Elasticity (physics)

Continuum mechanics

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Elasticity is a branch of physics which studies the properties of elastic materials. A material is said to be elastic if it deforms under stress (e.g., external forces), but then returns to its original shape when the stress is removed. The amount of deformation is called the strain.

Modeling elasticity

The elastic regime is characterized by a linear relationship between stress and strain, denoted linear elasticity. Good examples are a rubber band and a bouncing ball. This idea was first stated^[1] by Robert Hooke in 1675 as a Latin anagram^[2] "ceiiinossssttuv", whose solution he published in 1678 as "Ut tensio, sic vis" which means "As the extension, so the force." This linear relationship is called Hooke's law. The classic model of linear elasticity is the perfect spring. Although the general proportionality constant between stress and strain in three dimensions is a 4th order tensor, when considering simple situations of higher symmetry such as a rod in one dimensional loading, the relationship may often be reduced to applications of Hooke's law. Because most materials are only elastic under relatively small deformations, several assumptions are used to linearize the theory. Most importantly, higher order terms are generally discarded based on the small deformation assumption. In certain special cases, such as when considering a rubbery material, these assumptions may not be permissible. However, in general, elasticity refers to the linearized theory of the continuum stresses and strains.

Transitions to inelasticity

Above a certain stress known as the elastic limit or the yield strength of an elastic material, the relationship between stress and strain becomes nonlinear. Beyond this limit, the solid may deform irreversibly, exhibiting plasticity. A stress-strain curve is one tool for visualizing this transition. Furthermore, not only solids exhibit elasticity. Some non-Newtonian fluids, such as viscoelastic fluids, will also exhibit elasticity in certain conditions. In response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape. Under larger strains, or strains applied for longer periods of time, these fluids may start to flow, exhibiting viscosity.

See also

• Stiffness
• Elastic modulus
• Linear elasticity
• 3-D elasticity
• Pseudoelasticity

References

1. ^ http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml
2. ^ cf. his description of the catenary, which appeared in the preceding paragraph.

• W.J. Ibbetson (1887), An Elementary Treatise on the Mathematical Theory of Perfectly Elastic Solids, McMillan, London, p.162
• L.D. Landau, E.M. Lifshitz (1986), Course of Theoretical Physics: Theory of Elasticity Butterworth-Heinemann, ISBN 0-7506-2633-X
• J.E. Marsden, T.J. Hughes (1983), Mathematical Foundations of Elasticity, Dover, ISBN 0-486-67865-2
• P.C. Chou, N. J. Pagano (1992), Elasticity: Tensor, Dyadic, and Engineering Approaches, Dover, ISBN 0-486-66958-0
• R.W. Ogden (1997), Non-linear Elastic Deformation, Dover, ISBN 0-486-69648-0