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| Boilerpipe Text | Hooke's law: the force is proportional to the extension
Bourdon tubes
are based on Hooke's law. The force created by gas
pressure
inside the coiled metal tube above unwinds it by an amount proportional to the pressure.
The
balance wheel
at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period.
In
physics
,
Hooke's law
is an
empirical law
which states that the
force
(
F
) needed to extend or compress a
spring
by some distance (
x
)
scales linearly
with respect to that distance—that is,
F
s
=
kx
,
where
k
is a constant factor characteristic of the spring (i.e., its
stiffness
), and
x
is small compared to the total possible deformation of the spring.
The law is named after 17th-century British physicist
Robert Hooke
. He first stated the law in 1676 as a Latin
anagram
.
[
1
]
[
2
]
He published the solution of his anagram in 1678
[
3
]
as:
ut tensio, sic vis
("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. It is the fundamental principle behind the
spring scale
, the
manometer
, the
galvanometer
, and the
balance wheel
of the
mechanical clock
.
The equation holds in many situations where an
elastic
body is
deformed
. An elastic body or material for which this equation can be assumed is said to be
linear-elastic
or
Hookean
. Hooke's law is a
first-order linear approximation
to the real response of springs and other elastic bodies to applied forces. It fails once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those
elastic limits
are reached.
Definition
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.
In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials they are made of. For example, one can deduce that a
homogeneous
rod with uniform
cross section
will behave like a simple spring when stretched, with a stiffness
k
directly proportional to its cross-section area and inversely proportional to its length.
Linear springs
Elongation and compression of a spring
Consider a simple
helical
spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is
F
s
. Suppose that the spring has reached a state of
equilibrium
, where its length is not changing anymore. Let
x
be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that
or, equivalently,
where
k
is a positive real number, characteristic of the spring. A spring with spaces between the coils can be compressed, and the same formula holds for compression, with
F
s
and
x
both negative in that case.
[
4
]
Graphical derivation
According to this formula, the
graph
of the applied force
F
s
as a function of the displacement
x
will be a straight line passing through the
origin
, whose
slope
is
k
.
Hooke's law for a spring is also stated under the convention that
F
s
is the
restoring force
exerted by the spring on whatever is pulling its free end. In that case, the equation becomes
since the direction of the restoring force is opposite to that of the displacement.
Torsional springs
The
torsional
analog of Hooke's law applies to
torsional springs
. It states that the torque (τ) required to rotate an object is directly proportional to the angular displacement (θ) from the equilibrium position. It describes the relationship between the torque applied to an object and the resulting angular
deformation
due to torsion. Mathematically, it can be expressed as:
Where:
τ is the
torque
measured in Newton-meters or N·m.
k is the
torsional constant
(measured in N·m/radian), which characterizes the stiffness of the torsional spring or the resistance to angular displacement.
θ is the
angular displacement
(measured in radians) from the equilibrium position.
Just as in the linear case, this law shows that the torque is proportional to the angular displacement, and the negative sign indicates that the torque acts in a direction opposite to the angular displacement, providing a restoring force to bring the system back to equilibrium.
General "scalar" springs
Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative.
For example, when a block of rubber attached to two parallel plates is deformed by
shearing
, rather than stretching or compression, the shearing force
F
s
and the sideways displacement of the plates
x
obey Hooke's law (for small enough deformations).
Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight
F
placed at some intermediate point. The displacement
x
in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.
Vector formulation
In the case of a helical spring that is stretched or compressed along its
axis
, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if
F
s
and
x
are defined as
vectors
, Hooke's
equation
still holds and says that the force vector is the
elongation vector
multiplied by a fixed
scalar
.
General tensor form
Some elastic bodies will deform in one direction when subjected to a force with a different direction. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. In such cases, the
magnitude
of the displacement
x
will be proportional to the magnitude of the force
F
s
, as long as the direction of the latter remains the same (and its value is not too large); so the scalar version of Hooke's law
F
s
= −
kx
will hold. However, the force and displacement
vectors
will not be scalar multiples of each other, since they have different directions. Moreover, the ratio
k
between their magnitudes will depend on the direction of the vector
F
s
.
Yet, in such cases there is often a fixed
linear relation
between the force and deformation vectors, as long as they are small enough. Namely, there is a
function
κ
from vectors to vectors, such that
F
=
κ
(
X
)
, and
κ
(
α
X
1
+
β
X
2
) =
α
κ
(
X
1
) +
β
κ
(
X
2
)
for any real numbers
α
,
β
and any displacement vectors
X
1
,
X
2
. Such a function is called a (second-order)
tensor
.
With respect to an arbitrary
Cartesian coordinate system
, the force and displacement vectors can be represented by 3 × 1
matrices
of real numbers. Then the tensor
κ
connecting them can be represented by a 3 × 3 matrix
κ
of real coefficients, that, when
multiplied
by the displacement vector, gives the force vector:
That is,
for
i
= 1, 2, 3
. Therefore, Hooke's law
F
=
κ
X
can be said to hold also when
X
and
F
are vectors with variable directions, except that the stiffness of the object is a tensor
κ
, rather than a single real number
k
.
Hooke's law for continuous media
(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.
[
5
]
The stresses and strains of the material inside a
continuous
elastic material (such as a block of rubber, the wall of a
boiler
, or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name.
However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.
In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the
strain tensor
ε
(in lieu of the displacement
X
) and the
stress tensor
σ
(replacing the restoring force
F
). The analogue of Hooke's spring law for continuous media is then
where
c
is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the
stiffness tensor
or
elasticity tensor
. One may also write it as
where the tensor
s
, called the
compliance tensor
, represents the inverse of said linear map.
In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices
Being a linear mapping between the nine numbers
σ
ij
and the nine numbers
ε
kl
, the stiffness tensor
c
is represented by a matrix of
3 × 3 × 3 × 3 = 81
real numbers
c
ijkl
. Hooke's law then says that
where
i
,
j
= 1,2,3
.
All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor
ε
merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor
σ
specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor
c
, on the other hand, is a property of the material, and often depends on physical state variables such as temperature,
pressure
, and
microstructure
.
Due to the inherent symmetries of
σ
,
ε
, and
c
, only 21 elastic coefficients of the latter are independent.
[
6
]
This number can be further reduced by the symmetry of the material: 9 for an
orthorhombic
crystal, 5 for an
hexagonal
structure, and 3 for a
cubic
symmetry.
[
7
]
For
isotropic
media (which have the same physical properties in any direction),
c
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.
Analogous laws
Since Hooke's law is a simple proportionality between two quantities, its formulas and consequences are mathematically similar to those of many other physical laws, such as those describing the motion of
fluids
, or the
polarization
of a
dielectric
by an
electric field
.
In particular, the tensor equation
σ
=
cε
relating elastic stresses to strains is entirely similar to the equation
τ
=
με̇
relating the
viscous stress tensor
τ
and the
strain rate tensor
ε̇
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).
Units of measurement
In
SI units
, displacements are measured in meters (m), and forces in
newtons
(N or kg·m/s
2
). Therefore, the spring constant
k
, and each element of the tensor
κ
, is measured in newtons per meter (N/m), or kilograms per second squared (kg/s
2
).
For continuous media, each element of the stress tensor
σ
is a force divided by an area; it is therefore measured in units of pressure, namely
pascals
(Pa, or N/m
2
, or kg/(m·s
2
). The elements of the strain tensor
ε
are
dimensionless
(displacements divided by distances). Therefore, the entries of
c
ijkl
are also expressed in units of pressure.
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.
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.
Generalizations of Hooke's law for the case of
large deformations
is provided by models of
neo-Hookean solids
and
Mooney–Rivlin solids
.
Derived formulae
Tensional stress of a uniform bar
A rod of any
elastic
material may be viewed as a linear
spring
. The rod has length
L
and cross-sectional area
A
. Its
tensile stress
σ
is linearly proportional to its fractional extension or strain
ε
by the
modulus of elasticity
E
:
The modulus of elasticity may often be considered constant. In turn,
(that is, the fractional change in length), and since
it follows that:
The change in length may be expressed as
Spring energy
The potential energy
U
el
(
x
)
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. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative. Substituting
gives
This potential
U
el
can be visualized as a
parabola
on the
Ux
-plane such that
U
el
(
x
) =
1
/
2
kx
2
. As the spring is stretched in the positive
x
-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate:
Note that the change in the change in
U
is constant even when the displacement and acceleration are zero.
Relaxed force constants (generalized compliance constants)
Relaxed force constants (the inverse of generalized
compliance constants
) are uniquely defined for molecular systems, in contradistinction to the usual "rigid" force constants, and thus their use allows meaningful correlations to be made between force fields calculated for
reactants
,
transition states
, and products of a
chemical reaction
. Just as the
potential energy
can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed
compliance constants
. A direct method exists for calculating the compliance constant for any internal coordinate of a molecule, without the need to do the normal mode analysis.
[
8
]
The suitability of relaxed force constants (inverse compliance constants) as
covalent bond
strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.
[
9
]
Harmonic oscillator
A mass suspended by a spring is the classical example of a harmonic oscillator
A mass
m
attached to the end of a spring is a classic example of a
harmonic oscillator
. By pulling slightly on the mass and then releasing it, the system will be set in
sinusoidal
oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglect
friction
and the mass of the spring, the amplitude of the oscillation will remain constant; and its
frequency
f
will be independent of its amplitude, determined only by the mass and the stiffness of the spring:
This phenomenon made possible the construction of accurate
mechanical clocks
and watches that could be carried on ships and people's pockets.
Rotation in gravity-free space
If the mass
m
were attached to a spring with force constant
k
and rotating in free space, the spring tension (
F
t
) would supply the required
centripetal force
(
F
c
):
Since
F
t
=
F
c
and
x
=
r
, then:
Given that
ω
= 2π
f
, this leads to the same frequency equation as above:
Isotropic materials
For an analogous development for viscous fluids, see
Viscosity
.
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.
[
10
]
Thus in
index notation
:
where
δ
ij
is the
Kronecker delta
. In direct tensor notation:
where
I
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 modulus
and
G
is the
shear modulus
.
Using the relationships between the
elastic moduli
, 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
[
11
]
where
λ
=
K
−
2
/
3
G
=
c
1111
− 2
c
1212
and
μ
=
G
=
c
1212
are the
Lamé constants
,
I
is the second-rank identity tensor, and
I
is the symmetric part of the fourth-rank identity tensor. In index notation:
The inverse relationship is
[
12
]
Therefore, the compliance tensor in the relation
ε
=
s
:
σ
is
In terms of
Young's modulus
and
Poisson's ratio
, 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
Young's modulus
and
ν
is
Poisson's ratio
. (See
3-D elasticity
).
Derivation of Hooke's law in three dimensions
The three-dimensional form of Hooke's law can be derived using Poisson's ratio and the one-dimensional 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
ν
is Poisson's ratio and
E
is Young's modulus.
We get similar equations to the loads in directions 2 and 3,
and
Summing the three cases together (
ε
i
=
ε
i
′ +
ε
i
″ +
ε
i
‴
) we get
or by adding and subtracting one
νσ
and further we get by solving
σ
1
Calculating the sum
and substituting it to the equation solved for
σ
1
gives
where
μ
and
λ
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
γ
ij
= 2
ε
ij
is the
engineering shear strain
. The inverse relation may be written as
which can be simplified thanks to the Lamé constants:
In vector notation this becomes
where
I
is the identity tensor.
Plane stress
Under
plane stress
conditions,
σ
31
=
σ
13
=
σ
32
=
σ
23
=
σ
33
= 0
. In that case Hooke's law takes the form
In vector notation this becomes
The inverse relation is usually written in the reduced form
Plane strain
Under
plane strain
conditions,
ε
31
=
ε
13
=
ε
32
=
ε
23
=
ε
33
= 0
. In this case Hooke's law takes the form
Anisotropic materials
The symmetry of the
Cauchy stress tensor
(
σ
ij
=
σ
ji
) and the generalized Hooke's laws (
σ
ij
=
c
ijkl
ε
kl
) implies that
c
ijkl
=
c
jikl
. Similarly, the symmetry of the
infinitesimal strain tensor
implies that
c
ijkl
=
c
ijlk
. These symmetries are called the
minor symmetries
of the stiffness tensor
c
. This reduces the number of elastic constants from 81 to 36.
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 (
U
), then
The arbitrariness of the order of differentiation implies that
c
ijkl
=
c
klij
. These are called the
major symmetries
of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. 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 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 (
e
1
,
e
2
,
e
3
) as
Then the stiffness tensor (
c
) can be expressed as
and Hooke's law is written as
Similarly the compliance tensor (
s
) 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
[
13
]
where
l
ab
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
then
In addition, if the material is symmetric with respect to the transformation
[
L
]
then
Orthotropic materials
Orthotropic materials
have three
orthogonal
planes of symmetry
. If the basis vectors (
e
1
,
e
2
,
e
3
) are normals to the planes of symmetry then the coordinate transformation relations imply that
The inverse of this relation is commonly written as
[
14
]
[
page needed
]
where
E
i
is the
Young's modulus
along axis
i
G
ij
is the
shear modulus
in direction
j
on the plane whose normal is in direction
i
ν
ij
is the
Poisson's ratio
that corresponds to a contraction in direction
j
when an extension is applied in direction
i
.
Under
plane stress
conditions,
σ
zz
=
σ
zx
=
σ
yz
= 0
, 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
e
3
is the axis of symmetry, Hooke's law can be expressed as
More frequently, the
x
≡
e
1
axis is taken to be the axis of symmetry and the inverse Hooke's law is written as
[
15
]
Universal elastic anisotropy index
To grasp the degree of anisotropy of any class, a
universal elastic anisotropy index
(AU)
[
16
]
was formulated. It replaces the
Zener ratio
, which is suited for
cubic crystals
.
Thermodynamic basis
Linear deformations of elastic materials can be approximated as
adiabatic
. Under these conditions and for quasistatic processes the
first law of thermodynamics
for a deformed body can be expressed as
where
δU
is the increase in
internal energy
and
δW
is the
work
done by external forces. The work can be split into two terms
where
δW
s
is the work done by
surface forces
while
δW
b
is the work done by
body forces
. If
δ
u
is a
variation
of the displacement field
u
in the body, then the two external work terms can be expressed as
where
t
is the surface
traction
vector,
b
is the body force vector,
Ω
represents the body and
∂
Ω
represents its surface. Using the relation between the
Cauchy stress
and the surface traction,
t
=
n
·
σ
(where
n
is the unit outward normal to
∂
Ω
), 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
we have the following
From the definition of
strain
and from the equations of
equilibrium
we have
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,
U
0
=
U
0
(
ε
)
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
∂
U
0
/
∂
ε
is a linear function of
ε
, and can therefore be expressed as
where
c
is a fourth-rank tensor of material constants, also called the
stiffness tensor
. We can see why
c
must be a fourth-rank tensor by noting that, for a linear elastic material,
In index notation
The right-hand side constant requires four indices and is a fourth-rank 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-rank tensors.
See also
Acoustoelastic effect
Elastic potential energy
Laws of science
List of scientific laws named after people
Quadratic form
Series and parallel springs
Spring system
Simple harmonic motion of a mass on a spring
Sine wave
Solid mechanics
Spring pendulum
Notes
^
The anagram was given in alphabetical order,
ceiiinosssttuv
, representing
Ut tensio, sic vis
– "As the extension, so the force":
Petroski, Henry
(1996).
Invention by Design: How Engineers Get from Thought to Thing
. Cambridge, MA: Harvard University Press. p.
11
.
ISBN
978-0-674-46368-4
.
^
See
http://civil.lindahall.org/design.shtml
, where one can find also an anagram for
catenary
.
^
Robert Hooke
,
De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies
, London, 1678.
^
Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2016).
Sears and Zemansky's University Physics: With Modern Physics
(14th ed.). Pearson. p. 209.
^
Ushiba, Shota; Masui, Kyoko; Taguchi, Natsuo; Hamano, Tomoki; Kawata, Satoshi; Shoji, Satoru (2015).
"Size dependent nanomechanics of coil spring shaped polymer nanowires"
.
Scientific Reports
.
5
17152.
Bibcode
:
2015NatSR...517152U
.
doi
:
10.1038/srep17152
.
PMC
4661696
.
PMID
26612544
.
^
Belen'kii; Salaev (1988).
"Deformation effects in layer crystals"
.
Uspekhi Fizicheskikh Nauk
.
155
(5): 89.
doi
:
10.3367/UFNr.0155.198805c.0089
.
^
Mouhat, Félix; Coudert, François-Xavier (5 December 2014).
"Necessary and sufficient elastic stability conditions in various crystal systems"
.
Physical Review B
.
90
(22) 224104.
arXiv
:
1410.0065
.
Bibcode
:
2014PhRvB..90v4104M
.
doi
:
10.1103/PhysRevB.90.224104
.
ISSN
1098-0121
.
S2CID
54058316
.
^
Vijay Madhav, M.; Manogaran, S. (2009). "A relook at the compliance constants in redundant internal coordinates and some new insights".
J. Chem. Phys
.
131
(17):
174112–
174116.
Bibcode
:
2009JChPh.131q4112V
.
doi
:
10.1063/1.3259834
.
PMID
19895003
.
^
Ponomareva, Alla; Yurenko, Yevgen; Zhurakivsky, Roman; Van Mourik, Tanja; Hovorun, Dmytro (2012). "Complete conformational space of the potential HIV-1 reverse transcriptase inhibitors d4U and d4C. A quantum chemical study".
Phys. Chem. Chem. Phys
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14
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6795.
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doi
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10.1039/C2CP40290D
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Symon, Keith R. (1971). "Chapter 10".
Mechanics
. Reading, Massachusetts: Addison-Wesley.
ISBN
978-0-201-07392-8
.
^
Simo, J. C.; Hughes, T. J. R. (1998).
Computational Inelasticity
. Springer.
ISBN
978-0-387-97520-7
.
^
Milton, Graeme W. (2002).
The Theory of Composites
. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press.
ISBN
978-0-521-78125-1
.
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Slaughter, William S. (2001).
The Linearized Theory of Elasticity
. Birkhäuser.
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978-0-8176-4117-7
.
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Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. (1993).
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(5th ed.). Wiley.
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978-0-471-60009-1
.
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Tan, S. C. (1994).
Stress Concentrations in Laminated Composites
. Lancaster, PA: Technomic Publishing Company.
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978-1-56676-077-5
.
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Ranganathan, S.I.;
Ostoja-Starzewski, M.
(2008). "Universal Elastic Anisotropy Index".
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References
Hooke's law - The Feynman Lectures on Physics
Hooke's Law - Classical Mechanics - Physics - MIT OpenCourseWare
External links
JavaScript Applet demonstrating Springs and Hooke's law
JavaScript Applet demonstrating Spring Force
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two quantities among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part).
