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Stat. Mech. of Elasticity
 



Statistical Mechanics of Elasticity

Course Information 
Course Material 

Course Information

Lecturer

 

Prof. Dr. S. Govindjee

 

Teaching Assistant

 

Felix Hildebrand

 

Number

 

151-0420-00

 

Time and location

 

Summer semester

Lecture: Thursday 10-12 , CAB H57

Discussion section: Tuesday 12-13, CLA E4

 

Type/Credits

 

3G/4KP

 

Language

 

English

 

Level

 

Masters

 

Short Description

 

Introduction to statistical mechanics for engineers interested in the constitutive behavior of elastic continua. The primary systems of interest will be polymers and crystalline solids. Coverage will include an introduction to statistical mechanics, notions of ensembles, phase spaces, partition functions, derivation of constitutive relations, polymer chain statistics, polymer networks, harmonic and quasi-harmonic crystalline solids, limitations of classical methods and quantum mechanical influences.

 

Description

 

The theories of continuum mechanics form a solid foundation for the description of the deformation of many engineering systems. At the heart of the application of such frameworks is a description of the make up of the material -the constitutive model. In this regard, one can approach the specification from a phenomenological viewpoint, a mathematical viewpoint, and/or a physical viewpoint. Of great appeal is the notion of using information form detailed molecular and atomistic characterizations of materials to construct the constitutive relations. Statistical mechanics provides an interesting and powerful tool to effect such a procedure. This course is intended for students with a background in continuum mechanics that desire a firmer understanding of the atomistic aspects of the subject. The course will first cover a basic presentation of thermo-elasticity from a continuum viewpoint. Then fundamental concepts of classical statistical mechanics will be introduced such as Boltzmann's entropy, phase space averages and canonical distributions. Use will be made of Hamilton's formulation of mechanics in this regard. The special cases of isolated and weakly interacting systems will be defined and discussed throughly. These two presentations, continuum mechanics and statistical mechanics, will next be combined and corresponding notions from both descriptions will be identified and discussed. Particular emphasis will be placed on the statistical basis for continuum state functions and quantities derived from them. Applications of this framework will be made to the development of constitutive relations based on microscale information for a variety of systems: ideal and van der Walls gases, single polymer chains, elastomeric solids, and crystalline solids.

 

Contents
 

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
quasi-harmonic 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

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Course Material

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 non-dimensionality
5 Statistical Thermodynamics of Random Networks (Proc. R. Soc. London A, 351 (1666): 351-380 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

 

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15 Feb 2008 | Stephan Kaufmann | ZfM | ETH