DOCSLIB.ORG

Modeling and Characterization of the Elastic Behavior Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modeling and Characterization of the Elastic Behavior Of MODELING AND CHARACTERIZATION OF THE ELASTIC BEHAVIOR OF INTERFACES IN NANOSTRUCTURED MATERIALS: FROM AN ATOMISTIC DESCRIPTION TO A CONTINUUM APPROACH A Dissertation Presented to The Academic Faculty by Rémi Dingreville In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Mechanical Engineering Georgia Institute of Technology December 2007 MODELING AND CHARACTERIZATION OF THE ELASTIC BEHAVIOR OF INTERFACES IN NANOSTRUCTURED MATERIALS: FROM AN ATOMISTIC DESCRIPTION TO A CONTINUUM APPROACH Approved by: Dr. Jianmin Qu, Advisor Dr. Mo Li GWW School of Mechanical Engineering School of Materials Science & Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. David L. McDowell Dr. Elisa Riedo GWW School of Mechanical Engineering School of Physics Georgia Institute of Technology Georgia Institute of Technology Dr. Min Zhou GWW School of Mechanical Engineering Georgia Institute of Technology Date Approved: July 24 th , 2007 To my wife Stephani, my family and my friends. ACKNOWLEDGEMENTS This work is the accomplishment of both an individual and collective effort to find a path in the world of “nano ” and the meanders of interface theory. If one had to find the roots of this journey, I would probably cite two particular events. The first one would be the reading of an essay by Gleiter on nanocrystalline materials in Progress in Materials Science during my Masters study in France. I discovered a whole new world made of nanotubes and nanoribbons, a world ruled by abnormal behaviors. It seemed to me there were many interesting things to discover and issues to tackle. The second event occurred a few months earlier during an internship at Georgia Tech in the summer of 2000. That was the meeting of Professor Jianmin Qu who gave me the opportunity later on to join his research group for this work. The fact that, many years later, this dissertation reflects the achievements of our collaborative work on nanostructured materials, constitute for me a great honor and pride. But I also see this as a sign that, fundamental research is first and foremost a question of time and interpersonal relationships. I owe therefore a great deal of appreciation to Jianmin Qu who gave me the freedom to direct this research project according to my own observations and ideas, developing in me the ability to think creatively and independently. I also would like to thank him for his guidance, advice and patience especially in moments when disappointment might have overcome my enthusiasm. My thanks and gratitude then go Dr. Dave McDowell, Dr. Min Zhou, Dr. Mo Li and Dr. Elisa Riedo who serve as the members of my Ph.D. thesis reading committee and iv have the task of reviewing this work. My interaction with some of them over the past several years have undoubtfully added to the quality and relevance of this research. There are a number of people at Georgia Tech who also have contributed greatly to my experiences over the course of my doctorate. I would like to thank Douglas Spearot for introducing me to the “joys” of Molecular Dynamics and the technical explanations on the MD software WARP and PARADYN. Thank you also to Ambarish Kulkarni for introducing me to the “joys” of ionic systems and ZnO nanobelts. Even though we unsuccessfully spent many hours working on these systems, our collaboration greatly contributed to the quality of this work. I thank him for his enthusiasm, his availability and cheerful spirit. I would also like to stress the essential role of the other members of this research group who directly or indirectly contributed to this work by their availability, sense of humor, sense of welcome or own skills. Thanks to Marie Blanche Cornil, Jason