SE 394
Announcements
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(i)In future, if you wish to see repetition/explanation of some material covered in previous lectures, please email me beforehand.
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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: TThF 8-9am, T: W 8-9am
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
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1.Gurtin, M. E.,1982. An Introduction to Continuum Mechanics, Academic Press (Textbook).
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2.Gurtin, M. E.,1972. The Linear Theory of Elasticity, Handbuch der Physik, Vol. VIa/2, Springer.
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3.Truesdell, C., 1977. A first course in Rational Continuum Mechanics, Academic Press.
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4.Jog, C. S., 2007. Continuum Mechanics, Vol 1, Narosa.
Useful links
http://www1.mengr.tamu.edu/rbowen/
Course Summary
This course will introduce and develop the subject of mechanics in a continuum, specially elasticity, applicable to the circumstances when the material of a body experiences large (or finite) deformation including large displacement or large rotation. The main objective is rigorous introduction to continuum mechanics emphasizing solid mechanics and linear elasticity.
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(i)Introduction to the notion of a tensor and preparation of some background in tensor algebra, tensor analysis and tensor calculus. This part mainly deals with vectors and tensors with direct notation.
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(ii)Definition of motion of a body, in particular a deformable body. Discussion of the concept of force. Definition of Cauchy stress. Definition of a reference configuration. Definition of several other notions of stress and some ways of quantifying `strain' or `strain rate'. This part is kinematics and forces: deformations, rotations, strains and strain rates, force and stress, balance principles.
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(iii)Constitutive relations: Definition of frame indifference (symmetry of physical space) and material symmetry (symmetry of material space). Highlights on the role of a reference configuration in this subject. 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.
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(iv)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 anisotropic and isotropic cases. The topics may include stability; basic theorems of elastostatics and boundary-value problems.
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(v)Conclusion of the course with a discussion of some open problems.
Introduction to Continuum Mechanics
Some information on Lectures
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(I)Lecture 1-7 (): Sets and Mappings, Groups, Fields and Linear Spaces, Special cases of Real Linear Space and Complex Linear Space, Examples of Linear Spaces, Isomorphism, 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.
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(II)Lecture 8-13 (): Discussion on Body and Configuration, Reference Configuration, Deformation, Deformation Gradient, Rotation, Stretch and Strain Fields, Special Types of Deformation: Homogeneous Deformation, Rigid deformation, Isochoric Deformation, 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.
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(III)Lecture 14-17 (): 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. 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, Balance of Mass and Momenta and Power Expended Theorem on a Control Volume.
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(IV)Lecture 18-24 (): 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.
