In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials.
Mathematically, Hooke's law states that
When this holds, the behavior is said to be linear. If shown on a graph, the line should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the displacement (for example, when a spring is stretched to the left, it pulls back to the right).
Hooke's law is named after the 17th century British physicist Robert Hooke. He first stated this law in 1676 as a Latin anagram, whose solution he published in 1678 as Ut tensio, sic vis, meaning, "As the extension, so the force".
Objects that quickly regain their original shape after being deformed by a force, with the molecules or atoms of their material returning to the initial state of stable equilibrium, often obey Hooke's law.
We may view a rod of any elastic material as a linear spring. The rod has length L and cross-sectional area A. Its extension (strain) is linearly proportional to its tensile stress σ, by a constant factor, the inverse of its modulus of elasticity, E, hence,
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 aluminium, 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.
Rubber is generally regarded as a "non-hookean" material because its elasticity is stress dependent and sensitive to temperature and loading rate.
Applications of the law include spring operated weighing machines, stress analysis and modeling of materials.
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.
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
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 non-negative.)
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.
or as a frequency
When two springs are attached to a mass and compressed, the following table compares values of the springs.
|Comparison||In Series||In Parallel|
|Equivalent Spring Constant (Series)|
|Deriving keq in
the series case is a little trickier than in the parallel case.
Defining the equilibrium position of the block to be
x2, we'll be looking for an equation for the
force on the block that looks like:
To begin, we'll also define the equilibrium position of the point between the two springs to be x1.
The force on the block is
Meanwhile, the force on the point between the two springs is
Now, when the block is pushed so the springs are compressed and the system is allowed to come to equilibrium, the force between the springs must sum to zero, so with Fs = 0 we can solve for :
Now we just plug this back into (1):
Finally, the force on the block has been found:
So we can define everything in the parenthesis to be
Which can also be written:
|Equivalent Spring Constant (Parallel)|
|Both springs are touching the block in this case, and whatever
distance spring 1 is compressed has to be the same amount spring 2
The force on the block is then:
So the force on the block is
Which is why we can define the equivalent spring constant as
|In the case where two springs are in series, the magnitude of
the force of the springs on each other are equal:
For spring 1, x1 is the distance from equilibrium length, and for spring 2, x2 - x1 is the distance from its equilibrium length. So we can define
Plug these definitions into the force equation, and we'll get a relationship between the compressed distances for the in series case:
|For the series case, the ratio of energy
stored in springs is:
but a there is a relationship between a1 and a2 derived earlier, so we can plug that in:
For the parallel case,
because the compressed distance of the springs is the same, this simplifies to
The tensor is called the stiffness tensor. 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 are also in units of pressure.
The expression for generalized Hooke's law can be inverted to get a relation for the strain in terms of stress:
The tensor is called the compliance tensor.
(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 any 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. (Symon (1971) Ch. 10) Thus:
where δij is the Kronecker delta. 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:
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:
|Derivation of Hooke's law in 3D|
|The 3-D form of Hooke's law can be derived using Poisson's
ratio and the 1-D form of Hooke's law as follows.
Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3),
Summing the three cases together () we get
or by adding and subtracting one νσ
and further we get by solving σ1
Calculating the sum
and substituting it to the equation solved for σ1 gives
where μ and λ are the Lamé parameters. Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions.
The symmetry of the Cauchy stress tensor () and the generalized Hooke's laws () implies that . Similarly, the symmetry of the infinitesimal strain tensor implies that . These symmetries are called the minor symmetries of the stiffness tensor ().
If in addition, the stress-strain relation can be derived from a strain energy density functional (U), then
The arbitrariness of the order of differentiation implies that . These are called the major symmetries of the stiffness tensor. The major and minor symmetries indicate that the stiffness tensor has only 21 independent components.
It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called Voigt notation. To do this we take advantage of the symmetry of the stress and strain tensors and express them as six-dimensional vectors in an orthonormal coordinate system () as
Then the stiffness tensor () can be expressed as
and Hooke's law is written as
Similarly the compliance tensor () can be written as
If a linear elastic material is rotated from a reference configuration to another, then the material is symmetric with respect to the rotation if the components of the stiffness tensor in the rotated configuration are related to the components in the reference configuration by the relation 
where lab are the components of an orthogonal rotation matrix [L]. The same relation also holds for inversions.
In matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by
In addition, if the material is symmetric with respect to the transformation [L] then
The inverse of this relation is commonly written as
More frequently, the axis is taken to be the axis of symmetry and the inverse Hooke's law is written as 
where is the surface traction vector, is the body force vector, represents the body and represents its surface. Using the relation between the Cauchy stress and the surface traction, (where is the unit outward normal to ), we have
Converting the surface integral into a volume integral via the divergence theorem gives
Using the symmetry of the Cauchy stress and the identity
Hence we can write
and therefore the variation in the internal energy density is given by
An elastic material is defined as one in which the total internal energy is equal to the potential energy of the internal forces (also called the elastic strain energy). Therefore the internal energy density is a function of the strains, and the variation of the internal energy can be expressed as
Since the variation of strain is arbitrary, the stress-strain relation of an elastic material is given by
For a linear elastic material, the quantity is a linear function of , and can therefore be expressed as
where is a fourth-order tensor of material constants, also called the stiffness tensor.
From the name of Robert Hooke, English physicist
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The spring equation shows Hooke's Law with math.
= The length of a spring always changes by the same amount when it is pushed or pulled. The equation for that is:
The negative sign is because the direction of the force is always opposite of the direction that the spring was pushed or pulled. When x = 0, the spring how far it was when there was no push or pull on it. The spring is at the equilibrium position. There is a minus sign because the force of the spring does not go to the same direction as the spring moves. This equation only works on a linear spring. A linear spring is a spring that is only being pushed or pulled in one direction, like left or right or up or down.
Elastic potential energy is the energy saved in an object that is stretched, compressed (compression is pressing objects together), twisted or bent. For example, an arrow gets the elastic potential energy from the bow. When it leaves the bow, the potential energy turns into kinetic energy. The equation of the elastic potential energy is: