Hooke's law

Hooke's law accurately models the physical properties of common mechanical springs.

In physics, Hooke's law of elasticity is an approximation which states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials.

For systems that obey Hooke's law, the extension produced is proportional to the load:

$A=C(La+90)^{3}\$

where

x is the distance the spring is elongated by,

F is the restoring force exerted by the spring, and

k is the spring constant or force constant of the spring.

When this holds, we say that the spring is a linear spring.

Hooke's law mathematically comes from the fact that in most solids (and in most isolated molecules) atoms are in the state of stable equilibrium.

For many applications, a prismatic rod, with length L and cross sectional area A, can be treated as a linear spring. Its extension (strain) is linearly proportional to its tensile stress, σ by a constant factor, the inverse of its modulus of elasticity, E. Hence,

$\sigma =E\cdot \varepsilon$

or

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

Hooke's law is named after the 17^th century physicist Robert Hooke.

Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-elastic behavior in most engineering applications; 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.

Applications of the law include spring operated weighing machines, stress analysis and modeling of materials.

The spring equation

stress-strain curve for low-carbon steel. Hooke's law is only valid for the portion of the curve between the origin and the yield point.
1. Ultimate Strength
2. Yield Strength
3. Rupture
4. Strain hardening region
5. Necking region.

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 a spring constant, k, measured in force per length.

$F=-kx\,$

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 stored in a spring is given by

$U={1 \over 2}kx^{2}$

which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over distance. (Note that potential energy of a spring is always positive.)

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.

If a mass is attached to the end of such a spring and the system is bumped, it will oscilate with a natural frequency (or resonant angular (circular) frequency) of

$\omega _{n}={\sqrt {k \over m}}.$

Tensor expression of Hooke's Law

When working with a 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).

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

Due to the symmetry of the stress tensor, strain tensor, and stiffness tensor, only 21 elastic coefficients are independent.

As stress is measured in units of pressure and strain is dimensionless, the entries of c_ijkl are also in units of pressure.

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

Isotropic materials

(see viscosity for an analogous development for viscous fluids.)

Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the trace of any tensor is independent of coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor. Template:Ref harvard Thus:

$\varepsilon _{ij}=\left({\frac {1}{3}}\varepsilon _{kk}1_{ij}\right)+\left(\varepsilon _{ij}-{\frac {1}{3}}\varepsilon _{kk}1_{ij}\right)$

where $1_{ij}$ is the unit tensor. The first term on the right is the constant tensor, also known as the pressure, and the second term is the traceless symmetric tensor, also known as the shear tensor.

The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors:

$\sigma _{ij}=3K\left({\frac {1}{3}}\varepsilon _{kk}1_{ij}\right)+2G\left(\varepsilon _{ij}-{\frac {1}{3}}\varepsilon _{kk}1_{ij}\right)\,$

where K is the bulk modulus and G is the shear modulus.

Using the relationships between the elastic moduli, these equations may also be expressed in various other ways. For example, the strain may be expressed in terms of the stress tensor as:

$\varepsilon _{11}={\frac {1}{Y}}\left(\sigma _{11}-\nu (\sigma _{22}+\sigma _{33})\right)$

$\varepsilon _{22}={\frac {1}{Y}}\left(\sigma _{22}-\nu (\sigma _{11}+\sigma _{33})\right)$

$\varepsilon _{33}={\frac {1}{Y}}\left(\sigma _{33}-\nu (\sigma _{11}+\sigma _{22})\right)$

$\varepsilon _{12}={\frac {\sigma _{12}}{2G}}$

$\varepsilon _{13}={\frac {\sigma _{13}}{2G}}$

$\varepsilon _{23}={\frac {\sigma _{23}}{2G}}$

Where Y is the modulus of elasticity and $\nu$ is Poisson's ratio. (See 3-D Elasticity)

Zero-length springs

Main article: Zero-length spring

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

• Elastic
• Theory of elasticity
• Linear elasticity
• Young's modulus
• Scientific laws named after people
• 3-D Elasticity

External links

• A Biography of Lucien LaCoste, inventor of the zero-length spring
• Zero Length Springs in Seismometers
• Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0201073927.

References

• A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed