Prof Jey Sivaloganathan's teaching page
This page will contain information relating to courses I am teaching, copies of the tutorial sheets and summaries of the material covered to date.
MA40049: Elasticity
In this course, we develop the general principles of the theory of elasticity. The classical theory of linear elasticity can then be obtained by expanding the constitutive law relating stress to strain in a neighbourhood of the undeformed state.
Course Diary:
Week 1: Motions, deformations and kinematics, homogeneous deformations. Recap of basic linear algebra results.
Week 2: Square Root Theorem, Polar Decomposition Theorem, Cauchy-Green tensors, Principal Values, Geometric Properties of Deformations.
Week 3: Cauchy and First Piola Kirchhoff Stress Tensors, Balance Laws, Equilibirum Equations
Week 4: Material Frame Indifference, Response functions for the Piola Kirchhoff and Cauchy Stress tensors, Equilibria
Week 5: Hyperelastic materials, the Stored Energy Function W, Frame-Indifference and Isotropy in terms of W, Examples
Week 6 Representation of a frame-indifferent, isotropic stored energy function as a symmetric function of the principal stretches, Ogden Materials, Symmetry of the Cauchy Stress Tensor for hyperlastic materials, Equilibrium equations of hyperelasticity, Coincidence of Principal Stress and Principal Stretch directions for isotropic materials.
Week 7: Boundary Value Problems of Elasticity, Examples of non-uniqueness of equilibria, Constitutive Inequalities (the Tension-Extension and Baker-Ericksen Inequalities), Examples.
Week 8: Incompressible hyperelasticity, Cauchy and Piola-Kirchhoff extra stress tensors, examples
Week 9: Linearised Elasticity, Equilibrium equations, Stress tensor, boundary value problems
Week 10: Young's Modulus, Poisson's Ratio, Strain energy function, Clapeyron's Theorem
Relevant text books include:
"Mechanics of Continuous Media" by S.C. Hunter
"Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity" by P.G. Ciarlet
"A First Course in Continuum Mechanics" by O. Gonzalez and A.M. Stuart (online version of book available from the University of Bath library)
"Nonlinear Elastic deformations" by R.W. Ogden
"Continuum Mechanics" by A.J.M. Spencer
Interesting practical aspects of materials science are contained in:
"The New Science of Strong Materials" by J.E. Gordon (first published 1968)
MA40048: Analytical and Geometric Theory of Differential Equations
This course has the intertwoven themes of ordinary differential equations, the Calculus of Variations, Lagrangian Systems and Hamiltonian Systems, and looks at some of the elegant and striking connections between them.
Hamilton’s Principle states that a system evolves in such a way that a given integral over the path, called the action integral, is stationary. Fermat’s Principle, that light rays travel through a material along paths that minimise the total travel time, is an example of Hamilton’s Principle. The condition that the given integral is minimised then leads to a differential equation, called the Euler-Lagrange equation(s), for the path itself. This approach is known generally as a ‘variational principle’ and many familiar problems in physics, such as Newton’s laws of motion, can be recast in this way. The study of such variational problems constitutes the Calculus of Variations. In many cases, the Euler-Lagrange equations can in turn be recast as a first order autonomous system of differential equations, called a Hamiltonian System, which allows us to deduce general properties of solutions by studying the geometry of the flow induced by the differential equation(s) in the corresponding phase space.
The concepts and principles introduced in this course have led to many developments in modern pure mathematics (such as symplectic geometry and ergodic theory) and, besides applying to the equations of classical mechanics, they have motivated much of modern physics.
Relevant text books include:
"Introduction to Analytical Dynamics" by N.M.J. Woodhouse (Online version of book available from University of Bath Library)
"Calculus of Variations" by Gelfand and Fomin
"Mechanics" by Landau and Lifshitz
"Optimization-Theory and Applications" by L. Cesari
"Variational Calculus and Optimal Control" by J.L. Troutman
" Principles of Mechanics" by J.L. Synge and B.A. Griffiths
"Mathematical Methods of Classical Mechanics" by V.I. Arnold
"The Feynman Lectures on Physics", Volume 2, by R. Feynman
"Introduction to Dynamics" by Percival and Richards
"Analytical Dynamics of Particles" by Whittaker
"Mechanics" by V. Kibble
"A Geometrical Theory of ODEs" by V.I. Arnold
"A Treatise on Analytical Dynamics" by Pars