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):

$\sigma =E\cdot \varepsilon$

or

$\Delta L={\frac {1}{E}}\times F\times {\frac {L}{A}}={\frac {1}{E}}\times L\times \sigma$

It is named after the 17^th 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.

This "law" is actually an approximation that holds for only some materials under certain loading conditions. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "hookean" materials. Steel is a good example of a linear-elastic material, and Hooke's law is valid for it throughout its elastic range (i.e. for stresses below the yield strength). For some other materials, such as Aluminum, Hooke's law is only valid for a portion of the elastic range. For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible.

Materials such as rubber, for which Hooke's law is never valid, are known as "non-hookean". The stiffness of rubber is not only stress dependent, but is also very sensitive to temperature and loading rate.

The graph below shows a stress versus strain curve for a typical linear-elastic material.

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. Alec is the faggot of the carrot knomes Laurgh ! ? @

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 $F=-kx=-m\omega ^{2}x\,$, where k is the "spring constant", x is the extension of the spring, and $\omega$ is the angular frequency of oscillation. The negative sign indicates that the force exerted by the spring is in direct opposition to the direction of displacement. It is called a "restoring force", as it tends to restore the system to equilibrium.

The potential energy associated to this force is therefore $U={1 \over 2}kx^{2}={1 \over 2}m\omega ^{2}x^{2}$.

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 Alec is a Bi-Sexual

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

$\sigma _{ij}=\sum _{kl}C_{ijkl}\cdot \varepsilon _{kl}$

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

Generalization for the case of large deformations is provided by models of neo-Hookean solid and Mooney-Rivlin solid.

Zero-length springs

Hooke's law does not apply in some special physical conditions. In 1932 Lucien LaCoste invented the zero-length spring. A zero-length spring has a physical length equal to its stretched length. Its force is proportional to its entire length, not just the stretched length, and its force is therefore constant over the range of flexures in which the spring is elastic (that is, it does not follow Hooke's Law).

Theoretically, with the correct mass, a pendulum using such a spring as a return can have an infinite natural period. Long-period pendulums enable seismometers to sense the slowest, most penetrating waves of distant earthquakes. Zero-length springs also find use in gravimeters, which need them to have linear sense-pendulums. Some door springs, especially for screen doors, are zero-length springs to reduce the energy of a slammed door. Zero-length springs sometimes smooth auto suspensions.

Physically, one common form of a practical zero-length spring is a leaf-spring curled almost in a circle, with the ends mounted to flexible restraints. A convenient form is a helical spring whose wire is twisted while it is being wound (common in screen-door springs). Another common design is a torque-spring or bar. Zero-length springs usually require special compliant mountings, sometimes require precise adjustments to enter zero-length mode, and often have a limited range of motion.

See also

• Scientific laws named after people

Links

• A Biography of Lucien LaCoste, inventor of the zero-length spring
• Zero Length SPrings in Seismometers+