3D Formulae
Knowns
Bulk modulus
(
K
)
Young's modulus
(
E
)
Lamé's first parameter
(λ)
Shear modulus
(
G
)
Poisson's ratio
(
ν
)
P-wave modulus
(
M
)
Notes
(
K
,
E
)
3
K
(
1 +
6
K
/
E
− 9
K
)
E
/
3 −
E
/
3
K
1
/
2
−
E
/
6
K
3
K
+
E
/
3 −
E
/
3
K
(
K
, λ)
9
K
(
K
− λ)
/
3
K
− λ
3(
K
− λ)
/
2
λ
/
3
K
− λ
3
K
− 2λ
(
K
,
G
)
9
KG
/
3
K
+
G
K
−
2
G
/
3
3
K
− 2
G
/
6
K
+ 2
G
K
+
4
G
/
3
(
K
,
ν
)
3
K
(1 − 2
ν
)
3
Kν
/
1 +
ν
3
K
(1 − 2
ν
)
/
2(1 +
ν
)
3
K
(1 −
ν
)
/
1 +
ν
(
K
,
M
)
9
K
(
M
−
K
)
/
3
K
+
M
3
K
−
M
/
2
3(
M
−
K
)
/
4
3
K
−
M
/
3
K
+
M
(
E
, λ)
E
+ 3λ + R
/
6
E
− 3λ +
R
/
4
−
E
+
R
/
4λ
−
1
/
4
E
− λ +
R
/
2
R
= ±
(
E
2
+ 9λ
2
+ 2
E
λ
)
1
/
2
(
E
,
G
)
EG
/
3(3
G
−
E
)
G
(
E
− 2
G
)
/
3
G
−
E
E
/
2
G
− 1
G
(4
G
−
E
)
/
3
G
−
E
(
E
,
ν
)
E
/
3 − 6
ν
Eν
/
(1 +
ν
)(1 − 2
ν
)
E
/
2(1 +
ν
)
E
(1 −
ν
)
/
(1 +
ν
)(1 − 2
ν
)
(
E
,
M
)
3
M
−
E
+
S
/
6
M
−
E
+
S
/
4
3
M
+
E
−
S
/
8
E
+
S
/
4
M
−
1
/
4
S
= ±
(
E
2
+ 9M
2
− 10
E
M
)
1
/
2
(λ,
G
)
λ +
2
G
/
3
G
(3λ + 2
G
)
/
λ +
G
λ
/
2(λ +
G
)
λ + 2
G
(λ,
ν
)
λ
/
3
(
1 +
1
/
ν
)
λ
(
1
/
ν
− 2
ν
− 1
)
λ
(
1
/
2
ν
− 1
)
λ
(
1
/
ν
− 1
)
(λ,
M
)
M
+ 2λ
/
3
(
M
− λ)(
M
+2λ)
/
M
+ λ
M
− λ
/
2
λ
/
M
+ λ
(
G
,
ν
)
2
G
(1 +
ν
)
/
3 − 6
ν
2
G
(1 +
ν
)
2
G
ν
/
1 − 2
ν
2
G
(1 −
ν
)
/
1 − 2
ν
(
G
,
M
)
M
−
4
G
/
3
G
(3
M
− 4
G
)
/
M
−
G
M
− 2
G
M
− 2
G
/
2
M
− 2
G
(
ν
,
M
)
M
(1 +
ν
)
/
3(1 −
ν
)
M
(1 +
ν
)(1 − 2
ν
)
/
1 −
ν
M
ν
/
1 −
ν
M
(1 − 2
ν
)
/
2(1 −
ν
)
2D Formulae
Knowns
(
K
)
(
E
)
(λ)
(
G
)
(
ν
)
(
M
)
Notes
(
K
2D
,
E
2D
)
2
K
2D
(2
K
2D
−
E
2D
)
/
4
K
2D
−
E
2D
K
2D
E
2D
/
4
K
2D
−
E
2D
2
K
2D
−
E
2D
/
2
K
2D
4
K
2D
^2
/
4
K
2D
−
E
2D
(
K
2D
, λ
2D
)
4
K
2D
(
K
2D
− λ
2D
)
/
2
K
2D
− λ
2D
K
2D
− λ
2D
λ
2D
/
2
K
2D
− λ
2D
2
K
2D
− λ
2D
(
K
2D
,
G
2D
)
4
K
2D
G
2D
/
K
2D
+
G
2D
K
2D
−
G
2D
K
2D
−
G
2D
/
K
2D
+
G
2D
K
2D
+
G
2D
(
K
2D
,
ν
2D
)
2
K
2D
(1 −
ν
2D
)
2
K
2D
ν
2D
/
1 +
ν
2D
K
2D
(1 −
ν
2D
)
/
1 +
ν
2D
2
K
2D
/
1 +
ν
2D
(
E
2D
,
G
2D
)
E
2D
G
2D
/
4
G
2D
−
E
2D
2
G
2D
(
E
2D
− 2
G
2D
)
/
4
G
2D
−
E
2D
E
2D
/
2
G
2D
− 1
4
G
2D
^2
/
4
G
2D
−
E
2D
(
E
2D
,
ν
2D
)
E
2D
/
2(1 −
ν
2D
)
E
2D
ν
2D
/
(1 +
ν
2D
)(1 −
ν
2D
)
E
2D
/
2(1 +
ν
2D
)
E
2D
/
(1 +
ν
2D
)(1 −
ν
2D
)
(λ
2D
,
G
2D
)
λ
2D
+
G
2D
4
G
2D
(λ
2D
+
G
2D
)
/
λ
2D
+ 2
G
2D
λ
2D
/
λ
2D
+ 2
G
2D
λ
2D
+ 2
G
2D
(λ
2D
,
ν
2D
)
λ
2D
(1 +
ν
2D
)
/
2
ν
2D
λ
2D
(1 +
ν
2D
)(1 −
ν
2D
)
/
ν
2D
λ
2D
(1 −
ν
2D
)
/
2
ν
2D
λ
2D
/
ν
2D
(
G
2D
,
ν
2D
)
G
2D
(1 +
ν
2D
)
/
1 −
ν
2D
2
G
2D
(1 +
ν
2D
)
2
G
2D
ν
2D
/
1 −
ν
2D
2
G
2D
/
1 −
ν
2D
(
G
2D
,
M
2D
)
M
2D
−
G
2D
4
G
2D
(
M
2D
−
G
2D
)
/
M
2D
M
2D
− 2
G
2D
M
2D
− 2
G
2D
/
M
2D
|
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## Contents
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- [(Top)](https://en.wikipedia.org/wiki/Hooke%27s_law)
- [1 Definition](https://en.wikipedia.org/wiki/Hooke%27s_law#Definition)
Toggle Definition subsection
- [1\.1 Linear springs](https://en.wikipedia.org/wiki/Hooke%27s_law#Linear_springs)
- [1\.2 Torsional springs](https://en.wikipedia.org/wiki/Hooke%27s_law#Torsional_springs)
- [1\.3 General "scalar" springs](https://en.wikipedia.org/wiki/Hooke%27s_law#General_"scalar"_springs)
- [1\.4 Vector formulation](https://en.wikipedia.org/wiki/Hooke%27s_law#Vector_formulation)
- [1\.5 General tensor form](https://en.wikipedia.org/wiki/Hooke%27s_law#General_tensor_form)
- [1\.6 Hooke's law for continuous media](https://en.wikipedia.org/wiki/Hooke%27s_law#Hooke's_law_for_continuous_media)
- [2 Analogous laws](https://en.wikipedia.org/wiki/Hooke%27s_law#Analogous_laws)
- [3 Units of measurement](https://en.wikipedia.org/wiki/Hooke%27s_law#Units_of_measurement)
- [4 General application to elastic materials](https://en.wikipedia.org/wiki/Hooke%27s_law#General_application_to_elastic_materials)
- [5 Derived formulae](https://en.wikipedia.org/wiki/Hooke%27s_law#Derived_formulae)
Toggle Derived formulae subsection
- [5\.1 Tensional stress of a uniform bar](https://en.wikipedia.org/wiki/Hooke%27s_law#Tensional_stress_of_a_uniform_bar)
- [5\.2 Spring energy](https://en.wikipedia.org/wiki/Hooke%27s_law#Spring_energy)
- [5\.3 Relaxed force constants (generalized compliance constants)](https://en.wikipedia.org/wiki/Hooke%27s_law#Relaxed_force_constants_\(generalized_compliance_constants\))
- [5\.4 Harmonic oscillator](https://en.wikipedia.org/wiki/Hooke%27s_law#Harmonic_oscillator)
- [5\.5 Rotation in gravity-free space](https://en.wikipedia.org/wiki/Hooke%27s_law#Rotation_in_gravity-free_space)
- [6 Linear elasticity theory for continuous media](https://en.wikipedia.org/wiki/Hooke%27s_law#Linear_elasticity_theory_for_continuous_media)
Toggle Linear elasticity theory for continuous media subsection
- [6\.1 Isotropic materials](https://en.wikipedia.org/wiki/Hooke%27s_law#Isotropic_materials)
- [6\.1.1 Plane stress](https://en.wikipedia.org/wiki/Hooke%27s_law#Plane_stress)
- [6\.1.2 Plane strain](https://en.wikipedia.org/wiki/Hooke%27s_law#Plane_strain)
- [6\.2 Anisotropic materials](https://en.wikipedia.org/wiki/Hooke%27s_law#Anisotropic_materials)
- [6\.2.1 Matrix representation (stiffness tensor)](https://en.wikipedia.org/wiki/Hooke%27s_law#Matrix_representation_\(stiffness_tensor\))
- [6\.2.2 Change of coordinate system](https://en.wikipedia.org/wiki/Hooke%27s_law#Change_of_coordinate_system)
- [6\.2.3 Orthotropic materials](https://en.wikipedia.org/wiki/Hooke%27s_law#Orthotropic_materials)
- [6\.2.4 Transversely isotropic materials](https://en.wikipedia.org/wiki/Hooke%27s_law#Transversely_isotropic_materials)
- [6\.2.5 Universal elastic anisotropy index](https://en.wikipedia.org/wiki/Hooke%27s_law#Universal_elastic_anisotropy_index)
- [7 Thermodynamic basis](https://en.wikipedia.org/wiki/Hooke%27s_law#Thermodynamic_basis)
- [8 See also](https://en.wikipedia.org/wiki/Hooke%27s_law#See_also)
- [9 Notes](https://en.wikipedia.org/wiki/Hooke%27s_law#Notes)
- [10 References](https://en.wikipedia.org/wiki/Hooke%27s_law#References)
- [11 External links](https://en.wikipedia.org/wiki/Hooke%27s_law#External_links)
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# Hooke's law
69 languages
- [Alemannisch](https://als.wikipedia.org/wiki/Hookesches_Gesetz "Hookesches Gesetz – Alemannic")
- [አማርኛ](https://am.wikipedia.org/wiki/%E1%8B%A8%E1%88%81%E1%8A%AD_%E1%88%85%E1%8C%8D "የሁክ ህግ – Amharic")
- [العربية](https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D9%87%D9%88%D9%83 "قانون هوك – Arabic")
- [Asturianu](https://ast.wikipedia.org/wiki/Llei_d%27elasticid%C3%A1_de_Hooke "Llei d'elasticidá de Hooke – Asturian")
- [Azərbaycanca](https://az.wikipedia.org/wiki/Huk_qanunu "Huk qanunu – Azerbaijani")
- [Беларуская](https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83%D0%BA%D0%B0 "Закон Гука – Belarusian")
- [Български](https://bg.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%BD%D0%B0_%D0%A5%D1%83%D0%BA "Закон на Хук – Bulgarian")
- [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A7%81%E0%A6%95%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0 "হুকের সূত্র – Bangla")
- [Bosanski](https://bs.wikipedia.org/wiki/Hookeov_zakon "Hookeov zakon – Bosnian")
- [Català](https://ca.wikipedia.org/wiki/Llei_de_Hooke "Llei de Hooke – Catalan")
- [Čeština](https://cs.wikipedia.org/wiki/Hook%C5%AFv_z%C3%A1kon "Hookův zákon – Czech")
- [Чӑвашла](https://cv.wikipedia.org/wiki/%D0%93%D1%83%D0%BA_%D1%81%D0%B0%D0%BA%D0%BA%D1%83%D0%BD%C4%95 "Гук саккунĕ – Chuvash")
- [Dansk](https://da.wikipedia.org/wiki/Hookes_lov "Hookes lov – Danish")
- [Deutsch](https://de.wikipedia.org/wiki/Hookesches_Gesetz "Hookesches Gesetz – German")
- [Ελληνικά](https://el.wikipedia.org/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%A7%CE%BF%CF%85%CE%BA "Νόμος του Χουκ – Greek")
- [Esperanto](https://eo.wikipedia.org/wiki/Le%C4%9Do_de_Hooke "Leĝo de Hooke – Esperanto")
- [Español](https://es.wikipedia.org/wiki/Ley_de_elasticidad_de_Hooke "Ley de elasticidad de Hooke – Spanish")
- [Eesti](https://et.wikipedia.org/wiki/Hooke%27i_seadus "Hooke'i seadus – Estonian")
- [Euskara](https://eu.wikipedia.org/wiki/Hookeren_elastikotasun_legea "Hookeren elastikotasun legea – Basque")
- [فارسی](https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D9%87%D9%88%DA%A9 "قانون هوک – Persian")
- [Suomi](https://fi.wikipedia.org/wiki/Hooken_laki "Hooken laki – Finnish")
- [Français](https://fr.wikipedia.org/wiki/Loi_de_Hooke "Loi de Hooke – French")
- [Gaeilge](https://ga.wikipedia.org/wiki/Dl%C3%AD_Hooke "Dlí Hooke – Irish")
- [Galego](https://gl.wikipedia.org/wiki/Lei_de_Hooke "Lei de Hooke – Galician")
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From Wikipedia, the free encyclopedia
Force needed to pull a spring grows linearly with distance
For the KeiyaA album, see [Hooke's Law (album)](https://en.wikipedia.org/wiki/Hooke%27s_Law_\(album\) "Hooke's Law (album)").
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[](https://en.wikipedia.org/wiki/File:Hookes-law-springs.png)
Hooke's law: the force is proportional to the extension
[](https://en.wikipedia.org/wiki/File:Manometer_anim_02.gif)
[Bourdon tubes](https://en.wikipedia.org/wiki/Bourdon_tube "Bourdon tube") are based on Hooke's law. The force created by gas [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") inside the coiled metal tube above unwinds it by an amount proportional to the pressure.
[](https://en.wikipedia.org/wiki/File:Balancier_avec_ressort_spiral.png)
The [balance wheel](https://en.wikipedia.org/wiki/Balance_wheel "Balance wheel") at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period.
| |
|---|
| Part of a series on |
| [Continuum mechanics](https://en.wikipedia.org/wiki/Continuum_mechanics "Continuum mechanics") |
| J \= − D d φ d x {\\displaystyle J=-D{\\frac {d\\varphi }{dx}}} [Fick's laws of diffusion](https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion "Fick's laws of diffusion") |
| [Solid mechanics](https://en.wikipedia.org/wiki/Solid_mechanics "Solid mechanics") [Deformation](https://en.wikipedia.org/wiki/Deformation_\(physics\) "Deformation (physics)") [Elasticity](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)") [linear](https://en.wikipedia.org/wiki/Linear_elasticity "Linear elasticity") [Plasticity](https://en.wikipedia.org/wiki/Plasticity_\(physics\) "Plasticity (physics)") [Hooke's law]() [Stress](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") [Strain](https://en.wikipedia.org/wiki/Strain_\(mechanics\) "Strain (mechanics)") [Finite strain](https://en.wikipedia.org/wiki/Finite_strain_theory "Finite strain theory") [Infinitesimal strain](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory "Infinitesimal strain theory") [Compatibility](https://en.wikipedia.org/wiki/Compatibility_\(mechanics\) "Compatibility (mechanics)") [Bending](https://en.wikipedia.org/wiki/Bending "Bending") [Contact mechanics](https://en.wikipedia.org/wiki/Contact_mechanics "Contact mechanics") [frictional](https://en.wikipedia.org/wiki/Frictional_contact_mechanics "Frictional contact mechanics") [Material failure theory](https://en.wikipedia.org/wiki/Material_failure_theory "Material failure theory") [Fracture mechanics](https://en.wikipedia.org/wiki/Fracture_mechanics "Fracture mechanics") |
| [Fluids](https://en.wikipedia.org/wiki/Fluid "Fluid") |
| [Statics](https://en.wikipedia.org/wiki/Hydrostatics "Hydrostatics") **·** [Dynamics](https://en.wikipedia.org/wiki/Fluid_dynamics "Fluid dynamics") [Archimedes' principle](https://en.wikipedia.org/wiki/Archimedes%27_principle "Archimedes' principle") **·** [Bernoulli's principle](https://en.wikipedia.org/wiki/Bernoulli%27s_principle "Bernoulli's principle") [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations") [Poiseuille equation](https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation "Hagen–Poiseuille equation") **·** [Pascal's law](https://en.wikipedia.org/wiki/Pascal%27s_law "Pascal's law") [Viscosity](https://en.wikipedia.org/wiki/Viscosity "Viscosity") ([Newtonian](https://en.wikipedia.org/wiki/Newtonian_fluid "Newtonian fluid") **·** [non-Newtonian](https://en.wikipedia.org/wiki/Non-Newtonian_fluid "Non-Newtonian fluid")) [Buoyancy](https://en.wikipedia.org/wiki/Buoyancy "Buoyancy") **·** [Mixing](https://en.wikipedia.org/wiki/Mixing_\(process_engineering\) "Mixing (process engineering)") **·** [Pressure](https://en.wikipedia.org/wiki/Fluid_pressure "Fluid pressure") |
| [Liquids](https://en.wikipedia.org/wiki/Liquid "Liquid") |
| [Adhesion](https://en.wikipedia.org/wiki/Adhesion "Adhesion") [Capillary action](https://en.wikipedia.org/wiki/Capillary_action "Capillary action") [Chromatography](https://en.wikipedia.org/wiki/Chromatography "Chromatography") [Cohesion (chemistry)](https://en.wikipedia.org/wiki/Cohesion_\(chemistry\) "Cohesion (chemistry)") [Surface tension](https://en.wikipedia.org/wiki/Surface_tension "Surface tension") |
| [Gases](https://en.wikipedia.org/wiki/Gas "Gas") |
| [Atmosphere](https://en.wikipedia.org/wiki/Atmosphere "Atmosphere") [Boyle's law](https://en.wikipedia.org/wiki/Boyle%27s_law "Boyle's law") [Charles's law](https://en.wikipedia.org/wiki/Charles%27s_law "Charles's law") [Combined gas law](https://en.wikipedia.org/wiki/Combined_gas_law "Combined gas law") [Fick's law](https://en.wikipedia.org/wiki/Fick%27s_law "Fick's law") [Gay-Lussac's law](https://en.wikipedia.org/wiki/Gay-Lussac%27s_law "Gay-Lussac's law") [Graham's law](https://en.wikipedia.org/wiki/Graham%27s_law "Graham's law") |
| [Plasma](https://en.wikipedia.org/wiki/Plasma_\(physics\) "Plasma (physics)") |
| |
| [Viscoelasticity](https://en.wikipedia.org/wiki/Viscoelasticity "Viscoelasticity") [Rheometry](https://en.wikipedia.org/wiki/Rheometry "Rheometry") [Rheometer](https://en.wikipedia.org/wiki/Rheometer "Rheometer") |
| [Smart fluids](https://en.wikipedia.org/wiki/Smart_fluid "Smart fluid") |
| [Electrorheological](https://en.wikipedia.org/wiki/Electrorheological_fluid "Electrorheological fluid") [Magnetorheological](https://en.wikipedia.org/wiki/Magnetorheological_fluid "Magnetorheological fluid") [Ferrofluids](https://en.wikipedia.org/wiki/Ferrofluid "Ferrofluid") |
| Scientists [Bernoulli](https://en.wikipedia.org/wiki/Daniel_Bernoulli "Daniel Bernoulli") [Boyle](https://en.wikipedia.org/wiki/Robert_Boyle "Robert Boyle") [Cauchy](https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy "Augustin-Louis Cauchy") [Charles](https://en.wikipedia.org/wiki/Jacques_Charles "Jacques Charles") [Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") [Fick](https://en.wikipedia.org/wiki/Adolf_Eugen_Fick "Adolf Eugen Fick") [Gay-Lussac](https://en.wikipedia.org/wiki/Joseph_Louis_Gay-Lussac "Joseph Louis Gay-Lussac") [Graham](https://en.wikipedia.org/wiki/Thomas_Graham_\(chemist\) "Thomas Graham (chemist)") [Hooke](https://en.wikipedia.org/wiki/Robert_Hooke "Robert Hooke") [Newton](https://en.wikipedia.org/wiki/Isaac_Newton "Isaac Newton") [Navier](https://en.wikipedia.org/wiki/Claude-Louis_Navier "Claude-Louis Navier") [Noll](https://en.wikipedia.org/wiki/Walter_Noll "Walter Noll") [Pascal](https://en.wikipedia.org/wiki/Blaise_Pascal "Blaise Pascal") [Stokes](https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet "Sir George Stokes, 1st Baronet") [Truesdell](https://en.wikipedia.org/wiki/Clifford_Truesdell "Clifford Truesdell") |
| [v](https://en.wikipedia.org/wiki/Template:Continuum_mechanics "Template:Continuum mechanics") [t](https://en.wikipedia.org/wiki/Template_talk:Continuum_mechanics "Template talk:Continuum mechanics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Continuum_mechanics "Special:EditPage/Template:Continuum mechanics") |
In [physics](https://en.wikipedia.org/wiki/Physics "Physics"), **Hooke's law** is an [empirical law](https://en.wikipedia.org/wiki/Empirical_law "Empirical law") which states that the [force](https://en.wikipedia.org/wiki/Force "Force") (F) needed to extend or compress a [spring](https://en.wikipedia.org/wiki/Spring_\(device\) "Spring (device)") by some distance (x) [scales linearly](https://en.wikipedia.org/wiki/Proportionality_\(mathematics\)#Direct_proportionality "Proportionality (mathematics)") with respect to that distance—that is, *Fs* = *kx*, where k is a constant factor characteristic of the spring (i.e., its [stiffness](https://en.wikipedia.org/wiki/Stiffness "Stiffness")), and x is small compared to the total possible deformation of the spring.
The law is named after 17th-century British physicist [Robert Hooke](https://en.wikipedia.org/wiki/Robert_Hooke "Robert Hooke"). He first stated the law in 1676 as a Latin [anagram](https://en.wikipedia.org/wiki/Anagram "Anagram").[\[1\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-1)[\[2\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-2) He published the solution of his anagram in 1678[\[3\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-3) as: *ut tensio, sic vis* ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. It is the fundamental principle behind the [spring scale](https://en.wikipedia.org/wiki/Spring_scale "Spring scale"), the [manometer](https://en.wikipedia.org/wiki/Manometer "Manometer"), the [galvanometer](https://en.wikipedia.org/wiki/Galvanometer "Galvanometer"), and the [balance wheel](https://en.wikipedia.org/wiki/Balance_wheel "Balance wheel") of the [mechanical clock](https://en.wikipedia.org/wiki/Mechanical_clock "Mechanical clock").
The equation holds in many situations where an [elastic](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)") body is [deformed](https://en.wikipedia.org/wiki/Deformation_\(physics\) "Deformation (physics)"). An elastic body or material for which this equation can be assumed is said to be [linear-elastic](https://en.wikipedia.org/wiki/Linear_elasticity "Linear elasticity") or **Hookean**. Hooke's law is a [first-order linear approximation](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") to the real response of springs and other elastic bodies to applied forces. It fails once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those [elastic limits](https://en.wikipedia.org/wiki/Elastic_limit "Elastic limit") are reached.
## Definition
The modern [theory of elasticity](https://en.wikipedia.org/wiki/Theory_of_elasticity "Theory of elasticity") generalizes Hooke's law to say that the [strain](https://en.wikipedia.org/wiki/Deformation_\(mechanics\) "Deformation (mechanics)") (deformation) of an elastic object or material is proportional to the [stress](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") 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](https://en.wikipedia.org/wiki/Linear_map "Linear map") (a [tensor](https://en.wikipedia.org/wiki/Tensor "Tensor")) that can be represented by a [matrix](https://en.wikipedia.org/wiki/Matrix_\(mathematics\) "Matrix (mathematics)") of real numbers.
In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials they are made of. For example, one can deduce that a [homogeneous](https://en.wikipedia.org/wiki/Homogeneous "Homogeneous") rod with uniform [cross section](https://en.wikipedia.org/wiki/Cross_section_\(geometry\) "Cross section (geometry)") will behave like a simple spring when stretched, with a stiffness k directly proportional to its cross-section area and inversely proportional to its length.
### Linear springs
[](https://en.wikipedia.org/wiki/File:Spring-elongation-and-forces.svg)
Elongation and compression of a spring
Consider a simple [helical](https://en.wikipedia.org/wiki/Helix "Helix") spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is Fs. Suppose that the spring has reached a state of [equilibrium](https://en.wikipedia.org/wiki/Mechanical_equilibrium "Mechanical equilibrium"), where its length is not changing anymore. Let x be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that [F s = k x {\\displaystyle F\_{s}=kx}](https://en.wikipedia.org/w/index.php?title=Special:MathWikibase&qid=Q170282) or, equivalently, x \= F s k {\\displaystyle x={\\frac {F\_{s}}{k}}}  where k is a positive real number, characteristic of the spring. A spring with spaces between the coils can be compressed, and the same formula holds for compression, with Fs and x both negative in that case.[\[4\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-4)
[](https://en.wikipedia.org/wiki/File:Hooke%27s_Law_wikipedia.png)
Graphical derivation
According to this formula, the [graph](https://en.wikipedia.org/wiki/Graph_of_a_function "Graph of a function") of the applied force Fs as a function of the displacement x will be a straight line passing through the [origin](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates"), whose [slope](https://en.wikipedia.org/wiki/Slope "Slope") is k.
Hooke's law for a spring is also stated under the convention that Fs is the [restoring force](https://en.wikipedia.org/wiki/Restoring_force "Restoring force") exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F s \= − k x {\\displaystyle F\_{s}=-kx}  since the direction of the restoring force is opposite to that of the displacement.
