In
mechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
, and
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
,
Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law is a useful approximation are known as
linear-elasticLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
or "Hookean" materials. Hooke's law in simple terms says that
strainIn continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body...
is directly proportional to stress.
Mathematically, Hooke's law states that
where
- x is the displacement
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
of the spring's end from its equilibriumA standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
position (a distance, in SISi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units: meters);
- F is the restoring force exerted by the spring on that end (in SI units: N or kg·m/s2); and
- k is a constant called the rate or spring constant (in SI units: N/m or kg/s2).
When this holds, the behavior is said to be
linear. If shown on a graph, the line should show a
direct variationIn mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
. 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 HookeRobert Hooke FRS was an English natural philosopher, architect and polymath.His adult life comprised three distinct periods: as a scientific inquirer lacking money; achieving great wealth and standing through his reputation for hard work and scrupulous honesty following the great fire of 1666, but...
. He first stated this law in 1660 as a
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
anagramAn anagram is a type of word play, the result of rearranging the letters of a word or phrase to produce a new word or phrase, using all the original letters exactly once; e.g., orchestra = carthorse, A decimal point = I'm a dot in place, Tom Marvolo Riddle = I am Lord Voldemort. Someone who...
, whose solution he published in 1678 as
Ut tensio, sic vis, meaning, "As the extension, so the force".
General application to elastic materials
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
elasticIn physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
material as a linear
springA spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...
. 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,
or
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 strengthThe yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...
). 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 modelling of materials.
The 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 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 displacement. (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.
If a mass
m is attached to the end of such a spring, the system becomes a harmonic oscillator. It will oscillate with a
natural frequencyThe fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
given either as an
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
or as a natural frequency
This idealized description of spring mechanics works as long as the mass of the spring is very small compared to the mass
m, there is no significant friction on the system, and the spring is not overextended beyond its natural range (which can deform it permanently).
Multiple springs
When two springs are attached to a mass and compressed, the following table compares values of the springs.
| Comparison |
In Parallel |
In Series |
|
|
|
Equivalent
spring constant |
 |
 |
Compressed
distance |
 |
 |
Energy
stored |
 |
 |
Derivation
| Equivalent Spring Constant (Series) |
x2, we'll be looking for an equation for the force on the block that looks like:
-
To begin, we'll also define the displacement from 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  we can solve for  :
-
  
so
-
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) |
|
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
-
 |
| Compressed Distance |
- {>
|
|
 |
}
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:
-
 |
| Energy Stored |
For the series case, the ratio of energy stored in springs is:
-
but 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
-
 |
Tensor expression of Hooke's Law
When working with a three-dimensional stress state, a 4th order
tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
(
) containing 81 elastic coefficients must be defined to link the
stress tensorIn continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
(σ
ij) and the strain tensor
(
).
Expressed in terms of components with respect to an
orthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
, the generalized form of Hooke's law is written as (using the summation convention)
The tensor
is called the
stiffness tensor or the
elasticity tensor. Due to the symmetry of the stress tensor, strain tensor, and
stiffnessStiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...
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.
Generalization for the case of large deformations is provided by models of
neo-Hookean solidA 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. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, a the...
s and
Mooney-Rivlin solids.
Isotropic materials
(see
viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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
traceIn linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
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. Thus:
where
is the
Kronecker delta. In direct tensor notation
where
is the second-order identity tensor.
The first term on the right is the constant tensor, also known as the
volumetric strain tensor, and the second term is the traceless symmetric tensor, also known as the
deviatoric strain tensor or 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:
where
K is the
bulk modulusThe bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...
and
G is the
shear modulus.
Using the relationships between the
elastic moduliAn elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...
, these equations may also be expressed in various other ways. A common form of Hooke's law for isotropic materials, expressed in direct tensor notation, is
where
and
are the Lamé constants,
is the second-order identity tensor, and
is the symmetric part of the fourth-order identity tensor. In terms of components with respect to a Cartesian basis,
The inverse relationship is
Therefore the compliance tensor in the relation
is
In terms of
Young's modulusYoung's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...
and
Poisson's ratioPoisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....
, Hooke's law for isotropic materials can then be expressed as
This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is
where
E is the
modulus of elasticityYoung's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...
and
is
Poisson's ratioPoisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....
. (See 3-D elasticity).
| 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),
 , , ,
where  is the Poisson's ratioPoisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....
and  the Young Modulus.
We get similar equations to the loads in directions 2 and 3,
 , , ,
and
 , , .
Summing the three cases together ( ) we get
  
