To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials. Cartesian tensors are first introduced. This is followed by modelling of the mechanics of continuous media. The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity.

(LO1) Work confidently with index notation. Solve routine problems involving transformation of vectors, rotation matrices, routine matrix and vector algebra.

(LO2) Solve routine problems involving Cartesian tensors, their properties and routine tensor relations.

(LO3) Solve routine problems for deformed solids, calculate routine matrix characteristics for finite and infinitesimal deformations and kinematics of motion.

(LO4) Solve routine problems involving balance equations and analysis of stresses in a continuum.

(LO5) Solve routine problems for both viscous fluids and elastic solids.

(LO6) Use tensor calculus to establish different vector and matrix identities, analyse kinematics of motion, deformations and stresses in a continuum, derive linear constitutive equations, balance equations and solve both routine and unseen problems of continuum mechanics.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

Cartesian tensors. Transformation of components, symmetry and skew symmetry. Isotropic tensors.

Kinematics. Transformation of line elements, deformation gradient, Green strain. Linear strain measure.

Displacement, velocity, strain-rate.

Stress. Cauchy stress. Relation between traction vector and stress tensor.

Global balance laws. Equations of motion, boundary conditions.

Newtonian fluids. The constitutive law, uniform flow, Poiseuille flow, flow between rotating cylinders.

Linear elasticity. Field equations. Young''s modulus, Poisson''s ratio. Strain energy function. Betti''s reciprocal theorem.

Some simple problems of elastostatics. Expansion of a spherical shell, bulk modulus; deformation of a block under gravity; elementary bending solutions.Torsion of cylinders, Prandtl''s stress function.