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MA 5019: Mathematical Continuum Mechanics

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This course contains the essential conservation laws of continuum mechanics, i.e. mass, momentum and energy conservation. These laws are also satisfied for the classical Newtonian mechanics. Therefore the knowledge of distributions is appreciated, but is not required. Also elastic materials are introduced in physical coordinates. The entropy principle is defined as a differential inequality and the free energy inequality is considered as a special case.

As stated above, the knowledge of distributions is not necessary for successfully passing of the lecture, but distributions lead to a better understanding.

After this general part, we will focus on specific models, for example the following applications (only a limited number of applications has to be understood):

Lecture Schedule

Lecture   Date      Topic       Script
1. 22nd Oct      
      Overview / Introduction
    Mass and momentum
    (Definitions, Figure 1)
    Conservation laws



 Cont 
 Cont 
    Main principles
    Notations of derivatives

    Conservation law (I.1.1), General mass conservation (I.1.6)
    I.1.1 Representation of the divergence operator
    I.1.6 Exapmle: Mixture - Air
2. 23rd Oct      
      Conservation laws
    Distributions
   
 Cont 
 Cont 
 Cont 
    I.1.7 Relativity of velocity, I.1.8 Example (Mass conservation)
    I.2.1 Definition (Derivatives, Multiplication), I.2.2 Functions as distribution
    Conservation law in the distributional sense (I.2.2)
3. 29th Oct      
      Gravitation
   
 Cont 
 Cont 
 Cont 
    Newton's gravitation (I2.10), I.2.13 Gravitational field of a globe
    I.1.12 Theorem (Jump condition)
    I.2.14 Convergence to a mass point
4. 30th Oct      
      Conservation of momentum
   
 Cont 
 Cont 
 Cont 
    General mass-momentum equation (I3.1) and (I3.3)
    I.2.5 Moving mass point, I.2.6 Lemma (Mass equation)
    I.3.1 Mass point, I.3.2 Collision of mass points
5. 5th Nov      
      Conservation of momentum
   
 Cont 
 Cont 
 Cont 
    Momentum for planets
    I.3.3 Kepler’s laws of planetary motion
    Collection of mass points
6. 6th Nov      
      Conservation of momentum
    Flow problems
 Cont 
 Cont 
 Cont 
    I.3.4 Multiple mass points
    (Compressible) Navier-Stokes equations (I3.19)
    Incompressible Navier-Stokes equation (I3.24)
7. 12th Nov      
      Flow problems
   
 Cont 
 Cont 
    I.3.5 Centrifuge, I.3.6 Different Materials (2),(3)
    I.3.7 Poiseuille flow in a pipe
8. 13th Nov      
      Interfaces
   
 Cont 
 Cont 
 Cont 
    I.4.4 Gravity field of an incompressible planet
    General equations (I4.1)
    I.4.1 Principle of Archimendes
9. 19th Nov      
      Change of coordinates
   
 Cont 
 Cont 
 Cont 
    General transformation rule (I.5.1 Theorem, I.5.2 Property)
    Invariance of the divergence system with respect to Z (I5.11) (or (I5.8)
    Example: I.5.4 Cylindrical coordinates
10. 20th Nov      
     
   Reference coordinates
 Cont 
 Cont 
    Example: I.5.5. Air flow on the earth
    Reference coordinates (I6.2)
11. 26th Nov      
     Reference coordinates
   
   Observers transformations
 Cont 
 Cont 
 Cont 
 Cont 
    I.6.1 Lemma, I.6.2 Theorem: Mass and momentum equation
    Nonlinear elasticity (I6.9),(I6.10), Rigid bodies: Lemma I.6.4
    II.1.1 Galilei transformation, Newtonian transformation (II1.3)
    II.1.4 Linear approximation
12. 27th Nov      
     Observers transformations
   Objectivity
   
 Cont 
 Cont 
 Cont 
    Characterization of Newtonian transformation: II.1.6 Theorem
    II.3.2 Scalar equation, II.3.3 Objective tensors, II.3.4 Velocity (Definition)
    II.3.5 Mass equation, II.3.6 Gravitation law, II.3.7 Mass-momentum equation
13. 10th Dec      
     Objectivity of balance laws
   
   Constitutive relations
 Cont 
 Cont 
 Cont 
    II.3.7 Mass-momentum equation, II.3.9 Classical force (Definition)
    II.3.12 Mass-momentum-energy equation (Definition)
    II.4.1 Definition (Objective constitutive function), II.4.2 Example
14. 11th Dec      
     Constitutive relations
   
   
   
 Cont 
 Cont 
 Cont 
 Cont 
    II.4.3 Example, II.4.4 Inequality, II.4.5 Lemma (Objectivity of \hat{J})
    II.4.8 Lemma (Objective representation of \Pi)
    II.4.9 Lemma (Layered material), II.4.14. Lemma (Objective tensor)
    II.4.15. Constitutive function for liquids (without proof)
15. 17th Dec      
     Entropy
   
 Cont 
 Cont 
    III.1.1 Entropy principle, III.1.2 Property
    III.1.3 Example from gas theory
16. 18th Dec      
     Entropy
   
   
 Cont 
 Cont 
 Cont 
    III.1.4 Gibbs relation, 1.7 Example
    III.2.1 Energy identity, Mass-momentum-energy system (III2.5)
    III.2.2 Lemma
17. 7th Jan      
     Entropy
   
 Cont 
 Cont 
    III.2.3 Lemma
    III.2.4 Theorem: Residual inequality for Mass-momemtum-energy system
18. 8th Jan      
     Entropy
   Applications
 Cont 
 Cont 
    III.2.5 Theorem: Gases and liquids
    IV.1 Tidal period
19. 14th Jan      
     Applications  Cont      IV.1 Tidal period
20. 15th Jan      
     Applications  Cont      IV.1 Tidal period
21. 21st Jan      
     Applications  Cont      IV.2 Fluids and gases
22. 22nd Jan      
     Applications  Cont      IV.8 Navier-Stokes equation
23. 28th Jan      
     Applications  Cont      IV.9 Vorticity
24. 29th Jan      
     Applications  Cont      IV.9 Vorticity
25. 4th Feb      
     Applications  Cont      IV.15 Self-gravitation
26. 5th Feb      
     Applications  Cont      IV.10 Reaction-diffusion systems

Exercise Schedule

Exercise   Date      Content      
1. 29th Oct / 30th Oct  
      I.1.3 Plane polar coordinates, I.1.4 Plain divergence
2. 5th Nov / 6th Nov  
      Cylinder Coordinates Explanation distributions, I.1.10
3. 12th Nov / 13th Nov  
      Gravitational field of a hollow sphere and convergence result
    Script Exercises Mathematical Continuum Mechanics, see Section 6
4. 19th Nov / 20th Nov  
      Gravitational field of a sphere shell
    Exercise I.7.16: Free energy
    Exercise I.7.17: Earth's atmosphere
5. 26th Nov / 27th Nov  
      Low-pressure area
    Script Exercises Mathematical Continuum Mechanics, see Section 3: Cyclone
6. 10th Dec / 11th Dec  
      Exercise II.5.3
    Exercise II.5.6: Material derivative
7. 17th Dec / 18th Dec  
      II.4.11 Elastic body and objectivity
    II.4.12 Constitutive function for elastic bodies
8. 7th Jan / 8th Jan  
      Thermometer, III.2.6

Time and Location

Material