### Torsional springs
The [torsional](https://en.wikipedia.org/wiki/Torsional "Torsional") analog of Hooke's law applies to [torsional springs](https://en.wikipedia.org/wiki/Torsional_spring "Torsional spring"). It states that the torque (τ) required to rotate an object is directly proportional to the angular displacement (θ) from the equilibrium position. It describes the relationship between the torque applied to an object and the resulting angular [deformation](https://en.wikipedia.org/wiki/Deformation_\(physics\) "Deformation (physics)") due to torsion. Mathematically, it can be expressed as:
τ
\=
−
k
θ
{\\displaystyle \\tau =-k\\theta }
Where:
- τ is the [torque](https://en.wikipedia.org/wiki/Torque "Torque") measured in Newton-meters or N·m.
- k is the [torsional constant](https://en.wikipedia.org/wiki/Torsional_constant "Torsional constant") (measured in N·m/radian), which characterizes the stiffness of the torsional spring or the resistance to angular displacement.
- θ is the [angular displacement](https://en.wikipedia.org/wiki/Angular_displacement "Angular displacement") (measured in radians) from the equilibrium position.
Just as in the linear case, this law shows that the torque is proportional to the angular displacement, and the negative sign indicates that the torque acts in a direction opposite to the angular displacement, providing a restoring force to bring the system back to equilibrium.
### General "scalar" springs
Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative.
For example, when a block of rubber attached to two parallel plates is deformed by [shearing](https://en.wikipedia.org/wiki/Simple_shear "Simple shear"), rather than stretching or compression, the shearing force *Fs* and the sideways displacement of the plates x obey Hooke's law (for small enough deformations).
Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight F placed at some intermediate point. The displacement x in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.
### Vector formulation
In the case of a helical spring that is stretched or compressed along its [axis](https://en.wikipedia.org/wiki/Axial_symmetry "Axial symmetry"), the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if Fs and x are defined as [vectors](https://en.wikipedia.org/wiki/Vector_\(mathematics\) "Vector (mathematics)"), Hooke's [equation](https://en.wikipedia.org/wiki/Equation "Equation") still holds and says that the force vector is the [elongation vector](https://en.wikipedia.org/wiki/Displacement_\(vector\) "Displacement (vector)") multiplied by a fixed [scalar](https://en.wikipedia.org/wiki/Scalar_\(mathematics\) "Scalar (mathematics)").
### General tensor form
Some elastic bodies will deform in one direction when subjected to a force with a different direction. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. In such cases, the *magnitude* of the displacement x will be proportional to the magnitude of the force Fs, as long as the direction of the latter remains the same (and its value is not too large); so the scalar version of Hooke's law *Fs* = −*kx* will hold. However, the force and displacement *vectors* will not be scalar multiples of each other, since they have different directions. Moreover, the ratio k between their magnitudes will depend on the direction of the vector Fs.
Yet, in such cases there is often a fixed [linear relation](https://en.wikipedia.org/wiki/Linear_map "Linear map") between the force and deformation vectors, as long as they are small enough. Namely, there is a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") **κ** from vectors to vectors, such that **F** = ***κ***(**X**), and ***κ***(*α***X**1 + *β***X**2) = *α**κ***(**X**1) + *β**κ***(**X**2) for any real numbers α, β and any displacement vectors **X**1, **X**2. Such a function is called a (second-order) [tensor](https://en.wikipedia.org/wiki/Tensor "Tensor").
With respect to an arbitrary [Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates"), the force and displacement vectors can be represented by 3 × 1 [matrices](https://en.wikipedia.org/wiki/Matrix_\(mathematics\) "Matrix (mathematics)") of real numbers. Then the tensor **κ** connecting them can be represented by a 3 × 3 matrix **κ** of real coefficients, that, when [multiplied](https://en.wikipedia.org/wiki/Matrix_product "Matrix product") by the displacement vector, gives the force vector:
F \= \[ F 1 F 2 F 3 \] \= \[ κ 11 κ 12 κ 13 κ 21 κ 22 κ 23 κ 31 κ 32 κ 33 \] \[ X 1 X 2 X 3 \] \= κ X {\\displaystyle \\mathbf {F} \\,=\\,{\\begin{bmatrix}F\_{1}\\\\F\_{2}\\\\F\_{3}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}\\kappa \_{11}&\\kappa \_{12}&\\kappa \_{13}\\\\\\kappa \_{21}&\\kappa \_{22}&\\kappa \_{23}\\\\\\kappa \_{31}&\\kappa \_{32}&\\kappa \_{33}\\end{bmatrix}}{\\begin{bmatrix}X\_{1}\\\\X\_{2}\\\\X\_{3}\\end{bmatrix}}\\,=\\,{\\boldsymbol {\\kappa }}\\mathbf {X} } 
That is, F i \= κ i 1 X 1 \+ κ i 2 X 2 \+ κ i 3 X 3 {\\displaystyle F\_{i}=\\kappa \_{i1}X\_{1}+\\kappa \_{i2}X\_{2}+\\kappa \_{i3}X\_{3}}  for *i* = 1, 2, 3. Therefore, Hooke's law **F** = ***κ*X** can be said to hold also when **X** and **F** are vectors with variable directions, except that the stiffness of the object is a tensor **κ**, rather than a single real number k.
### Hooke's law for continuous media
Main article: [Linear elasticity](https://en.wikipedia.org/wiki/Linear_elasticity "Linear elasticity")
[](https://en.wikipedia.org/wiki/File:Hookes_law_nanoscale.jpg)
(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.[\[5\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-5)
The stresses and strains of the material inside a [continuous](https://en.wikipedia.org/wiki/Continuum_mechanics "Continuum mechanics") elastic material (such as a block of rubber, the wall of a [boiler](https://en.wikipedia.org/wiki/Boiler "Boiler"), or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name.
However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.
In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the [strain tensor](https://en.wikipedia.org/wiki/Strain_tensor "Strain tensor") **ε** (in lieu of the displacement **X**) and the [stress tensor](https://en.wikipedia.org/wiki/Cauchy_stress_tensor "Cauchy stress tensor") **σ** (replacing the restoring force **F**). The analogue of Hooke's spring law for continuous media is then σ \= c ε , {\\displaystyle {\\boldsymbol {\\sigma }}=\\mathbf {c} {\\boldsymbol {\\varepsilon }},}  where **c** is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the [stiffness tensor](https://en.wikipedia.org/wiki/Stiffness_tensor "Stiffness tensor") or [elasticity tensor](https://en.wikipedia.org/wiki/Elasticity_tensor "Elasticity tensor"). One may also write it as ε \= s σ , {\\displaystyle {\\boldsymbol {\\varepsilon }}=\\mathbf {s} {\\boldsymbol {\\sigma }},}  where the tensor **s**, called the [compliance tensor](https://en.wikipedia.org/wiki/Stiffness_tensor "Stiffness tensor"), represents the inverse of said linear map.
In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices
ε \= \[ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 \] ; σ \= \[ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 \] {\\displaystyle {\\boldsymbol {\\varepsilon }}\\,=\\,{\\begin{bmatrix}\\varepsilon \_{11}&\\varepsilon \_{12}&\\varepsilon \_{13}\\\\\\varepsilon \_{21}&\\varepsilon \_{22}&\\varepsilon \_{23}\\\\\\varepsilon \_{31}&\\varepsilon \_{32}&\\varepsilon \_{33}\\end{bmatrix}}\\,;\\qquad {\\boldsymbol {\\sigma }}\\,=\\,{\\begin{bmatrix}\\sigma \_{11}&\\sigma \_{12}&\\sigma \_{13}\\\\\\sigma \_{21}&\\sigma \_{22}&\\sigma \_{23}\\\\\\sigma \_{31}&\\sigma \_{32}&\\sigma \_{33}\\end{bmatrix}}} 
Being a linear mapping between the nine numbers *σij* and the nine numbers *εkl*, the stiffness tensor **c** is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers *cijkl*. Hooke's law then says that σ i j \= ∑ k \= 1 3 ∑ l \= 1 3 c i j k l ε k l {\\displaystyle \\sigma \_{ij}=\\sum \_{k=1}^{3}\\sum \_{l=1}^{3}c\_{ijkl}\\varepsilon \_{kl}}  where *i*,*j* = 1,2,3.
All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor **ε** merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor **σ** specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor **c**, on the other hand, is a property of the material, and often depends on physical state variables such as temperature, [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure"), and [microstructure](https://en.wikipedia.org/wiki/Microstructure "Microstructure").
Due to the inherent symmetries of **σ**, **ε**, and **c**, only 21 elastic coefficients of the latter are independent.[\[6\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-6) This number can be further reduced by the symmetry of the material: 9 for an [orthorhombic](https://en.wikipedia.org/wiki/Orthorhombic_crystal_system "Orthorhombic crystal system") crystal, 5 for an [hexagonal](https://en.wikipedia.org/wiki/Hexagonal_crystal_family "Hexagonal crystal family") structure, and 3 for a [cubic](https://en.wikipedia.org/wiki/Cubic_crystal_system "Cubic crystal system") symmetry.[\[7\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-7) For [isotropic](https://en.wikipedia.org/wiki/Isotropic "Isotropic") media (which have the same physical properties in any direction), **c** can be reduced to only two independent numbers, the [bulk modulus](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") K and the [shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus") G, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.
## Analogous laws
Since Hooke's law is a simple proportionality between two quantities, its formulas and consequences are mathematically similar to those of many other physical laws, such as those describing the motion of [fluids](https://en.wikipedia.org/wiki/Fluid "Fluid"), or the [polarization](https://en.wikipedia.org/wiki/Ionic_polarization "Ionic polarization") of a [dielectric](https://en.wikipedia.org/wiki/Dielectric "Dielectric") by an [electric field](https://en.wikipedia.org/wiki/Electric_field "Electric field").
In particular, the tensor equation **σ** = **cε** relating elastic stresses to strains is entirely similar to the equation **τ** = **με̇** relating the [viscous stress tensor](https://en.wikipedia.org/wiki/Viscous_stress_tensor "Viscous stress tensor") **τ** and the [strain rate tensor](https://en.wikipedia.org/wiki/Strain_rate_tensor "Strain rate tensor") **ε̇** in flows of [viscous](https://en.wikipedia.org/wiki/Viscosity "Viscosity") fluids; although the former pertains to [static](https://en.wikipedia.org/wiki/Statics "Statics") stresses (related to *amount* of deformation) while the latter pertains to [dynamical](https://en.wikipedia.org/wiki/Dynamics_\(physics\) "Dynamics (physics)") stresses (related to the *rate* of deformation).
## Units of measurement
In [SI units](https://en.wikipedia.org/wiki/International_System_of_Units "International System of Units"), displacements are measured in meters (m), and forces in [newtons](https://en.wikipedia.org/wiki/Newton_\(unit\) "Newton (unit)") (N or kg·m/s2). Therefore, the spring constant k, and each element of the tensor **κ**, is measured in newtons per meter (N/m), or kilograms per second squared (kg/s2).
For continuous media, each element of the stress tensor **σ** is a force divided by an area; it is therefore measured in units of pressure, namely [pascals](https://en.wikipedia.org/wiki/Pascal_\(unit\) "Pascal (unit)") (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensor **ε** are [dimensionless](https://en.wikipedia.org/wiki/Dimensionless "Dimensionless") (displacements divided by distances). Therefore, the entries of cijkl are also expressed in units of pressure.
## General application to elastic materials
[](https://en.wikipedia.org/wiki/File:Stress_v_strain_A36_2.svg)
[Stress–strain curve](https://en.wikipedia.org/wiki/Stress%E2%80%93strain_curve "Stress–strain curve") for low-carbon steel, showing the relationship between the [stress](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") (force per unit area) and [strain](https://en.wikipedia.org/wiki/Deformation_\(mechanics\) "Deformation (mechanics)") (resulting compression/stretching, known as deformation). Hooke's law is only valid for the portion of the curve between the origin and the yield point (2).
1. [Ultimate strength](https://en.wikipedia.org/wiki/Ultimate_tensile_strength "Ultimate tensile strength")
2. [Yield strength](https://en.wikipedia.org/wiki/Yield_\(engineering\) "Yield (engineering)") (yield point)
3. Rupture
4. [Strain hardening](https://en.wikipedia.org/wiki/Strain_hardening "Strain hardening") region
5. [Necking](https://en.wikipedia.org/wiki/Necking_\(engineering\) "Necking (engineering)") region
1. Apparent stress (*F*/*A*0)
2. Actual stress (*F*/*A*)
(
- [view](https://en.wikipedia.org/wiki/Template:Stress%E2%80%93strain_plot "Template:Stress–strain plot")
- [talk](https://en.wikipedia.org/wiki/Template_talk:Stress%E2%80%93strain_plot "Template talk:Stress–strain plot")
- [edit](https://en.wikipedia.org/wiki/Special:EditPage/Template:Stress%E2%80%93strain_plot "Special:EditPage/Template:Stress–strain plot")
)
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.
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](https://en.wikipedia.org/wiki/Yield_\(engineering\) "Yield (engineering)")). 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](https://en.wikipedia.org/wiki/Proportional_limit "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.
Generalizations of Hooke's law for the case of [large deformations](https://en.wikipedia.org/wiki/Finite_strain_theory "Finite strain theory") is provided by models of [neo-Hookean solids](https://en.wikipedia.org/wiki/Neo-Hookean_solid "Neo-Hookean solid") and [Mooney–Rivlin solids](https://en.wikipedia.org/wiki/Mooney%E2%80%93Rivlin_solid "Mooney–Rivlin solid").
## Derived formulae
### Tensional stress of a uniform bar
A rod of any [elastic](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)") material may be viewed as a linear [spring](https://en.wikipedia.org/wiki/Spring_\(device\) "Spring (device)"). The rod has length L and cross-sectional area A. Its [tensile stress](https://en.wikipedia.org/wiki/Tensile_stress "Tensile stress") σ is linearly proportional to its fractional extension or strain ε by the [modulus of elasticity](https://en.wikipedia.org/wiki/Modulus_of_elasticity "Modulus of elasticity") E: σ \= E ε . {\\displaystyle \\sigma =E\\varepsilon .} 
The modulus of elasticity may often be considered constant. In turn, ε \= Δ L L {\\displaystyle \\varepsilon ={\\frac {\\Delta L}{L}}}  (that is, the fractional change in length), and since σ \= F A , {\\displaystyle \\sigma ={\\frac {F}{A}}\\,,}  it follows that:
ε \= σ E \= F A E . {\\displaystyle \\varepsilon ={\\frac {\\sigma }{E}}={\\frac {F}{AE}}\\,.} 
The change in length may be expressed as
Δ L \= ε L \= F L A E . {\\displaystyle \\Delta L=\\varepsilon L={\\frac {FL}{AE}}\\,.} 
### Spring energy
The potential energy *U*el(*x*) stored in a spring is given by U e l ( x ) \= 1 2 k x 2 {\\displaystyle U\_{\\mathrm {el} }(x)={\\tfrac {1}{2}}kx^{2}}  which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative. Substituting x \= F / k {\\displaystyle x=F/k}  gives U e l ( F ) \= F 2 2 k . {\\displaystyle U\_{\\mathrm {el} }(F)={\\frac {F^{2}}{2k}}.} 
This potential *U*el can be visualized as a [parabola](https://en.wikipedia.org/wiki/Parabola "Parabola") on the Ux\-plane such that *U*el(*x*) = 1/2*kx*2. As the spring is stretched in the positive x\-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate: d 2 U e l d x 2 \= k . {\\displaystyle {\\frac {d^{2}U\_{\\mathrm {el} }}{dx^{2}}}=k\\,.}  Note that the change in the change in U is constant even when the displacement and acceleration are zero.
### Relaxed force constants (generalized compliance constants)
Relaxed force constants (the inverse of generalized [compliance constants](https://en.wikipedia.org/wiki/Compliance_Constants "Compliance Constants")) are uniquely defined for molecular systems, in contradistinction to the usual "rigid" force constants, and thus their use allows meaningful correlations to be made between force fields calculated for [reactants](https://en.wikipedia.org/wiki/Reactant "Reactant"), [transition states](https://en.wikipedia.org/wiki/Transition_state "Transition state"), and products of a [chemical reaction](https://en.wikipedia.org/wiki/Chemical_reaction "Chemical reaction"). Just as the [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy") can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed [compliance constants](https://en.wikipedia.org/wiki/Compliance_constant "Compliance constant"). A direct method exists for calculating the compliance constant for any internal coordinate of a molecule, without the need to do the normal mode analysis.[\[8\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-8) The suitability of relaxed force constants (inverse compliance constants) as [covalent bond](https://en.wikipedia.org/wiki/Covalent_bond "Covalent bond") strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.[\[9\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-9)
### Harmonic oscillator
See also: [Harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator "Harmonic oscillator")
[](https://en.wikipedia.org/wiki/File:Mass-spring-system.png)
A mass suspended by a spring is the classical example of a harmonic oscillator
A mass m attached to the end of a spring is a classic example of a [harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator "Harmonic oscillator"). By pulling slightly on the mass and then releasing it, the system will be set in [sinusoidal](https://en.wikipedia.org/wiki/Sine_wave "Sine wave") oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglect [friction](https://en.wikipedia.org/wiki/Friction "Friction") and the mass of the spring, the amplitude of the oscillation will remain constant; and its [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") f will be independent of its amplitude, determined only by the mass and the stiffness of the spring: f \= 1 2 π k m {\\displaystyle f={\\frac {1}{2\\pi }}{\\sqrt {\\frac {k}{m}}}}  This phenomenon made possible the construction of accurate [mechanical clocks](https://en.wikipedia.org/wiki/Mechanical_clock "Mechanical clock") and watches that could be carried on ships and people's pockets.
### Rotation in gravity-free space
If the mass m were attached to a spring with force constant k and rotating in free space, the spring tension (*F*t) would supply the required [centripetal force](https://en.wikipedia.org/wiki/Centripetal_force "Centripetal force") (*F*c):
F t \= k x ; F c \= m ω 2 r {\\displaystyle F\_{\\mathrm {t} }=kx\\,;\\qquad F\_{\\mathrm {c} }=m\\omega ^{2}r}  Since *F*t = *F*c and *x* = *r*, then: k \= m ω 2 {\\displaystyle k=m\\omega ^{2}}  Given that *ω* = 2π*f*, this leads to the same frequency equation as above: f \= 1 2 π k m {\\displaystyle f={\\frac {1}{2\\pi }}{\\sqrt {\\frac {k}{m}}}} 
## Linear elasticity theory for continuous media
See also: [Elasticity tensor](https://en.wikipedia.org/wiki/Elasticity_tensor "Elasticity tensor")
Note: the [Einstein summation convention](https://en.wikipedia.org/wiki/Einstein_summation_convention "Einstein summation convention") of summing on repeated indices is used below.
### Isotropic materials
For an analogous development for viscous fluids, see [Viscosity](https://en.wikipedia.org/wiki/Viscosity "Viscosity").
[Isotropic materials](https://en.wikipedia.org/wiki/Isotropic_solid "Isotropic solid") 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](https://en.wikipedia.org/wiki/Trace_\(linear_algebra\) "Trace (linear algebra)") 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.[\[10\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-10) Thus in [index notation](https://en.wikipedia.org/wiki/Ricci_calculus "Ricci calculus"):
ε i j \= ( 1 3 ε k k δ i j ) \+ ( ε i j − 1 3 ε k k δ i j ) {\\displaystyle \\varepsilon \_{ij}=\\left({\\tfrac {1}{3}}\\varepsilon \_{kk}\\delta \_{ij}\\right)+\\left(\\varepsilon \_{ij}-{\\tfrac {1}{3}}\\varepsilon \_{kk}\\delta \_{ij}\\right)}  where δij is the [Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta "Kronecker delta"). In direct tensor notation: ε \= vol ( ε ) \+ dev ( ε ) ; vol ( ε ) \= 1 3 tr ( ε ) I ; dev ( ε ) \= ε − vol ( ε ) {\\displaystyle {\\boldsymbol {\\varepsilon }}=\\operatorname {vol} ({\\boldsymbol {\\varepsilon }})+\\operatorname {dev} ({\\boldsymbol {\\varepsilon }})\\,;\\qquad \\operatorname {vol} ({\\boldsymbol {\\varepsilon }})={\\tfrac {1}{3}}\\operatorname {tr} ({\\boldsymbol {\\varepsilon }})~\\mathbf {I} \\,;\\qquad \\operatorname {dev} ({\\boldsymbol {\\varepsilon }})={\\boldsymbol {\\varepsilon }}-\\operatorname {vol} ({\\boldsymbol {\\varepsilon }})} 
where **I** is the second-order identity tensor.
The first term on the right is the constant tensor, also known as the **[volumetric strain tensor](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory#Volumetric_strain "Infinitesimal strain theory")**, and the second term is the traceless symmetric tensor, also known as the **[deviatoric strain tensor](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory#Strain_deviator_tensor "Infinitesimal strain theory")** 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:
σ i j \= 3 K ( 1 3 ε k k δ i j ) \+ 2 G ( ε i j − 1 3 ε k k δ i j ) ; σ \= 3 K vol ( ε ) \+ 2 G dev ( ε ) {\\displaystyle \\sigma \_{ij}=3K\\left({\\tfrac {1}{3}}\\varepsilon \_{kk}\\delta \_{ij}\\right)+2G\\left(\\varepsilon \_{ij}-{\\tfrac {1}{3}}\\varepsilon \_{kk}\\delta \_{ij}\\right)\\,;\\qquad {\\boldsymbol {\\sigma }}=3K\\operatorname {vol} ({\\boldsymbol {\\varepsilon }})+2G\\operatorname {dev} ({\\boldsymbol {\\varepsilon }})}  where K is the [bulk modulus](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") and G is the [shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus").