or by adding and subtracting one 
  
and further we get by solving 
 .
Calculating the sum
 
and substituting it to the equation solved for  gives
 , ,
where  and  are the Lamé parameters.
Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions. |
In matrix form, Hooke's law for isotropic materials can be written as
where
is the
engineering shear strain.
The inverse relation may be written as
which expression can be simplified thanks to the Lamé constants :
Plane stress Hooke's law
Under plane stress conditions
. In that case Hooke's law takes the form
The inverse relation is usually written in the reduced form
Anisotropic materials
The symmetry of the
Cauchy stress tensorIn continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
(
) 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, since the displacement gradient and the Cauchy stress are work conjugate, the stress-strain relation can be derived from a strain energy density functional (
), 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.
Matrix representation (stiffness tensor)
It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called
Voigt notationIn mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by...
. 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
Change of coordinate system
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
are the components of an
orthogonal rotation matrixIn linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
. The same relation also holds for inversions.
In matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by
then
In addition, if the material is symmetric with respect to the transformation
then
Orthotropic materials
Orthotropic materialAn orthotropic material has two or three mutually orthogonal twofold axes of rotational symmetry so that its mechanical properties are, in general, different along each axis. Orthotropic materials are thus anisotropic; their properties depend on the direction in which they are measured...
s have three orthogonal planes of symmetry. If the basis vectors (
) are normals to the planes of symmetry then the coordinate transformation relations imply that
The inverse of this relation is commonly written as
where
is the
Young's modulusYoung's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...
along axis
is the
shear modulus in direction
on the plane whose normal is in direction
is the Poisson's ratio that corresponds to a contraction in direction
when an extension is applied in direction
.
Under
plane stress conditions,
, Hooke's law for an orthotropic material takes the form
The inverse relation is
The transposed form of the above stiffness matrix is also often used.
Transversely isotropic materials
A transversely isotropic material is symmetric with respect to a rotation about an axis of symmetry. For such a material, if
is the axis of symmetry, Hooke's law can be expressed as
More frequently, the
axis is taken to be the axis of symmetry and the inverse Hooke's law is written as
Thermodynamic basis of Hooke's law
Linear deformations of elastic materials can be approximated as adiabatic. Under these conditions and for quasistatic processes the
first law of thermodynamicsThe first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...
for a deformed body can be expressed as
where
is the increase in
internal energyIn thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
and
is the work done by external forces. The work can be split into two terms
where
is the work done by
surface forceSurface force denoted fs is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed in to two perpendicular components: pressure and stress forces....
s while
is the work done by
body forceA body force is a force that acts throughout the volume of a body, in contrast to contact forces.Gravity and electromagnetic forces are examples of body forces. Centrifugal and Coriolis forces can also be viewed as body forces.This can be put into contrast to the classical definition of surface...
s. If
is a
variation- Physics :* Magnetic variation, difference between magnetic north and true north, measured as an angle* Variation , any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon- Mathematics :* Bounded variation...
of the displacement field
in the body, then the two external work terms can be expressed 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 theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
gives
Using the symmetry of the Cauchy stress and the identity
we have
From the definition of strain and from the equations of
equilibriumLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
we have
Hence we can write
and therefore the variation in the
internal energyIn thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
density is given by
An
elasticIn physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
material is defined as one in which the total internal energy is equal to the
potential energyIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
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. We can see why
must be a fourth-order tensor by noting that, for a linear elastic material,
In index notation
Clearly, the right hand side constant requires four indices and is a fourth-order quantity. We can also see that this quantity must be a tensor because it is a linear transformation that takes the strain tensor to the stress tensor. We can also show that the constant obeys the tensor transformation rules for fourth order tensors.
See also
- Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
- Elastic limit
- Elastic modulus
An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...
- Elastic potential energy
- Infinitesimal strain theory
- List of scientific laws named after people
- Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
- Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
- Spring system
In engineering and physics, a spring system or spring network is a model of physics described as a graph with a position at each vertex and a spring of given stiffness and length along each edge. This generalizes Hooke's law to higher dimensions. This simple model can be used to solve the pose of...
- Poisson's ratio
Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....
- Simple harmonic motion of a mass on a spring
- Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
- Solid mechanics
Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...
- Stress (mechanics)
- Spring pendulum
A spring pendulum is a physical system where a mass is connected to a spring so that the resulting motion contains elements of a simple pendulum as well as a spring. The system is much more complex than a simple pendulum because the properties of the spring adds an extra dimension of freedom to...
External links