Course Catalogue

View Specification Document | Reading List

This course introduces students to the general mathematical model of three-dimensional deformable bodies, and explores a particular sub-branch, namely, the theory of linearly elastic bodies (representing a generalisation of Hooke's elastic springs).

34 x 1 hr lectures and 12 x 1 hr tutorials in a semester

Mandatory Entry Requirements

3H Mechanics of Rigid and Deformable Bodies (MATHS4078)

3H Mathematical Methods (MATHS4075)

Recommended Entry Requirements

3H Methods of Complex Analysis (MATHS4076)

Assessment

90% Examination, 10% Coursework.

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

The aim of this course is to define the mathematical apparatus needed for modelling deformable bodies, with a special emphasis on the closely related area of linear elasticity. The former aspect will demand an extension of the classical concepts from Newtonian mechanics and the use of vector and tensor calculus; the latter involves setting up and solving various boundary-value problems that describe simple physical phenomena such as stretching of a thin elastic sheet, the torsion of shafts of various cross-sections, bending of bars, and so on.

By the end of this course students will be able to:

a) Use differential operators (del, div, etc) with vector and tensor functions; manipulate tensor products and second-order tensors in Cartesian and polar coordinates;

b) Use Nanson's formula connecting area elements derive, and deduce various transport formulae; 

c) Derive the equations of motion of a deformable body;

d) Understand the notion of stress tensor, and find surface tractions and principal stresses; determine

whether given tensors are objective; 

e) Understand the differences between constitutive laws, and define what is meant by an isotropic tensor

function;

f) Be able to state various linear constitutive laws for elastic materials;

g) Be able to derive the Lame-Navier system of PDE's and use the compatibility conditions

to obtain the Beltrami-Michell equations;

h) Understand the mathematical arguments which lead to the uniqueness of the displacement

or stress fields in linear elasticity;

i) Explain the role played by the Saint-Venant's Principle in setting up and solving boundary-value

problems;

j) Solve a variety of representative problems related to plane stress, plane strain or anti-plane

strain situations;

k) Describe the torsion of beams of circular and polygonal cross-sections and explain the role

of the Prandtl stress function in this context; solve simple torsion problems for a range of different

cross-sections;

l) Be able to apply their knowledge of solution techniques to solve classical problems, to appreciate the complications associated with more realistic problems, and to discuss the limitations of the mathematical techniques.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.