Mayeur, Narashiman Swaminathan, Min Pei, Janine Johnson, and many others. Thank you also to Cecilia Jones the administrative assistant of this group. On a lighter note, I would like to thank my friends for their cheerful support over the years, may they be oversea or in the building next door on campus. Therefore thank you to the “manioules” Baptiste Anti, Sébastien Doulet, Marcel Raybaud, Laurent Licht, Shane Elipot, Sylvain Rigal; my high school friends Joffrey and Stéphanie Desrousseaux, Geoffrey Denans, Guillaume Poignet, Frédéric Olech; and the “French GT mafia” Matthieu Masquelet, Diane de Zelicourt, Franklin Genin, Léon Phan, Gilles Eggenspieler, Frédéric Bouygue, Rémy Prosper Magnier, Kaoruko Magnier Watanabe, Igor “Butchy” Vilfan and all the others I forgot. v Finally, I do not forget my parents, Philippe and Annie, my grand-parents, Jacques and Muguette, my families, the Dingrevilles, the Stephens and the Swanns, and my wife Stephani whose patience and support were tested many times. Thank you to all of you from the bottom of my heart. vi TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iv LIST OF TABLES xi LIST OF FIGURES xiii NOMENCLATURE xvii SUMMARY xxvi 1 INTRODUCTION 1 1.1 Motivations 1 1.2 Dissertation Objectives and Goals 7 1.3 Dissertation Structure 13 2 ATOMISTIC SIMULATIONS 19 2.1 Introduction 19 2.2 Nuts and Bolts of Atomistic Simulation 22 2.2.1 Periodic Boundary Conditions 22 2.2.2 Control of the System: Role of Statistical Ensembles 24 2.3 Molecular Statics 27 2.4 Interatomic Potentials 30 2.4.1 Interatomic Potentials for F.C.C metals: the Embedded Atom Method (EAM) 31 3 ELASTIC DESCRIPTION OF BULK PHASES 37 3.1 Introduction 37 3.2 Definition of the Elastic Constants 39 3.2.1 Elastic Strain and Complementary Energy Representations 39 vii 3.2.2 Second and Third Order Elastic Constants of Cubic Crystals 41 3.2.3 Second and Third Order Elastic Constants of Isotropic Materials 43 3.3 A Method of Computing the Elastic Constants 44 3.4 Elastic Constants for Single Crystals 48 3.5 Elastic Constants of Isotropic Aggregates 51 3.5.1 Voigt and Reuss-type Estimates of the Elastic Constants of an Isotropic Aggregate 51 3.5.2 Semi Consistent Estimates of the Elastic Constants of an Isotropic Aggregate 54 3.6 Summary and Conclusions 56 4 ELASTIC DESCRIPTION OF FREE SURFACES AND INTERFACES 65 4.1 Introduction 66 4.2 Surface Free Energy and Surface Stress 70 4.2.1 Dividing Surface 70 4.2.2 Definition of Interfacial Excess Energy 72 4.2.3 Surface Stress and Surface Strain 77 4.2.4 Surface Elasticity, Generalized Shuttleworth Relation 77 4.2.5 Isotropic Bimaterials 79 4.3 Semi-analytical Method to Evaluate Surface Properties 83 4.3.1 Free Surface 84 4.3.2 Bicrystal Interface 88 4.3.2.1 Atomic Level Mapping 89 4.3.2.2 Total Energy of the Atomic Assembly 90 4.3.2.3 Atomic Level Stress 91 4.3.2.4 Finding the Internal Relaxations 92 4.3.2.5 Interface Elastic Properties 93 viii 4.4 Surface Elastic Properties of Cu, Ni, Ag and Pd Free Surfaces 95 4.4.1 Computational Framework and Results 95 4.4.2 Surface Elastic Properties of Transition Metals 98 4.4.3 Surface Relaxation 101 4.5 Surface Elastic Properties for Grain Boundaries in Cu Bicrystals 103 4.5.1 Computational Framework and Results 103 4.5.2 Atomic Level Moduli for Grain Boundaries 105 4.5.3 Interface Internal Relaxation 108 4.6 Summary and Conclusions 111 5 FROM AN ATOMISTIC DESCRIPTION TO A CONTINUUM FRAMEWORK: SIZE-DEPENDENT ELASTICITY 138 5.1 Introduction 138 5.2 Effective Modulus of Nanoparticle 140 5.2.1 Special Cases 146 5.2.1.1 Thin Films 146 5.2.1.2 Thin Wire of Square Cross-section 149 5.2.1.3 Spherical Particles 151 5.2.2 Atomistic Calculation for Computing the Effective Elastic Constants of Nanoparticles 152 5.2.3 Numerical Results for Thin Films, Nanowires and Nanospheres 156 5.2.4 Influence of Non-Linear