## Continuum Mechanics

 Unit code: MATH65061 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

None

Students are not permitted to take, for credit, MATH45061 in an undergraduate programme and then MATH65061 in a postgraduate programme at the University of Manchester, as the courses are identical.

#### 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.

#### 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.

#### Learning outcomes

On successful completion of this course unit students will

• Be able to formulate governing equations for a variety of problems in continuum mechanics.
• Understand the relationship between the general theory and its specialisation to the equations of linear elasticity and incompressible Newtonian fluid mechanics.
• Solve simple problems in continuum mechanics analytically.
• Be aware of certain numerical techniques that can be applied to problems in continuum mechanics.

#### Assessment methods

• Other - 25%
• Written exam - 75%

#### Assessment Further Information

• Mid-semester coursework: 25%
• End of semester examination: three hours weighting 75%

#### 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.

• 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

#### Feedback methods

Tutorials represent the principal forum for feedback, providing an opportunity for students' work on example sheet questions and the coursework to be discussed. There are two coursework assignments in the form of extended calculations that both test understanding and provide opportunities for further feedback. Students can also get feedback on their understanding directly from the lecturer by making an appointment, for example during the lecturer's office hours.

#### Study hours

• Lectures - 27 hours
• Tutorials - 6 hours
• Independent study hours - 117 hours

#### Teaching staff

Andrew Hazel - Unit coordinator