Lecture Notes on The Mechanics of Elastic Solids Volume I: A Brief Review of Some Mathematical Preliminaries Version 1.1 Rohan Abeyaratne Quentin Berg Professor of Mechanics Department of Mechanical Engineering MIT Copyright c Rohan Abeyaratne, 1987 All rights reserved. http://web.mit.edu/abeyaratne/lecture notes.html December 2, 2006 2 3 Electronic Publication Rohan Abeyaratne Quentin Berg Professor of Mechanics Department of Mechanical Engineering 77 Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright c by Rohan Abeyaratne, 1987 All rights reserved Abeyaratne, Rohan, 1952- Lecture Notes on The Mechanics of Elastic Solids. Volume I: A Brief Review of Some Math- ematical Preliminaries / Rohan Abeyaratne { 1st Edition { Cambridge, MA: ISBN-13: 978-0-9791865-0-9 ISBN-10: 0-9791865-0-1 QC Please send corrections, suggestions and comments to [email protected] Updated 17 April 2014 4 i Dedicated with admiration and affection to Matt Murphy and the miracle of science, for the gift of renaissance. iii PREFACE The Department of Mechanical Engineering at MIT offers a series of graduate level sub- jects on the Mechanics of Solids and Structures which include: 2.071: Mechanics of Solid Materials, 2.072: Mechanics of Continuous Media, 2.074: Solid Mechanics: Elasticity, 2.073: Solid Mechanics: Plasticity and Inelastic Deformation, 2.075: Advanced Mechanical Behavior of Materials, 2.080: Structural Mechanics, 2.094: Finite Element Analysis of Solids and Fluids, 2.095: Molecular Modeling and Simulation for Mechanics, and 2.099: Computational Mechanics of Materials. Over the years, I have had the opportunity to regularly teach the second and third of these subjects, 2.072 and 2.074 (formerly known as 2.083), and the current three volumes are comprised of the lecture notes I developed for them. The first draft of these notes was produced in 1987 and they have been corrected, refined and expanded on every following occasion that I taught these classes. The material in the current presentation is still meant to be a set of lecture notes, not a text book. It has been organized as follows: Volume I: A Brief Review of Some Mathematical Preliminaries Volume II: Continuum Mechanics Volume III: Elasticity My appreciation for mechanics was nucleated by Professors Douglas Amarasekara and Munidasa Ranaweera of the (then) University of Ceylon, and was subsequently shaped and grew substantially under the influence of Professors James K. Knowles and Eli Sternberg of the California Institute of Technology. I have been most fortunate to have had the opportunity to apprentice under these inspiring and distinctive scholars. I would especially like to acknowledge a great many illuminating and stimulating interactions with my mentor, colleague and friend Jim Knowles, whose influence on me cannot be overstated. I am also indebted to the many MIT students who have given me enormous fulfillment and joy to be part of their education. My understanding of elasticity as well as these notes have also benefitted greatly from many useful conversations with Kaushik Bhattacharya, Janet Blume, Eliot Fried, Morton E. iv Gurtin, Richard D. James, Stelios Kyriakides, David M. Parks, Phoebus Rosakis, Stewart Silling and Nicolas Triantafyllidis, which I gratefully acknowledge. Volume I of these notes provides a collection of essential definitions, results, and illus- trative examples, designed to review those aspects of mathematics that will be encountered in the subsequent volumes. It is most certainly not meant to be a source for learning these topics for the first time. The treatment is concise, selective and limited in scope. For exam- ple, Linear Algebra is a far richer subject than the treatment here, which is limited to real 3-dimensional Euclidean vector spaces. The topics covered in Volumes II and III are largely those one would expect to see covered in such a set of lecture notes. Personal taste has led me to include a few special (but still well-known) topics. Examples of this include sections on the statistical mechanical theory of polymer chains and the lattice theory of crystalline solids in the discussion of constitutive theory in Volume II; and sections on the so-called Eshelby problem and the effective behavior of two-phase materials in Volume III. There are a number of Worked Examples at the end of each chapter which are an essential part of the notes. Many of these examples either provide, more details, or a proof, of a result that had been quoted previously in the text; or it illustrates a general concept; or it establishes a result that will be used subsequently (possibly in a later volume). The content of these notes are entirely classical, in the best sense of the word, and none of the material here is original. I have drawn on