# Hooke's law

spring constantforce constantelasticity tensorHookeanHooke’s lawElastic constantsforce constantslinear elasticreturn springsspring constants
Hooke's law is a law of physics that states that the force (F) needed to extend or compress a spring by some distance x scales linearly with respect to that distance.wikipedia
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### Spring (device)

Hooke's law is a law of physics that states that the force (F) needed to extend or compress a spring by some distance x scales linearly with respect to that distance.
In 1676 British physicist Robert Hooke postulated Hooke's law, which states that the force a spring exerts is proportional to its extension.

### Deformation (mechanics)

straindeformationshear strain
The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it.
The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials.

### Elasticity (physics)

elasticityelasticelasticity theory
Hooke's equation holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar.
If the material is isotropic, the linearized stress–strain relationship is called Hooke's law, which is often presumed to apply up to the elastic limit for most metals or crystalline materials whereas nonlinear elasticity is generally required to model large deformations of rubbery materials even in the elastic range.

### Stiffness

flexibilityrigidityrigid
That is: F_s=kx, where k is a constant factor characteristic of the spring: its stiffness, and x is small compared to the total possible deformation of the spring.
A description including all possible stretch and shear parameters is given by the elasticity tensor.

### Force

forcesattractiveelastic force
Hooke's law is a law of physics that states that the force (F) needed to extend or compress a spring by some distance x scales linearly with respect to that distance.
Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs.

### Anagram

anagrammaticanagram solveranagrammatically
He first stated the law in 1676 as a Latin anagram.
When Robert Hooke discovered Hooke's law in 1660, he first published it in anagram form,, for ut tensio, sic vis (Latin: as the extension, so the force).

### Spring scale

spring balancespring scales
It is also the fundamental principle behind the spring scale, the manometer, and the balance wheel of the mechanical clock.
It works by Hooke's Law, which states that the force needed to extend a spring is proportional to the distance that spring is extended from its rest position.

### Robert Hooke

HookeDr Robert HookeHooke, Robert
The law is named after 17th-century British physicist Robert Hooke.
In 1660, Hooke discovered the law of elasticity which bears his name and which describes the linear variation of tension with extension in an elastic spring.

### Tensor

tensorsorderclassical treatment of tensors
However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a linear map (a tensor) that can be represented by a matrix of real numbers.
The stress tensor and strain tensor are both second-order tensor fields, and are related in a general linear elastic material by a fourth-order elasticity tensor field.

### Viscosity

viscouskinematic viscositydynamic viscosity
in flows of viscous fluids; although the former pertains to static stresses (related to amount of deformation) while the latter pertains to dynamical stresses (related to the rate of deformation).
For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium.

### Balance wheel

foliotbalancecompensation balance
It is also the fundamental principle behind the spring scale, the manometer, and the balance wheel of the mechanical clock.
A balance wheel's period of oscillation T in seconds, the time required for one complete cycle (two beats), is determined by the wheel's moment of inertia I in kilogram-meter 2 and the stiffness (spring constant) of its balance spring κ in newton-meters per radian:

### Shear modulus

modulus of rigidityshear moduliRigidity modulus
can be reduced to only two independent numbers, the bulk modulus K and the shear modulus G, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.
In this case, one may need to use the full tensor-expression of the elastic constants, rather than a single scalar value.

### Neo-Hookean solid

Neo-Hookeanneo-Hookean model
Generalizations of Hooke's law for the case of large deformations is provided by models of neo-Hookean solids and Mooney–Rivlin solids.
A neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations.

### Bulk modulus

bulkbulk modulicompressibility
can be reduced to only two independent numbers, the bulk modulus K and the shear modulus G, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.
For a complex anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law.

### Yield (engineering)

yield strengthyield stressyield
Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.

### Young's modulus

Young’s modulusmodulustensile modulus
:In terms of Young's modulus and Poisson's ratio, Hooke's law for isotropic materials can then be expressed as
At near-zero stress and strain, the stress–strain curve is linear, and the relationship between stress and strain is described by Hooke's law that states stress is proportional to strain.

### Restoring force

Hooke's law for a spring is often stated under the convention that F_s is the restoring force exerted by the spring on whatever is pulling its free end.
The amount of force can be determined by multiplying the spring constant of the spring by the amount of stretch, also known as the Hooke's Law.

### Transverse isotropy

Transversely isotropicThomsen parametersBackus upscaling
A transversely isotropic material is symmetric with respect to a rotation about an axis of symmetry.
This type of material exhibits hexagonal symmetry (though technically this ceases to be true for tensors of rank 6 and higher), so the number of independent constants in the (fourth-rank) elasticity tensor are reduced to 5 (from a total of 21 independent constants in the case of a fully anisotropic solid).

### Poisson's ratio

Poisson ratioPoisson’s ratioPoisson effect
:In terms of Young's modulus and Poisson's ratio, Hooke's law for isotropic materials can then be expressed as
Thus it is possible to generalize Hooke's Law (for compressive forces) into three dimensions:

### Linear elasticity

elastic waveselastic wavelinear elastic
An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.
In isotropic media, the stiffness tensor gives the relationship between the stresses (resulting internal stresses) and the strains (resulting deformations).

### Acoustoelastic effect

This relationship is commonly known as the generalised Hooke's law.

### Scientific law

laws of physicsphysical lawlaws of nature
Ohm's law only applies to linear networks, Newton's law of universal gravitation only applies in weak gravitational fields, the early laws of aerodynamics such as Bernoulli's principle do not apply in case of compressible flow such as occurs in transonic and supersonic flight, Hooke's law only applies to strain below the elastic limit, Boyle's law applies with perfect accuracy only to the ideal gas, etc. These laws remain useful, but only under the conditions where they apply.

### Zener ratio

universal elastic anisotropy index
It replaces the Zener ratio, which is suited for cubic crystals.
where C_{ij} refers to Elastic constants in Voigt notation.

### Spring system

This generalizes Hooke's law to higher dimensions.

### Lamé parameters

Lamé constantsLamé's first parameter\lambda
are the Lamé constants,
In homogeneous and isotropic materials, these define Hooke's law in 3D,