Course unit details:
Continuum Mechanics
Unit code | MATH65062 |
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Credit rating | 15 |
Unit level | FHEQ level 7 – master's degree or fourth year of an integrated master's degree |
Teaching period(s) | Semester 2 |
Offered by | Department of Mathematics |
Available as a free choice unit? | No |
Overview
This unit describes the fundamental theory of continuum mechanics in a unified mathematical framework. The unit will cover the formulation of governing conservation and balance laws in generalised coordinates in both Eulerian and Lagrangian viewpoints. Specific examples of constitutive modelling will be developed via the theories of nonlinear and linear elasticity together with those of compressible and incompressible fluid mechanics.
Pre/co-requisites
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Aims
The course unit concerns the formulation and solution of problems in continuum mechanics (solid and fluid mechanics) from a modern unified perspective. The aims are (i) to introduce students to the general analytic machinery of tensor calculus, variational principles and conservation laws in order to formulate governing equations; (ii) to illustrate the principles of constitutive modelling; and (iii) to make students aware of some exact, approximate and numerical methods to solve the resulting equations.
Learning outcomes
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Syllabus
- Introduction [4]: Vectors, tensors, co- and contra-variant transformation laws, invariance concepts, metric tensor, tensor calculus, divergence theorem.
- Kinematics [4]: Deformation maps, Lagrangean and Eulerian viewpoints, displacement, velocity and acceleration, material derivative, strain measures, strain invariants, deformation rates, Reynolds transport theorem.
- Forces, momentum & stress [3]: The continuum hypothesis, linear and angular momenta, stress tensors, equations of equilibrium.
- Conservation and Balance Laws & Thermodynamics [3]: Conservation of mass and energy, balance of linear and angular momenta, work conjugacy, temperature and heat, first and second laws of thermodynamics, Clausius--Duhem inequality.
- Constitutive Modelling [3]: Introduction to constitutive relationships, axiom of objectivity, objective deformation rates, constitutive modelling for an ideal gas.
- Elasticity [5]: Constitutive modelling for thermoelastic materials, Hyperelastic materials, strain energy function, homogeneous, isotropic materials, incompressibility constraints, example analytic solutions, boundary conditions, linear thermoelasticity and reduction to Navier--Lame equations.
- Fluid Mechanics [5]: Constitutive modelling for fluids, isotropic fluids, Newtonian and Reiner--Rivlin fluids, example analytic solutions, boundary conditions, reduction to Navier--Stokes equations.
Assessment methods
Method | Weight |
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Other | 20% |
Written exam | 80% |
- Mid-semester coursework: 20%
- End of semester examination: weighting 80%
Feedback methods
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Recommended reading
- Spencer, A.J.M, Continuum Mechanics, Dover
- Gonzalez, O. and Stuart, A.M., A first course in continuum mechanics, CUP
- Irgens, F., Continuum Mechanics, Springer
Study hours
Scheduled activity hours | |
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Lectures | 12 |
Tutorials | 12 |
Independent study hours | |
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Independent study | 126 |
Teaching staff
Staff member | Role |
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John Gray | Unit coordinator |