Mekanik, KTH
Advanced Topics in Continuum Mechanics 5C5125, 4p
Graduate course spring 1999
Approximately 6 double hours in the first half of the spring
This course is a continuation of the undergraduate course in continuum mechanics that I have been given regularly at KTH.
List of contents
 There is a natural dividing line in continuum mechanics between conservative (or elastic as is the term in continuum mechanics) and dissipative materials. The most systematic and powerful approach for the elastic case is that of Hamilton's principle. The material properties are described by one scalar function, the Lagrangian density. The action, to be varied, also contains the relevant boundary conditions.

There is an interesting variant of kinematics called inverse kinematics. The position in the reference configuration is then considered to be a function of present position and time, X=X(x, t). A distinct advantage of this formulation is that X is then a function of space and time like all other fields, e.g., the electromagnetic field. 

For dissipative materials the situation is more complicated than for elastic materials. In the simplest cases dissipation is caused by gradients in the temperature and velocity fields. The theory is well developed for small (linear) dissipation and is closely related to what is often called linear theory of irreversible thermodynamics. Physically, this means that locally, on a molecular scale, the material is almost at equilibrium, the small deviations being associated with dissipation.

Instead of a variational principle, the basis is the Clausius-Duhem inequality, a local reformulation of the second law of thermodynamics. This is reformulated as an inequality, which has to be satisfied identically by all processes and thus puts severe restrictions on the possible material functions. A typical result here is that in a Newtonian fluid, shear viscosity has to be non-negative. Similarly, volume viscosity has to have values in a well-defined range.

The first topic to be covered is that of curvilinear coordinates. In the undergraduate course I have been using a moving orthonormal frame of the kind met in elementary mechanics course for polar coordinates. Another method of wide use is that of a coordinate frame, where one has to distinguish between co- and contravariant components of tensors. Both methods have advantages and disadvantages. I will consider the general case containing both.

The kinetic theory of gases gives some insights into the limitations of the linear dissipation theory. In particular, it is found that the entropy current contains terms other than those assumed in the Clausius-Duhem inequality.

Finally, som aspects of wave propagation will be treated.


The first meeting will take place March 2 at 13-15 in the seminar room at the Department of Mechanics, Osquars backe 18. We will then decide on the remaining schedule.

Schedule decided so far
 
Tuesday March 2 at 13-15
Thursday March 11 at 13-15
Tuesday March 16 at 13-15. Home assignment 1 to be handed in
Tuesday March 23 at 13-15. Home assignment 2 to be handed in
Tuesday April 13 at 13-15.  Home assignment 3 to be handed in
 Tuesday April 20 at 13-15.  Home assignment 4 to be handed in
 
 

There will be home assignments. They can be downloaded as postscript files, clicking on "Home assignment..." above. - There will also be an oral examination, which is scheduled in weeks 16-17.
  


A necessary background for this course is the undergraduate course on continuum mechanics or some similar course.
 

 For information, please contact Lars Söderholm, lhs@mech.kth.se.

This page will be updated to contain more detailed information
 
 

Last updated March 16, 1999.