a number of sources over the years as I prepared my lectures. I cannot recall every source I have used but certainly they include those listed at the end of each chapter. In a more general sense the broad approach and philosophy taken has been influenced by: Volume I: A Brief Review of Some Mathematical Preliminaries I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall, 1963. J.K. Knowles, Linear Vector Spaces and Cartesian Tensors, Oxford University Press, New York, 1997. Volume II: Continuum Mechanics P. Chadwick, Continuum Mechanics: Concise Theory and Problems, Dover,1999. J.L. Ericksen, Introduction to the Thermodynamics of Solids, Chapman and Hall, 1991. M.E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981. J. K. Knowles and E. Sternberg, (Unpublished) Lecture Notes for AM136: Finite Elas- ticity, California Institute of Technology, Pasadena, CA 1978. v C. Truesdell and W. Noll, The nonlinear field theories of mechanics, in Handb¨uchder Physik, Edited by S. Fl¨ugge, Volume III/3, Springer, 1965. Volume IIII: Elasticity M.E. Gurtin, The linear theory of elasticity, in Mechanics of Solids - Volume II, edited by C. Truesdell, Springer-Verlag, 1984. J. K. Knowles, (Unpublished) Lecture Notes for AM135: Elasticity, California Institute of Technology, Pasadena, CA, 1976. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, 1944. S. P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill, 1987. The following notation will be used consistently in Volume I: Greek letters will denote real numbers; lowercase boldface Latin letters will denote vectors; and uppercase boldface Latin letters will denote linear transformations. Thus, for example, α; β; γ::: will denote scalars (real numbers); a; b; c; ::: will denote vectors; and A; B; C; ::: will denote linear transforma- tions. In particular, \o" will denote the null vector while \0" will denote the null linear transformation. As much as possible this notation will also be used in Volumes II and III though there will be some lapses (for reasons of tradition). vi Contents 1 Matrix Algebra and Indicial Notation 1 1.1 Matrix algebra . .1 1.2 Indicial notation . .5 1.3 Summation convention . .7 1.4 Kronecker delta . .9 1.5 The alternator or permutation symbol . 10 1.6 Worked Examples and Exercises. 11 2 Vectors and Linear Transformations 19 2.1 Vectors . 20 2.1.1 Euclidean point space . 22 2.2 Linear Transformations. 22 2.3 Worked Examples and Exercises. 28 3 Components of Tensors. Cartesian Tensors 45 3.1 Components of a vector in a basis. 45 3.2 Components of a linear transformation in a basis. 47 3.3 Components in two bases. 49 vii viii CONTENTS 3.4 Scalar-valued functions of linear transformations. Determinant, trace, scalar- product and norm. 51 3.5 Cartesian Tensors . 54 3.6 Worked Examples and Exercises. 56 4 Symmetry: Groups of Linear Transformations 73 4.1 Symmetry Transformations: an example in two-dimensions. 74 4.2 Symmetry Transformations: an example in three-dimensions. 75 4.3 Symmetry Transformations: lattices. 78 4.4 Groups of Linear Transformations. 79 4.5 Worked Examples and Exercises. 80 4.6 Invariance. Representation Theorems. 83 4.6.1 Symmetry group of a function. 84 4.6.2 Isotropic scalar-valued functions. Invariants. 86 4.6.3 Isotropic tensor-valued functions. 89 4.6.4 Anisotropic invariance. 91 4.7 Worked Examples and Exercises. 93 5 Calculus of Vector and Tensor Fields 107 5.1 Notation and definitions. 107 5.2 Integral theorems . 109 5.3 Localization . 110 5.4 Worked Examples and Exercises. 111 6 Orthogonal Curvilinear Coordinates 121 6.1 Introductory Remarks . 121 6.2 General Orthogonal Curvilinear Coordinates . 124 CONTENTS ix 6.2.1 Coordinate transformation. Inverse transformation. 124 6.2.2 Metric coefficients, scale moduli. 126 6.2.3 Inverse partial derivatives . 128 6.2.4 Components of @e^i=@x^j in the local basis (e^1; e^2; e^3).......... 129 6.3 Transformation of Basic Tensor Relations . 130 6.3.1 Gradient of a scalar field . 130 6.3.2 Gradient of a vector field . 131 6.3.3 Divergence of a vector field . 132 6.3.4 Laplacian of a scalar field . 132 6.3.5 Curl of a vector field . 132 6.3.6 Divergence of a symmetric 2-tensor field . 133 6.3.7 Differential elements of volume . 134 6.3.8 Differential elements of area . 134 6.4 Some Examples of Orthogonal Curvilinear Coordinate Systems . 135 6.5 Worked Examples and Exercises. 135 7 Calculus of Variations 145 7.1 Introduction. 145 7.2 Brief review of calculus. 148 7.3 The basic idea: necessary conditions for a minimum: δF = 0; δ2F 0. 150 ≥ 7.4 Application of the necessary condition δF = 0 to the basic problem. Euler equation. 152 7.4.1 The basic problem. Euler equation. 152 7.4.2 An example. The Brachistochrone Problem. 154 7.4.3 A Formalism for Deriving the Euler Equation .