Using the relationships between the [elastic moduli](https://en.wikipedia.org/wiki/Elastic_modulus "Elastic modulus"), 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 [\[11\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Simo98-11)
σ \= λ tr ( ε ) I \+ 2 μ ε \= c : ε ; c \= λ I ⊗ I \+ 2 μ I {\\displaystyle {\\boldsymbol {\\sigma }}=\\lambda \\operatorname {tr} ({\\boldsymbol {\\varepsilon }})\\mathbf {I} +2\\mu {\\boldsymbol {\\varepsilon }}={\\mathsf {c}}:{\\boldsymbol {\\varepsilon }}\\,;\\qquad {\\mathsf {c}}=\\lambda \\mathbf {I} \\otimes \\mathbf {I} +2\\mu {\\mathsf {I}}}  where *λ* = *K* − 2/3*G* = *c*1111 − 2*c*1212 and *μ* = *G* = *c*1212 are the [Lamé constants](https://en.wikipedia.org/wiki/Lam%C3%A9_constants "Lamé constants"), **I** is the second-rank identity tensor, and **I** is the symmetric part of the fourth-rank identity tensor. In index notation: σ i j \= λ ε k k δ i j \+ 2 μ ε i j \= c i j k l ε k l ; c i j k l \= λ δ i j δ k l \+ μ ( δ i k δ j l \+ δ i l δ j k ) {\\displaystyle \\sigma \_{ij}=\\lambda \\varepsilon \_{kk}~\\delta \_{ij}+2\\mu \\varepsilon \_{ij}=c\_{ijkl}\\varepsilon \_{kl}\\,;\\qquad c\_{ijkl}=\\lambda \\delta \_{ij}\\delta \_{kl}+\\mu \\left(\\delta \_{ik}\\delta \_{jl}+\\delta \_{il}\\delta \_{jk}\\right)} 
The inverse relationship is[\[12\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Milton02-12)
ε \= 1 2 μ σ − λ 2 μ ( 3 λ \+ 2 μ ) tr ( σ ) I \= 1 2 G σ \+ ( 1 9 K − 1 6 G ) tr ( σ ) I {\\displaystyle {\\boldsymbol {\\varepsilon }}={\\frac {1}{2\\mu }}{\\boldsymbol {\\sigma }}-{\\frac {\\lambda }{2\\mu (3\\lambda +2\\mu )}}\\operatorname {tr} ({\\boldsymbol {\\sigma }})\\mathbf {I} ={\\frac {1}{2G}}{\\boldsymbol {\\sigma }}+\\left({\\frac {1}{9K}}-{\\frac {1}{6G}}\\right)\\operatorname {tr} ({\\boldsymbol {\\sigma }})\\mathbf {I} } 
Therefore, the compliance tensor in the relation **ε** = **s** : **σ** is
s \= − λ 2 μ ( 3 λ \+ 2 μ ) I ⊗ I \+ 1 2 μ I \= ( 1 9 K − 1 6 G ) I ⊗ I \+ 1 2 G I {\\displaystyle {\\mathsf {s}}=-{\\frac {\\lambda }{2\\mu (3\\lambda +2\\mu )}}\\mathbf {I} \\otimes \\mathbf {I} +{\\frac {1}{2\\mu }}{\\mathsf {I}}=\\left({\\frac {1}{9K}}-{\\frac {1}{6G}}\\right)\\mathbf {I} \\otimes \\mathbf {I} +{\\frac {1}{2G}}{\\mathsf {I}}} 
In terms of [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") and [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio"), Hooke's law for isotropic materials can then be expressed as
ε i j \= 1 E ( σ i j − ν ( σ k k δ i j − σ i j ) ) ; ε \= 1 E ( σ − ν ( tr ( σ ) I − σ ) ) \= 1 \+ ν E σ − ν E tr ( σ ) I {\\displaystyle \\varepsilon \_{ij}={\\frac {1}{E}}{\\big (}\\sigma \_{ij}-\\nu (\\sigma \_{kk}\\delta \_{ij}-\\sigma \_{ij}){\\big )}\\,;\\qquad {\\boldsymbol {\\varepsilon }}={\\frac {1}{E}}{\\big (}{\\boldsymbol {\\sigma }}-\\nu (\\operatorname {tr} ({\\boldsymbol {\\sigma }})\\mathbf {I} -{\\boldsymbol {\\sigma }}){\\big )}={\\frac {1+\\nu }{E}}{\\boldsymbol {\\sigma }}-{\\frac {\\nu }{E}}\\operatorname {tr} ({\\boldsymbol {\\sigma }})\\mathbf {I} } 
This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is ε 11 \= 1 E ( σ 11 − ν ( σ 22 \+ σ 33 ) ) ε 22 \= 1 E ( σ 22 − ν ( σ 11 \+ σ 33 ) ) ε 33 \= 1 E ( σ 33 − ν ( σ 11 \+ σ 22 ) ) ε 12 \= 1 2 G σ 12 ; ε 13 \= 1 2 G σ 13 ; ε 23 \= 1 2 G σ 23 {\\displaystyle {\\begin{aligned}\\varepsilon \_{11}&={\\frac {1}{E}}{\\big (}\\sigma \_{11}-\\nu (\\sigma \_{22}+\\sigma \_{33}){\\big )}\\\\\\varepsilon \_{22}&={\\frac {1}{E}}{\\big (}\\sigma \_{22}-\\nu (\\sigma \_{11}+\\sigma \_{33}){\\big )}\\\\\\varepsilon \_{33}&={\\frac {1}{E}}{\\big (}\\sigma \_{33}-\\nu (\\sigma \_{11}+\\sigma \_{22}){\\big )}\\\\\\varepsilon \_{12}&={\\frac {1}{2G}}\\sigma \_{12}\\,;\\qquad \\varepsilon \_{13}={\\frac {1}{2G}}\\sigma \_{13}\\,;\\qquad \\varepsilon \_{23}={\\frac {1}{2G}}\\sigma \_{23}\\end{aligned}}}  where E is [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") and ν is [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio"). (See [3-D elasticity](https://en.wikipedia.org/wiki/3-D_elasticity "3-D elasticity")).
**Derivation of Hooke's law in three dimensions**
The three-dimensional form of Hooke's law can be derived using Poisson's ratio and the one-dimensional 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), ε 1 ′ \= 1 E σ 1 , ε 2 ′ \= − ν E σ 1 , ε 3 ′ \= − ν E σ 1 , {\\displaystyle {\\begin{aligned}\\varepsilon \_{1}'&={\\frac {1}{E}}\\sigma \_{1}\\,,\\\\\\varepsilon \_{2}'&=-{\\frac {\\nu }{E}}\\sigma \_{1}\\,,\\\\\\varepsilon \_{3}'&=-{\\frac {\\nu }{E}}\\sigma \_{1}\\,,\\end{aligned}}}  where ν is Poisson's ratio and E is Young's modulus.
We get similar equations to the loads in directions 2 and 3, ε 1 ″ \= − ν E σ 2 , ε 2 ″ \= 1 E σ 2 , ε 3 ″ \= − ν E σ 2 , {\\displaystyle {\\begin{aligned}\\varepsilon \_{1}''&=-{\\frac {\\nu }{E}}\\sigma \_{2}\\,,\\\\\\varepsilon \_{2}''&={\\frac {1}{E}}\\sigma \_{2}\\,,\\\\\\varepsilon \_{3}''&=-{\\frac {\\nu }{E}}\\sigma \_{2}\\,,\\end{aligned}}}  and ε 1 ‴ \= − ν E σ 3 , ε 2 ‴ \= − ν E σ 3 , ε 3 ‴ \= 1 E σ 3 . {\\displaystyle {\\begin{aligned}\\varepsilon \_{1}'''&=-{\\frac {\\nu }{E}}\\sigma \_{3}\\,,\\\\\\varepsilon \_{2}'''&=-{\\frac {\\nu }{E}}\\sigma \_{3}\\,,\\\\\\varepsilon \_{3}'''&={\\frac {1}{E}}\\sigma \_{3}\\,.\\end{aligned}}} 
Summing the three cases together (*εi* = *εi*′ + *εi*″ + *εi*‴) we get ε 1 \= 1 E ( σ 1 − ν ( σ 2 \+ σ 3 ) ) , ε 2 \= 1 E ( σ 2 − ν ( σ 1 \+ σ 3 ) ) , ε 3 \= 1 E ( σ 3 − ν ( σ 1 \+ σ 2 ) ) , {\\displaystyle {\\begin{aligned}\\varepsilon \_{1}&={\\frac {1}{E}}{\\big (}\\sigma \_{1}-\\nu (\\sigma \_{2}+\\sigma \_{3}){\\big )}\\,,\\\\\\varepsilon \_{2}&={\\frac {1}{E}}{\\big (}\\sigma \_{2}-\\nu (\\sigma \_{1}+\\sigma \_{3}){\\big )}\\,,\\\\\\varepsilon \_{3}&={\\frac {1}{E}}{\\big (}\\sigma \_{3}-\\nu (\\sigma \_{1}+\\sigma \_{2}){\\big )}\\,,\\end{aligned}}}  or by adding and subtracting one νσ ε 1 \= 1 E ( ( 1 \+ ν ) σ 1 − ν ( σ 1 \+ σ 2 \+ σ 3 ) ) , ε 2 \= 1 E ( ( 1 \+ ν ) σ 2 − ν ( σ 1 \+ σ 2 \+ σ 3 ) ) , ε 3 \= 1 E ( ( 1 \+ ν ) σ 3 − ν ( σ 1 \+ σ 2 \+ σ 3 ) ) , {\\displaystyle {\\begin{aligned}\\varepsilon \_{1}&={\\frac {1}{E}}{\\big (}(1+\\nu )\\sigma \_{1}-\\nu (\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3}){\\big )}\\,,\\\\\\varepsilon \_{2}&={\\frac {1}{E}}{\\big (}(1+\\nu )\\sigma \_{2}-\\nu (\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3}){\\big )}\\,,\\\\\\varepsilon \_{3}&={\\frac {1}{E}}{\\big (}(1+\\nu )\\sigma \_{3}-\\nu (\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3}){\\big )}\\,,\\end{aligned}}}  and further we get by solving *σ*1 σ 1 \= E 1 \+ ν ε 1 \+ ν 1 \+ ν ( σ 1 \+ σ 2 \+ σ 3 ) . {\\displaystyle \\sigma \_{1}={\\frac {E}{1+\\nu }}\\varepsilon \_{1}+{\\frac {\\nu }{1+\\nu }}(\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3})\\,.} 
Calculating the sum ε 1 \+ ε 2 \+ ε 3 \= 1 E ( ( 1 \+ ν ) ( σ 1 \+ σ 2 \+ σ 3 ) − 3 ν ( σ 1 \+ σ 2 \+ σ 3 ) ) \= 1 − 2 ν E ( σ 1 \+ σ 2 \+ σ 3 ) σ 1 \+ σ 2 \+ σ 3 \= E 1 − 2 ν ( ε 1 \+ ε 2 \+ ε 3 ) {\\displaystyle {\\begin{aligned}\\varepsilon \_{1}+\\varepsilon \_{2}+\\varepsilon \_{3}&={\\frac {1}{E}}{\\big (}(1+\\nu )(\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3})-3\\nu (\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3}){\\big )}={\\frac {1-2\\nu }{E}}(\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3})\\\\\\sigma \_{1}+\\sigma \_{2}+\\sigma \_{3}&={\\frac {E}{1-2\\nu }}(\\varepsilon \_{1}+\\varepsilon \_{2}+\\varepsilon \_{3})\\end{aligned}}}  and substituting it to the equation solved for *σ*1 gives σ 1 \= E 1 \+ ν ε 1 \+ E ν ( 1 \+ ν ) ( 1 − 2 ν ) ( ε 1 \+ ε 2 \+ ε 3 ) \= 2 μ ε 1 \+ λ ( ε 1 \+ ε 2 \+ ε 3 ) , {\\displaystyle {\\begin{aligned}\\sigma \_{1}&={\\frac {E}{1+\\nu }}\\varepsilon \_{1}+{\\frac {E\\nu }{(1+\\nu )(1-2\\nu )}}(\\varepsilon \_{1}+\\varepsilon \_{2}+\\varepsilon \_{3})\\\\&=2\\mu \\varepsilon \_{1}+\\lambda (\\varepsilon \_{1}+\\varepsilon \_{2}+\\varepsilon \_{3})\\,,\\end{aligned}}}  where μ and λ are the [Lamé parameters](https://en.wikipedia.org/wiki/Lam%C3%A9_parameters "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 \[ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 \] \= \[ ε 11 ε 22 ε 33 γ 23 γ 13 γ 12 \] \= 1 E \[ 1 − ν − ν 0 0 0 − ν 1 − ν 0 0 0 − ν − ν 1 0 0 0 0 0 0 2 \+ 2 ν 0 0 0 0 0 0 2 \+ 2 ν 0 0 0 0 0 0 2 \+ 2 ν \] \[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 \] {\\displaystyle {\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\\\varepsilon \_{33}\\\\2\\varepsilon \_{23}\\\\2\\varepsilon \_{13}\\\\2\\varepsilon \_{12}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\\\varepsilon \_{33}\\\\\\gamma \_{23}\\\\\\gamma \_{13}\\\\\\gamma \_{12}\\end{bmatrix}}\\,=\\,{\\frac {1}{E}}{\\begin{bmatrix}1&-\\nu &-\\nu &0&0&0\\\\-\\nu &1&-\\nu &0&0&0\\\\-\\nu &-\\nu &1&0&0&0\\\\0&0&0&2+2\\nu &0&0\\\\0&0&0&0&2+2\\nu &0\\\\0&0&0&0&0&2+2\\nu \\end{bmatrix}}{\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{33}\\\\\\sigma \_{23}\\\\\\sigma \_{13}\\\\\\sigma \_{12}\\end{bmatrix}}}  where *γij* = 2*εij* is the **engineering shear strain**. The inverse relation may be written as \[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 \] \= E ( 1 \+ ν ) ( 1 − 2 ν ) \[ 1 − ν ν ν 0 0 0 ν 1 − ν ν 0 0 0 ν ν 1 − ν 0 0 0 0 0 0 1 − 2 ν 2 0 0 0 0 0 0 1 − 2 ν 2 0 0 0 0 0 0 1 − 2 ν 2 \] \[ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 \] {\\displaystyle {\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{33}\\\\\\sigma \_{23}\\\\\\sigma \_{13}\\\\\\sigma \_{12}\\end{bmatrix}}\\,=\\,{\\frac {E}{(1+\\nu )(1-2\\nu )}}{\\begin{bmatrix}1-\\nu &\\nu &\\nu &0&0&0\\\\\\nu &1-\\nu &\\nu &0&0&0\\\\\\nu &\\nu &1-\\nu &0&0&0\\\\0&0&0&{\\frac {1-2\\nu }{2}}&0&0\\\\0&0&0&0&{\\frac {1-2\\nu }{2}}&0\\\\0&0&0&0&0&{\\frac {1-2\\nu }{2}}\\end{bmatrix}}{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\\\varepsilon \_{33}\\\\2\\varepsilon \_{23}\\\\2\\varepsilon \_{13}\\\\2\\varepsilon \_{12}\\end{bmatrix}}}  which can be simplified thanks to the Lamé constants: \[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 \] \= \[ 2 μ \+ λ λ λ 0 0 0 λ 2 μ \+ λ λ 0 0 0 λ λ 2 μ \+ λ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ \] \[ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 \] {\\displaystyle {\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{33}\\\\\\sigma \_{23}\\\\\\sigma \_{13}\\\\\\sigma \_{12}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}2\\mu +\\lambda &\\lambda &\\lambda &0&0&0\\\\\\lambda &2\\mu +\\lambda &\\lambda &0&0&0\\\\\\lambda &\\lambda &2\\mu +\\lambda &0&0&0\\\\0&0&0&\\mu &0&0\\\\0&0&0&0&\\mu &0\\\\0&0&0&0&0&\\mu \\end{bmatrix}}{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\\\varepsilon \_{33}\\\\2\\varepsilon \_{23}\\\\2\\varepsilon \_{13}\\\\2\\varepsilon \_{12}\\end{bmatrix}}}  In vector notation this becomes \[ σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33 \] \= 2 μ \[ ε 11 ε 12 ε 13 ε 12 ε 22 ε 23 ε 13 ε 23 ε 33 \] \+ λ I ( ε 11 \+ ε 22 \+ ε 33 ) {\\displaystyle {\\begin{bmatrix}\\sigma \_{11}&\\sigma \_{12}&\\sigma \_{13}\\\\\\sigma \_{12}&\\sigma \_{22}&\\sigma \_{23}\\\\\\sigma \_{13}&\\sigma \_{23}&\\sigma \_{33}\\end{bmatrix}}\\,=\\,2\\mu {\\begin{bmatrix}\\varepsilon \_{11}&\\varepsilon \_{12}&\\varepsilon \_{13}\\\\\\varepsilon \_{12}&\\varepsilon \_{22}&\\varepsilon \_{23}\\\\\\varepsilon \_{13}&\\varepsilon \_{23}&\\varepsilon \_{33}\\end{bmatrix}}+\\lambda \\mathbf {I} \\left(\\varepsilon \_{11}+\\varepsilon \_{22}+\\varepsilon \_{33}\\right)}  where **I** is the identity tensor.
#### Plane stress
Under [plane stress](https://en.wikipedia.org/wiki/Plane_stress#Plane_stress "Plane stress") conditions, *σ*31 = *σ*13 = *σ*32 = *σ*23 = *σ*33 = 0. In that case Hooke's law takes the form \[ σ 11 σ 22 σ 12 \] \= E 1 − ν 2 \[ 1 ν 0 ν 1 0 0 0 1 − ν 2 \] \[ ε 11 ε 22 2 ε 12 \] {\\displaystyle {\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{12}\\end{bmatrix}}\\,=\\,{\\frac {E}{1-\\nu ^{2}}}{\\begin{bmatrix}1&\\nu &0\\\\\\nu &1&0\\\\0&0&{\\frac {1-\\nu }{2}}\\end{bmatrix}}{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\2\\varepsilon \_{12}\\end{bmatrix}}} 
In vector notation this becomes \[ σ 11 σ 12 σ 12 σ 22 \] \= E 1 − ν 2 ( ( 1 − ν ) \[ ε 11 ε 12 ε 12 ε 22 \] \+ ν I ( ε 11 \+ ε 22 ) ) {\\displaystyle {\\begin{bmatrix}\\sigma \_{11}&\\sigma \_{12}\\\\\\sigma \_{12}&\\sigma \_{22}\\end{bmatrix}}\\,=\\,{\\frac {E}{1-\\nu ^{2}}}\\left((1-\\nu ){\\begin{bmatrix}\\varepsilon \_{11}&\\varepsilon \_{12}\\\\\\varepsilon \_{12}&\\varepsilon \_{22}\\end{bmatrix}}+\\nu \\mathbf {I} \\left(\\varepsilon \_{11}+\\varepsilon \_{22}\\right)\\right)} 
The inverse relation is usually written in the reduced form \[ ε 11 ε 22 2 ε 12 \] \= 1 E \[ 1 − ν 0 − ν 1 0 0 0 2 \+ 2 ν \] \[ σ 11 σ 22 σ 12 \] {\\displaystyle {\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\2\\varepsilon \_{12}\\end{bmatrix}}\\,=\\,{\\frac {1}{E}}{\\begin{bmatrix}1&-\\nu &0\\\\-\\nu &1&0\\\\0&0&2+2\\nu \\end{bmatrix}}{\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{12}\\end{bmatrix}}} 
#### Plane strain
Under [plane strain](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory#Plane_strain "Infinitesimal strain theory") conditions, *ε*31 = *ε*13 = *ε*32 = *ε*23 = *ε*33 = 0. In this case Hooke's law takes the form \[ σ 11 σ 22 σ 12 \] \= E ( 1 \+ ν ) ( 1 − 2 ν ) \[ 1 − ν ν 0 ν 1 − ν 0 0 0 1 − 2 ν 2 \] \[ ε 11 ε 22 2 ε 12 \] {\\displaystyle {\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{12}\\end{bmatrix}}\\,=\\,{\\frac {E}{(1+\\nu )(1-2\\nu )}}{\\begin{bmatrix}1-\\nu &\\nu &0\\\\\\nu &1-\\nu &0\\\\0&0&{\\frac {1-2\\nu }{2}}\\end{bmatrix}}{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\2\\varepsilon \_{12}\\end{bmatrix}}} 
### Anisotropic materials
The symmetry of the [Cauchy stress tensor](https://en.wikipedia.org/wiki/Stress_\(physics\) "Stress (physics)") (*σij* = *σji*) and the generalized Hooke's laws (*σij* = *cijklεkl*) implies that *cijkl* = *cjikl*. Similarly, the symmetry of the [infinitesimal strain tensor](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory "Infinitesimal strain theory") implies that *cijkl* = *cijlk*. These symmetries are called the **minor symmetries** of the stiffness tensor **c**. This reduces the number of elastic constants from 81 to 36.