Elastic Behavior of the Bulk Core and Surface Elasticity 159 5.2.4.1 Parameters Influencing the Size-Dependence of Nanowires161 5.2.4.2 Parameters Influencing the Size-Dependence of Thin Films162 5.3 Eshelby nano-inclusion problem 164 5.3.1 Interphase vs. Interface 164 ix 5.3.2 Mesoscopic Interfacial Conditions 166 5.3.2.1 Kinematic Interfacial Conditions: Displacement Fields Near the Interphase 167 5.3.2.2 Kinetic Interfacial Conditions: Traction Across the Interface 171 5.3.3 Inhomogeneity problem 172 5.4 Summary and Conclusions 176 6 CONCLUSIONS AND RECOMMENDATIONS 196 6.1 Summary of Significant Contributions 196 6.2 Recommendations for Future Work 204 APPENDIX A: TAYLOR EXPANSION OF THE EAM POTENTIAL 207 APPENDIX B: “T” STRESS DECOMPOSTION 210 APPENDIX C: ATOMIC LEVEL STRESS AND INTERNAL RELAXATION 213 APPENDIX D: COORDINATE TRANSFORMATION 221 APPENDIX E: BULK AND SURFACE ELASTICITY TENSORS 223 APPENDIX F: EFFECTIVE ELASTIC PROPERTIES OF NANOPARTICLES: SPECIAL CASES 225 REFERENCES 229 x LIST OF TABLES Page Table 2.1: Elastic constants (in 100 GPa) and surface energy for (111), (100) and (110) surfaces (in J.m -2) from EAM potential validation 36 Table 3.1: Interatomic lattice spacing (in Å) 60 Table 3.2: Second order elastic constants for single crystals (in 100 GPa) 60 Table 3.3: Third order elastic constants for single crystals (in 100 GPa) calculated from analytical method 61 Table 3.4: Elastic constants of single crystals calculated by the strain meshing of the energy (in 100GPa) 62 Table 3.5: Voigt estimates of the SOEC and TOEC for isotropic polycrystalline aggregates (in 100 GPa) 62 Table 3.6: Reuss estimates of the SOEC and TOEC for isotropic polycrystalline aggregates (in 100 GPa) 63 Table 3.7: Hill estimates of the SOEC and TOEC for isotropic polycrystalline aggregates (in 100 GPa) 63 Table 3.8: Self-Consistent estimates of the SOE and TOE constants for isotropic polycrystalline aggregates (in 100 GPa) 64 Table 4.1: Calculated surface elastic properties of low-index surfaces for several FCC metals.
Load more

Recommended publications
  • An Approximate Nodal Is Developed to Calculate the Change of •Laatio Constants Induced by Point Defect* in Hep Metals
    1 - INTBOPUCTIOB the elastic conatants.aa well aa othernmechanieal pro partita of* IC/79/lW irradiated materiale (are vary sensitive to tne oonoantration of i- INTERNAL REPORT (Limited distribution) rradiation produced point defeota.One of the firat eatimatee of thla effect waa done by Dienea [l"J who aiaply averaged over the whole la- International Atomic Energy Agency ttice the locally changed interatoalo bonds due to the pxesense of" and the defect.With tola nodal ha predioted an inoreaee of the alaatle United Nations Educational Scientific and Cultural Organization constant* of about 10* par atonic f of interatitlala in Ou and a da. oreaee of l]t par at. % of vacanolea. INTERNATIONAL CENTRE FOE THEORETICAL PHYSICS Later on,experlaental etudiea by Konlg at al.[2]and wanal lilgar* very large decrease a of about 5O}( per at.)t of Vrenlcal dafaota.fha theory waa than iaproved in order to relate the change of a la* tic con a tan t a to the defect lnduoed change of force oonatanta and tno equivalent aethoda were devalopedi the energy-»athod of ludwig [4] CHANGE OF ELASTIC CONSTANTS and the t-aatrix method of Slllot et al.C ?]. INDUCED BY FOIMT DEFECTS IN hep CRYSTALS * Iheoretioal eetlnates for oublo oryetala have been oarrie* out by Ludwig[4] for the caae of vacancleafby Piatorlueld for intarati- Carlos Tome •• tiala and by Sederloaa et al.t?] for duabell interotitiala. International Centre for Theoretical Physics, Trieste, Italy. Re thaoretloal work baa been done ao far for hexagonal cryatmlaj and the experimental neaeurentanta (available only for Kg) ax* eona- ABSTRACT what crude t8,9i,ayan though in the last few.
