Lecturer: Thomas Hudson

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 100% 3 hour written examination

Formal registration prerequisites: None

Assumed knowledge: This module assumes knowledge of various aspects of first and second year core maths material. Modules from other departments may also cover the necessary background. We list where the relevant material can be found for Maths and joint degree students.

• MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices: an understanding of vectors and matrices, including the ability to compute eigenvalues and eigenvectors
• MA263 Multivariate Analysis: knowledge of multivariable calculus, including partial derivatives, divergence, curl and accompanying integral theorems
• MA146 Methods of Mathematical Modelling 1 or MA147 Mathematical Methods of Modelling 1 or MA133 Differential Equations: an ability to compute explicit solutions to various ODEs
• MA141 Analysis 1 or MA140 Mathematical Analysis 1 or MA142 Calculus 1: an appreciation of continuity and the ability to take limits

Useful background: MA3G1 Theory of PDEs

Synergies: The third year modules listed under "Useful Background" would go well alongside this module. Fourth year modules which would also synergise well are:

• MA4L2 Statistical Mechanics
• MA4L0 Advanced Topics in Fluids
• MA4M2 Mathematics of Inverse Problems

Some background on the theory of ODEs and PDEs would also be useful, as covered in Additional Resources

Content: The modelling and simulation of fluids and solids is an important area of study in applied mathematics and engineering. Necessary for such studies is a fundamental understanding of the basic principles of continuum mechanics and thermodynamics. This course focuses on the foundational theory behind these models, and closely follows the text "A First Course in Continuum Mechanics'' by Gonzalez and Stuart. Throughout, our focus is on building general theoretical understanding over solving particular cases of the equations. An indicative module outline is as follows: we begin by discussing tensor algebras and their calculus. We then use these tools to build equations which govern both fluids and solids. If time permit, we will proceed to discuss some commonly used models governing isothermal solids.

Aims: To provide understanding of the theoretical foundation of PDE models of solids and fluids.

Objectives: By the end of the module the student should be able to:

• Manipulate and perform calculations with tensors.
• Explain the concepts of stress and strain.
• State physical balance laws in integral form and derive their local versions as PDEs.
• Use their understanding to interpret the results of theoretical calculations.

Book(s):

Oscar Gonzalez, Andrew Stuart, A First Course in Continuum Mechanics, Cambridge University Press, 2008.