

Statistical Mechanics of ElasticityCourse Information

Week 
Lecture Topics 
1 
Introduction to Hamiltonian mechanics. 
2 
Statistics in statistical mechanics, Phase functions and time averages. 
3 
Phase space dynamics of isolated systems, weakly interacting systems. 
4 
Canonical distributions. 
5 
Concepts of temperature, local equilibrium processes, phase functions for generalized forces. 
6 
First and second laws of thermodynamics. 
7 
Partition function relations, continuum formulations of nonuniform processes. 
8 
Equipartition and alternative definitions of entropy, applications to gases. 
9 
Crystal elasticity, Bravais lattices, harmonic and quasiharmonic approximations to crystals. 
10 
Taking advantage of periodicity to calculate dispersion relations in crystals: The dynamical matrix. 
11 
Rubber elasticity of single chains in 1D. 
12 
Rubber elasticity of single chains in 3D. 
13 
Rubber elasticity of networks. 
Course Format/Time
2 hours lecture and 1 hour discussion/problem section
Exam
(Sessionsprüfung) Oral 30 minutes
Testatbedingung
Completion of 80% of homework assignments
Primary Goal
To provide a modern introduction to the application of statistical mechanics to the determination of constitutive relations for elastic solids.
Reading Materials
Required: Statistical Mechanics of Elasticity, J.H.Weiner, Dover Press, 2002 (or Wiley Press 1983).
Recommended: Crystals, Defects and Microstructures, Rob Phillips, Cambridge University Press, 2001
Theoretical background: Mathematical Foundations of Statistical Mechanics, A.I. Khininchin, Dover Press, 1960 (first edition 1949)
Foundations: Elementary Principles of Statistical Mechanics, J.W. Gibbs, Ox Bow Press, 1981 (first edition 1902).
Polymer chains: Statistical Mechanics of Chain Molecules, P. J. Flory, Hanser Publishers, 1988
Polymer networks: Structures and properties of rubberlike networks, B. Erman and J.E. Mark, Oxford University Press, 1997
Reading Assignments
Unless otherwise noted, reading assignments refer to Weiner (see above)
Week 
Lecture Topics 
Reading 
1 
Introduction to Hamiltonian mechanics. 
2.1  2.3 
2 
Statistics in statistical mechanics, Phase functions and time averages. Ergodicity. Phase space dynamics of isolated systems. 
2.4  2.5 
3 
The microcanonical distribution. Weakly interacting systems. The canonical distribution. 
2.5  2.8 
4 
Empirical Temperature. Quasistatic Processes. Microscopic interpretation of the first and second law of thermodynamics 
3.1  3.6 
5 
Partition function relations, Application to examples. 
3.7 
6  Fluctuations, entropy from the information theory viewpoint  3.8  3.12 
7  Introduction to crystal lattices  4.1  4.2 
8  Thermoelastic stress  strain relations in crystalline solids: Quasiharmonic approximation  4.3  4.8 
9  Simplified phonon calculation  4.9 
10  Rubber elasticity of single polymer chains in 1D.  5.1  5.4 
11  Rubber elasticity of single polymer chains in 3D. Stress and strain ensemble.  5.5 Flory: Chapters I and VII 
12  Rubber elasticity of polymer networks.  6.10 & 5.6 
[Top]
Additional Reading/Handouts
No. 
Handout (pdf) 
1 
Notes on the derivation of the microcanonical ensemble 
2  Foundations of Statistical Mechanics (Penrose, O. 1979), detailed review covering the theoretical foundations 
3  The Ergodic Hypothesis (Patrascioiu, A. 1987), less mathematical but instructive discussion of the ergodic hypothesis 
4  Notes on discrete states and nondimensionality 
5  Statistical Thermodynamics of Random Networks (Proc. R. Soc. London A, 351 (1666): 351380 1976) 
Discussion Section
Week 
Handout (pdf) 
Additional downloads  Solution 
1 
Handout  Incomplete Matlab Files  
2 
Handout    
3 
Handout  harmonicOscillator.nb  
4 
Handout    
5 
Handout  B2T.nb  
6 
Handout    
7 
Handout  plotLattice.m  
8 + 9 
Handout    
10 
Handout  Incomplete Matlab Files  
11 
Handout    
12 
[Top]