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 (U), then σ i j \= ∂ U ∂ ε i j ⟹ c i j k l \= ∂ 2 U ∂ ε i j ∂ ε k l . {\\displaystyle \\sigma \_{ij}={\\frac {\\partial U}{\\partial \\varepsilon \_{ij}}}\\quad \\implies \\quad c\_{ijkl}={\\frac {\\partial ^{2}U}{\\partial \\varepsilon \_{ij}\\partial \\varepsilon \_{kl}}}\\,.}  The arbitrariness of the order of differentiation implies that *cijkl* = *cklij*. These are called the **major symmetries** of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. 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 notation](https://en.wikipedia.org/wiki/Voigt_notation "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 (**e**1,**e**2,**e**3) as \[ σ \] \= \[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 \] ≡ \[ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 \] ; \[ ε \] \= \[ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 \] ≡ \[ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 \] {\\displaystyle \[{\\boldsymbol {\\sigma }}\]\\,=\\,{\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{33}\\\\\\sigma \_{23}\\\\\\sigma \_{13}\\\\\\sigma \_{12}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}\\sigma \_{1}\\\\\\sigma \_{2}\\\\\\sigma \_{3}\\\\\\sigma \_{4}\\\\\\sigma \_{5}\\\\\\sigma \_{6}\\end{bmatrix}}\\,;\\qquad \[{\\boldsymbol {\\varepsilon }}\]\\,=\\,{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\\\varepsilon \_{33}\\\\2\\varepsilon \_{23}\\\\2\\varepsilon \_{13}\\\\2\\varepsilon \_{12}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}\\varepsilon \_{1}\\\\\\varepsilon \_{2}\\\\\\varepsilon \_{3}\\\\\\varepsilon \_{4}\\\\\\varepsilon \_{5}\\\\\\varepsilon \_{6}\\end{bmatrix}}} ![{\\displaystyle \[{\\boldsymbol {\\sigma }}\]\\,=\\,{\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{33}\\\\\\sigma \_{23}\\\\\\sigma \_{13}\\\\\\sigma \_{12}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}\\sigma \_{1}\\\\\\sigma \_{2}\\\\\\sigma \_{3}\\\\\\sigma \_{4}\\\\\\sigma \_{5}\\\\\\sigma \_{6}\\end{bmatrix}}\\,;\\qquad \[{\\boldsymbol {\\varepsilon }}\]\\,=\\,{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\\\varepsilon \_{33}\\\\2\\varepsilon \_{23}\\\\2\\varepsilon \_{13}\\\\2\\varepsilon \_{12}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}\\varepsilon \_{1}\\\\\\varepsilon \_{2}\\\\\\varepsilon \_{3}\\\\\\varepsilon \_{4}\\\\\\varepsilon \_{5}\\\\\\varepsilon \_{6}\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d84c34fc9efc62922b42a33f888656c62d794b) Then the stiffness tensor (**c**) can be expressed as \[ c \] \= \[ c 1111 c 1122 c 1133 c 1123 c 1131 c 1112 c 2211 c 2222 c 2233 c 2223 c 2231 c 2212 c 3311 c 3322 c 3333 c 3323 c 3331 c 3312 c 2311 c 2322 c 2333 c 2323 c 2331 c 2312 c 3111 c 3122 c 3133 c 3123 c 3131 c 3112 c 1211 c 1222 c 1233 c 1223 c 1231 c 1212 \] ≡ \[ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 \] {\\displaystyle \[{\\mathsf {c}}\]\\,=\\,{\\begin{bmatrix}c\_{1111}\&c\_{1122}\&c\_{1133}\&c\_{1123}\&c\_{1131}\&c\_{1112}\\\\c\_{2211}\&c\_{2222}\&c\_{2233}\&c\_{2223}\&c\_{2231}\&c\_{2212}\\\\c\_{3311}\&c\_{3322}\&c\_{3333}\&c\_{3323}\&c\_{3331}\&c\_{3312}\\\\c\_{2311}\&c\_{2322}\&c\_{2333}\&c\_{2323}\&c\_{2331}\&c\_{2312}\\\\c\_{3111}\&c\_{3122}\&c\_{3133}\&c\_{3123}\&c\_{3131}\&c\_{3112}\\\\c\_{1211}\&c\_{1222}\&c\_{1233}\&c\_{1223}\&c\_{1231}\&c\_{1212}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}C\_{11}\&C\_{12}\&C\_{13}\&C\_{14}\&C\_{15}\&C\_{16}\\\\C\_{12}\&C\_{22}\&C\_{23}\&C\_{24}\&C\_{25}\&C\_{26}\\\\C\_{13}\&C\_{23}\&C\_{33}\&C\_{34}\&C\_{35}\&C\_{36}\\\\C\_{14}\&C\_{24}\&C\_{34}\&C\_{44}\&C\_{45}\&C\_{46}\\\\C\_{15}\&C\_{25}\&C\_{35}\&C\_{45}\&C\_{55}\&C\_{56}\\\\C\_{16}\&C\_{26}\&C\_{36}\&C\_{46}\&C\_{56}\&C\_{66}\\end{bmatrix}}} ![{\\displaystyle \[{\\mathsf {c}}\]\\,=\\,{\\begin{bmatrix}c\_{1111}\&c\_{1122}\&c\_{1133}\&c\_{1123}\&c\_{1131}\&c\_{1112}\\\\c\_{2211}\&c\_{2222}\&c\_{2233}\&c\_{2223}\&c\_{2231}\&c\_{2212}\\\\c\_{3311}\&c\_{3322}\&c\_{3333}\&c\_{3323}\&c\_{3331}\&c\_{3312}\\\\c\_{2311}\&c\_{2322}\&c\_{2333}\&c\_{2323}\&c\_{2331}\&c\_{2312}\\\\c\_{3111}\&c\_{3122}\&c\_{3133}\&c\_{3123}\&c\_{3131}\&c\_{3112}\\\\c\_{1211}\&c\_{1222}\&c\_{1233}\&c\_{1223}\&c\_{1231}\&c\_{1212}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}C\_{11}\&C\_{12}\&C\_{13}\&C\_{14}\&C\_{15}\&C\_{16}\\\\C\_{12}\&C\_{22}\&C\_{23}\&C\_{24}\&C\_{25}\&C\_{26}\\\\C\_{13}\&C\_{23}\&C\_{33}\&C\_{34}\&C\_{35}\&C\_{36}\\\\C\_{14}\&C\_{24}\&C\_{34}\&C\_{44}\&C\_{45}\&C\_{46}\\\\C\_{15}\&C\_{25}\&C\_{35}\&C\_{45}\&C\_{55}\&C\_{56}\\\\C\_{16}\&C\_{26}\&C\_{36}\&C\_{46}\&C\_{56}\&C\_{66}\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85c8bf05adff9dcaec56f4863dd039fae5986a79)
and Hooke's law is written as
\[ σ \] \= \[ C \] \[ ε \] or σ i \= C i j ε j . {\\displaystyle \[{\\boldsymbol {\\sigma }}\]=\[{\\mathsf {C}}\]\[{\\boldsymbol {\\varepsilon }}\]\\qquad {\\text{or}}\\qquad \\sigma \_{i}=C\_{ij}\\varepsilon \_{j}\\,.} ![{\\displaystyle \[{\\boldsymbol {\\sigma }}\]=\[{\\mathsf {C}}\]\[{\\boldsymbol {\\varepsilon }}\]\\qquad {\\text{or}}\\qquad \\sigma \_{i}=C\_{ij}\\varepsilon \_{j}\\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0315b5cfc25f83e499fadf8ce4921e11340f8e) Similarly the compliance tensor (**s**) can be written as \[ s \] \= \[ s 1111 s 1122 s 1133 2 s 1123 2 s 1131 2 s 1112 s 2211 s 2222 s 2233 2 s 2223 2 s 2231 2 s 2212 s 3311 s 3322 s 3333 2 s 3323 2 s 3331 2 s 3312 2 s 2311 2 s 2322 2 s 2333 4 s 2323 4 s 2331 4 s 2312 2 s 3111 2 s 3122 2 s 3133 4 s 3123 4 s 3131 4 s 3112 2 s 1211 2 s 1222 2 s 1233 4 s 1223 4 s 1231 4 s 1212 \] ≡ \[ S 11 S 12 S 13 S 14 S 15 S 16 S 12 S 22 S 23 S 24 S 25 S 26 S 13 S 23 S 33 S 34 S 35 S 36 S 14 S 24 S 34 S 44 S 45 S 46 S 15 S 25 S 35 S 45 S 55 S 56 S 16 S 26 S 36 S 46 S 56 S 66 \] {\\displaystyle \[{\\mathsf {s}}\]\\,=\\,{\\begin{bmatrix}s\_{1111}\&s\_{1122}\&s\_{1133}&2s\_{1123}&2s\_{1131}&2s\_{1112}\\\\s\_{2211}\&s\_{2222}\&s\_{2233}&2s\_{2223}&2s\_{2231}&2s\_{2212}\\\\s\_{3311}\&s\_{3322}\&s\_{3333}&2s\_{3323}&2s\_{3331}&2s\_{3312}\\\\2s\_{2311}&2s\_{2322}&2s\_{2333}&4s\_{2323}&4s\_{2331}&4s\_{2312}\\\\2s\_{3111}&2s\_{3122}&2s\_{3133}&4s\_{3123}&4s\_{3131}&4s\_{3112}\\\\2s\_{1211}&2s\_{1222}&2s\_{1233}&4s\_{1223}&4s\_{1231}&4s\_{1212}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}S\_{11}\&S\_{12}\&S\_{13}\&S\_{14}\&S\_{15}\&S\_{16}\\\\S\_{12}\&S\_{22}\&S\_{23}\&S\_{24}\&S\_{25}\&S\_{26}\\\\S\_{13}\&S\_{23}\&S\_{33}\&S\_{34}\&S\_{35}\&S\_{36}\\\\S\_{14}\&S\_{24}\&S\_{34}\&S\_{44}\&S\_{45}\&S\_{46}\\\\S\_{15}\&S\_{25}\&S\_{35}\&S\_{45}\&S\_{55}\&S\_{56}\\\\S\_{16}\&S\_{26}\&S\_{36}\&S\_{46}\&S\_{56}\&S\_{66}\\end{bmatrix}}} ![{\\displaystyle \[{\\mathsf {s}}\]\\,=\\,{\\begin{bmatrix}s\_{1111}\&s\_{1122}\&s\_{1133}&2s\_{1123}&2s\_{1131}&2s\_{1112}\\\\s\_{2211}\&s\_{2222}\&s\_{2233}&2s\_{2223}&2s\_{2231}&2s\_{2212}\\\\s\_{3311}\&s\_{3322}\&s\_{3333}&2s\_{3323}&2s\_{3331}&2s\_{3312}\\\\2s\_{2311}&2s\_{2322}&2s\_{2333}&4s\_{2323}&4s\_{2331}&4s\_{2312}\\\\2s\_{3111}&2s\_{3122}&2s\_{3133}&4s\_{3123}&4s\_{3131}&4s\_{3112}\\\\2s\_{1211}&2s\_{1222}&2s\_{1233}&4s\_{1223}&4s\_{1231}&4s\_{1212}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}S\_{11}\&S\_{12}\&S\_{13}\&S\_{14}\&S\_{15}\&S\_{16}\\\\S\_{12}\&S\_{22}\&S\_{23}\&S\_{24}\&S\_{25}\&S\_{26}\\\\S\_{13}\&S\_{23}\&S\_{33}\&S\_{34}\&S\_{35}\&S\_{36}\\\\S\_{14}\&S\_{24}\&S\_{34}\&S\_{44}\&S\_{45}\&S\_{46}\\\\S\_{15}\&S\_{25}\&S\_{35}\&S\_{45}\&S\_{55}\&S\_{56}\\\\S\_{16}\&S\_{26}\&S\_{36}\&S\_{46}\&S\_{56}\&S\_{66}\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34760b2d8ef86f720051aebe5a45a65b312bcab6)
#### 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[\[13\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Slaughter-13)
c p q r s \= l p i l q j l r k l s l c i j k l {\\displaystyle c\_{pqrs}=l\_{pi}l\_{qj}l\_{rk}l\_{sl}c\_{ijkl}}  where lab are the components of an [orthogonal rotation matrix](https://en.wikipedia.org/wiki/Orthogonal_matrix "Orthogonal 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
\[ e i ′ \] \= \[ L \] \[ e i \] {\\displaystyle \[\\mathbf {e} \_{i}'\]=\[L\]\[\\mathbf {e} \_{i}\]} ![{\\displaystyle \[\\mathbf {e} \_{i}'\]=\[L\]\[\\mathbf {e} \_{i}\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/213d0bb55cc1da894c855871790e09d78635c17b)
then
C i j ε i ε j \= C i j ′ ε i ′ ε j ′ . {\\displaystyle C\_{ij}\\varepsilon \_{i}\\varepsilon \_{j}=C\_{ij}'\\varepsilon '\_{i}\\varepsilon '\_{j}\\,.}  In addition, if the material is symmetric with respect to the transformation \[*L*\] then C i j \= C i j ′ ⟹ C i j ( ε i ε j − ε i ′ ε j ′ ) \= 0 . {\\displaystyle C\_{ij}=C'\_{ij}\\quad \\implies \\quad C\_{ij}(\\varepsilon \_{i}\\varepsilon \_{j}-\\varepsilon '\_{i}\\varepsilon '\_{j})=0\\,.} 
#### Orthotropic materials
Main article: [Orthotropic material](https://en.wikipedia.org/wiki/Orthotropic_material "Orthotropic material")
[Orthotropic materials](https://en.wikipedia.org/wiki/Orthotropic_material "Orthotropic material") have three [orthogonal](https://en.wikipedia.org/wiki/Orthogonal "Orthogonal") [planes of symmetry](https://en.wikipedia.org/wiki/Plane_of_symmetry "Plane of symmetry"). If the basis vectors (**e**1,**e**2,**e**3) are normals to the planes of symmetry then the coordinate transformation relations imply that
\[ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 \] \= \[ C 11 C 12 C 13 0 0 0 C 12 C 22 C 23 0 0 0 C 13 C 23 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66 \] \[ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 \] {\\displaystyle {\\begin{bmatrix}\\sigma \_{1}\\\\\\sigma \_{2}\\\\\\sigma \_{3}\\\\\\sigma \_{4}\\\\\\sigma \_{5}\\\\\\sigma \_{6}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}C\_{11}\&C\_{12}\&C\_{13}&0&0&0\\\\C\_{12}\&C\_{22}\&C\_{23}&0&0&0\\\\C\_{13}\&C\_{23}\&C\_{33}&0&0&0\\\\0&0&0\&C\_{44}&0&0\\\\0&0&0&0\&C\_{55}&0\\\\0&0&0&0&0\&C\_{66}\\end{bmatrix}}{\\begin{bmatrix}\\varepsilon \_{1}\\\\\\varepsilon \_{2}\\\\\\varepsilon \_{3}\\\\\\varepsilon \_{4}\\\\\\varepsilon \_{5}\\\\\\varepsilon \_{6}\\end{bmatrix}}}  The inverse of this relation is commonly written as[\[14\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Boresi-14)\[*[page needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources "Wikipedia:Citing sources")*\] \[ ε x x ε y y ε z z 2 ε y z 2 ε z x 2 ε x y \] \= \[ 1 E x − ν y x E y − ν z x E z 0 0 0 − ν x y E x 1 E y − ν z y E z 0 0 0 − ν x z E x − ν y z E y 1 E z 0 0 0 0 0 0 1 G y z 0 0 0 0 0 0 1 G z x 0 0 0 0 0 0 1 G x y \] \[ σ x x σ y y σ z z σ y z σ z x σ x y \] {\\displaystyle {\\begin{bmatrix}\\varepsilon \_{xx}\\\\\\varepsilon \_{yy}\\\\\\varepsilon \_{zz}\\\\2\\varepsilon \_{yz}\\\\2\\varepsilon \_{zx}\\\\2\\varepsilon \_{xy}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}{\\frac {1}{E\_{x}}}&-{\\frac {\\nu \_{yx}}{E\_{y}}}&-{\\frac {\\nu \_{zx}}{E\_{z}}}&0&0&0\\\\-{\\frac {\\nu \_{xy}}{E\_{x}}}&{\\frac {1}{E\_{y}}}&-{\\frac {\\nu \_{zy}}{E\_{z}}}&0&0&0\\\\-{\\frac {\\nu \_{xz}}{E\_{x}}}&-{\\frac {\\nu \_{yz}}{E\_{y}}}&{\\frac {1}{E\_{z}}}&0&0&0\\\\0&0&0&{\\frac {1}{G\_{yz}}}&0&0\\\\0&0&0&0&{\\frac {1}{G\_{zx}}}&0\\\\0&0&0&0&0&{\\frac {1}{G\_{xy}}}\\\\\\end{bmatrix}}{\\begin{bmatrix}\\sigma \_{xx}\\\\\\sigma \_{yy}\\\\\\sigma \_{zz}\\\\\\sigma \_{yz}\\\\\\sigma \_{zx}\\\\\\sigma \_{xy}\\end{bmatrix}}}  where
- Ei is the [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") along axis i
- Gij is the [shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus") in direction j on the plane whose normal is in direction i
- νij is the [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio") that corresponds to a contraction in direction j when an extension is applied in direction i.
Under *plane stress* conditions, *σzz* = *σzx* = *σyz* = 0, Hooke's law for an orthotropic material takes the form \[ ε x x ε y y 2 ε x y \] \= \[ 1 E x − ν y x E y 0 − ν x y E x 1 E y 0 0 0 1 G x y \] \[ σ x x σ y y σ x y \] . {\\displaystyle {\\begin{bmatrix}\\varepsilon \_{xx}\\\\\\varepsilon \_{yy}\\\\2\\varepsilon \_{xy}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}{\\frac {1}{E\_{x}}}&-{\\frac {\\nu \_{yx}}{E\_{y}}}&0\\\\-{\\frac {\\nu \_{xy}}{E\_{x}}}&{\\frac {1}{E\_{y}}}&0\\\\0&0&{\\frac {1}{G\_{xy}}}\\end{bmatrix}}{\\begin{bmatrix}\\sigma \_{xx}\\\\\\sigma \_{yy}\\\\\\sigma \_{xy}\\end{bmatrix}}\\,.}  The inverse relation is \[ σ x x σ y y σ x y \] \= 1 1 − ν x y ν y x \[ E x ν y x E x 0 ν x y E y E y 0 0 0 G x y ( 1 − ν x y ν y x ) \] \[ ε x x ε y y 2 ε x y \] . {\\displaystyle {\\begin{bmatrix}\\sigma \_{xx}\\\\\\sigma \_{yy}\\\\\\sigma \_{xy}\\end{bmatrix}}\\,=\\,{\\frac {1}{1-\\nu \_{xy}\\nu \_{yx}}}{\\begin{bmatrix}E\_{x}&\\nu \_{yx}E\_{x}&0\\\\\\nu \_{xy}E\_{y}\&E\_{y}&0\\\\0&0\&G\_{xy}(1-\\nu \_{xy}\\nu \_{yx})\\end{bmatrix}}{\\begin{bmatrix}\\varepsilon \_{xx}\\\\\\varepsilon \_{yy}\\\\2\\varepsilon \_{xy}\\end{bmatrix}}\\,.}  The transposed form of the above stiffness matrix is also often used.
#### Transversely isotropic materials
A [transversely isotropic](https://en.wikipedia.org/wiki/Transversely_isotropic "Transversely isotropic") material is symmetric with respect to a rotation about an [axis of symmetry](https://en.wikipedia.org/wiki/Axis_of_symmetry "Axis of symmetry"). For such a material, if **e**3 is the axis of symmetry, Hooke's law can be expressed as \[ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 \] \= \[ C 11 C 12 C 13 0 0 0 C 12 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 11 − C 12 2 \] \[ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 \] {\\displaystyle {\\begin{bmatrix}\\sigma \_{1}\\\\\\sigma \_{2}\\\\\\sigma \_{3}\\\\\\sigma \_{4}\\\\\\sigma \_{5}\\\\\\sigma \_{6}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}C\_{11}\&C\_{12}\&C\_{13}&0&0&0\\\\C\_{12}\&C\_{11}\&C\_{13}&0&0&0\\\\C\_{13}\&C\_{13}\&C\_{33}&0&0&0\\\\0&0&0\&C\_{44}&0&0\\\\0&0&0&0\&C\_{44}&0\\\\0&0&0&0&0&{\\frac {C\_{11}-C\_{12}}{2}}\\end{bmatrix}}{\\begin{bmatrix}\\varepsilon \_{1}\\\\\\varepsilon \_{2}\\\\\\varepsilon \_{3}\\\\\\varepsilon \_{4}\\\\\\varepsilon \_{5}\\\\\\varepsilon \_{6}\\end{bmatrix}}} 
More frequently, the *x* ≡ **e**1 axis is taken to be the axis of symmetry and the inverse Hooke's law is written as [\[15\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Tan-15) \[ ε x x ε y y ε z z 2 ε y z 2 ε z x 2 ε x y \] \= \[ 1 E x − ν y x E y − ν z x E z 0 0 0 − ν x y E x 1 E y − ν z y E z 0 0 0 − ν x z E x − ν y z E y 1 E z 0 0 0 0 0 0 1 G y z 0 0 0 0 0 0 1 G x z 0 0 0 0 0 0 1 G x y \] \[ σ x x σ y y σ z z σ y z σ z x σ x y \] {\\displaystyle {\\begin{bmatrix}\\varepsilon \_{xx}\\\\\\varepsilon \_{yy}\\\\\\varepsilon \_{zz}\\\\2\\varepsilon \_{yz}\\\\2\\varepsilon \_{zx}\\\\2\\varepsilon \_{xy}\\end{bmatrix}}\\,=\\,{\\begin{bmatrix}{\\frac {1}{E\_{x}}}&-{\\frac {\\nu \_{yx}}{E\_{y}}}&-{\\frac {\\nu \_{zx}}{E\_{z}}}&0&0&0\\\\-{\\frac {\\nu \_{xy}}{E\_{x}}}&{\\frac {1}{E\_{y}}}&-{\\frac {\\nu \_{zy}}{E\_{z}}}&0&0&0\\\\-{\\frac {\\nu \_{xz}}{E\_{x}}}&-{\\frac {\\nu \_{yz}}{E\_{y}}}&{\\frac {1}{E\_{z}}}&0&0&0\\\\0&0&0&{\\frac {1}{G\_{yz}}}&0&0\\\\0&0&0&0&{\\frac {1}{G\_{xz}}}&0\\\\0&0&0&0&0&{\\frac {1}{G\_{xy}}}\\\\\\end{bmatrix}}{\\begin{bmatrix}\\sigma \_{xx}\\\\\\sigma \_{yy}\\\\\\sigma \_{zz}\\\\\\sigma \_{yz}\\\\\\sigma \_{zx}\\\\\\sigma \_{xy}\\end{bmatrix}}} 
#### Universal elastic anisotropy index
To grasp the degree of anisotropy of any class, a **universal elastic anisotropy index** (AU)[\[16\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-16) was formulated. It replaces the [Zener ratio](https://en.wikipedia.org/wiki/Zener_ratio "Zener ratio"), which is suited for [cubic crystals](https://en.wikipedia.org/wiki/Cubic_crystal_system "Cubic crystal system").
