The question was raised in class as to what the appropriate equilibrium condition for a column of fluid at rest should be. Specifically, given we expect a hydrostatic gradient in pressure with height, whether  the chemical potential must be the same throughout the column was questioned. Here are my first thoughts. In brief, I assert that  the chemical potential must be everywhere identical, and that the pv term is balanced, at every height in the column, by the potential energy conferred by position in a gravitational field.

These are slides of poroelasticity and diffusion in elastic solids for final presentation based on ES241 notes.

These slides are based on an on-going paper written by Wei Hong, Xuanhe Zhao and Zhigang Suo and Suo's talk in ucsb.

Attached is my final presentation.

Here are the slides for my final presentation for ES 241.  During the presentation, a few suggestions were made, which I plan to follow up on.  Please check back here or subscribe for updates.

Here are some slides I made on the subject of "Pressure and Chemical Potential" for the final meeting of Prof. Zhigang Suo's ES 241 class in the Spring of 2009.

In the field of Solid Mechanics, Harvard has a sequence of 5 graduate courses:

• ES 240 Solid Mechanics
• ES 241 Advanced Elasticity
• ES 242r Solid Mechanics:  Advanced Seminar
• ES 246 Plasticity
• ES 247 Fracture Mechanics

The first course goes over linear elasticity, finite element method, vibration, waves, viscoelasticity, as well as some ideas of finite deformation.

Please see the attached slides on electric potential, deformation and polarization.

Please see attachment for ES 241 final presentation on heat conduction and the Boltzmann distribution.

Please see attachment for ES241 final presentation--general theory of finite deformation

The notes on finite deformation have been divided into two parts: special cases and general theory (http://imechanica.org/node/538). In class I start with special cases, and then sketch the general theory. But the two parts can be read in any order.

So far we have used the fundamental postulate to study experimental phenomena by following an algorithm. For a given phenomenon, we construct an isolated system with an internal variable. The isolated system has a whole set of quantum states. Associated with each value of the internal variable, the isolated system flips among a subset of the quantum states. The fundamental postulate implies that the internal variable evolves in time, from one value corresponding to a subset of the quantum states to another value corresponding to a subset of a larger number of quantum states. After a long time, the internal variable attains an equilibrium value, corresponding to a subset of the largest number of quantum states.

For a system in thermal contact with the rest of the world, we have described three quantities: entropy, energy, and temperature. We have also described the idea of a constraint internal to the system, and associated this constraint to an internal variable.

The system can be isolated at a particular value of energy. For such an isolated system, of all values of the internal variable, the most probable value maximizes entropy. We will paraphrase this statement under two different conditions, either when the entropy is fixed, or when the temperature is fixed. Under these conditions, the system is no longer isolated. Consequently, we need to maximize or minimize quantities other than entropy.

Update on 23 May 2009:  I'm adding links to the slides as they are uploaded.

The final exam will take the form of a pedagogical workshop. We have 8 students taking the class for credit. I have divided the lecture notes into 8 parts as follows.

A sponge is an elastic solid with connected pores. When immersed in water, the sponge absorbs water. When a saturated sponge is squeezed, water will come out. More generally, the subject is known as diffusion in elastic solids, or elasticity of fluid-infiltrated porous solids, or poroelasticity. The theory has been applied to diverse phenomena. Here are a few examples.

• A homogeneous field in a parallel-plate capacitor
• Particles and places
• A field of stress
• A field of electric displacement
• Helmholtz function
• Invariance under rigid-body rotation
• Materials laws expressed in true fields
• Nonpolar material
• Isotropic material
• Electrical Gibbs function
• Fluid dielectrics
• Solid dielectrics
• Coulomb attraction between the two electrodes in a parallel-plate capacitor.
• A lateral force in a parallel-plate capacitor
• Rupture of a charged sphere
• Piezoelectric actuators and sensors

• Electric charge
• Movements of charged particles
• Elastic dielectric
• Work done by a battery and by a weight
• Electromechanical coupling
• Conservative system
• Experimental determination of electric potential
• Lagendre transformation
• parallel-plate capacitor

Return to the outline of Statistical Mechanics

• A system that can exchange particles with the rest of the world
• Chemical potential
• Ideal gas
• Experimental determination of chemical potential
• Lagendre transformation
• Ideal gas once more
• Experimental determination of chemical potential
• A system in contact with a reservoir of energy, volume and particles
• A kinetic model

Return to the outline of Statistical Mechanics

So far we have been mainly concerned with systems of a single independent variable: energy (http://imechanica.org/node/4878). We now consider a system of two independent variables: energy and volume. A thermodynamic model of the system is prescribed by entropy as a function of energy and volume.

The partial derivatives of the function give the temperature and the pressure. This fact leads to an experimental procedure to determine the function for a given system.

The laws of ideal gases and osmosis are derived. The two phenomena illustrate entropic elasticity.

Spring 2011, Tuesday and Thursday 10:00 am - 11:30 am, Cruft Lab 309. First meeting of the class:  25 January 2011

• Instructor: Zhigang Suo, 617-495-3789, suo@seas.harvard.edu, Pierce Hall 309.
• Office hour: Tuesday 2:00 pm - 3:00 pm.
• No textbook is required.

This is a second graduate course in solid mechanics.  The course builds on elements of thermodynamics, and explores coupled mechanical, thermal, electrical and chemical actions.  The course draws heavily upon phenomena in soft active materials.

This page is updated for ES 241 taught in Spring 2011. See also

The notes on finite deformation have been divided into two parts: special cases (http://imechanica.org/node/5065) and general theory. In class I start with special cases, and then sketch the general theory. But the two parts can be read in any order.

Subject to loads, a body deforms. We would like to develop a theory to evolve this deformation in time. In continuum mechanics, we model the body by a field of particles, and update the positions of the particles by using an equation of motion. We formulate the equation of motion by mixing the following ingredients:

We have described two great principles of our world: the fundamental postulate and the conservation of energy. The former is the foundation of thermodynamics, as we have learned in a previous lecture. The latter is not specific to thermodynamics: we borrow the concept of energy—along with the principle of the conservation of energy—from other branches of science, such as mechanics and electrodynamics. Both principles are abstracted from many empirical observations.

Of our world the following facts are known:

• An isolated system has a set of quantum states.
• The isolated system ceaselessly flips from one quantum state to another.
• A system isolated for a long time is equally probable to be in any one of its quantum states.

Thus, an isolated system behaves like a fair die. The following notes remind you what an isolated system is, and translate the theory of probability of rolling a fair die to the thermodynamics of an isolated system.