MA4J1 Continuum Mechanics
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 / MA149 Linear Algebra (formerly MA106 Linear Algebra): an understanding of vectors and matrices, including the ability to compute eigenvalues and eigenvectors
- MA144 Methods of Mathematical Modelling 2 / MA145 Mathematical Methods and Modelling 2 (formerly MA134 Geometry and Motion) and MA263 Multivariable Analysis (formerly MA259 Multivariable Calculus): knowledge of multivariable calculus, including partial derivatives, divergence, curl and accompanying integral theorems
- MA146 Methods of Mathematical Modelling 1 / MA147 Mathematical Methods and Modelling 1 (formerly MA133 Differential Equations or MA113 Differential Equations A): an ability to compute explicit solutions to various ODEs
- MA139 Analysis 2 / MA152 Mathematical Analysis 2 (formerly MA131 Analysis or MA137 Mathematical Analysis): an appreciation of continuity and the ability to take limits
Useful background: MA3D1 Fluid Dynamics or MA3J4 Mathematical Modelling with PDE will serve as useful background to the modelling aspect of this module. Some background on the theory of ODEs and PDEs would also be useful, as covered in MA254 Theory of ODEs and 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:
Leads to: The following modules have this module listed as assumed knowledge or useful background:
- MA269 Asymptotics and Integral Transforms
- MA3D1 Fluid Dynamics
- MA3J4 Mathematical Modelling with PDE all provide useful aspects background to the modelling aspect of this module.
Some background on the theory of ODEs and PDEs would also be useful, as covered in MA254 Theory of ODEs and MA3G1 Theory of Partial Differential Equations.
Content: The modelling and simulation of fluids and solids with significant coupling and thermal effects 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 follow 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 tensors algebra and their calculus. We then use these tools to build equations which govern both fluids and solids, we will proceed to derive some commonly used models governing isothermal fluids and 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.