## Thermodynamic basis
Linear deformations of elastic materials can be approximated as [adiabatic](https://en.wikipedia.org/wiki/Adiabatic "Adiabatic"). Under these conditions and for quasistatic processes the [first law of thermodynamics](https://en.wikipedia.org/wiki/First_law_of_thermodynamics "First law of thermodynamics") for a deformed body can be expressed as δ W \= δ U {\\displaystyle \\delta W=\\delta U}  where δU is the increase in [internal energy](https://en.wikipedia.org/wiki/Internal_energy "Internal energy") and δW is the [work](https://en.wikipedia.org/wiki/Work_\(physics\) "Work (physics)") done by external forces. The work can be split into two terms δ W \= δ W s \+ δ W b {\\displaystyle \\delta W=\\delta W\_{\\mathrm {s} }+\\delta W\_{\\mathrm {b} }}  where *δW*s is the work done by [surface forces](https://en.wikipedia.org/wiki/Surface_force "Surface force") while *δW*b is the work done by [body forces](https://en.wikipedia.org/wiki/Body_force "Body force"). If *δ***u** is a [variation](https://en.wikipedia.org/wiki/Calculus_of_variations "Calculus of variations") of the displacement field **u** in the body, then the two external work terms can be expressed as δ W s \= ∫ ∂ Ω t ⋅ δ u d S ; δ W b \= ∫ Ω b ⋅ δ u d V {\\displaystyle \\delta W\_{\\mathrm {s} }=\\int \_{\\partial \\Omega }\\mathbf {t} \\cdot \\delta \\mathbf {u} \\,dS\\,;\\qquad \\delta W\_{\\mathrm {b} }=\\int \_{\\Omega }\\mathbf {b} \\cdot \\delta \\mathbf {u} \\,dV}  where **t** is the surface [traction](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") vector, **b** is the body force vector, Ω represents the body and ∂*Ω* represents its surface. Using the relation between the [Cauchy stress](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") and the surface traction, **t** = **n** · **σ** (where **n** is the unit outward normal to ∂*Ω*), we have δ W \= δ U \= ∫ ∂ Ω ( n ⋅ σ ) ⋅ δ u d S \+ ∫ Ω b ⋅ δ u d V . {\\displaystyle \\delta W=\\delta U=\\int \_{\\partial \\Omega }(\\mathbf {n} \\cdot {\\boldsymbol {\\sigma }})\\cdot \\delta \\mathbf {u} \\,dS+\\int \_{\\Omega }\\mathbf {b} \\cdot \\delta \\mathbf {u} \\,dV\\,.}  Converting the [surface integral](https://en.wikipedia.org/wiki/Surface_integral "Surface integral") into a [volume integral](https://en.wikipedia.org/wiki/Volume_integral "Volume integral") via the [divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem "Divergence theorem") gives δ U \= ∫ Ω ( ∇ ⋅ ( σ ⋅ δ u ) \+ b ⋅ δ u ) d V . {\\displaystyle \\delta U=\\int \_{\\Omega }{\\big (}\\nabla \\cdot ({\\boldsymbol {\\sigma }}\\cdot \\delta \\mathbf {u} )+\\mathbf {b} \\cdot \\delta \\mathbf {u} {\\big )}\\,dV\\,.}  Using the symmetry of the Cauchy stress and the identity ∇ ⋅ ( a ⋅ b ) \= ( ∇ ⋅ a ) ⋅ b \+ 1 2 ( a T : ∇ b \+ a : ( ∇ b ) T ) {\\displaystyle \\nabla \\cdot (\\mathbf {a} \\cdot \\mathbf {b} )=(\\nabla \\cdot \\mathbf {a} )\\cdot \\mathbf {b} +{\\tfrac {1}{2}}\\left(\\mathbf {a} ^{\\mathsf {T}}:\\nabla \\mathbf {b} +\\mathbf {a} :(\\nabla \\mathbf {b} )^{\\mathsf {T}}\\right)}  we have the following
δ U \= ∫ Ω ( σ : 1 2 ( ∇ δ u \+ ( ∇ δ u ) T ) \+ ( ∇ ⋅ σ \+ b ) ⋅ δ u ) d V . {\\displaystyle \\delta U=\\int \_{\\Omega }\\left({\\boldsymbol {\\sigma }}:{\\tfrac {1}{2}}\\left(\\nabla \\delta \\mathbf {u} +(\\nabla \\delta \\mathbf {u} )^{\\mathsf {T}}\\right)+\\left(\\nabla \\cdot {\\boldsymbol {\\sigma }}+\\mathbf {b} \\right)\\cdot \\delta \\mathbf {u} \\right)\\,dV\\,.}  From the definition of [strain](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory "Infinitesimal strain theory") and from the equations of [equilibrium](https://en.wikipedia.org/wiki/Linear_elasticity "Linear elasticity") we have δ ε \= 1 2 ( ∇ δ u \+ ( ∇ δ u ) T ) ; ∇ ⋅ σ \+ b \= 0 . {\\displaystyle \\delta {\\boldsymbol {\\varepsilon }}={\\tfrac {1}{2}}\\left(\\nabla \\delta \\mathbf {u} +(\\nabla \\delta \\mathbf {u} )^{\\mathsf {T}}\\right)\\,;\\qquad \\nabla \\cdot {\\boldsymbol {\\sigma }}+\\mathbf {b} =\\mathbf {0} \\,.}  Hence we can write δ U \= ∫ Ω σ : δ ε d V {\\displaystyle \\delta U=\\int \_{\\Omega }{\\boldsymbol {\\sigma }}:\\delta {\\boldsymbol {\\varepsilon }}\\,dV}  and therefore the variation in the [internal energy](https://en.wikipedia.org/wiki/Internal_energy "Internal energy") density is given by δ U 0 \= σ : δ ε . {\\displaystyle \\delta U\_{0}={\\boldsymbol {\\sigma }}:\\delta {\\boldsymbol {\\varepsilon }}\\,.}  An [elastic](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)") material is defined as one in which the total internal energy is equal to the [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy") of the internal forces (also called the **elastic strain energy**). Therefore, the internal energy density is a function of the strains, *U*0 = *U*0(**ε**) and the variation of the internal energy can be expressed as δ U 0 \= ∂ U 0 ∂ ε : δ ε . {\\displaystyle \\delta U\_{0}={\\frac {\\partial U\_{0}}{\\partial {\\boldsymbol {\\varepsilon }}}}:\\delta {\\boldsymbol {\\varepsilon }}\\,.}  Since the variation of strain is arbitrary, the stress–strain relation of an elastic material is given by σ \= ∂ U 0 ∂ ε . {\\displaystyle {\\boldsymbol {\\sigma }}={\\frac {\\partial U\_{0}}{\\partial {\\boldsymbol {\\varepsilon }}}}\\,.}  For a linear elastic material, the quantity ∂*U*0/∂**ε** is a linear function of **ε**, and can therefore be expressed as σ \= c : ε {\\displaystyle {\\boldsymbol {\\sigma }}={\\mathsf {c}}:{\\boldsymbol {\\varepsilon }}}  where **c** is a fourth-rank tensor of material constants, also called the **stiffness tensor**. We can see why **c** must be a fourth-rank tensor by noting that, for a linear elastic material, ∂ ∂ ε σ ( ε ) \= constant \= c . {\\displaystyle {\\frac {\\partial }{\\partial {\\boldsymbol {\\varepsilon }}}}{\\boldsymbol {\\sigma }}({\\boldsymbol {\\varepsilon }})={\\text{constant}}={\\mathsf {c}}\\,.}  In index notation ∂ σ i j ∂ ε k l \= constant \= c i j k l . {\\displaystyle {\\frac {\\partial \\sigma \_{ij}}{\\partial \\varepsilon \_{kl}}}={\\text{constant}}=c\_{ijkl}\\,.} 
The right-hand side constant requires four indices and is a fourth-rank 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-rank tensors.
## See also
- [Acoustoelastic effect](https://en.wikipedia.org/wiki/Acoustoelastic_effect "Acoustoelastic effect")
- [Elastic potential energy](https://en.wikipedia.org/wiki/Elastic_potential_energy "Elastic potential energy")
- [Laws of science](https://en.wikipedia.org/wiki/Laws_of_science "Laws of science")
- [List of scientific laws named after people](https://en.wikipedia.org/wiki/List_of_scientific_laws_named_after_people "List of scientific laws named after people")
- [Quadratic form](https://en.wikipedia.org/wiki/Quadratic_form "Quadratic form")
- [Series and parallel springs](https://en.wikipedia.org/wiki/Series_and_parallel_springs "Series and parallel springs")
- [Spring system](https://en.wikipedia.org/wiki/Spring_system "Spring system")
- [Simple harmonic motion of a mass on a spring](https://en.wikipedia.org/wiki/Simple_harmonic_motion#Mass_on_a_spring "Simple harmonic motion")
- [Sine wave](https://en.wikipedia.org/wiki/Sine_wave "Sine wave")
- [Solid mechanics](https://en.wikipedia.org/wiki/Solid_mechanics "Solid mechanics")
- [Spring pendulum](https://en.wikipedia.org/wiki/Spring_pendulum "Spring pendulum")
## Notes
1. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-1)**
The anagram was given in alphabetical order, *ceiiinosssttuv*, representing *Ut tensio, sic vis* – "As the extension, so the force":
[Petroski, Henry](https://en.wikipedia.org/wiki/Henry_Petroski "Henry Petroski") (1996). [*Invention by Design: How Engineers Get from Thought to Thing*](https://archive.org/details/inventionbydesig00petr). Cambridge, MA: Harvard University Press. p. [11](https://archive.org/details/inventionbydesig00petr/page/11). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-674-46368-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-674-46368-4 "Special:BookSources/978-0-674-46368-4")
.
2. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-2)** See <http://civil.lindahall.org/design.shtml>, where one can find also an anagram for [catenary](https://en.wikipedia.org/wiki/Catenary "Catenary").
3. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-3)** [Robert Hooke](https://en.wikipedia.org/wiki/Robert_Hooke "Robert Hooke"), *De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies*, London, 1678.
4. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-4)**
Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2016). *Sears and Zemansky's University Physics: With Modern Physics* (14th ed.). Pearson. p. 209.
5. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-5)**
Ushiba, Shota; Masui, Kyoko; Taguchi, Natsuo; Hamano, Tomoki; Kawata, Satoshi; Shoji, Satoru (2015). ["Size dependent nanomechanics of coil spring shaped polymer nanowires"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4661696). *Scientific Reports*. **5** 17152. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2015NatSR...517152U](https://ui.adsabs.harvard.edu/abs/2015NatSR...517152U). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/srep17152](https://doi.org/10.1038%2Fsrep17152). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [4661696](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4661696). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [26612544](https://pubmed.ncbi.nlm.nih.gov/26612544).
6. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-6)**
Belen'kii; Salaev (1988). ["Deformation effects in layer crystals"](https://doi.org/10.3367%2FUFNr.0155.198805c.0089). *Uspekhi Fizicheskikh Nauk*. **155** (5): 89. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3367/UFNr.0155.198805c.0089](https://doi.org/10.3367%2FUFNr.0155.198805c.0089).
7. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-7)**
Mouhat, Félix; Coudert, François-Xavier (5 December 2014). ["Necessary and sufficient elastic stability conditions in various crystal systems"](https://link.aps.org/doi/10.1103/PhysRevB.90.224104). *Physical Review B*. **90** (22) 224104. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1410\.0065](https://arxiv.org/abs/1410.0065). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2014PhRvB..90v4104M](https://ui.adsabs.harvard.edu/abs/2014PhRvB..90v4104M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevB.90.224104](https://doi.org/10.1103%2FPhysRevB.90.224104). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1098-0121](https://search.worldcat.org/issn/1098-0121). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [54058316](https://api.semanticscholar.org/CorpusID:54058316).
8. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-8)**
Vijay Madhav, M.; Manogaran, S. (2009). "A relook at the compliance constants in redundant internal coordinates and some new insights". *J. Chem. Phys*. **131** (17): 174112–174116\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2009JChPh.131q4112V](https://ui.adsabs.harvard.edu/abs/2009JChPh.131q4112V). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1063/1.3259834](https://doi.org/10.1063%2F1.3259834). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [19895003](https://pubmed.ncbi.nlm.nih.gov/19895003).
9. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-9)**
Ponomareva, Alla; Yurenko, Yevgen; Zhurakivsky, Roman; Van Mourik, Tanja; Hovorun, Dmytro (2012). "Complete conformational space of the potential HIV-1 reverse transcriptase inhibitors d4U and d4C. A quantum chemical study". *Phys. Chem. Chem. Phys*. **14** (19): 6787–6795\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2012PCCP...14.6787P](https://ui.adsabs.harvard.edu/abs/2012PCCP...14.6787P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1039/C2CP40290D](https://doi.org/10.1039%2FC2CP40290D). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [22461011](https://pubmed.ncbi.nlm.nih.gov/22461011).
10. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-10)**
Symon, Keith R. (1971). "Chapter 10". *Mechanics*. Reading, Massachusetts: Addison-Wesley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-201-07392-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-07392-8 "Special:BookSources/978-0-201-07392-8")
.
11. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Simo98_11-0)**
Simo, J. C.; Hughes, T. J. R. (1998). *Computational Inelasticity*. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-97520-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97520-7 "Special:BookSources/978-0-387-97520-7")
.
12. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Milton02_12-0)**
Milton, Graeme W. (2002). *The Theory of Composites*. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-78125-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-78125-1 "Special:BookSources/978-0-521-78125-1")
.
13. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Slaughter_13-0)**
Slaughter, William S. (2001). *The Linearized Theory of Elasticity*. Birkhäuser. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-8176-4117-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8176-4117-7 "Special:BookSources/978-0-8176-4117-7")
.
14. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Boresi_14-0)**
Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. (1993). *Advanced Mechanics of Materials* (5th ed.). Wiley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-471-60009-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-60009-1 "Special:BookSources/978-0-471-60009-1")
.
15. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Tan_15-0)**
Tan, S. C. (1994). *Stress Concentrations in Laminated Composites*. Lancaster, PA: Technomic Publishing Company. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-56676-077-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-56676-077-5 "Special:BookSources/978-1-56676-077-5")
.
16. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-16)**
Ranganathan, S.I.; [Ostoja-Starzewski, M.](https://en.wikipedia.org/wiki/Martin_Ostoja-Starzewski "Martin Ostoja-Starzewski") (2008). "Universal Elastic Anisotropy Index". *Physical Review Letters*. **101** (5): 055504–1–4. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2008PhRvL.101e5504R](https://ui.adsabs.harvard.edu/abs/2008PhRvL.101e5504R). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevLett.101.055504](https://doi.org/10.1103%2FPhysRevLett.101.055504). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [18764407](https://pubmed.ncbi.nlm.nih.gov/18764407). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [6668703](https://api.semanticscholar.org/CorpusID:6668703).
## References
- [Hooke's law - The Feynman Lectures on Physics](https://feynmanlectures.caltech.edu/II_38.html#Ch38-S1)
- [Hooke's Law - Classical Mechanics - Physics - MIT OpenCourseWare](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/pages/week-2-newtons-laws/7-4-hookes-law/)
## External links
- [JavaScript Applet demonstrating Springs and Hooke's law](https://www.compadre.org/Physlets/mechanics/illustration5_4.cfm)
- [JavaScript Applet demonstrating Spring Force](https://www.compadre.org/Physlets/mechanics/ex5_3.cfm)
| [v](https://en.wikipedia.org/wiki/Template:Elastic_moduli "Template:Elastic moduli") [t](https://en.wikipedia.org/wiki/Template_talk:Elastic_moduli "Template talk:Elastic moduli") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Elastic_moduli "Special:EditPage/Template:Elastic moduli")[Elastic moduli](https://en.wikipedia.org/wiki/Elastic_modulus "Elastic modulus") for homogeneous [isotropic](https://en.wikipedia.org/wiki/Isotropic "Isotropic") materials |
|---|
| [Bulk modulus](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") (*K*) [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") (*E*) [Lamé's first parameter](https://en.wikipedia.org/wiki/Lam%C3%A9_parameters "Lamé parameters") (λ) [Shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus") (*G*, *μ*) [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio") (*ν*) [P-wave modulus](https://en.wikipedia.org/wiki/P-wave_modulus "P-wave modulus") (*M*) |
| 3D Formulae | | | | | | | |
|---|---|---|---|---|---|---|---|
| Knowns | [Bulk modulus](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") (*K*) | [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") (*E*) | [Lamé's first parameter](https://en.wikipedia.org/wiki/Lam%C3%A9_parameters "Lamé parameters") (λ) | [Shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus") (*G*) | [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio") (*ν*) | [P-wave modulus](https://en.wikipedia.org/wiki/P-wave_modulus "P-wave modulus") (*M*) | Notes |
| (*K*, *E*) | | | 3*K*(1 + 6*K*/*E* − 9*K*) | *E*/ 3 − *E*/3*K* | 1/2 − *E*/6*K* | 3*K* + *E*/ 3 − *E*/3*K* | |
| (*K*, λ) | | 9*K*(*K* − λ)/3*K* − λ | | 3(*K* − λ)/2 | λ/3*K* − λ | 3*K* − 2λ | |
| (*K*, *G*) | | 9*KG*/3*K* + *G* | *K* − 2*G*/3 | | 3*K* − 2*G*/6*K* + 2*G* | *K* + 4*G*/3 | |
| (*K*, *ν*) | | 3*K*(1 − 2*ν*) | 3*Kν*/1 + *ν* | 3*K*(1 − 2*ν*)/2(1 + *ν*) | | 3*K*(1 − *ν*)/1 + *ν* | |
| (*K*, *M*) | | 9*K*(*M* − *K*)/3*K* + *M* | 3*K* − *M*/2 | 3(*M* − *K*)/4 | 3*K* − *M*/3*K* + *M* | | |
| (*E*, λ) | *E* + 3λ + R/6 | | | *E* − 3λ + *R*/4 | − *E* + *R*/4λ − 1/4 | *E* − λ + *R*/2 | *R* = ±(*E*2 + 9λ2 + 2*E*λ)1/2 |
| (*E*, *G*) | *EG*/3(3*G* − *E*) | | *G*(*E* − 2*G*)/3*G* − *E* | | *E*/2*G* − 1 | *G*(4*G* − *E*)/3*G* − *E* | |
| (*E*, *ν*) | *E*/3 − 6*ν* | | *Eν*/(1 + *ν*)(1 − 2*ν*) | *E*/2(1 + *ν*) | | *E*(1 − *ν*)/(1 + *ν*)(1 − 2*ν*) | |
| (*E*, *M*) | 3*M* − *E* + *S*/6 | | *M* − *E* + *S*/4 | 3*M* + *E* − *S*/8 | *E* + *S*/4*M* − 1/4 | | *S* = ±(*E*2 + 9M2 − 10*E*M)1/2 |
| (λ, *G*) | λ + 2*G*/3 | *G*(3λ + 2*G*)/λ + *G* | | | λ/2(λ + *G*) | λ + 2*G* | |
| (λ, *ν*) | λ/3(1 + 1/*ν*) | λ(1/*ν* − 2*ν* − 1) | | λ(1/2*ν* − 1) | | λ(1/*ν* − 1) | |
| (λ, *M*) | *M* + 2λ/3 | (*M* − λ)(*M*\+2λ)/*M* + λ | | *M* − λ/2 | λ/*M* + λ | | |
| (*G*, *ν*) | 2*G*(1 + *ν*)/3 − 6*ν* | 2*G*(1 + *ν*) | 2 *G* *ν*/1 − 2*ν* | | | 2*G*(1 − *ν*)/1 − 2*ν* | |
| (*G*, *M*) | *M* − 4*G*/3 | *G*(3*M* − 4*G*)/*M* − *G* | *M* − 2*G* | | *M* − 2*G*/2*M* − 2*G* | | |
| (*ν*, *M*) | *M*(1 + *ν*)/3(1 − *ν*) | *M*(1 + *ν*)(1 − 2*ν*)/1 − *ν* | *M* *ν*/1 − *ν* | *M*(1 − 2*ν*)/2(1 − *ν*) | | | |
| 2D Formulae | | | | | | | |
| Knowns | (*K*) | (*E*) | (λ) | (*G*) | (*ν*) | (*M*) | Notes |
| (*K*2D, *E*2D) | | | 2*K*2D(2*K*2D − *E*2D)/4*K*2D − *E*2D | *K*2D*E*2D/4*K*2D − *E*2D | 2*K*2D − *E*2D/2*K*2D | 4*K*2D^2/4*K*2D − *E*2D | |
| (*K*2D, λ2D) | | 4*K*2D(*K*2D − λ2D)/2*K*2D − λ2D | | *K*2D − λ2D | λ2D/2*K*2D − λ2D | 2*K*2D − λ2D | |
| (*K*2D, *G*2D) | | 4*K*2D*G*2D/*K*2D + *G*2D | *K*2D − *G*2D | | *K*2D − *G*2D/*K*2D + *G*2D | *K*2D + *G*2D | |
| (*K*2D, *ν*2D) | | 2*K*2D(1 − *ν*2D) | 2*K*2D*ν*2D/1 + *ν*2D | *K*2D(1 − *ν*2D)/1 + *ν*2D | | 2*K*2D/1 + *ν*2D | |
| (*E*2D, *G*2D) | *E*2D*G*2D/4*G*2D − *E*2D | | 2*G*2D(*E*2D − 2*G*2D)/4*G*2D − *E*2D | | *E*2D/2*G*2D − 1 | 4*G*2D^2/4*G*2D − *E*2D | |
| (*E*2D, *ν*2D) | *E*2D/2(1 − *ν*2D) | | *E*2D*ν*2D/(1 + *ν*2D)(1 − *ν*2D) | *E*2D/2(1 + *ν*2D) | | *E*2D/(1 + *ν*2D)(1 − *ν*2D) | |
| (λ2D, *G*2D) | λ2D + *G*2D | 4*G*2D(λ2D + *G*2D)/λ2D + 2*G*2D | | | λ2D/λ2D + 2*G*2D | λ2D + 2*G*2D | |
| (λ2D, *ν*2D) | λ2D(1 + *ν*2D)/2*ν*2D | λ2D(1 + *ν*2D)(1 − *ν*2D)/*ν*2D | | λ2D(1 − *ν*2D)/2*ν*2D | | λ2D/*ν*2D | |
| (*G*2D, *ν*2D) | *G*2D(1 + *ν*2D)/1 − *ν*2D | 2*G*2D(1 + *ν*2D) | 2 *G*2D *ν*2D/1 − *ν*2D | | | 2*G*2D/1 − *ν*2D | |
| (*G*2D, *M*2D) | *M*2D − *G*2D | 4*G*2D(*M*2D − *G*2D)/*M*2D | *M*2D − 2*G*2D | | *M*2D − 2*G*2D/*M*2D | | |
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Hooke's law
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| Readable Markdown | [](https://en.wikipedia.org/wiki/File:Hookes-law-springs.png)
Hooke's law: the force is proportional to the extension
[](https://en.wikipedia.org/wiki/File:Manometer_anim_02.gif)
[Bourdon tubes](https://en.wikipedia.org/wiki/Bourdon_tube "Bourdon tube") are based on Hooke's law. The force created by gas [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") inside the coiled metal tube above unwinds it by an amount proportional to the pressure.
[](https://en.wikipedia.org/wiki/File:Balancier_avec_ressort_spiral.png)
The [balance wheel](https://en.wikipedia.org/wiki/Balance_wheel "Balance wheel") at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period.