    [Show full text]
  • Gravitation in the Surface Tension Model of Spacetime
    IARD 2018 IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1239 (2019) 012010 doi:10.1088/1742-6596/1239/1/012010 Gravitation in the surface tension model of spacetime H A Perko1 1Office 14, 140 E. 4th Street, Loveland, CO, USA 80537 E-mail: [email protected] Abstract. A mechanical model of spacetime was introduced at a prior conference for describing perturbations of stress, strain, and displacement within a spacetime exhibiting surface tension. In the prior work, equations governing spacetime dynamics described by the model show some similarities to fundamental equations of quantum mechanics. Similarities were identified between the model and equations of Klein-Gordon, Schrödinger, Heisenberg, and Weyl. The introduction did not explain how gravitation arises within the model. In this talk, the model will be summarized, corrected, and extended for comparison with general relativity. An anisotropic elastic tensor is proposed as a constitutive relation between stress energy and curvature instead of the traditional Einstein constant. Such a relation permits spatial geometric terms in the mechanical model to resemble quantum mechanics while temporal terms and the overall structure of tensor equations remain consistent with general relativity. This work is in its infancy; next steps are to show how the anisotropic tensor affects cosmological predictions and to further explore if geometry and quantum mechanics can be related in more than just appearance. 1. Introduction The focus of this research is to find a mechanism by which spacetime might curl, warp, or re-configure at small scales to provide a geometrical explanation for quantum mechanics while remaining consistent with gravity and general relativity.
    [Show full text]
  • Arxiv:1704.01012V1 [Cond-Mat.Mtrl-Sci] 1 Apr 2017 Is the Second Most Important Material After Silicon
    Symmetry and Piezoelectricity: Evaluation of α-Quartz coefficients C. Tannous Laboratoire des Sciences et Techniques de l'Information, de la Communication et de la Connaissance, UMR-6285 CNRS, Brest Cedex3, FRANCE Piezoelectric coefficients of α-Quartz are derived from symmetry arguments based on Neumann's Principle with three different methods: Fumi, Landau-Lifshitz and Royer-Dieulesaint. While Fumi method is tedious and Landau-Lifshitz requires additional physical principles to evaluate the piezo- electric coefficients, Royer-Dieulesaint is the most elegant and most efficient of the three techniques. PACS numbers: 77.65.-j, 77.65.Bn, 77.84.-s Keywords: Piezoelectricity, piezoelectric constants, piezoelectric materials I. INTRODUCTION AND MOTIVATION Physics students are exposed to various types of symmetry [1] and conservation laws in Graduate/Undergraduate Mechanics and Electromagnetism with Lorentz transformation and Gauge symmetries, in Graduate/Undergraduate Quantum Mechanics during the study of Atoms and Molecules. In undergraduate courses such as Special Relativity, Lorentz Transformation is used to unify symmetries between Mechanics and Electromagnetism. In Graduate High Energy Physics, the CPT theorem where C denotes charge conjugation (Q ! −Q), P is parity (r ! −r) and T is time reversal (t ! −t) as well as Gauge symmetry (Ai ! Ai + @iχ) provide an important insight into the role of symmetry in the building blocks of matter and unification of fundamental forces and interaction between particles. Graduate/undergraduate Solid State Physics provide a direct illustration of how Crystal Symmetry plays a fun- damental role in the determination of physical constants and transport coefficients as well as conservation and sim- plification of physical laws. The relation between symmetry and dispersion relations through Kramers theorem (T symmetry) is another example of the power of symmetry in Solid State physics.
