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Continuum Mechanics - MIT

Continuum Mechanics Volume II of Lecture Notes on the Mechanics of Solids Rohan Abeyaratne Quentin Berg Professor of Mechanics MIT Department of Mechanical engineering and Director SMART Center Singapore MIT Alliance for Research and technology Copyright c Rohan Abeyaratne, 1988. All rights reserved. 11 May 2012. 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, 1988. All rights reserved Abeyaratne, Rohan, Continuum Mechanics , Volume II of Lecture Notes on the Mechanics of Solids. / Rohan Abeyaratne 1st Edition Cambridge, MA and Singapore: ISBN-13: 978-0-9791865-1-6. ISBN-10: 0-9791865-1-X. QC. Please send corrections, suggestions and comments to Updated 28 May 2020. 4. i Dedicated to Pods and Nangi for their gifts of love and presence. iii NOTE TO READER. I had hoped to finalize this second set of notes an year or two after publishing Volume I.

Department of Mechanical Engineering 77 Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA ... In a more general sense the broad approach and philosophy taken has been in uenced by: ... Introduction to the Thermodynamics of Solids, Chapman and Hall, 1991. M.E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981

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1 Continuum Mechanics Volume II of Lecture Notes on the Mechanics of Solids Rohan Abeyaratne Quentin Berg Professor of Mechanics MIT Department of Mechanical engineering and Director SMART Center Singapore MIT Alliance for Research and technology Copyright c Rohan Abeyaratne, 1988. All rights reserved. 11 May 2012. 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, 1988. All rights reserved Abeyaratne, Rohan, Continuum Mechanics , Volume II of Lecture Notes on the Mechanics of Solids. / Rohan Abeyaratne 1st Edition Cambridge, MA and Singapore: ISBN-13: 978-0-9791865-1-6. ISBN-10: 0-9791865-1-X. QC. Please send corrections, suggestions and comments to Updated 28 May 2020. 4. i Dedicated to Pods and Nangi for their gifts of love and presence. iii NOTE TO READER. I had hoped to finalize this second set of notes an year or two after publishing Volume I.

2 Of this series back in 2007. However I have been distracted by various other interesting tasks and it has sat on a back-burner. Since I continue to receive email requests for this second set of notes, I am now making Volume II available even though it is not as yet complete. In addition, it has been cleaned-up at a far more rushed pace than I would have liked. In the future, I hope to sufficiently edit my notes on Viscoelastic Fluids and Microme- chanical Models of Viscoelastic Fluids so that they may be added to this volume; and if I. ever get around to it, a chapter on the mechanical response of materials that are affected by electromagnetic fields. I would be most grateful if the reader would please inform me of any errors in the notes by emailing me at v PREFACE. During the period 1986 - 2008, the Department of Mechanical engineering at MIT offered a series of graduate level subjects on the Mechanics of Solids and Structures that included: : Mechanics of Solid Materials, : Mechanics of Continuous Media, : Solid Mechanics : Elasticity, : Solid Mechanics : Plasticity and Inelastic Deformation, : Advanced Mechanical Behavior of Materials, : Structural Mechanics , : Finite Element Analysis of Solids and Fluids, : Molecular Modeling and Simulation for Mechanics , and : Computational Mechanics of Materials.

3 Over the years, I have had the opportunity to regularly teach the second and third of these subjects, and (formerly known as ), and the current four volumes are comprised of the lecture notes I developed for them. First drafts of these notes were produced in 1987 (Volumes I and IV) and 1988 (Volumes II) and they have been corrected, refined and expanded on every subsequent 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: A Brief Introduction to Finite Elasticity Volume IV: Elasticity This is Volume II. 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.

4 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 the innumerable illuminating and stimulating interactions with my mentor, colleague and friend the late Jim Knowles. His influence on vi 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. I am deeply grateful for, and to, Curtis Almquist SSJE, friend and companion. My understanding of elasticity as well as these notes have benefitted greatly from many useful conversations with Kaushik Bhattacharya, Janet Blume, Eliot Fried, Morton E. 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.

5 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 in Volume I, 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 these include sections on the statistical mechanical theory of polymer chains and the lattice theory of crystalline solids in the discussion of constitutive relations in Volume II, as well as several initial-boundary value problems designed to illustrate various nonlinear phenomena also 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 and Exercises at the end of each chapter which are an essential part of the notes.