In [physics](https://en.wikipedia.org/wiki/Physics "Physics"), **Hooke's law** is an [empirical law](https://en.wikipedia.org/wiki/Empirical_law "Empirical law") which states that the [force](https://en.wikipedia.org/wiki/Force "Force") (F) needed to extend or compress a [spring](https://en.wikipedia.org/wiki/Spring_\(device\) "Spring (device)") by some distance (x) [scales linearly](https://en.wikipedia.org/wiki/Proportionality_\(mathematics\)#Direct_proportionality "Proportionality (mathematics)") with respect to that distance—that is, *Fs* = *kx*, where k is a constant factor characteristic of the spring (i.e., its [stiffness](https://en.wikipedia.org/wiki/Stiffness "Stiffness")), and x is small compared to the total possible deformation of the spring.
The law is named after 17th-century British physicist [Robert Hooke](https://en.wikipedia.org/wiki/Robert_Hooke "Robert Hooke"). He first stated the law in 1676 as a Latin [anagram](https://en.wikipedia.org/wiki/Anagram "Anagram").[\[1\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-1)[\[2\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-2) He published the solution of his anagram in 1678[\[3\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-3) as: *ut tensio, sic vis* ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. It is the fundamental principle behind the [spring scale](https://en.wikipedia.org/wiki/Spring_scale "Spring scale"), the [manometer](https://en.wikipedia.org/wiki/Manometer "Manometer"), the [galvanometer](https://en.wikipedia.org/wiki/Galvanometer "Galvanometer"), and the [balance wheel](https://en.wikipedia.org/wiki/Balance_wheel "Balance wheel") of the [mechanical clock](https://en.wikipedia.org/wiki/Mechanical_clock "Mechanical clock").
The equation holds in many situations where an [elastic](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)") body is [deformed](https://en.wikipedia.org/wiki/Deformation_\(physics\) "Deformation (physics)"). An elastic body or material for which this equation can be assumed is said to be [linear-elastic](https://en.wikipedia.org/wiki/Linear_elasticity "Linear elasticity") or **Hookean**. Hooke's law is a [first-order linear approximation](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") to the real response of springs and other elastic bodies to applied forces. It fails once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those [elastic limits](https://en.wikipedia.org/wiki/Elastic_limit "Elastic limit") are reached.
Definition
The modern [theory of elasticity](https://en.wikipedia.org/wiki/Theory_of_elasticity "Theory of elasticity") generalizes Hooke's law to say that the [strain](https://en.wikipedia.org/wiki/Deformation_\(mechanics\) "Deformation (mechanics)") (deformation) of an elastic object or material is proportional to the [stress](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") 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](https://en.wikipedia.org/wiki/Linear_map "Linear map") (a [tensor](https://en.wikipedia.org/wiki/Tensor "Tensor")) that can be represented by a [matrix](https://en.wikipedia.org/wiki/Matrix_\(mathematics\) "Matrix (mathematics)") of real numbers.
In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials they are made of. For example, one can deduce that a [homogeneous](https://en.wikipedia.org/wiki/Homogeneous "Homogeneous") rod with uniform [cross section](https://en.wikipedia.org/wiki/Cross_section_\(geometry\) "Cross section (geometry)") will behave like a simple spring when stretched, with a stiffness k directly proportional to its cross-section area and inversely proportional to its length.
Linear springs
[](https://en.wikipedia.org/wiki/File:Spring-elongation-and-forces.svg)
Elongation and compression of a spring
Consider a simple [helical](https://en.wikipedia.org/wiki/Helix "Helix") spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is Fs. Suppose that the spring has reached a state of [equilibrium](https://en.wikipedia.org/wiki/Mechanical_equilibrium "Mechanical equilibrium"), where its length is not changing anymore. Let x be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that [](https://en.wikipedia.org/w/index.php?title=Special:MathWikibase&qid=Q170282) or, equivalently,  where k is a positive real number, characteristic of the spring. A spring with spaces between the coils can be compressed, and the same formula holds for compression, with Fs and x both negative in that case.[\[4\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-4)
[](https://en.wikipedia.org/wiki/File:Hooke%27s_Law_wikipedia.png)
Graphical derivation
According to this formula, the [graph](https://en.wikipedia.org/wiki/Graph_of_a_function "Graph of a function") of the applied force Fs as a function of the displacement x will be a straight line passing through the [origin](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates"), whose [slope](https://en.wikipedia.org/wiki/Slope "Slope") is k.
Hooke's law for a spring is also stated under the convention that Fs is the [restoring force](https://en.wikipedia.org/wiki/Restoring_force "Restoring force") exerted by the spring on whatever is pulling its free end. In that case, the equation becomes  since the direction of the restoring force is opposite to that of the displacement.
Torsional springs
The [torsional](https://en.wikipedia.org/wiki/Torsional "Torsional") analog of Hooke's law applies to [torsional springs](https://en.wikipedia.org/wiki/Torsional_spring "Torsional spring"). It states that the torque (τ) required to rotate an object is directly proportional to the angular displacement (θ) from the equilibrium position. It describes the relationship between the torque applied to an object and the resulting angular [deformation](https://en.wikipedia.org/wiki/Deformation_\(physics\) "Deformation (physics)") due to torsion. Mathematically, it can be expressed as:
Where:
- τ is the [torque](https://en.wikipedia.org/wiki/Torque "Torque") measured in Newton-meters or N·m.
- k is the [torsional constant](https://en.wikipedia.org/wiki/Torsional_constant "Torsional constant") (measured in N·m/radian), which characterizes the stiffness of the torsional spring or the resistance to angular displacement.
- θ is the [angular displacement](https://en.wikipedia.org/wiki/Angular_displacement "Angular displacement") (measured in radians) from the equilibrium position.
Just as in the linear case, this law shows that the torque is proportional to the angular displacement, and the negative sign indicates that the torque acts in a direction opposite to the angular displacement, providing a restoring force to bring the system back to equilibrium.
General "scalar" springs
Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative.
For example, when a block of rubber attached to two parallel plates is deformed by [shearing](https://en.wikipedia.org/wiki/Simple_shear "Simple shear"), rather than stretching or compression, the shearing force *Fs* and the sideways displacement of the plates x obey Hooke's law (for small enough deformations).
Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight F placed at some intermediate point. The displacement x in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.
Vector formulation
In the case of a helical spring that is stretched or compressed along its [axis](https://en.wikipedia.org/wiki/Axial_symmetry "Axial symmetry"), the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if Fs and x are defined as [vectors](https://en.wikipedia.org/wiki/Vector_\(mathematics\) "Vector (mathematics)"), Hooke's [equation](https://en.wikipedia.org/wiki/Equation "Equation") still holds and says that the force vector is the [elongation vector](https://en.wikipedia.org/wiki/Displacement_\(vector\) "Displacement (vector)") multiplied by a fixed [scalar](https://en.wikipedia.org/wiki/Scalar_\(mathematics\) "Scalar (mathematics)").
General tensor form
Some elastic bodies will deform in one direction when subjected to a force with a different direction. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. In such cases, the *magnitude* of the displacement x will be proportional to the magnitude of the force Fs, as long as the direction of the latter remains the same (and its value is not too large); so the scalar version of Hooke's law *Fs* = −*kx* will hold. However, the force and displacement *vectors* will not be scalar multiples of each other, since they have different directions. Moreover, the ratio k between their magnitudes will depend on the direction of the vector Fs.
Yet, in such cases there is often a fixed [linear relation](https://en.wikipedia.org/wiki/Linear_map "Linear map") between the force and deformation vectors, as long as they are small enough. Namely, there is a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") **κ** from vectors to vectors, such that **F** = ***κ***(**X**), and ***κ***(*α***X**1 + *β***X**2) = *α**κ***(**X**1) + *β**κ***(**X**2) for any real numbers α, β and any displacement vectors **X**1, **X**2. Such a function is called a (second-order) [tensor](https://en.wikipedia.org/wiki/Tensor "Tensor").
With respect to an arbitrary [Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates"), the force and displacement vectors can be represented by 3 × 1 [matrices](https://en.wikipedia.org/wiki/Matrix_\(mathematics\) "Matrix (mathematics)") of real numbers. Then the tensor **κ** connecting them can be represented by a 3 × 3 matrix **κ** of real coefficients, that, when [multiplied](https://en.wikipedia.org/wiki/Matrix_product "Matrix product") by the displacement vector, gives the force vector:
That is,  for *i* = 1, 2, 3. Therefore, Hooke's law **F** = ***κ*X** can be said to hold also when **X** and **F** are vectors with variable directions, except that the stiffness of the object is a tensor **κ**, rather than a single real number k.
Hooke's law for continuous media
[](https://en.wikipedia.org/wiki/File:Hookes_law_nanoscale.jpg)
(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.[\[5\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-5)
The stresses and strains of the material inside a [continuous](https://en.wikipedia.org/wiki/Continuum_mechanics "Continuum mechanics") elastic material (such as a block of rubber, the wall of a [boiler](https://en.wikipedia.org/wiki/Boiler "Boiler"), or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name.
However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.
In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the [strain tensor](https://en.wikipedia.org/wiki/Strain_tensor "Strain tensor") **ε** (in lieu of the displacement **X**) and the [stress tensor](https://en.wikipedia.org/wiki/Cauchy_stress_tensor "Cauchy stress tensor") **σ** (replacing the restoring force **F**). The analogue of Hooke's spring law for continuous media is then  where **c** is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the [stiffness tensor](https://en.wikipedia.org/wiki/Stiffness_tensor "Stiffness tensor") or [elasticity tensor](https://en.wikipedia.org/wiki/Elasticity_tensor "Elasticity tensor"). One may also write it as  where the tensor **s**, called the [compliance tensor](https://en.wikipedia.org/wiki/Stiffness_tensor "Stiffness tensor"), represents the inverse of said linear map.
In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices
Being a linear mapping between the nine numbers *σij* and the nine numbers *εkl*, the stiffness tensor **c** is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers *cijkl*. Hooke's law then says that  where *i*,*j* = 1,2,3.
All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor **ε** merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor **σ** specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor **c**, on the other hand, is a property of the material, and often depends on physical state variables such as temperature, [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure"), and [microstructure](https://en.wikipedia.org/wiki/Microstructure "Microstructure").
Due to the inherent symmetries of **σ**, **ε**, and **c**, only 21 elastic coefficients of the latter are independent.[\[6\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-6) This number can be further reduced by the symmetry of the material: 9 for an [orthorhombic](https://en.wikipedia.org/wiki/Orthorhombic_crystal_system "Orthorhombic crystal system") crystal, 5 for an [hexagonal](https://en.wikipedia.org/wiki/Hexagonal_crystal_family "Hexagonal crystal family") structure, and 3 for a [cubic](https://en.wikipedia.org/wiki/Cubic_crystal_system "Cubic crystal system") symmetry.[\[7\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-7) For [isotropic](https://en.wikipedia.org/wiki/Isotropic "Isotropic") media (which have the same physical properties in any direction), **c** can be reduced to only two independent numbers, the [bulk modulus](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") K and the [shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus") G, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.
Analogous laws
Since Hooke's law is a simple proportionality between two quantities, its formulas and consequences are mathematically similar to those of many other physical laws, such as those describing the motion of [fluids](https://en.wikipedia.org/wiki/Fluid "Fluid"), or the [polarization](https://en.wikipedia.org/wiki/Ionic_polarization "Ionic polarization") of a [dielectric](https://en.wikipedia.org/wiki/Dielectric "Dielectric") by an [electric field](https://en.wikipedia.org/wiki/Electric_field "Electric field").
In particular, the tensor equation **σ** = **cε** relating elastic stresses to strains is entirely similar to the equation **τ** = **με̇** relating the [viscous stress tensor](https://en.wikipedia.org/wiki/Viscous_stress_tensor "Viscous stress tensor") **τ** and the [strain rate tensor](https://en.wikipedia.org/wiki/Strain_rate_tensor "Strain rate tensor") **ε̇** in flows of [viscous](https://en.wikipedia.org/wiki/Viscosity "Viscosity") fluids; although the former pertains to [static](https://en.wikipedia.org/wiki/Statics "Statics") stresses (related to *amount* of deformation) while the latter pertains to [dynamical](https://en.wikipedia.org/wiki/Dynamics_\(physics\) "Dynamics (physics)") stresses (related to the *rate* of deformation).
Units of measurement
In [SI units](https://en.wikipedia.org/wiki/International_System_of_Units "International System of Units"), displacements are measured in meters (m), and forces in [newtons](https://en.wikipedia.org/wiki/Newton_\(unit\) "Newton (unit)") (N or kg·m/s2). Therefore, the spring constant k, and each element of the tensor **κ**, is measured in newtons per meter (N/m), or kilograms per second squared (kg/s2).
For continuous media, each element of the stress tensor **σ** is a force divided by an area; it is therefore measured in units of pressure, namely [pascals](https://en.wikipedia.org/wiki/Pascal_\(unit\) "Pascal (unit)") (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensor **ε** are [dimensionless](https://en.wikipedia.org/wiki/Dimensionless "Dimensionless") (displacements divided by distances). Therefore, the entries of cijkl are also expressed in units of pressure.
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.
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](https://en.wikipedia.org/wiki/Yield_\(engineering\) "Yield (engineering)")). 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](https://en.wikipedia.org/wiki/Proportional_limit "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.
Generalizations of Hooke's law for the case of [large deformations](https://en.wikipedia.org/wiki/Finite_strain_theory "Finite strain theory") is provided by models of [neo-Hookean solids](https://en.wikipedia.org/wiki/Neo-Hookean_solid "Neo-Hookean solid") and [Mooney–Rivlin solids](https://en.wikipedia.org/wiki/Mooney%E2%80%93Rivlin_solid "Mooney–Rivlin solid").
Derived formulae
Tensional stress of a uniform bar
A rod of any [elastic](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)") material may be viewed as a linear [spring](https://en.wikipedia.org/wiki/Spring_\(device\) "Spring (device)"). The rod has length L and cross-sectional area A. Its [tensile stress](https://en.wikipedia.org/wiki/Tensile_stress "Tensile stress") σ is linearly proportional to its fractional extension or strain ε by the [modulus of elasticity](https://en.wikipedia.org/wiki/Modulus_of_elasticity "Modulus of elasticity") E: 
The modulus of elasticity may often be considered constant. In turn,  (that is, the fractional change in length), and since  it follows that:
The change in length may be expressed as
Spring energy
The potential energy *U*el(*x*) 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. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative. Substituting  gives 
This potential *U*el can be visualized as a [parabola](https://en.wikipedia.org/wiki/Parabola "Parabola") on the Ux\-plane such that *U*el(*x*) = 1/2*kx*2. As the spring is stretched in the positive x\-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate:  Note that the change in the change in U is constant even when the displacement and acceleration are zero.
Relaxed force constants (generalized compliance constants)
Relaxed force constants (the inverse of generalized [compliance constants](https://en.wikipedia.org/wiki/Compliance_Constants "Compliance Constants")) are uniquely defined for molecular systems, in contradistinction to the usual "rigid" force constants, and thus their use allows meaningful correlations to be made between force fields calculated for [reactants](https://en.wikipedia.org/wiki/Reactant "Reactant"), [transition states](https://en.wikipedia.org/wiki/Transition_state "Transition state"), and products of a [chemical reaction](https://en.wikipedia.org/wiki/Chemical_reaction "Chemical reaction"). Just as the [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy") can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed [compliance constants](https://en.wikipedia.org/wiki/Compliance_constant "Compliance constant"). A direct method exists for calculating the compliance constant for any internal coordinate of a molecule, without the need to do the normal mode analysis.[\[8\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-8) The suitability of relaxed force constants (inverse compliance constants) as [covalent bond](https://en.wikipedia.org/wiki/Covalent_bond "Covalent bond") strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.[\[9\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-9)
Harmonic oscillator
[](https://en.wikipedia.org/wiki/File:Mass-spring-system.png)
A mass suspended by a spring is the classical example of a harmonic oscillator
A mass m attached to the end of a spring is a classic example of a [harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator "Harmonic oscillator"). By pulling slightly on the mass and then releasing it, the system will be set in [sinusoidal](https://en.wikipedia.org/wiki/Sine_wave "Sine wave") oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglect [friction](https://en.wikipedia.org/wiki/Friction "Friction") and the mass of the spring, the amplitude of the oscillation will remain constant; and its [frequency](https://en.wikipedia.org/wiki/Frequency "Frequency") f will be independent of its amplitude, determined only by the mass and the stiffness of the spring:  This phenomenon made possible the construction of accurate [mechanical clocks](https://en.wikipedia.org/wiki/Mechanical_clock "Mechanical clock") and watches that could be carried on ships and people's pockets.
Rotation in gravity-free space
If the mass m were attached to a spring with force constant k and rotating in free space, the spring tension (*F*t) would supply the required [centripetal force](https://en.wikipedia.org/wiki/Centripetal_force "Centripetal force") (*F*c):
 Since *F*t = *F*c and *x* = *r*, then:  Given that *ω* = 2π*f*, this leads to the same frequency equation as above: 
Isotropic materials
For an analogous development for viscous fluids, see [Viscosity](https://en.wikipedia.org/wiki/Viscosity "Viscosity").
[Isotropic materials](https://en.wikipedia.org/wiki/Isotropic_solid "Isotropic solid") 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](https://en.wikipedia.org/wiki/Trace_\(linear_algebra\) "Trace (linear algebra)") 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.[\[10\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-10) Thus in [index notation](https://en.wikipedia.org/wiki/Ricci_calculus "Ricci calculus"):
 where δij is the [Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta "Kronecker delta"). In direct tensor notation: 
where **I** is the second-order identity tensor.
The first term on the right is the constant tensor, also known as the **[volumetric strain tensor](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory#Volumetric_strain "Infinitesimal strain theory")**, and the second term is the traceless symmetric tensor, also known as the **[deviatoric strain tensor](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory#Strain_deviator_tensor "Infinitesimal strain theory")** 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 modulus](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") and G is the [shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus").
Using the relationships between the [elastic moduli](https://en.wikipedia.org/wiki/Elastic_modulus "Elastic modulus"), 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 [\[11\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Simo98-11)
 where *λ* = *K* − 2/3*G* = *c*1111 − 2*c*1212 and *μ* = *G* = *c*1212 are the [Lamé constants](https://en.wikipedia.org/wiki/Lam%C3%A9_constants "Lamé constants"), **I** is the second-rank identity tensor, and **I** is the symmetric part of the fourth-rank identity tensor. In index notation: 
The inverse relationship is[\[12\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Milton02-12)
Therefore, the compliance tensor in the relation **ε** = **s** : **σ** is
In terms of [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") and [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio"), 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 [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") and ν is [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio"). (See [3-D elasticity](https://en.wikipedia.org/wiki/3-D_elasticity "3-D elasticity")).
**Derivation of Hooke's law in three dimensions**
The three-dimensional form of Hooke's law can be derived using Poisson's ratio and the one-dimensional 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 ν is Poisson's ratio and E is Young's modulus.
We get similar equations to the loads in directions 2 and 3,  and 
Summing the three cases together (*εi* = *εi*′ + *εi*″ + *εi*‴) we get  or by adding and subtracting one νσ  and further we get by solving *σ*1 
Calculating the sum  and substituting it to the equation solved for *σ*1 gives  where μ and λ are the [Lamé parameters](https://en.wikipedia.org/wiki/Lam%C3%A9_parameters "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 *γij* = 2*εij* is the **engineering shear strain**. The inverse relation may be written as  which can be simplified thanks to the Lamé constants:  In vector notation this becomes  where **I** is the identity tensor.
Plane stress
Under [plane stress](https://en.wikipedia.org/wiki/Plane_stress#Plane_stress "Plane stress") conditions, *σ*31 = *σ*13 = *σ*32 = *σ*23 = *σ*33 = 0. In that case Hooke's law takes the form 
In vector notation this becomes 
The inverse relation is usually written in the reduced form 
Plane strain
Under [plane strain](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory#Plane_strain "Infinitesimal strain theory") conditions, *ε*31 = *ε*13 = *ε*32 = *ε*23 = *ε*33 = 0. In this case Hooke's law takes the form 
Anisotropic materials
The symmetry of the [Cauchy stress tensor](https://en.wikipedia.org/wiki/Stress_\(physics\) "Stress (physics)") (*σij* = *σji*) and the generalized Hooke's laws (*σij* = *cijklεkl*) implies that *cijkl* = *cjikl*. Similarly, the symmetry of the [infinitesimal strain tensor](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory "Infinitesimal strain theory") implies that *cijkl* = *cijlk*. These symmetries are called the **minor symmetries** of the stiffness tensor **c**. This reduces the number of elastic constants from 81 to 36.