    [Show full text]
  • A Manifold Learning Approach to Data-Driven Computational Mechanics
    A Manifold Learning Approach to Data-Driven Computational Mechanics Ruben Ibanez, Emmanuelle Abisset-Chavanne, Jose Vicente Aguado, David Gonzalez, Elías Cueto, Francisco Chinesta To cite this version: Ruben Ibanez, Emmanuelle Abisset-Chavanne, Jose Vicente Aguado, David Gonzalez, Elías Cueto, et al.. A Manifold Learning Approach to Data-Driven Computational Mechanics. 13e colloque national en calcul des structures, Université Paris-Saclay, May 2017, Giens, Var, France. hal-01926477 HAL Id: hal-01926477 https://hal.archives-ouvertes.fr/hal-01926477 Submitted on 19 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. CSMA 2017 13ème Colloque National en Calcul des Structures 15-19 Mai 2017, Presqu’île de Giens (Var) A Manifold Learning Approach to Data-Driven Computational Me- chanics R. Ibañez1, E. Abisset-Chavanne1, J.V. Aguado1, D. Gonzalez2, E. Cueto2, F. Chinesta1 1 ICI Institute, Ecole Centrale Nantes, {Ruben.Ibanez-Pinillo,Emmanuelle.Abisset-Chavanne,Jose.Aguado-Lopez,Francisco.Chinesta}@ec-nantes.fr 2 Aragon Institute of Engineering Research, Universidad de Zaragoza, Spain, {gonzal;ecueto}@unizar.es Résumé — Standard simulation in classical mechanics is based on the use of two very different types of equations.
    [Show full text]
  • Irreducible Matrix Resolution of the Elasticity Tensor for Symmetry Systems
    Irreducible matrix resolution of the elasticity tensor for symmetry systems Yakov Itin Inst. Mathematics, Hebrew Univ. of Jerusalem, Givat Ram, Jerusalem 91904, Israel, and Jerusalem College of Technology, Jerusalem 91160, Israel, email: [email protected] December 11, 2018, file ElastMatrix1.tex Abstract In linear elasticity, a fourth order elasticity (stiffness) tensor of 21 independent com- ponents completely describes deformation properties of a material. Due to Voigt, this tensor is conventionally represented by a 6 × 6 symmetric matrix. This classical matrix representation does not conform with the irreducible decomposition of the elasticity tensor. In this paper, we construct two alternative matrix representations. The 3 × 7 matrix representation is in a correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional repre- sentation of the elasticity tensor by three 3 × 3 matrices is suitable for description the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation. Key index words: anisotropic elasticity tensor, irreducible decomposition, matrix repre- sentation. Mathematics Subject Classification: 74B05, 15A69, 15A72. Contents arXiv:1812.03367v1 [physics.class-ph] 8 Dec 2018 1 Introduction2 2 Matrix representations3 2.1 Voigt’s representation . .4 2.2 An alternative representation . .4 1 Y. Itin Irreducible matrix resolution of the elasticity tensor 2 3 Irreducible decomposition5 3.1 GL(3)-decomposition . .5 3.2 SO(3; R)-decomposition . .8 3.2.1 Cauchy part . .8 3.2.2 Non-Cauchy part . 10 3.3 Irreducible decomposition .