6 Many of these examples provide more details; or the proof of a result that had been quoted previously in the text; or illustrates a general concept; or 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 Gelfand and Fomin, Calculus of Variations, Prentice Hall, 1963. vii 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. Ericksen, Introduction to the thermodynamics of Solids, Chapman and Hall, 1991.

7 Gurtin, An Introduction to Continuum Mechanics , Academic Press, 1981. Gurtin, E. Fried and L. Anand, The Mechanics and thermodynamics of Con- tinua, Cambridge University Press, 2010. J. K. Knowles and E. Sternberg, (Unpublished) Lecture Notes for AM136: Finite Elas- ticity, California institute of technology , Pasadena, CA 1978. C. Truesdell and W. Noll, The nonlinear field theories of Mechanics , in Handbu ch der Physik, Edited by S. Flu gge, Volume III/3, Springer, 1965. Volume III: Elasticity 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 Goodier, Theory of Elasticity, McGraw-Hill, 1987. The following notation will be used in Volume II though there will be some lapses (for reasons of tradition): Greek letters will denote real numbers; lowercase boldface Latin letters will denote vectors; and uppercase boldface Latin letters will denote linear transformations.

8 Thus, for example, , , .. will denote scalars (real numbers); x, y, z, .. will denote vectors;. and X, Y, Z, .. will denote linear transformations. In particular, o will denote the null vector while 0 will denote the null linear transformation. One result of this notational convention is that we will not use the uppercase bold letter X to denote the position vector of a particle in the reference configuration. Instead we use the lowercase boldface letters x and y to denote the positions of a particle in the reference and current configurations. viii Contents 1 Some Preliminary Notions 1. Bodies and Configurations.. 2. Reference Configuration.. 4. Description of Physical Quantities: Spatial and Referential (or Eulerian and Lagrangian) forms.. 6. Eulerian and Lagrangian Spatial Derivatives.. 7. Motion of a Body.. 9. Eulerian and Lagrangian Time Derivatives.. 10. A Part of a Body.. 11. Extensive Properties and their Densities.. 12. 2 Kinematics: Deformation 15.

9 Deformation .. 16. Deformation Gradient Tensor. Deformation in the Neighborhood of a Particle. 18. Some Special Deformations.. 21. Transformation of Length, Orientation, Angle, Volume and Area.. 25. Change of Length and Orientation.. 26. Change of Angle.. 27. Change of Volume.. 28. ix x CONTENTS. Change of Area.. 29. Rigid Deformation.. 30. Decomposition of Deformation Gradient Tensor into a Rotation and a Stretch. 32. Strain.. 36. Linearization.. 39. Worked Examples and Exercises.. 42. 3 Kinematics: Motion 63. Motion.. 64. Rigid Motions.. 65. Velocity Gradient, Stretching and Spin Tensors.. 66. Rate of Change of Length, Orientation, and Volume.. 68. Rate of Change of Length and Orientation.. 68. Rate of Change of Angle.. 70. Rate of Change of Volume.. 71. Rate of Change of Area and Orientation.. 71. Current Configuration as Reference Configuration.. 74. Worked Examples and Exercises .. 78. Transport Equations.. 86. Change of Observer. Objective Physical Quantities.

10 89. Convecting and Co-Rotating Bases and Rates.. 93. Linearization.. 95. Worked Examples and Exercises.. 96. 4 Mechanical Balance Laws and Field Equations 101. Introduction .. 102. CONTENTS xi Conservation of Mass.. 104. Force.. 105. The Balance of Momentum Principles.. 110. A Consequence of Linear Momentum Balance: Stress.. 110. Field Equations Associated with the Momentum Balance Principles.. 114. Principal Stresses.. 116. Formulation of Mechanical Principles with Respect to a Reference Stress Power.. 125. A Work-Energy Identity.. 125. Work Conjugate Stress-Strain Pairs.. 126. Linearization.. 127. Objectivity of Mechanical Quantities.. 128. Worked Examples and Exercises.. 129. 5 Thermodynamic Balance Laws and Field Equations 147. The First Law of thermodynamics .. 147. The Second Law of thermodynamics .. 149. Formulation of Thermodynamic Principles with Respect to a Reference Con- figuration.. 152. Summary.. 153. Objectivity of Thermomechanical Quantities.


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