of intracardiac lead bending stress and motion characteristics of RVA versus RVOS pacing leads using an innovative 3-D reconstruction technique, 2003 Summer Bioengineering Conference, June 25-29, Sonesta Beach Resort in Key Biscayne, Florida. 9㧚 E. Homma, Y. Tatsumi et al, 2003, Long-term in vivo testing of the totally implantable artificial heart system with newly energy convert at National Cardiovascular Center”, 2003 Summer Bioengineering Conference, June 25-29, Sonesta Beach Resort in Key Biscayne, Florida. 10㧚 Y.C. Fung, 1993, Biomechanics: Mechanical Properties of Living Tissues, Springer Verlag, , New York. 11㧚 M.P. Nash, P.J. Hunter, 2000, Computational mechanics of the heart, Journal of Elasticity, 61, 113-141.

Author Index H. Balke 3-14 H. Berger 121-130 N. Bohn 121-130 P. Bürmann 3-14 C.Q. Chen 49-56 Y. Chen 164-174 Y.H. Chen 107-111 273-290 D. Chopp D.N. Fang 15-21 F. Fang 32-39 P. Feng 207-216 X.Q. Feng 112-120 U. Gabbert 121-130 C.F. Gao 87-95 D. Gross 22-31 B. Gu 112-120 L.H. Han 227-236 S. Hao 273-290 Q. Jiang 68-84 F. Jin 175-184, 185-193 S. Kari 121-130 3-14 H. Kessler K. Kishimoto 175-184, 185-193 Z.B. Kuang 68-84 131-143 M. Kuna S. Lentzen 154-163 F.X. Li 15-21 X.E. Liu 239-252 Q.H. Lu 261-270 T.J. Lu 227-236 H.S. Luo 32-39 D.C. Lupascu 40-48 N.K. Ma 291-307 Y.W. Mai 217-226 Y. Mochimaru 175-184 B. Moran 273-290 R. Mueller 22-31 N. Noda 87-95 Z.C. Ou 107-111 Z. Qian 185-193 144-153 Q.H. Qin R.K.N.D. Rajapakse 164-174

G.X. Ren R. Rodriguez J. Rödel R. Schmidt Y.P. Shen Q. P. Sun M. Urago Q. Wan Y.P. Wan B. Wang Z. Wang K. Watanabe I. Westram C. H. Woo W.Y. Yan F. Yang W. Yang S.W. Yu F.C. Zhang H. Zhang T.Y. Zhang X.M. Zhang Y. Zhao Y. Zheng X.J. Zheng Z. Zhong L.C. Zhu W.B. Zhu

261-270 121-130 40-48 154-163 49-56 207-217 175-184 49-56 253-260 57-67 185-193 194-204 40-48 57-67 217-226 291-307 32-39, 291-307 112-120 32-39 261-270 96-106 291-307 291-307 57-67 239-252 253-260 261-270 261-270

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P. Pedersen and M.P. endsøe (eds.) IUTAM Symposium on Synthesis in Bio Solid Mechanics. Proceedings of the T M ymposium held in openhagen, Denmark. 1 0 7 23 5615 2 .K. grawal and . . Fabien Optimization of Dynamic Systems. 1 0 7 23 5681 0 . arpinteri Nonlinear Crack Models for Nonmetallic Materials. 1 0 7 23 5750 7 F. Pfeifer (ed.) IUTAM Symposium on Unilateral Multibody Contacts. Proceedings of the T M ymposium held in Munich, ermany. 1 0 7 23 6030 3 . avendelis and M. Zakrzhevsky (eds.) IUTAM/IFToMM Symposium on Synthesis of Nonlinear Dynamical Systems. Proceedings of the T M/ FToMM ymposium held in Riga, atvia. 2000 0 7 23 6106 7 . P. Merlet Parallel Robots. 2000 0 7 23 6308 6 .T. Pindera Techniques of Tomographic Isodyne Stress Analysis. 2000 0 7 23 6388 4 . . Maugin, R. Drouot and F. idoroff (eds.) Continuum Thermomechanics. The rt and cience of Modelling Material ehaviour. 2000 0 7 23 6407 4 . Van Dao and . . Kreuzer (eds.) IUTAM Symposium on Recent Developments in Non-linear Oscillations of Mechanical Systems. 2000 0 7 23 6470 8 .D. kbarov and . . uz Mechanics of Curved Composites. 2000 0 7 23 6477 5 M. . Rubin Cosserat Theories: Shells, Rods and Points. 2000 0 7 23 648 . Pellegrino and .D. uest (eds.) IUTAM-IASS Symposium on Deployable Structures: Theory and Applications. Proceedings of the T M ymposium held in ambridge, .K., 6– eptember 1 8. 2000 0 7 23 6516 .D. Rosato and D. . lackmore (eds.) IUTAM Symposium on Segregation in Granular Flows. Proceedings of the T M ymposium held in ape May, , . . ., une 5–10, 1 . 2000 0 7 23 6547 . agarde (ed.) IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics. Proceedings of the T M ymposium held in Futuroscope, Poitiers, France, ugust 31– eptember 4, 1 8. 2000 0 7 23 6604 2 D. eichert and . Maier (eds.) Inelastic Analysis of Structures under Variable Loads. Theory and ngineering pplications. 2000 0 7 23 6645 T. . huang and . . Rudnicki (eds.) Multiscale Deformation and Fracture in Materials and Structures. The ames R. Rice 60th nniversary Volume. 2001 0 7 23 6718 . arayanan and R. . yengar (eds.) IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. Proceedings of the T M ymposium held in Madras, hennai, ndia, 4–8 anuary 1 0 7 23 6733 2 . Murakami and . hno (eds.) IUTAM Symposium on Creep in Structures. Proceedings of the T M ymposium held in agoya, apan, 3 7 pril 2000. 2001 0 7 23 6737 5 . hlers (ed.) IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials. Proceedings of the T M ymposium held at the niversity of tuttgart, ermany, eptember 5 10, 1 . 2001 0 7 23 6766 D. Durban, D. ivoli and . . immonds (eds.) Advances in the Mechanis of Plates and Shells The Avinoam Libai Anniversary Volume. 2001 0 7 23 6785 5 . abbert and . . Tzou (eds.) IUTAM Symposium on Smart Structures and Structonic Systems. Proceedings of the T M ymposium held in Magdeburg, ermany, 26–2 eptember 2000. 2001 0 7 23 6 68 8

