Solid Mechanics and Its Applications Petia Dineva Dietmar Gross Ralf Müller Tsviatko Rangelov Dynamic Fracture of Piezoelectric Materials Solution of Time-Harmonic Problems via BIEM Solid Mechanics and Its Applications Volume 212 Series editor G. M. L. Gladwell, Elmira, Canada For furthervolumes: http://www.springer.com/series/6557 Aims and Scope Thefundamentalquestionsarisinginmechanicsare:Why?,How?,andHowmuch? The aim of this series is to provide lucid accounts written by authoritative researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof 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,kinematicsanddynamicsofrigidandelasticbodies:vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and mem- branes; 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 aremonographsdefiningthecurrentstateofthefield;othersareaccessibletofinal year undergraduates; but essentially the emphasis is on readability and clarity. Petia Dineva Dietmar Gross • Ralf Müller Tsviatko Rangelov • Dynamic Fracture of Piezoelectric Materials Solution of Time-Harmonic Problems via BIEM 123 Petia Dineva RalfMüller Department of SolidMechanics Department of Mechanical andProcess Instituteof Mechanics Engineering Bulgarian Academyof Sciences Instituteof AppliedMechanics Sofia TU Kaiserslautern Bulgaria Kaiserslautern Germany Dietmar Gross Division ofSolid Mechanics Tsviatko Rangelov TU Darmstadt Department of Differential Equations Darmstadt and Mathematical Physics Germany Instituteof Mathematics andInformatics Bulgarian Academyof Sciences Sofia Bulgaria ISSN 0925-0042 ISSN 2214-7764 (electronic) ISBN 978-3-319-03960-2 ISBN 978-3-319-03961-9 (eBook) DOI 10.1007/978-3-319-03961-9 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013957124 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The aim of the book is to reveal the potential of the Boundary Integral Equation Methodasanefficientcomputationaltoolfortreatingwavepropagationproblems in homogeneous and smoothly inhomogeneous piezoelectric solids with defects like cracks and holes. Theinterdisciplinarycharacterofthestudyisbasedoncontinuumandfracture mechanics,theoryofwavepropagation,non-destructiveevaluation,computational mechanics and mathematical physics in their pure theoretical and applied sense. Themainresultsandcontributions are thecoupledelectro-mechanicalmodels, thecomputationalmethod,itsvalidationandsimulationsrevealingdifferenteffects useful for the engineering design and practice. Themainideas,mechanicalmodels,computationaltoolsandsimulationresults aredesignedformasterdegreestudents,Ph.D.studentsandresearcheswholiketo specialize in the field of dynamic computational fracture mechanics and its con- nections with wave propagation theory and continuum mechanics. The authors are indebted to all who have contributed to this book. Special thanks go to Tatiana Parkhomenko, MSc, who helped to prepare the final figures. Petia Dineva Dietmar Gross Ralf Müller Tsviatko Rangelov v Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Part I Theoretical Basics 2 Piezoelectric Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Short Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Types of Piezoelectric Materials . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Physical Peculiarities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Field Equations in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.1 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Field Equations in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.1 In-plane Piezoelectric Equations. . . . . . . . . . . . . . . . . . 23 2.5.2 Anti-plane Piezoelectric Equations . . . . . . . . . . . . . . . . 25 2.6 2D Domains with Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.1 Wave Propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.2 Fracture Mechanics Approach . . . . . . . . . . . . . . . . . . . 27 2.6.3 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . 29 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Fundamental Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 State of the Art. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Static Fundamental Solutions . . . . . . . . . . . . . . . . . . . . 34 3.1.2 Dynamic Fundamental Solutions. . . . . . . . . . . . . . . . . . 35 3.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.1 Fourier Transform in R2 . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.2 Radon Transform in R2 . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.3 Fundamental Solution of ODE . . . . . . . . . . . . . . . . . . . 40 3.3 Anti-plane Piezoelectric Case . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.1 Uncoupled Anti-plane Case . . . . . . . . . . . . . . . . . . . . . 42 3.3.2 Coupled Anti-plane Case. . . . . . . . . . . . . . . . . . . . . . . 45 vii viii Contents 3.4 In-plane Piezoelectric Case. . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 Uncoupled In-plane Case. . . . . . . . . . . . . . . . . . . . . . . 47 3.4.2 Coupled In-plane Case. . . . . . . . . . . . . . . . . . . . . . . . . 52 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4 Numerical Realization by BIEM. . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1 Introduction Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Traction BIEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.1 Bounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.2 Unbounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Numerical Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.1 Discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.2 Numerical Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.3 Solution of the Integrals. . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Programme Code, Material Constants . . . . . . . . . . . . . . . . . . . 76 4.4.1 Programme Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4.2 Material Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Part II Homogeneous PEM 5 Steady-State Problems in a Cracked Anisotropic Domain . . . . . . . 81 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Statement of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3.1 Incident Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3.2 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6 2D Wave Scattering by Cracks in a Piezoelectric Plane. . . . . . . . . 93 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.1 Validation Example. . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.2 Parametric Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7 Piezoelectric Cracked Finite Solids Under Time-Harmonic Loading 105 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.2 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Contents ix 7.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.3.1 Validation Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.3.2 Parametric Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8 Dynamic Crack Interaction in Piezoelectric and Anisotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.2 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.3 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.3.1 Piezoelectric Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.3.2 Anisotropic Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9 Different Electric Boundary Conditions . . . . . . . . . . . . . . . . . . . . 133 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.2.1 In-plane Crack Problem. . . . . . . . . . . . . . . . . . . . . . . . 136 9.2.2 Anti-plane Crack Problem . . . . . . . . . . . . . . . . . . . . . . 138 9.3 Numerical Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Part III Functionally Graded PEM 10 In-plane Crack Problems in Functionally Graded Piezoelectric Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 10.2 Formulation of the BVP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 10.3 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 10.3.1 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . 152 10.3.2 Non-hypersingular Traction BIEM . . . . . . . . . . . . . . . . 154 10.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 11 Functionally Graded Piezoelectric Media with a Single Anti-plane Crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 11.2 Formulation of the BVP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 11.3 Inhomogeneity and Fundamental Solution . . . . . . . . . . . . . . . . 168 x Contents 11.4 Numerical Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 11.4.1 Incident SH-Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . 173 11.4.2 Validation Example. . . . . . . . . . . . . . . . . . . . . . . . . . . 174 11.4.3 Parametric Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 12 Multiple Anti-plane Cracks in Quadratically Inhomogeneous Piezoelectric Finite Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 12.2 Statement of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 12.3 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 12.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 13 Anti-plane Cracks in Exponentially Inhomogeneous Finite Piezoelectric Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 13.2 Statement of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 13.3 Non-hypersingular BIEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 13.4 Numerical Solution and Results. . . . . . . . . . . . . . . . . . . . . . . . 204 13.4.1 Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 204 13.4.2 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 14 Exponentially Inhomogeneous Piezoelectric Solid with a Circular Anti-plane Hole. . . . . . . . . . . . . . . . . . . . . . . . . . 215 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 14.2 Statement of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 14.3 Exponential Material Inhomogeneity . . . . . . . . . . . . . . . . . . . . 218 14.3.1 Electro-Mechanical Load. . . . . . . . . . . . . . . . . . . . . . . 219 14.3.2 BIEM Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 14.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 14.4.1 Numerical Solution Procedure . . . . . . . . . . . . . . . . . . . 222 14.4.2 Validation Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 14.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231