SE 394

 

Announcements

  1. (i)End semester exam: 19 Nov 2008, 4-7pm. L9

  2. (ii)A copy of all lectures notes on tensor algebra and handout on tensor analysis as well as written material on deformation and motion should remain with you during future lectures. So that any result related to these topics can be referred by you immediately.

  3. (iii)One copy of the text book "An Introduction To Continuum Mechanics" by Morton E. Gurtin has been reserved and kept in the Reserve Section of the library.  Besides this, the details of title and no. of copies (of books in the reserve section besides the textbook) are: “Continuum Mechanics: Concise Theory And Problems” (1 copy) by P. Chadwick; “A First Course In Rational Continuum Mechanics”(1 copy) by Truesdell, C; “Continuum Mechanics” (2 copies) by Spencer,A. J. M.

  4. (iv)In future, if you wish to see repetition/explanation of some material covered in previous lectures, please email me beforehand.




Course details

Course No: SE 394

Instructor: Basant Lal Sharma

Department: Mechanical Engineering

Units: (L-T-P-D-U) 3-0-0-0-4

Schedule: LEC: MWF 12:00-13:00pm L6

Prerequisite: Basic course in integral and differential calculus of functions of a real variable, basic course in linear algebra and differential geometry.

Note: The course is offered and structured as an (open) elective for postgraduate students and advanced undergraduate students. Anyone else interested may take permission of the instructor.




MY CONTACT

Email: bls at iitk.ac.in

Phone: 6173

Office: FB 356




Books

  1. 1.Gurtin, M. E.,1982. An Introduction to Continuum Mechanics, Academic Press (Textbook).

  2. 2.Gurtin, M. E.,1972. The Linear Theory of Elasticity, Handbuch der Physik, Vol. VIa/2, Springer.

  3. 3.Truesdell, C., 1977. A first course in Rational Continuum Mechanics, Academic Press.

  4. 4.Jog, C. S., 2007. Continuum Mechanics, Vol 1, Narosa.




Useful links


http://www1.mengr.tamu.edu/rbowen/


Wikipedia, ScienceWorld, Imeccanica

 

Course Summary


Rigorous introduction to continuum mechanics emphasizing solid mechanics and linear elasticity. Course topics include vectors and tensors: tensor algebra and tensor analysis with direct notation; kinematics: deformations, rotations, strains and strain rates; force and stress; balance principles; frame indifference; material symmetry; constitutive relations; three-dimensional elasticity; finite elasticity; small-strain theory; linear elasticity: anisotropic and isotropic, topics include linearization and stability; basic theorems of elastostatics and boundary-value problems.


Additional topics (for example: guest lectures) may also include rate-dependent materials, and materials with internal state variables, linear viscoelasticity, thermoelasticity, Newtonian fluids; constitutive inequalities; the entropy inequality; conservation laws; exact solution of special problems; singular solutions in linear elasticity; plane stress, plane strain, anti-plane shear, airy stress functions; cracks and dislocations; classical plasticity.


This course will introduce and develop the subject of continuum mechanics, specially, elasticity and applicable to the circumstances when the material of a body experiences large (or finite) deformation including large displacement or large rotation. In their generality we will also discuss the issue of formulating kinematics (motion), balance laws (force) and material constitution (matter).


  1. (i)Introduction to the notion of a tensor and preparation of some background in tensor algebra and analysis. A brief exposure to function spaces and functional analysis.

  2. (ii)Definition of motion of a body, in particular a deformable body. Definition of a reference configuration. Discussion of the concept of force. Definition of Cauchy stress. Definition of several other notions of stress and some ways of quantifying ‘strain’, ‘strain rate’, other rates of deformation. Discussion on balance laws and states of equilibrium.

  3. (iii)Definition of objectivity (symmetry of physical space) and material symmetry (symmetry of material space). Discussion of certain crystalline symmetries that may be incorporated through an elastic model. Highlights on the role of a reference configuration in this subject.

  4. (iv)Definition of an elastic material. Discussion on the special case of an isotropic elastic material. Introduction to the three dimensional theory of hyperelasticity. Exposure to some useful results in the case of an isotropic hyperelastic material. Discussion on the issues of nonuniqueness and stability in hyperelastic materials.

  5. (v)Derivation of linear elasticity as a result of linearization of fully nonlinear theory discussed so far. Discussion on certain limitations of such linearization procedure. Discussion on the elasticity tensor and the special case of an isotropic elastic material and some other useful symmetries of materials.

