Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title CARTESIAN TENSORS AND MATHEMATICAL MODELS OF SOLIDS AND VISCOUS FLUIDS
Code MATH324
Coordinator Prof N Movchan
Mathematical Sciences
Nvm@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 6 FHEQ First Semester 15

Aims

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.


Learning Outcomes

After completing the module, students should be able to understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, and apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.


Syllabus

1

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.


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101; MATH102; MATH103  

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

 

Programme(s) (including Year of Study) to which this module is available on an optional basis:

Programme:G100 Year:3 Programme:G101 Year:3,4 Programme:GG14 Year:3 Programme:G110 Year:3 Programme:G1F7 Year:3 Programme:G1N2 Year:3 Programme:G1R9 Year:4 Programme:G1X3 Year:3 Programme:GG13 Year:3 Programme:GN11 Year:3 Programme:F344 Year:3,4 Programme:FGH1 Year:3,4 Programme:FG31 Year:3 Programme:GL11 Year:3 Programme:GR11 Year:4 Programme:GV15 Year:3 Programme:BCG0 Year:3 Programme:L000 Year:3 Programme:Y001 Year:3 Programme:MMAS Year:1

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Written Exam  2.5 hours  Second semester  100  Standard University Policy    Assessment 1 Notes (applying to all assessments) Full marks will be awarded for complete answers to FIVE questions. Only the best FIVE answers will be taken into account.  
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes