Elasticity and molecular properties

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A material becomes deformed when a stress is applied to it. Its elasticity is the material's ability to return to its original form once the stress is removed. This way the deformation of the material occurs follows Hooke's law: it is proportional to the stress up until a certain point, which is called the elastic limit. Any additional stress applied after the elastic limit has been reached, the material becomes permanently deformed -- it cannot return to its original shape. When stresses are so large that Hooke's law is no longer satisfied, we are talking about the nonlinear region. In the linear region, where Hooke's law does apply, we see that the graph of elasticity versus stress is linear. After the elastic limit it becomes curved and decisively non-linear.

All materials have an elastic region, although for hard systems this regions is typically very small. After a stress is applied, the resulting strain is related to the extent with which atoms move away from their equilibrium states. Atomic adjustments are localized in systems such as crystalline solids can only return to their original shape if they were very slightly deformed. Some polymeric materials, however, can be largely deformed without breaking. A classic example of this hyperelastic behavior is rubber (just think about how much you can stretch a rubber band before it breaks or becomes permanently stretched out!) The polymer chains in rubber are highly linked together and rotate around one another’s bonds when stress is applied. Rubber-like materials also become more deformed in the presence of a small stress as compared to non-rubber-like materials. Rubbers also releases heat when stretched and contracts when heated. Some common rubber and rubber-like compounds are:

polyvinyl chloride,
polystyrene , and
polymethyl methacrylate.

Each of these materials have long-chain molecules with freely rotation links, weak secondary forces between molecules, and interlocking of molecules to form a three-dimensional network (ie. there is cross linkage). In essence, in order to have a high degree of elasticity, a material must be able to take up a large variety of statistical conformations and be in sync at long ranges. It is interesting to note that at a certain temperature, rubbers go through a second order phase transition to enter a glassy state.

This glass transition temperature can be significant since the overall system properties differ greatly above and below this temperature. (http://www.ejpau.media.pl/volume8/issue4/art-69.html)

Strain in tension Eqn.gif,Strain in shear Eqn.gif,Definition of stress.png Tension and Shear.png
Linear elastic Eqns.png Stress vs Strain.png
            Elasticity as EnergyDensity.png

Since work is required to stretch a solid and the modulus is an energy per unit volume; or the modulus must be related to molecular interactions.

"Meaning" of elasticity

For small stresses on solids or for short times on liquid: Linear modulus Eqn.png
Energy stored per unit volume: Work of Elasticity.png
G has units of energy/volume. For "soft" matter: Elasticity wrt kT.png

Spring model of intemolecular interaction

Crystal model.png Crystal as spring model.png Spring model extended.png
L-J Potential energy.png Potential energy to elasticity.png Spring models.png

The multiple-spring model is combined with the potential energy model to calculate a molecular model of elasticity: Spring model to elasticity.png

Material Bond enery Distance Modulus
Metals 100kT About 1 Ang 10^12 Pa
Soft materials 1kT About 10 nm 10^3 Pa

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