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. vanov, V. heshkov and M. atova Polymer Composite Materials – Interface Phenomena & Processes. 2001 0 7 23 7008 2 R. . McPhedran, . . otten and . . icorovici (eds.) IUTAM Symposium on Mechanical and Electromagnetic Waves in Structured Media. Proceedings of the T M ymposium held in ydney, , ustralia, 18 22 anuari 1 . 2001 0 7 23 7038 4 D. . otiropoulos (ed.) IUTAM Symposium on Mechanical Waves for Composite Structures Characterization. Proceedings of the T M ymposium held in hania, rete, reece, une 14 17, 2000. 2001 0 7 23 7164 V.M. lexandrov and D. . Pozharskii Three-Dimensional Contact Problems. 2001 0 7 23 7165 8 .P. Dempsey and . . hen (eds.) IUTAM Symposium on Scaling Laws in Ice Mechanics and Ice Dynamics. Proceedings of the T M ymposium held in Fairbanks, laska, . . ., 13 16 une 2000. 2001 1 4020 0171 1 . Kirsch Design-Oriented Analysis of Structures. nified pproach. 2002 1 4020 0443 5 . Preumont Vibration Control of Active Structures. n ntroduction (2nd dition). 2002 1 4020 04 6 6 . . Karihaloo (ed.) IUTAM Symposium on Analytical and Computational Fracture Mechanics of Non-Homogeneous Materials. Proceedings of the T M ymposium held in ardiff, .K., 18 22 une 2001. 2002 1 4020 0510 5 .M. an and . enaroya Nonlinear and Stochastic Dynamics of Compliant Offshore Structures. 2002 1 4020 0573 3 .M. inkov Boundary Integral Equations in Elasticity Theory. 2002 1 4020 0574 1 .P. ebedev, . . Vorovich and .M. . ladwell Functional Analysis. pplications in Me chanics and nverse Problems (2nd dition). 2002 1 4020 0667 5 Pb 1 4020 0756 6 .P. un (ed.) IUTAM Symposium on Mechanics of Martensitic Phase Transformation in Solids. Proceedings of the T M ymposium held in ong Kong, hina, 11 15 une 2001. 2002 1 4020 0741 8 M. . Mun al (ed.) IUTAM Symposium on Designing for Quietness. Proceedings of the T M ymposium held in angkok, ndia, 12 14 December 2000. 2002 1 4020 0765 5 . . . Martins and M.D.P. Monteiro Mar ues (eds.) Contact Mechanics. Proceedings of the 3rd ontact Mechanics nternational ymposium, Praia da onsola¸ca˜ o, Peniche, Portugal, 17 21 une 2001. 2002 1 4020 0811 2 .R. Drew and . Pellegrino (eds.) New Approaches to Structural Mechanics, Shells and Biological Structures. 2002 1 4020 0862 7 .R. Vinson and R. . ierakowski The Behavior of Structures Composed of Composite Materials. econd dition. 2002 1 4020 0 04 6 ot yet published. .R. arber Elasticity. econd dition. 2002 b 1 4020 0 64 Pb 1 4020 0 66 6 . Miehe (ed.) IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Proceedings of the T M ymposium held in tuttgart, ermany, 20 24 ugust 2001. 2003 1 4020 1170

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. P. Merlet Parallel Robots. 2006 1 4020 4132 2 . . . Meier and K.R. reenivasan (eds.) IUTAM Symposium on One Hundred Years of Boundary Layer Research. Proceedings of the T M ymposium held at D R o¨ ttingen, ermany, ugust 12–14, 2004. 2006 1 4020 414 7 . lbrich and . u¨ nthner (eds.) IUTAM Symposium on Vibration Control of Nonlinear Mechanisms and Structures. 2006 1 4020 4160 8 . ibrescu and . ong Thin-Walled Composite Beams. Theory and pplication. 2006 1 4020 3457 1 . en Dor, . Dubinsky and T. lperin Applied High-Speed Plate Penetration Dynamics. 2006 1 4020 3452 0 . Markenscoff and . upta (eds.) Collected Works of J. D. Eshelby. Mechanics and Defects and eterogeneities. 2006 1 4020 4416 R. . nidle and .P. vans (eds.) IUTAM Symposium on Elastohydrodynamics and Microelastohydrodynamics. Proceedings of the T M ymposium held in ardiff, K, 1–3 eptember, 2004. 2006 1 4020 4532 8 T. adowski (ed.) IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials. Proceedings of the T M ymposium held in Kazimierz Dolny, Poland, 23–27 May 2005. 2006 1 4020 4565 4 . Preumont Mechatronics. Dynamics of lectromechanical and Piezoelectric ystems. 2006 1 4020 46 5 2 M.P. endsoe, . lhoff and . igmund (eds.) IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. tatus and Perspectives. 2006 1 4020 472 0

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