    [Show full text]
  • APPLIED ELASTICITY, 2Nd Edition Matrix and Tensor Analysis of Elastic Continua
    APPLIED ELASTICITY, 2nd Edition Matrix and Tensor Analysis of Elastic Continua Talking of education, "People have now a-days" (said he) "got a strange opinion that every thing should be taught by lectures. Now, I cannot see that lectures can do so much good as reading the books from which the lectures are taken. I know nothing that can be best taught by lectures, expect where experiments are to be shewn. You may teach chymestry by lectures. — You might teach making of shoes by lectures.' " James Boswell: Lifeof Samuel Johnson, 1766 [1709-1784] ABOUT THE AUTHOR In 1947 John D Renton was admitted to a Reserved Place (entitling him to free tuition) at King Edward's School in Edgbaston, Birmingham which was then a Grammar School. After six years there, followed by two doing National Service in the RAF, he became an undergraduate in Civil Engineering at Birmingham University, and obtained First Class Honours in 1958. He then became a research student of Dr A H Chilver (now Lord Chilver) working on the stability of space frames at Fitzwilliam House, Cambridge. Part of the research involved writing the first computer program for analysing three-dimensional structures, which was used by the consultants Ove Arup in their design project for the roof of the Sydney Opera House. He won a Research Fellowship at St John's College Cambridge in 1961, from where he moved to Oxford University to take up a teaching post at the Department of Engineering Science in 1963. This was followed by a Tutorial Fellowship to St Catherine's College in 1966.
    [Show full text]
  • Cambridge University Press 978-0-521-64250-7 — Group Theory with Applications in Chemical Physics Patrick Jacobs Index More Information
    Cambridge University Press 978-0-521-64250-7 — Group Theory with Applications in Chemical Physics Patrick Jacobs Index More Information Index Abelian group 2 central extension 336, 337, 367 acoustic mode 393 centralizer 14, 19, 434 active representation 23 centre 19 adiabatic potential 173 character 74, 99 adjoint 54 character system 74 of an operator 102 character tables 76–78, 80, 447 adjoint matrix 418 character vector 259 algebra of turns 228 characteristic equation 420, 441 ambivalent class 435 charge overlap 107 angular momentum 184, 189 charge transfer 178 anharmonicity 160 chemical bond 106 constant of 160 class 5, 19 antibonding orbitals 106, 125 class algebra 439 antiferromagnetic crystal 265 class constants 436 antilinear operator 252 class property 440 antipole 223 Clebsch–Gordan series 209, 277, 385 antisymmetrical direct product 100 closed shell 172 À2 antisymmetrizing operator 141 closo Bn Hn 51 antiunitary operator 252, 267, 405 closure 1, 393 associated Legendre functions 194 co-factor 413 associative 2, 220 coincidence 162 axial groups 82 column matrix 415 axial tensor 283 combination bands 160 axial vector 82 commutation relations (CRs) 131, 187 compatibility relations 362 basic domain 331 complementary IR 303 basis 53, 96 complementary minor 413 of a lattice 308 complementary operators 265 basis functions, construction of 97 complex conjugate 218 bcc see body-centred cubic complex conjugation operator 253 benzene 104, 109, 174 complex number 218 Bethe 80, 150, 151 complex plane 219 bilateral binary (BB) 232 complex
    [Show full text]
  • Continuum Mechanics Using Mathematica® Fundamentals, Methods, and Applications
    Modeling and Simulation in Science, Engineering and Technology Antonio Romano Addolorata Marasco Continuum Mechanics using Mathematica® Fundamentals, Methods, and Applications Second Edition Modeling and Simulation in Science, Engineering and Technology Series Editor Nicola Bellomo Politecnico di Torino Tor ino, Italy Editorial Advisory Board K.J. Bathe P. Koumoutsakos Department of Mechanical Engineering Computational Science & Engineering Massachusetts Institute of Technology Laboratory Cambridge, MA, USA ETH Zürich Zürich, Switzerland M. Chaplain Division of Mathematics H.G. Othmer University of Dundee Department of Mathematics Dundee, Scotland, UK University of Minnesota Minneapolis, MN, USA P. Degond Department of Mathematics, K.R. Rajagopal Imperial College London, Department of Mechanical Engineering London, United Kingdom Texas A&M University College Station, TX, USA A. Deutsch Center for Information Services T.E. Tezduyar and High-Performance Computing Department of Mechanical Engineering & Technische Universität Dresden Materials Science Dresden, Germany Rice University Houston, TX, USA M.A. Herrero Departamento de Matematica Aplicada A. Tosin Universidad Complutense de Madrid Istituto per le Applicazioni del Calcolo Madrid, Spain “M. Picone” Consiglio Nazionale delle Ricerche Roma, Italy More information about this series at http://www.springer.com/series/4960 Antonio Romano • Addolorata Marasco Continuum Mechanics using MathematicaR Fundamentals, Methods, and Applications Second Edition Antonio Romano Addolorata Marasco Department
