# Hooke's law

In physics, Hooke's law of elasticity states that if a force (F) is applied to an elastic spring or prismatic rod (with length L and cross section A), its extension is linearly proportional to its tensile stress σ and modulus of elasticity (E):

ΔL = 1/E × F × L/A = 1/E × L × σ

It is named after the 17th century physicist Robert Hooke, who initially published it as the anagram ceiiinosssttuv, which he later revealed to mean ut tensio sic vis, or as the extension, the force.

The law holds up to a limit, called the elastic limit, or limit of elasticity, after which the metal will enter a condition of 'yield' and the spring will suffer plastic deformation up to the plastic limit or limit of plasticity, after which it will eventually break if the force is further increased (see tensile strength).

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Stress-strain1.png
Image:stress-strain1.png

Applications of the law include spring operated weighing machines. Originally the law applied only to stretched springs, but subject to physical constraints it also applies to compression springs.

## Spring equation

The most commonly encountered form of Hooke's law is probably the spring equation, which relates the force exerted by a spring to the distance it is stretched by [itex]F=-kx\, [itex], where k is the "spring constant" and x is the extension of the spring. The negative sign indicates that the force exerted by the spring is in direct opposition to the direction of displacement.

The potential energy associated to this force is therefore [itex]U={1\over2}kx^2[itex].

This potential can be visualized as a parabola on the U-x plane. As the spring is stretched in the positive x-direction, the potential energy increases (the same thing happens as the spring is compressed). The corresponding point on the potential energy curve is higher than that corresponding to the equilibrium position (x=0). The tendency for the spring is to therefore decrease its potential energy by returning to its equilibrium (unstretched) position, just as a ball rolls downhill to decrease its gravitational potential energy.

## Generalized Hooke's law

When working a with three-dimensional stress state, a 4th order tensor (Cijkl) containing 81 elastic coefficients must be defined to link the stress tensorij) and the strain tensor (or Green tensor) (εkl).

[itex]\sigma_{ij} = \sum_{kl} C_{ijkl} \cdot \varepsilon_{kl}[itex]

Actually, due to the symmetry of the stress and strain tensor, only 36 elastic coefficients are independent.

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