Hooke's law

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.

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.

Hooke's equation holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.

Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail 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.

On the other hand, Hooke's law is an accurate approximation for most solid bodies, as long as the forces and deformations are small enough. For this reason, Hooke's law is extensively used in all branches of science and engineering, and is the foundation of many disciplines such as seismology, molecular mechanics and acoustics. It is also the fundamental principle behind the spring scale, the manometer, the galvanometer, and the balance wheel of the mechanical clock.

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.

Formal definition

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

$${\displaystyle F_{s}=kx}$$

 

or, equivalently,

$${\displaystyle x={\frac {F_{s}}{k}}}$$

 

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 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

$${\displaystyle F_{s}=-kx}$$

 

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:

${\displaystyle \tau =-k\theta }$ 

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(αX₁ + βX₂) = ακ(X₁) + βκ(X₂) for any real numbers α, β and any displacement vectors X₁, X₂. 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:

$${\displaystyle \mathbf {F} \,=\,{\begin{bmatrix}F_{1}\\F_{2}\\F_{3}\end{bmatrix}}\,=\,{\begin{bmatrix}\kappa _{11}&\kappa _{12}&\kappa _{13}\\\kappa _{21}&\kappa _{22}&\kappa _{23}\\\kappa _{31}&\kappa _{32}&\kappa _{33}\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}\,=\,{\boldsymbol {\kappa }}\mathbf {X} }$$

 

That is,

$${\displaystyle F_{i}=\kappa _{i1}X_{1}+\kappa _{i2}X_{2}+\kappa _{i3}X_{3}}$$

 

for i = 1, 2, 3. Therefore, Hooke's law F = κX 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

 

(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

$${\displaystyle {\boldsymbol {\sigma }}=\mathbf {c} {\boldsymbol {\varepsilon }},}$$

 

where c is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the stiffness tensor or elasticity tensor. One may also write it as

$${\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {s} {\boldsymbol {\sigma }},}$$

 

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

$${\displaystyle {\boldsymbol {\varepsilon }}\,=\,{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\,;\qquad {\boldsymbol {\sigma }}\,=\,{\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}$$

 

Being a linear mapping between the nine numbers 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

$${\displaystyle \sigma _{ij}=\sum _{k=1}^{3}\sum _{l=1}^{3}c_{ijkl}\varepsilon _{kl}}$$

 

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, 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/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 (Pa, or N/m², or kg/(m·s²). 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

 

Stress–strain curve for low-carbon steel, showing the relationship between the stress (force per unit area) and strain (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
2. Yield strength (yield point)
3. Rupture
4. Strain hardening region
5. Necking region

1. Apparent stress (F/A₀)
2. Actual stress (F/A)

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:

$${\displaystyle \sigma =E\varepsilon .}$$

 

The modulus of elasticity may often be considered constant. In turn,

$${\displaystyle \varepsilon ={\frac {\Delta L}{L}}}$$

 

(that is, the fractional change in length), and since

$${\displaystyle \sigma ={\frac {F}{A}}\,,}$$

 

it follows that:

$${\displaystyle \varepsilon ={\frac {\sigma }{E}}={\frac {F}{AE}}\,.}$$

 

The change in length may be expressed as

$${\displaystyle \Delta L=\varepsilon L={\frac {FL}{AE}}\,.}$$

 

Spring energy

The potential energy U_el(x) stored in a spring is given by

$${\displaystyle U_{\mathrm {el} }(x)={\tfrac {1}{2}}kx^{2}}$$

 

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 ${\displaystyle x=F/k}$  gives

$${\displaystyle U_{\mathrm {el} }(F)={\frac {F^{2}}{2k}}.}$$

 

This potential U_el can be visualized as a parabola on the Ux-plane such that U_el(x) = 1/2kx². 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:

$${\displaystyle {\frac {d^{2}U_{\mathrm {el} }}{dx^{2}}}=k\,.}$$

 

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) 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

See also: 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:

$${\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}$$

 

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):

$${\displaystyle F_{\mathrm {t} }=kx\,;\qquad F_{\mathrm {c} }=m\omega ^{2}r}$$

 

Since F_t = F_c and x = r, then:

$${\displaystyle k=m\omega ^{2}}$$

 

Given that ω = 2πf, this leads to the same frequency equation as above:

$${\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}$$

 

Linear elasticity theory for continuous media

See also: Elasticity tensor

Note: the Einstein summation convention of summing on repeated indices is used below.

