Module Specification

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.
1. Module Title CARTESIAN TENSORS AND MATHEMATICAL MODELS OF SOLIDS AND VISCOUS FLUIDS
2. Module Code MATH324
3. Year Session 2023-24
4. Originating Department Mathematical Sciences
5. Faculty Fac of Science & Engineering
6. Semester First Semester
7. CATS Level Level 6 FHEQ
8. CATS Value 15
9. Member of staff with responsibility for the module
Professor N Movchan Mathematical Sciences Nvm@liverpool.ac.uk
10. Module Moderator
11. Other Contributing Departments  
12. Other Staff Teaching on this Module
Professor A Movchan Mathematical Sciences Abm@liverpool.ac.uk
13. Board of Studies
14. Mode of Delivery
15. Location Main Liverpool City Campus
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other TOTAL
16. Study Hours 36

  12

      48
17.

Private Study

102
18.

TOTAL HOURS

150
 
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other
19. Timetable (if known)            
 
20. Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH102 CALCULUS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra
21. Modules for which this module is a pre-requisite:

 
22. Co-requisite modules:

 
23. Linked Modules:

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

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

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

27. 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.

 
28. Learning Outcomes
 

(LO1) To understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations.

 

(LO2) To apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

 

(S1) Problem solving skills

 

(S2) Numeracy

 

(S3) Adaptability

 
29. Teaching and Learning Strategies
 

Material is presented during lectures (3 hours per week). Tutorials (1 hour per week) are used for consolidation and practice, and for help with individual questions.

 
30. Syllabus
   

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.

 
31. Recommended Texts
  Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
 

Assessment

32. EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  Final assessment 120 60
33. CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
  Homework 1 0 10
  Homework 2 0 10
  Homework 3 0 10
  Homework 4 0 10