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ES 241

Zhigang Suo's picture


These notes may serve as a reminder of tensor algebra.  The notes supplement the course on advanced elasticity.  Several books are listed at the end of the notes.

Update on 17 April.  I am now breaking up the notes on tensors, cleaning them up, and posting them as sections on linear algebra.

Zhigang Suo's picture

Elasticity of rubber-like materials

In the notes on the general theory of finite deformation, we have left the free energy function unspecified. The notes here describe free energy function commonly used to describe the elasticity of rubber-like materials.  These notes are part of a course on advanced elasticity

Zhigang Suo's picture


These notes are part of a course on advanced elasticity.  The notes recall several phenomena where both elasticity and surface energy are significant, including

  • Griffith crack
  • Adhesion of flexible structures
  • Wafer bonding
  • Contraction of a soft elastic sheet 

The notes also contain a formulation of combined surface energy and elasticity of finite deformation.  

Zhigang Suo's picture

Theory of dielectric elastomers

In response to a stimulus, a soft material deforms, and the deformation provides a function. We call such a material a soft active material (SAM). This review focuses on one class of soft active materials: dielectric elastomers. Subject to a voltage, a membrane of a dielectric elastomer reduces thickness and expands area, possibly straining over 100%. The phenomenon is being developed as transducers for broad applications, including soft robots, adaptive optics, Braille displays, and electric generators.

Tony Rockwell's picture

Addendum To Pressure and Chemical Potential - a question on hydrostatics

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.

Cai Shengqiang's picture

Poroelasticity and diffusion in elastic solids

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

Yuhang Hu's picture

advanced elasticity 2009 slides (polyelectrolyte gels)

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

Final presentation

Attached is my final presentation.

Final presentation slides

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.

Tony Rockwell's picture

Slides on Pressure and Chemical Potential

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.

Zhigang Suo's picture

A course on Advanced Elasticity, with emphasis on thermodynamics and soft active materials

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

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

Zhigang Suo's picture

Finite Deformation: Special Cases

The notes on finite deformation have been divided into two parts: special cases and general theory (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.

Zhigang Suo's picture

Free Energy

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.

Zhigang Suo's picture

ES 241 Advanced Elasticity Final Examination

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.

Zhigang Suo's picture


  • Free energy and generalized coordinate. Equilibrium and stability
  • Control parameter
  • Configurational transitions of two types
  • Critical point of configurational transition of the second type. Bifurcation analysis
  • Behavior near a critical point. Post-bifurcation analysis
  • Load-displacement relation near a critical point
  • Koiter's theory of imperfection sensitivity
  • A family of systems of many degrees of freedom
  • Mode of bifurcation
  • Vibration in the neighborhood of an equilibrium configuration
Zhigang Suo's picture

Complex variable methods

  • What types of PDEs can be solved using complex variable methods
  • Anti-plane shear
  • Elements of a function of a complex variable (contour integral, analytic continuation, conformal mapping)
  • Line force
  • Screw dislocation
  • Crack
  • Circular hole
  • Elliptic hole
  • Plemelj formulas
  • Riemann-Hilbert problem
  • Crack interacting with a point singularity
  • In-plane deformation
  • Dundurs parameters
  • Interfacial cracks
  • Anisotropic materials. Stroh formalism
Zhigang Suo's picture

Homework on finite deformation

To the students of ES 241:

Although finite deformation was introduced in ES 240 (Solid Mechanics), finite deformation is a building block of ES 241. To review the subject, please go over a set of problems compiled by Jim Rice. If you need a reference, see my outline of finite deformation, where you can also find a short list of textbooks.

Zhigang Suo's picture

Poroelasticity, or migration of matter in elastic solids

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.


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