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:

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

 

where δ_ij is the Kronecker delta. In direct tensor notation:

$${\displaystyle {\boldsymbol {\varepsilon }}=\operatorname {vol} ({\boldsymbol {\varepsilon }})+\operatorname {dev} ({\boldsymbol {\varepsilon }})\,;\qquad \operatorname {vol} ({\boldsymbol {\varepsilon }})={\tfrac {1}{3}}\operatorname {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} \,;\qquad \operatorname {dev} ({\boldsymbol {\varepsilon }})={\boldsymbol {\varepsilon }}-\operatorname {vol} ({\boldsymbol {\varepsilon }})}$$

 

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, 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:

$${\displaystyle \sigma _{ij}=3K\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+2G\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)\,;\qquad {\boldsymbol {\sigma }}=3K\operatorname {vol} ({\boldsymbol {\varepsilon }})+2G\operatorname {dev} ({\boldsymbol {\varepsilon }})}$$

 

where K is the bulk modulus 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] ${\displaystyle {\boldsymbol {\sigma }}=\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}\,;\qquad {\mathsf {c}}=\lambda \mathbf {I} \otimes \mathbf {I} +2\mu {\mathsf {I}}}$  where λ = K − 2/3G = c₁₁₁₁ − 2c₁₂₁₂ and μ = G = c₁₂₁₂ 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:

$${\displaystyle \sigma _{ij}=\lambda \varepsilon _{kk}~\delta _{ij}+2\mu \varepsilon _{ij}=c_{ijkl}\varepsilon _{kl}\,;\qquad c_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu \left(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}\right)}$$

 

The inverse relationship is^[12]

$${\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2\mu }}{\boldsymbol {\sigma }}-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} ={\frac {1}{2G}}{\boldsymbol {\sigma }}+\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }$$

 

Therefore, the compliance tensor in the relation ε = s : σ is

$${\displaystyle {\mathsf {s}}=-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2\mu }}{\mathsf {I}}=\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2G}}{\mathsf {I}}}$$

 

In terms of Young's modulus and Poisson's ratio, Hooke's law for isotropic materials can then be expressed as

$${\displaystyle \varepsilon _{ij}={\frac {1}{E}}{\big (}\sigma _{ij}-\nu (\sigma _{kk}\delta _{ij}-\sigma _{ij}){\big )}\,;\qquad {\boldsymbol {\varepsilon }}={\frac {1}{E}}{\big (}{\boldsymbol {\sigma }}-\nu (\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} -{\boldsymbol {\sigma }}){\big )}={\frac {1+\nu }{E}}{\boldsymbol {\sigma }}-{\frac {\nu }{E}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }$$

 

This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is

$${\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{E}}{\big (}\sigma _{11}-\nu (\sigma _{22}+\sigma _{33}){\big )}\\\varepsilon _{22}&={\frac {1}{E}}{\big (}\sigma _{22}-\nu (\sigma _{11}+\sigma _{33}){\big )}\\\varepsilon _{33}&={\frac {1}{E}}{\big (}\sigma _{33}-\nu (\sigma _{11}+\sigma _{22}){\big )}\\\varepsilon _{12}&={\frac {1}{2G}}\sigma _{12}\,;\qquad \varepsilon _{13}={\frac {1}{2G}}\sigma _{13}\,;\qquad \varepsilon _{23}={\frac {1}{2G}}\sigma _{23}\end{aligned}}}$$

 

where E is Young's modulus 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),

$${\displaystyle {\begin{aligned}\varepsilon _{1}'&={\frac {1}{E}}\sigma _{1}\,,\\\varepsilon _{2}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\\\varepsilon _{3}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\end{aligned}}}$$

 

where ν is Poisson's ratio and E is Young's modulus.

We get similar equations to the loads in directions 2 and 3,

$${\displaystyle {\begin{aligned}\varepsilon _{1}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\\\varepsilon _{2}''&={\frac {1}{E}}\sigma _{2}\,,\\\varepsilon _{3}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\end{aligned}}}$$

 

and

$${\displaystyle {\begin{aligned}\varepsilon _{1}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{2}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{3}'''&={\frac {1}{E}}\sigma _{3}\,.\end{aligned}}}$$

 

Summing the three cases together (ε_i = ε_i′ + ε_i″ + ε_i‴) we get

$${\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}\sigma _{1}-\nu (\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}\sigma _{2}-\nu (\sigma _{1}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}){\big )}\,,\end{aligned}}}$$

 

or by adding and subtracting one νσ

$${\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{1}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{2}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\end{aligned}}}$$

 

and further we get by solving σ₁

$${\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {\nu }{1+\nu }}(\sigma _{1}+\sigma _{2}+\sigma _{3})\,.}$$

 

Calculating the sum

$${\displaystyle {\begin{aligned}\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )(\sigma _{1}+\sigma _{2}+\sigma _{3})-3\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}={\frac {1-2\nu }{E}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\\\sigma _{1}+\sigma _{2}+\sigma _{3}&={\frac {E}{1-2\nu }}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\end{aligned}}}$$

 

and substituting it to the equation solved for σ₁ gives

$${\displaystyle {\begin{aligned}\sigma _{1}&={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {E\nu }{(1+\nu )(1-2\nu )}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\\&=2\mu \varepsilon _{1}+\lambda (\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\,,\end{aligned}}}$$

 

where μ and λ are the Lamé parameters.

