# Hooke's law

**spring constantforce constantelasticity tensorHookeanElastic constantsforce constantslinear elasticreturn springsspring constantsAs the extension, so the force**

Hooke's law is a law of physics that states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance.wikipedia

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### 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. 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 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, a musician plucking a string of a guitar, and the filling of a party balloon.

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 is a constant factor characteristic of the spring: its stiffness, and 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

**forcesattractiveforce vector**

Hooke's law is a law of physics that states that the force needed to extend or compress a spring by some distance 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.

### 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.

### Anagram

**anagrammaticanagram solveranagrams**

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, ceiiinosssttuv, for ut tensio, sic vis (Latin: as the tension, so the force).

### 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

**tensorsorderrank**

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. 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.

| E.g. elasticity tensor

### Shear modulus

**modulus of rigidityRigidity modulusshear moduli**

can be reduced to only two independent numbers, the bulk modulus and the shear modulus, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.

All of them arise in the generalized Hooke's law:

### Balance wheel

**foliotcompensation balanceauxiliary temperature compensation**

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:

### Viscosity

**viscousdynamic viscositykinematic 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.

### 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

**bulk moduliAdiabatic Bulk modulusbulk**

can be reduced to only two independent numbers, the bulk modulus and the shear modulus, 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.

Proportionality limit: Up to this amount of stress, stress is proportional to strain (Hooke's law), so the stress–strain graph is a straight line, and the gradient will be equal to the elastic modulus of the material.

### Spring (device)

**springspringsspring-loaded**

Hooke's law is a law of physics that states that the force needed to extend or compress a spring by some distance 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.

### Young's modulus

**modulusYoung’s ModulusTensile 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

**restoring forces**

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.

### Transverse isotropy

**transversely isotropic**

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).

### Linear elasticity

**elastic waveslinear elastic3-D elasticity**

An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.

For elastic materials, Hooke's law represents the material behavior and relates the unknown stresses and strains.

### Poisson's ratio

**Poisson’s RatioPoisson contractionsPoisson 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:

### Acoustoelastic effect

Acoustoelastic effect

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

### Zener ratio

It replaces the Zener ratio, which is suited for cubic crystals.

where C_{ij} refers to Elastic constants in Voigt notation.

### Scientific law

**lawlawsempirical law**

Laws of science

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, etc. These laws remain useful, but only under the conditions where they apply.

### Spring system

Spring system

This generalizes Hooke's law to higher dimensions.

### Lamé parameters

**Lamé constants\lambdaelastic properties**

are the Lamé constants,

In homogeneous and isotropic materials, these define Hooke's law in 3D,