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 (U), then  The arbitrariness of the order of differentiation implies that *cijkl* = *cklij*. These are called the **major symmetries** of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. 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 notation](https://en.wikipedia.org/wiki/Voigt_notation "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 (**e**1,**e**2,**e**3) as ![{\\displaystyle \[{\\boldsymbol {\\sigma }}\]\\,=\\,{\\begin{bmatrix}\\sigma \_{11}\\\\\\sigma \_{22}\\\\\\sigma \_{33}\\\\\\sigma \_{23}\\\\\\sigma \_{13}\\\\\\sigma \_{12}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}\\sigma \_{1}\\\\\\sigma \_{2}\\\\\\sigma \_{3}\\\\\\sigma \_{4}\\\\\\sigma \_{5}\\\\\\sigma \_{6}\\end{bmatrix}}\\,;\\qquad \[{\\boldsymbol {\\varepsilon }}\]\\,=\\,{\\begin{bmatrix}\\varepsilon \_{11}\\\\\\varepsilon \_{22}\\\\\\varepsilon \_{33}\\\\2\\varepsilon \_{23}\\\\2\\varepsilon \_{13}\\\\2\\varepsilon \_{12}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}\\varepsilon \_{1}\\\\\\varepsilon \_{2}\\\\\\varepsilon \_{3}\\\\\\varepsilon \_{4}\\\\\\varepsilon \_{5}\\\\\\varepsilon \_{6}\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d84c34fc9efc62922b42a33f888656c62d794b) Then the stiffness tensor (**c**) can be expressed as ![{\\displaystyle \[{\\mathsf {c}}\]\\,=\\,{\\begin{bmatrix}c\_{1111}\&c\_{1122}\&c\_{1133}\&c\_{1123}\&c\_{1131}\&c\_{1112}\\\\c\_{2211}\&c\_{2222}\&c\_{2233}\&c\_{2223}\&c\_{2231}\&c\_{2212}\\\\c\_{3311}\&c\_{3322}\&c\_{3333}\&c\_{3323}\&c\_{3331}\&c\_{3312}\\\\c\_{2311}\&c\_{2322}\&c\_{2333}\&c\_{2323}\&c\_{2331}\&c\_{2312}\\\\c\_{3111}\&c\_{3122}\&c\_{3133}\&c\_{3123}\&c\_{3131}\&c\_{3112}\\\\c\_{1211}\&c\_{1222}\&c\_{1233}\&c\_{1223}\&c\_{1231}\&c\_{1212}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}C\_{11}\&C\_{12}\&C\_{13}\&C\_{14}\&C\_{15}\&C\_{16}\\\\C\_{12}\&C\_{22}\&C\_{23}\&C\_{24}\&C\_{25}\&C\_{26}\\\\C\_{13}\&C\_{23}\&C\_{33}\&C\_{34}\&C\_{35}\&C\_{36}\\\\C\_{14}\&C\_{24}\&C\_{34}\&C\_{44}\&C\_{45}\&C\_{46}\\\\C\_{15}\&C\_{25}\&C\_{35}\&C\_{45}\&C\_{55}\&C\_{56}\\\\C\_{16}\&C\_{26}\&C\_{36}\&C\_{46}\&C\_{56}\&C\_{66}\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85c8bf05adff9dcaec56f4863dd039fae5986a79)
and Hooke's law is written as
![{\\displaystyle \[{\\boldsymbol {\\sigma }}\]=\[{\\mathsf {C}}\]\[{\\boldsymbol {\\varepsilon }}\]\\qquad {\\text{or}}\\qquad \\sigma \_{i}=C\_{ij}\\varepsilon \_{j}\\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0315b5cfc25f83e499fadf8ce4921e11340f8e) Similarly the compliance tensor (**s**) can be written as ![{\\displaystyle \[{\\mathsf {s}}\]\\,=\\,{\\begin{bmatrix}s\_{1111}\&s\_{1122}\&s\_{1133}&2s\_{1123}&2s\_{1131}&2s\_{1112}\\\\s\_{2211}\&s\_{2222}\&s\_{2233}&2s\_{2223}&2s\_{2231}&2s\_{2212}\\\\s\_{3311}\&s\_{3322}\&s\_{3333}&2s\_{3323}&2s\_{3331}&2s\_{3312}\\\\2s\_{2311}&2s\_{2322}&2s\_{2333}&4s\_{2323}&4s\_{2331}&4s\_{2312}\\\\2s\_{3111}&2s\_{3122}&2s\_{3133}&4s\_{3123}&4s\_{3131}&4s\_{3112}\\\\2s\_{1211}&2s\_{1222}&2s\_{1233}&4s\_{1223}&4s\_{1231}&4s\_{1212}\\end{bmatrix}}\\,\\equiv \\,{\\begin{bmatrix}S\_{11}\&S\_{12}\&S\_{13}\&S\_{14}\&S\_{15}\&S\_{16}\\\\S\_{12}\&S\_{22}\&S\_{23}\&S\_{24}\&S\_{25}\&S\_{26}\\\\S\_{13}\&S\_{23}\&S\_{33}\&S\_{34}\&S\_{35}\&S\_{36}\\\\S\_{14}\&S\_{24}\&S\_{34}\&S\_{44}\&S\_{45}\&S\_{46}\\\\S\_{15}\&S\_{25}\&S\_{35}\&S\_{45}\&S\_{55}\&S\_{56}\\\\S\_{16}\&S\_{26}\&S\_{36}\&S\_{46}\&S\_{56}\&S\_{66}\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34760b2d8ef86f720051aebe5a45a65b312bcab6)
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[\[13\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Slaughter-13)
 where lab are the components of an [orthogonal rotation matrix](https://en.wikipedia.org/wiki/Orthogonal_matrix "Orthogonal 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
![{\\displaystyle \[\\mathbf {e} \_{i}'\]=\[L\]\[\\mathbf {e} \_{i}\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/213d0bb55cc1da894c855871790e09d78635c17b)
then
 In addition, if the material is symmetric with respect to the transformation \[*L*\] then 
Orthotropic materials
[Orthotropic materials](https://en.wikipedia.org/wiki/Orthotropic_material "Orthotropic material") have three [orthogonal](https://en.wikipedia.org/wiki/Orthogonal "Orthogonal") [planes of symmetry](https://en.wikipedia.org/wiki/Plane_of_symmetry "Plane of symmetry"). If the basis vectors (**e**1,**e**2,**e**3) are normals to the planes of symmetry then the coordinate transformation relations imply that
 The inverse of this relation is commonly written as[\[14\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Boresi-14)\[*[page needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources "Wikipedia:Citing sources")*\]  where
- Ei is the [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") along axis i
- Gij is the [shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus") in direction j on the plane whose normal is in direction i
- νij is the [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio") that corresponds to a contraction in direction j when an extension is applied in direction i.
Under *plane stress* conditions, *σzz* = *σzx* = *σyz* = 0, 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](https://en.wikipedia.org/wiki/Transversely_isotropic "Transversely isotropic") material is symmetric with respect to a rotation about an [axis of symmetry](https://en.wikipedia.org/wiki/Axis_of_symmetry "Axis of symmetry"). For such a material, if **e**3 is the axis of symmetry, Hooke's law can be expressed as 
More frequently, the *x* ≡ **e**1 axis is taken to be the axis of symmetry and the inverse Hooke's law is written as [\[15\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-Tan-15) 
Universal elastic anisotropy index
To grasp the degree of anisotropy of any class, a **universal elastic anisotropy index** (AU)[\[16\]](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_note-16) was formulated. It replaces the [Zener ratio](https://en.wikipedia.org/wiki/Zener_ratio "Zener ratio"), which is suited for [cubic crystals](https://en.wikipedia.org/wiki/Cubic_crystal_system "Cubic crystal system").
Thermodynamic basis
Linear deformations of elastic materials can be approximated as [adiabatic](https://en.wikipedia.org/wiki/Adiabatic "Adiabatic"). Under these conditions and for quasistatic processes the [first law of thermodynamics](https://en.wikipedia.org/wiki/First_law_of_thermodynamics "First law of thermodynamics") for a deformed body can be expressed as  where δU is the increase in [internal energy](https://en.wikipedia.org/wiki/Internal_energy "Internal energy") and δW is the [work](https://en.wikipedia.org/wiki/Work_\(physics\) "Work (physics)") done by external forces. The work can be split into two terms  where *δW*s is the work done by [surface forces](https://en.wikipedia.org/wiki/Surface_force "Surface force") while *δW*b is the work done by [body forces](https://en.wikipedia.org/wiki/Body_force "Body force"). If *δ***u** is a [variation](https://en.wikipedia.org/wiki/Calculus_of_variations "Calculus of variations") of the displacement field **u** in the body, then the two external work terms can be expressed as  where **t** is the surface [traction](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") vector, **b** is the body force vector, Ω represents the body and ∂*Ω* represents its surface. Using the relation between the [Cauchy stress](https://en.wikipedia.org/wiki/Stress_\(mechanics\) "Stress (mechanics)") and the surface traction, **t** = **n** · **σ** (where **n** is the unit outward normal to ∂*Ω*), we have  Converting the [surface integral](https://en.wikipedia.org/wiki/Surface_integral "Surface integral") into a [volume integral](https://en.wikipedia.org/wiki/Volume_integral "Volume integral") via the [divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem "Divergence theorem") gives  Using the symmetry of the Cauchy stress and the identity  we have the following
 From the definition of [strain](https://en.wikipedia.org/wiki/Infinitesimal_strain_theory "Infinitesimal strain theory") and from the equations of [equilibrium](https://en.wikipedia.org/wiki/Linear_elasticity "Linear elasticity") we have  Hence we can write  and therefore the variation in the [internal energy](https://en.wikipedia.org/wiki/Internal_energy "Internal energy") density is given by  An [elastic](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)") material is defined as one in which the total internal energy is equal to the [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy") of the internal forces (also called the **elastic strain energy**). Therefore, the internal energy density is a function of the strains, *U*0 = *U*0(**ε**) 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 ∂*U*0/∂**ε** is a linear function of **ε**, and can therefore be expressed as  where **c** is a fourth-rank tensor of material constants, also called the **stiffness tensor**. We can see why **c** must be a fourth-rank tensor by noting that, for a linear elastic material,  In index notation 
The right-hand side constant requires four indices and is a fourth-rank 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-rank tensors.
See also
- [Acoustoelastic effect](https://en.wikipedia.org/wiki/Acoustoelastic_effect "Acoustoelastic effect")
- [Elastic potential energy](https://en.wikipedia.org/wiki/Elastic_potential_energy "Elastic potential energy")
- [Laws of science](https://en.wikipedia.org/wiki/Laws_of_science "Laws of science")
- [List of scientific laws named after people](https://en.wikipedia.org/wiki/List_of_scientific_laws_named_after_people "List of scientific laws named after people")
- [Quadratic form](https://en.wikipedia.org/wiki/Quadratic_form "Quadratic form")
- [Series and parallel springs](https://en.wikipedia.org/wiki/Series_and_parallel_springs "Series and parallel springs")
- [Spring system](https://en.wikipedia.org/wiki/Spring_system "Spring system")
- [Simple harmonic motion of a mass on a spring](https://en.wikipedia.org/wiki/Simple_harmonic_motion#Mass_on_a_spring "Simple harmonic motion")
- [Sine wave](https://en.wikipedia.org/wiki/Sine_wave "Sine wave")
- [Solid mechanics](https://en.wikipedia.org/wiki/Solid_mechanics "Solid mechanics")
- [Spring pendulum](https://en.wikipedia.org/wiki/Spring_pendulum "Spring pendulum")
Notes
1. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-1)**
The anagram was given in alphabetical order, *ceiiinosssttuv*, representing *Ut tensio, sic vis* – "As the extension, so the force":
[Petroski, Henry](https://en.wikipedia.org/wiki/Henry_Petroski "Henry Petroski") (1996). [*Invention by Design: How Engineers Get from Thought to Thing*](https://archive.org/details/inventionbydesig00petr). Cambridge, MA: Harvard University Press. p. [11](https://archive.org/details/inventionbydesig00petr/page/11). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-674-46368-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-674-46368-4 "Special:BookSources/978-0-674-46368-4")
.
2. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-2)** See <http://civil.lindahall.org/design.shtml>, where one can find also an anagram for [catenary](https://en.wikipedia.org/wiki/Catenary "Catenary").
3. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-3)** [Robert Hooke](https://en.wikipedia.org/wiki/Robert_Hooke "Robert Hooke"), *De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies*, London, 1678.
4. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-4)**
Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2016). *Sears and Zemansky's University Physics: With Modern Physics* (14th ed.). Pearson. p. 209.
5. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-5)**
Ushiba, Shota; Masui, Kyoko; Taguchi, Natsuo; Hamano, Tomoki; Kawata, Satoshi; Shoji, Satoru (2015). ["Size dependent nanomechanics of coil spring shaped polymer nanowires"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4661696). *Scientific Reports*. **5** 17152. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2015NatSR...517152U](https://ui.adsabs.harvard.edu/abs/2015NatSR...517152U). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/srep17152](https://doi.org/10.1038%2Fsrep17152). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [4661696](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4661696). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [26612544](https://pubmed.ncbi.nlm.nih.gov/26612544).
6. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-6)**
Belen'kii; Salaev (1988). ["Deformation effects in layer crystals"](https://doi.org/10.3367%2FUFNr.0155.198805c.0089). *Uspekhi Fizicheskikh Nauk*. **155** (5): 89. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3367/UFNr.0155.198805c.0089](https://doi.org/10.3367%2FUFNr.0155.198805c.0089).
7. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-7)**
Mouhat, Félix; Coudert, François-Xavier (5 December 2014). ["Necessary and sufficient elastic stability conditions in various crystal systems"](https://link.aps.org/doi/10.1103/PhysRevB.90.224104). *Physical Review B*. **90** (22) 224104. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1410\.0065](https://arxiv.org/abs/1410.0065). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2014PhRvB..90v4104M](https://ui.adsabs.harvard.edu/abs/2014PhRvB..90v4104M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevB.90.224104](https://doi.org/10.1103%2FPhysRevB.90.224104). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1098-0121](https://search.worldcat.org/issn/1098-0121). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [54058316](https://api.semanticscholar.org/CorpusID:54058316).
8. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-8)**
Vijay Madhav, M.; Manogaran, S. (2009). "A relook at the compliance constants in redundant internal coordinates and some new insights". *J. Chem. Phys*. **131** (17): 174112–174116\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2009JChPh.131q4112V](https://ui.adsabs.harvard.edu/abs/2009JChPh.131q4112V). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1063/1.3259834](https://doi.org/10.1063%2F1.3259834). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [19895003](https://pubmed.ncbi.nlm.nih.gov/19895003).
9. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-9)**
Ponomareva, Alla; Yurenko, Yevgen; Zhurakivsky, Roman; Van Mourik, Tanja; Hovorun, Dmytro (2012). "Complete conformational space of the potential HIV-1 reverse transcriptase inhibitors d4U and d4C. A quantum chemical study". *Phys. Chem. Chem. Phys*. **14** (19): 6787–6795\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2012PCCP...14.6787P](https://ui.adsabs.harvard.edu/abs/2012PCCP...14.6787P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1039/C2CP40290D](https://doi.org/10.1039%2FC2CP40290D). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [22461011](https://pubmed.ncbi.nlm.nih.gov/22461011).
10. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-10)**
Symon, Keith R. (1971). "Chapter 10". *Mechanics*. Reading, Massachusetts: Addison-Wesley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-201-07392-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-07392-8 "Special:BookSources/978-0-201-07392-8")
.
11. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Simo98_11-0)**
Simo, J. C.; Hughes, T. J. R. (1998). *Computational Inelasticity*. Springer. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-97520-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97520-7 "Special:BookSources/978-0-387-97520-7")
.
12. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Milton02_12-0)**
Milton, Graeme W. (2002). *The Theory of Composites*. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-78125-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-78125-1 "Special:BookSources/978-0-521-78125-1")
.
13. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Slaughter_13-0)**
Slaughter, William S. (2001). *The Linearized Theory of Elasticity*. Birkhäuser. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-8176-4117-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8176-4117-7 "Special:BookSources/978-0-8176-4117-7")
.
14. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Boresi_14-0)**
Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. (1993). *Advanced Mechanics of Materials* (5th ed.). Wiley. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-471-60009-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-60009-1 "Special:BookSources/978-0-471-60009-1")
.
15. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-Tan_15-0)**
Tan, S. C. (1994). *Stress Concentrations in Laminated Composites*. Lancaster, PA: Technomic Publishing Company. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-56676-077-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-56676-077-5 "Special:BookSources/978-1-56676-077-5")
.
16. **[^](https://en.wikipedia.org/wiki/Hooke%27s_law#cite_ref-16)**
Ranganathan, S.I.; [Ostoja-Starzewski, M.](https://en.wikipedia.org/wiki/Martin_Ostoja-Starzewski "Martin Ostoja-Starzewski") (2008). "Universal Elastic Anisotropy Index". *Physical Review Letters*. **101** (5): 055504–1–4. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2008PhRvL.101e5504R](https://ui.adsabs.harvard.edu/abs/2008PhRvL.101e5504R). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/PhysRevLett.101.055504](https://doi.org/10.1103%2FPhysRevLett.101.055504). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [18764407](https://pubmed.ncbi.nlm.nih.gov/18764407). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [6668703](https://api.semanticscholar.org/CorpusID:6668703).
References
- [Hooke's law - The Feynman Lectures on Physics](https://feynmanlectures.caltech.edu/II_38.html#Ch38-S1)
- [Hooke's Law - Classical Mechanics - Physics - MIT OpenCourseWare](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/pages/week-2-newtons-laws/7-4-hookes-law/)
External links
- [JavaScript Applet demonstrating Springs and Hooke's law](https://www.compadre.org/Physlets/mechanics/illustration5_4.cfm)
- [JavaScript Applet demonstrating Spring Force](https://www.compadre.org/Physlets/mechanics/ex5_3.cfm)
| 3D Formulae | | | | | | | |
|---|---|---|---|---|---|---|---|
| Knowns | [Bulk modulus](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") (*K*) | [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus "Young's modulus") (*E*) | [Lamé's first parameter](https://en.wikipedia.org/wiki/Lam%C3%A9_parameters "Lamé parameters") (λ) | [Shear modulus](https://en.wikipedia.org/wiki/Shear_modulus "Shear modulus") (*G*) | [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio "Poisson's ratio") (*ν*) | [P-wave modulus](https://en.wikipedia.org/wiki/P-wave_modulus "P-wave modulus") (*M*) | Notes |
| (*K*, *E*) | | | 3*K*(1 + 6*K*/*E* − 9*K*) | *E*/3 − *E*/3*K* | 1/2 − *E*/6*K* | 3*K* + *E*/3 − *E*/3*K* | |
| (*K*, λ) | | 9*K*(*K* − λ)/3*K* − λ | | 3(*K* − λ)/2 | λ/3*K* − λ | 3*K* − 2λ | |
| (*K*, *G*) | | 9*KG*/3*K* + *G* | *K* − 2*G*/3 | | 3*K* − 2*G*/6*K* + 2*G* | *K* + 4*G*/3 | |
| (*K*, *ν*) | | 3*K*(1 − 2*ν*) | 3*Kν*/1 + *ν* | 3*K*(1 − 2*ν*)/2(1 + *ν*) | | 3*K*(1 − *ν*)/1 + *ν* | |
| (*K*, *M*) | | 9*K*(*M* − *K*)/3*K* + *M* | 3*K* − *M*/2 | 3(*M* − *K*)/4 | 3*K* − *M*/3*K* + *M* | | |
| (*E*, λ) | *E* + 3λ + R/6 | | | *E* − 3λ + *R*/4 | − *E* + *R*/4λ − 1/4 | *E* − λ + *R*/2 | *R* = ±(*E*2 + 9λ2 + 2*E*λ)1/2 |
| (*E*, *G*) | *EG*/3(3*G* − *E*) | | *G*(*E* − 2*G*)/3*G* − *E* | | *E*/2*G* − 1 | *G*(4*G* − *E*)/3*G* − *E* | |
| (*E*, *ν*) | *E*/3 − 6*ν* | | *Eν*/(1 + *ν*)(1 − 2*ν*) | *E*/2(1 + *ν*) | | *E*(1 − *ν*)/(1 + *ν*)(1 − 2*ν*) | |
| (*E*, *M*) | 3*M* − *E* + *S*/6 | | *M* − *E* + *S*/4 | 3*M* + *E* − *S*/8 | *E* + *S*/4*M* − 1/4 | | *S* = ±(*E*2 + 9M2 − 10*E*M)1/2 |
| (λ, *G*) | λ + 2*G*/3 | *G*(3λ + 2*G*)/λ + *G* | | | λ/2(λ + *G*) | λ + 2*G* | |
| (λ, *ν*) | λ/3(1 + 1/*ν*) | λ(1/*ν* − 2*ν* − 1) | | λ(1/2*ν* − 1) | | λ(1/*ν* − 1) | |
| (λ, *M*) | *M* + 2λ/3 | (*M* − λ)(*M*\+2λ)/*M* + λ | | *M* − λ/2 | λ/*M* + λ | | |
| (*G*, *ν*) | 2*G*(1 + *ν*)/3 − 6*ν* | 2*G*(1 + *ν*) | 2 *G* *ν*/1 − 2*ν* | | | 2*G*(1 − *ν*)/1 − 2*ν* | |
| (*G*, *M*) | *M* − 4*G*/3 | *G*(3*M* − 4*G*)/*M* − *G* | *M* − 2*G* | | *M* − 2*G*/2*M* − 2*G* | | |
| (*ν*, *M*) | *M*(1 + *ν*)/3(1 − *ν*) | *M*(1 + *ν*)(1 − 2*ν*)/1 − *ν* | *M* *ν*/1 − *ν* | *M*(1 − 2*ν*)/2(1 − *ν*) | | | |
| 2D Formulae | | | | | | | |
| Knowns | (*K*) | (*E*) | (λ) | (*G*) | (*ν*) | (*M*) | Notes |
| (*K*2D, *E*2D) | | | 2*K*2D(2*K*2D − *E*2D)/4*K*2D − *E*2D | *K*2D*E*2D/4*K*2D − *E*2D | 2*K*2D − *E*2D/2*K*2D | 4*K*2D^2/4*K*2D − *E*2D | |
| (*K*2D, λ2D) | | 4*K*2D(*K*2D − λ2D)/2*K*2D − λ2D | | *K*2D − λ2D | λ2D/2*K*2D − λ2D | 2*K*2D − λ2D | |
| (*K*2D, *G*2D) | | 4*K*2D*G*2D/*K*2D + *G*2D | *K*2D − *G*2D | | *K*2D − *G*2D/*K*2D + *G*2D | *K*2D + *G*2D | |
| (*K*2D, *ν*2D) | | 2*K*2D(1 − *ν*2D) | 2*K*2D*ν*2D/1 + *ν*2D | *K*2D(1 − *ν*2D)/1 + *ν*2D | | 2*K*2D/1 + *ν*2D | |
| (*E*2D, *G*2D) | *E*2D*G*2D/4*G*2D − *E*2D | | 2*G*2D(*E*2D − 2*G*2D)/4*G*2D − *E*2D | | *E*2D/2*G*2D − 1 | 4*G*2D^2/4*G*2D − *E*2D | |
| (*E*2D, *ν*2D) | *E*2D/2(1 − *ν*2D) | | *E*2D*ν*2D/(1 + *ν*2D)(1 − *ν*2D) | *E*2D/2(1 + *ν*2D) | | *E*2D/(1 + *ν*2D)(1 − *ν*2D) | |
| (λ2D, *G*2D) | λ2D + *G*2D | 4*G*2D(λ2D + *G*2D)/λ2D + 2*G*2D | | | λ2D/λ2D + 2*G*2D | λ2D + 2*G*2D | |
| (λ2D, *ν*2D) | λ2D(1 + *ν*2D)/2*ν*2D | λ2D(1 + *ν*2D)(1 − *ν*2D)/*ν*2D | | λ2D(1 − *ν*2D)/2*ν*2D | | λ2D/*ν*2D | |
| (*G*2D, *ν*2D) | *G*2D(1 + *ν*2D)/1 − *ν*2D | 2*G*2D(1 + *ν*2D) | 2 *G*2D *ν*2D/1 − *ν*2D | | | 2*G*2D/1 − *ν*2D | |
| (*G*2D, *M*2D) | *M*2D − *G*2D | 4*G*2D(*M*2D − *G*2D)/*M*2D | *M*2D − 2*G*2D | | *M*2D − 2*G*2D/*M*2D | | | |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Hooke%27s_law s443 |