Advanced Mathematical Methods in Elasticity

Wedi cwblhau'r modiwl dylai'r myfyrwyr fedru:

1. Give the definitions of strain and stress and explain the relations connecting them
2. Manipulate tensors
3. Formulate simple problems in 2D elasticity and provide appropriate boundary conditions
4. Demonstrate the ability to employ coordinate systems appropriate to the situation being modelled
5. Derive analytic solutions to one and two-dimensional problems in linear elasticity
6. Solve representative problems using Green's functions, integral transforms and the integral equation method

The aim of this module is to introduce the mathematical theory of linear elasticity in its 2D formulation. It will start from a broad framework in which a model of the deformation of a solid body undergoing external loading is derived by analyzing the internal stresses and the related strain deformation. The 2D theory will be developed as a system of differential equations and applied to selected practical problems. The analytical techniques required to tackle such problems will be presented. Green's functions will be introduced as a means to generate precise analytical solutions using the theory of integral equations.

• Introduction: Vectors, Matrices, Tensors, Tensor algebra.
• Analysis of strain. The infinitesimal strain tensor. Strain compatibility.
• The traction vector and the stress tensor. Equations of motion.
• Stress-strain relations for elastic and linearly elastic materials; isotropic materials.
• Formulation of boundary value problems of linear elasto-statics.
• One-dimensional problems. A selection of soluble problems (which are effectively one-dimensional) in Cartesian, cylindrical polars and spherical polar coordinates.
• Two dimensional problems in theory of elasticity.
• Torsion. Laplace’s equation and methods of its solution.
• Plane strain problems. Theory of plane strain, Airy stress function. A selection of soluble two-dimensional problems using plane-strain theory.
• Green’s functions in 1D. Construction for constant coefficient ODEs and Sturm Liouville problems. Applications to the steady state heat equation and wave equation.
• Green’s functions in 2 and 3D. Steady state heat equation and potential flow problems (Laplace) and time-harmonic wave equation (Helmholtz).
• Integral equations. Degenerate (separable) kernels and solution method. General integral equation types. Neumann series and iterated kernels. Volterra and Fredholm equations, Fredholm theory.

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 7