# MPhys Physics with Astrophysics / Course details

Year of entry: 2024

## Course unit details:Elasticity

Unit code MATH35021 10 Level 3 Semester 1 Department of Mathematics No

### Overview

This course unit gives an introduction to the linearised theory of elasticity. A typical problem of the subject is as follows: Suppose an elastic body (e.g. an underground oil pipe) is subjected to some loading on its outer surface. What is the stress distribution which is generated throughout the body? Does this stress distribution have unexpectedly large values which might lead to failure? The subject is developed, and particular problems solved, from a mathematical standpoint.

### Aims

This course aims to:(i) Introduce students to the mathematical theory of linear elasticity, (ii) Develop and apply the theory of linear elasticity to a number of practical problems in solid mechanics, (iii) Introduce a range of analytical techniques to solve the differential equations that arise in problems involving linear elasticity

### Learning outcomes

On successful completion of this course unit students will be able to:

• identify, calculate and provide physical interpretations for displacements, strains, rotation tensors, stresses and tractions for given deformations of linear elastic materials,
• distinguish between rigid-body motions and strains and use the compatibility equations to establish the validity of strain fields,
• use constitutive laws to relate strain and stress fields,
• solve the Navier-Lamé equations given specific functional forms for the displacement field,
• solve two-dimensional problems in linear elasticity (plane strain or plane stress) using the Airy stress function or related approaches,
• formulate, solve and provide physical interpretations for the solutions of boundary value problems in linear elasticity.

### Syllabus

• Analysis of strain [6]: the infinitesimal strain tensor, derivation and interpretation; maximum normal strain; strain invariants; equations of compatibility of strain.
• Analysis of stress [2]: the traction vector and the stress tensor; maximum normal stress. Stress equations of motion and their linearisation.
• Constitutive equations [1] stress-strain relations. Elastic and linearly elastic materials; isotropic materials.
• Governing Equations [1]: Navier's equation of motion for the displacement vector; equations of compatibility of stress for an isotropic materials in equilibrium (Beltrami-Michell equations).
• Formulation of boundary value problems of linear elastostatics [12]:  One-dimensional problems. A selection of soluble problems (which are effectively one-dimensional) in Cartesian, cylindrical polar or spherical polar coordinates. St. Venant's principle.  Plane strain problems. Theory of plane strain, Airy stress function. A selection of soluble two-dimensional problems using plane-strain theory.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework: weighting 20%
• End of semester examination: weighting 80%

### Feedback methods

Examples classes provide an opportunity for students' work on the weekly examples sheets to be discussed and the lecturer will provide formative feedback on their understanding. Students can also get formative feedback from the lecturer outside of examples classes by making an appointment, for example during the lecturer's office hours. The coursework tests understanding and provides summative feedback.

The course does not follow one particular book. A good book, covering most of the course, is

• P.L. Gould & Y. Feng Introduction to Linear Elasticity, 4th Edition, Springer, 2018

This book, and many others on the theory of elasticity can be found at 531.38 in the John Rylands University Library.

### Study hours

Scheduled activity hours
Lectures 11
Tutorials 11
Independent study hours
Independent study 78

### Teaching staff

Staff member Role
Tom Shearer Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.