# Mathematics / Applied Mathematics MSc

# 5E: Continuum Mechanics and Elasticity MATHS5066

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 2**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

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).

#### Timetable

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

#### Requirements of Entry

Mandatory Entry Requirements

3H Mechanics of Rigid and Deformable Bodies (MATHS4078)

3H Mathematical Methods (MATHS4075)

Recommended Entry Requirements

3H Methods of Complex Analysis (MATHS4076)

#### Excluded Courses

4H: Continuum Mechanics and Elasticity (MATHS4100)

#### Assessment

Assessment

90% Examination, 10% Coursework.

Reassessment

MSc students will have an opportunity to resit.

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** No

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.

#### Course Aims

The aim of this course is to define the mathematical apparatus needed for modelling deformable bodies, with a special emphasis on the closely related areas of linear elasticity and tensor calculus in curvilinear coordinates. The continuum theory part of the course will aim at presenting the relevant equations in a general framework by using curvilinear coordinates and the concepts of covariant/contravariant derivatives. In the second part of the course several fundamental aspects of linear elasticity will be pursued in considerable depth. Connections with other areas of mathematics such as potential theory and complex analysis will be established, and their relevance to solving practical problems will be illustrated with many interesting examples.

#### Intended Learning Outcomes of Course

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 curvilinear systems of coordinates;

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

(c) Derive the equations of motion of a deformable body and state these equations in tensor form using arbitrary curvilinear coordinates;

(d) Define the notion of a stress tensor and answer a range of questions on its properties, in particular, to be able to: find surface tractions and principal stresses; determine whether given tensors are objective;

(e) Delineate the differences between constitutive laws, and define what is meant by an isotropic tensor function;

(f) Formulate and solve basic equilibrium problems in elasticity;

(g) State various linear constitutive laws for elastic materials; demonstrate an understanding of the role of the strong ellipticity condition for the existence of physically realistic solutions in hyperelastic materials;

(h) Derive the Lame-Navier system of PDE's and use the compatibility conditions to obtain the Beltrami-Michell equations;

(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) Derive The Airy function, and use it to solve a range of problems;

(l) 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.

#### Minimum Requirement for Award of Credits

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