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 by Robert Hooke in 1675 as a Latin anagram 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 elastic only 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
References
- 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
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Last updated on Sunday September 28, 2008 at 13:18:05 PDT (GMT -0700)
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