    [Show full text]
  • Valley Degeneracy in Biaxially Strained Aluminum Arsenide Quantum Wells
    PHYSICAL REVIEW B 84, 125319 (2011) Valley degeneracy in biaxially strained aluminum arsenide quantum wells S. Prabhu-Gaunkar,1 S. Birner,2 S. Dasgupta,2 C. Knaak,2 and M. Grayson1 1Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois 60208, USA 2Walter Schottky Institut, Technische Universitat¨ Munchen,¨ Garching DE-85748, Germany (Received 13 September 2010; revised manuscript received 19 May 2011; published 16 September 2011) This paper describes a complete analytical formalism for calculating electron subband energy and degeneracy in strained multivalley quantum wells grown along any orientation with explicit results for AlAs quantum wells (QWs). In analogy to the spin index, the valley degree of freedom is justified as a pseudospin index due to the vanishing intervalley exchange integral. A standardized coordinate transformation matrix is defined to transform between the conventional-cubic-cell basis and the QW transport basis whereby effective mass tensors, valley vectors, strain matrices, anisotropic strain ratios, piezoelectric fields, and scattering vectors are all defined in their respective bases. The specific cases of (001)-, (110)-, and (111)-oriented aluminum arsenide (AlAs) QWs are examined, as is the unconventional (411) facet, which is of particular importance in AlAs literature. Calculations of electron confinement and strain for the (001), (110), and (411) facets determine the critical well width for crossover from double- to single-valley degeneracy in each system. The biaxial Poisson ratio is calculated for the high-symmetry lower Miller index (001)-, (110)-, and (111)-oriented QWs. An additional shear-strain component arises in the higher Miller index (411)-oriented QWs and we define and solve for a shear-to-biaxial strain ratio.
    [Show full text]
  • Graphical Representation of the Generalized Hooke's Law Izei®Ei H
    TECHNISCHE MECHANIK, Band 21, Heft 2, (2001), 145-158 Manuskripteingang: 30. Januar 2001 Graphical Representation of the Generalized Hooke’s Law T. Böhlke, C. Brüggemann The anisotropic linear elastic behavior of single crystals can be described equivalently by a 4th-0rder elasticity tensor or two functions E(d) and K These functions represent Young’s modulus and a generalized bulk modulus as functions of the tensile direction d in a tension test. In the present paper three— and two—dimensional graphical representations of Young’s modulus and the generalized bulk modulus are given for single crystals belonging to one of the following symmetry groups: monoclinic, rhombic, trigonal, tetragonal, hemagonal, and cubic symmetry. 1 Introduction The generalized Hooke’s law is the geometrical and physical linear relation between stress and strain of anisotropic elastic solids. It is one of the oldest and best known constitutive relations in continuum mechanics. One dimensional formulations have been discussed first in the 17th century. Over a long time the scientific community agreed to differ about the most general form in linear elasticity. Finally Voigt (1850-1919) answered the open question how many constants have to be determined in the isotropic and anisotropic case. He proved by experiments Green’s (1793-1841) hypothesis that in the isotropic case two and in the general anisotropic hyperelastic case 21 constants have to be determined. Based on Hooke’s law structures are analyzed in all branches of mechanical and civil engineering. In most cases the isotropic version of Hooke’s law is used. But in the last decades more and more anisotropies have been taken into account, which are especially important if the component parts are laminated or made from single crystals.
