(Relevant portion of Chapter I & II of book by I.S. Sokolnikoff). SECTION-III (Two Questions) Stress quadric of Cauchy, Principal stress and invariants. Maximum normal and shear stresses. Mohr’s circles, examples of stress. Equations of Elasticity : Generalised Hooks Law, Anisotropic symmetries, Homogeneous isotropic medium. (Relevant portion of Chapter II & III of book by I.S. Sokolnikoff).
SECTION- IV (Two Questions) Elasticity moduli for Isotropic media. Equilibrium and dynamic equations for an isotropic elastic solid. Strain energy function and its connection with Hooke’s Law, Uniqueness of solution. Beltrami-Michell compatibility equations. Clapeyrom’s theorem. Saint-Venant's principle. (Relevant portion of Chapter III of book by I.S.Sokolnikoff). Books: 1. I.S. Sokolnikoff, Mathematical Theory of Elasticity, Tata-McGraw Hill Publishing Company Ltd., New Delhi, 1977. 2. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity Dover Publications, New York. 3. Y.C. Fung. Foundations of Solid Mechanics, Prentice Hall, New Delhi, 1965. 4. D.S. Chandrasekharaiah and L. Debnath, Continuum Mechanics, Academic Press, 1994. 5. Shanti Narayan, Text Book of Cartesian Tensor, S. Chand & Co., 1950. 6. S. Timeshenki and N. Goodier. Theory of Elasticity, McGraw Hill, New York, 1970. 7. I.H. Shames, Introduction to Solid Mechanics, Prentice Hall, New Delhi, 1975.
SEMESTER-III MM-503 (opt. ii) Difference Equations-I Examination Hours : 3 Hours Max. Marks : 100 (External Theory Exam. Marks:80 + Internal Assessment Marks:20) NOTE : The examiner is requested to set nine questions in all taking two questions from each section and one compulsory question. The compulsory question will consist of eight parts and will be distributed over the whole syllabus. The candidate is required to attempt five questions selecting at least one from each section and the compulsory question. SECTION-I (Two Questions) Introduction,the difference calculus: The difference operator,falling factorial power r t ,binomial coefficient r t , summation, definition, properties and examples, Abel’s summation formula, Generating functions, Euler’s summation formula, Bernoulli polynomials and examples, approximate summation. SECTION-II (Two Questions) Linear Difference Equation: First order linear equations, general results for linear equations, solution of linear difference equation with constant coefficients and with variable coefficients, Non-Linear Equations that can be linearized, applications. SECTION-III (Two Questions) Stability Theory : Initial value Problems for Linear systems, eigen values, eigen vectors and spectral radius, Caylay-Hamilton Theorem, Putzer algorithm. Solution of nonhomogeneous system with initial conditions, Stability of linear systems, stable subspace theorem and example. Stability of non-linear system, Chaotic behaviour.
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