  6. (vi)Conclusion of the course with a discussion of some open problems.

  7. (vii)*There may be a few guest research seminars in this course, given by various professors, in order to motivate this subject. A few lectures or reading assignments may be devoted to some interesting results in the theory of nonlinear elasticity which find application in diverse areas of engineering and sciences.

Course flyerSE394_2008_files/SE394.pdfshapeimage_7_link_0

Introduction to Continuum Mechanics

Some information on Lectures


  1. (I)Lecture 0 (28 Jul): Quiz for certain preliminaries, Overview of Course

  2. (II)Lecture 1-12 (30 Jul, 1, 4, 6, 8, 11, 13, 18, 20, 22, 23, 25, 27 Aug): Sets and Mappings, Fields and Linear Spaces, Special cases of Real Linear Space and Complex Linear Space, Examples of Linear Spaces, Isomorphism,  General Linear Group, Multilinear Transformation, Metric Space, Cauchy Convergence, Compact Set, Real Inner Product Space, Complex Inner Product Space, Orthonormal Basis, Orthogonal Complement, Cauchy Schwarz Inequality, Hilbert Space, Banach Space, Adjoint Transformation, Hermitian and Skew Hermitian Transformation, Real Euclidean Space, Complex Euclidean Space, Representation Theorem for Linear Forms, Tensor Product of Vectors, Trace, Dot Product of Tensor Products, Tensors of arbitrary order, Cross Product of two vectors, Cartesian Tensors, Invariants of second order tensors, Determinant, Cayley Hamilton Theorem, Eigenvalues and Eigenvectors, The Characteristic Polynomial, Invariant Subspaces, Special Second Order Tensors:  Symmetric, Skew Symmetric, Orthogonal, Square Roots, Polar Decomposition Theorem, The Spectral Theorem, Differentiation and Integration of Tensors, Product rule and Chain rule of differentiation, Green's Theorem, Exponential Function. [MIDSEM I]

  3. (III)Lecture 11-22 (1, 3, 5, 8, 10, 12, 15, 17, 19 Sep): Discussion on Euclidean Point Space, Body and Configuration, Reference Configuration, Deformation, Deformation Gradient, Rotation, Stretch and Strain Fields, Special Types of Deformation: Homogeneous Deformation, Rigid deformation, Isochoric Deformation. Tutorial on Tensor Analysis, Differentiation and Integration. Continuous Family of Deformations or Motion, Velocity, Acceleration, and Derivatives of Material/Spatial Fields, Spin Rate, Stretch Rate, and Circulation, Pathline, Streamline, and Vortexline, Special Types of Motion: Steady Motion, Rigid Motion, Isochoric Motion, and Motion with Acceleration as the Gradient of a Potential. Using a Configuration during the Motion as Reference Configuration.

  4. (IV)Lecture 23-26 (22, 24, 26, 29 Sep, 13, 13 Oct): Mass, Conservation of Mass, Linear Momentum and Angular Momentum, Conservation of Linear Momentum and Angular Momentum, Force, Cauchy’s Hypothesis, Balance of Linear Momentum and Angular Momentum, Cauchy Stress, Local Balance of Mass, Linear Momentum and Angular Momentum, Power Expended Theorem.

  5. [MIDSEM II]

  6. Balance of Mass and Momenta and Power Expended Theorem on a Control Volume, Piola Transform, First Piola-Kirchhoff Stress, Second Piola-Kirchhoff Stress, Balance of Mass and Momenta and Power Expended Theorem on Reference Configuration.

  7. (V)Lecture 27-40 (15, 15, 17, 20, 22, 31 Oct, 3, 5, 7, 10, 12, 14 Nov): Frame, Change of Frame, Frame indifferent scalar, vector, and tensor fields, Frame indifferent kinematic fields, Galilean change of frame, Constitutive assumptions, Dynamical Process, Definition of a Material Body, Constitutive relation, Postulates of Determinism, Local action, Material frame indifference, Simple material body, Material Symmetry, Isotropy, Solid, Fluid, Fluid Crystal, Simple fluids: Ideal fluid, Elastic fluid, Newtonian fluid, Non-newtonian fluid, Simple Solids: Elastic body, hyperelastic body, linear elastic body, linear viscoelastic body, non-linear viscoelastic body.