ES 241

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.

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

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.  

These notes are part of a graduate course on advanced elasticity.

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.

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.

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.

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.

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.

• 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

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.