    [Show full text]
  • V “Type” Notation and Voigt Or Mandel Matrices
    T June 6, 2015 9:08 pm A F Tensors R a n n o n D c a B r R e b e c 1 n 2 Vm “type” notation and Voigt or Mandel matrices 3 4 This section defines a special “Vn ” tensor classification notation. Any tensor of type 5 m 6 n n n 7 Vm will be seen to have m components. Furthermore, any tensor of type Vm is also of 8 1 n 9 type VN where N= m . This latter identification helps us to convert indicial formulas 10 11 involving 3u 3 tensor matrices into matrix formulas involving 9u 1 arrays. Similarly, 12 13 indicial formulas involving 3 3 3 third-order tensor matrices may be alternatively cast 14 u u 15 in terms of 3 9 , 9 3 or even 27 1 arrays, depending on what is most convenient for 16 u u u 17 18 actual computations using linear algebra libraries in modern computing environments. 19 20 th st 21 Scalars are often called 0 -order tensors. Vectors are sometimes called 1 -order ten- 22 th n * 23 sors. In general, an n order engineering tensor has 3 components, and we say that 24 n 2 25 these tensors are of type V3 . For example, stress is a second-order tensor, so it has 3 26 27 (nine) components. Stress is symmetric, so these components are not all independent, but 28 29 that doesn’t change the fact that there are still nine components. 30 31 32 When solving a problem for which all tensors have symmetry with respect to some 2D 33 34 35 plane embedded in 3D space, it is conventional to set up the basis so that the third basis 36 37 vector points perpendicular to that plane.
    [Show full text]
  • Module 3 Constitutive Equations
    Module 3 Constitutive Equations Learning Objectives • Understand basic stress-strain response of engineering materials. • Quantify the linear elastic stress-strain response in terms of tensorial quantities and in particular the fourth-order elasticity or stiffness tensor describing Hooke's Law. • Understand the relation between internal material symmetries and macroscopic anisotropy, as well as the implications on the structure of the stiffness tensor. • Quantify the response of anisotropic materials to loadings aligned as well as rotated with respect to the material principal axes with emphasis on orthotropic and transversely- isotropic materials. • Understand the nature of temperature effects as a source of thermal expansion strains. • Quantify the linear elastic stress and strain tensors from experimental strain-gauge measurements. • Quantify the linear elastic stress and strain tensors resulting from special material loading conditions. 3.1 Linear elasticity and Hooke's Law Readings: Reddy 3.4.1 3.4.2 BC 2.6 Consider the stress strain curve σ = f() of a linear elastic material subjected to uni-axial stress loading conditions (Figure 3.1). 45 46 MODULE 3. CONSTITUTIVE EQUATIONS σ ˆ 1 2 ψ = 2 Eǫ E 1 ǫ Figure 3.1: Stress-strain curve for a linear elastic material subject to uni-axial stress σ (Note that this is not uni-axial strain due to Poisson effect) In this expression, E is Young's modulus. Strain Energy Density For a given value of the strain , the strain energy density (per unit volume) = ^(), is defined as the area under the curve. In this case, 1 () = E2 2 We note, that according to this definition, @ ^ σ = = E @ In general, for (possibly non-linear) elastic materials: @ ^ σij = σij() = (3.1) @ij Generalized Hooke's Law Defines the most general linear relation among all the components of the stress and strain tensor σij = Cijklkl (3.2) In this expression: Cijkl are the components of the fourth-order stiffness tensor of material properties or Elastic moduli.
    [Show full text]