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

$${\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &-\nu &0&0&0\\-\nu &1&-\nu &0&0&0\\-\nu &-\nu &1&0&0&0\\0&0&0&2+2\nu &0&0\\0&0&0&0&2+2\nu &0\\0&0&0&0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}}$$

 

where γ_ij = 2ε_ij is the engineering shear strain. The inverse relation may be written as

$${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &\nu &0&0&0\\\nu &1-\nu &\nu &0&0&0\\\nu &\nu &1-\nu &0&0&0\\0&0&0&{\frac {1-2\nu }{2}}&0&0\\0&0&0&0&{\frac {1-2\nu }{2}}&0\\0&0&0&0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}$$

 

which can be simplified thanks to the Lamé constants:

$${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}2\mu +\lambda &\lambda &\lambda &0&0&0\\\lambda &2\mu +\lambda &\lambda &0&0&0\\\lambda &\lambda &2\mu +\lambda &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}$$

 

In vector notation this becomes

$${\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{12}&\sigma _{22}&\sigma _{23}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}\,=\,2\mu {\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{12}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{13}&\varepsilon _{23}&\varepsilon _{33}\end{bmatrix}}+\lambda \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\right)}$$

 

where I is the identity tensor.

Plane stress

Under plane stress conditions, σ₃₁ = σ₁₃ = σ₃₂ = σ₂₃ = σ₃₃ = 0. In that case Hooke's law takes the form

$${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&{\frac {1-\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}$$

 

In vector notation this becomes

$${\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{12}&\sigma _{22}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}\left((1-\nu ){\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{12}&\varepsilon _{22}\end{bmatrix}}+\nu \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}\right)\right)}$$

 

The inverse relation is usually written in the reduced form

$${\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &0\\-\nu &1&0\\0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}}$$

 

Plane strain

Under plane strain conditions, ε₃₁ = ε₁₃ = ε₃₂ = ε₂₃ = ε₃₃ = 0. In this case Hooke's law takes the form

$${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &0\\\nu &1-\nu &0\\0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}$$

 

Anisotropic materials

The symmetry of the Cauchy stress tensor (σ_ij = σ_ji) and the generalized Hooke's laws (σ_ij = c_ijklε_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

$${\displaystyle \sigma _{ij}={\frac {\partial U}{\partial \varepsilon _{ij}}}\quad \implies \quad c_{ijkl}={\frac {\partial ^{2}U}{\partial \varepsilon _{ij}\partial \varepsilon _{kl}}}\,.}$$

 

The arbitrariness of the order of differentiation implies that 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 (e₁,e₂,e₃) as

$${\displaystyle [{\boldsymbol {\sigma }}]\,=\,{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,;\qquad [{\boldsymbol {\varepsilon }}]\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}$$

 

Then the stiffness tensor (c) can be expressed as

$${\displaystyle [{\mathsf {c}}]\,=\,{\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_{1131}&c_{1112}\\c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}}$$

 

and Hooke's law is written as

$${\displaystyle [{\boldsymbol {\sigma }}]=[{\mathsf {C}}][{\boldsymbol {\varepsilon }}]\qquad {\text{or}}\qquad \sigma _{i}=C_{ij}\varepsilon _{j}\,.}$$

 

Similarly the compliance tensor (s) can be written as

$${\displaystyle [{\mathsf {s}}]\,=\,{\begin{bmatrix}s_{1111}&s_{1122}&s_{1133}&2s_{1123}&2s_{1131}&2s_{1112}\\s_{2211}&s_{2222}&s_{2233}&2s_{2223}&2s_{2231}&2s_{2212}\\s_{3311}&s_{3322}&s_{3333}&2s_{3323}&2s_{3331}&2s_{3312}\\2s_{2311}&2s_{2322}&2s_{2333}&4s_{2323}&4s_{2331}&4s_{2312}\\2s_{3111}&2s_{3122}&2s_{3133}&4s_{3123}&4s_{3131}&4s_{3112}\\2s_{1211}&2s_{1222}&2s_{1233}&4s_{1223}&4s_{1231}&4s_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}}$$

 

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]

$${\displaystyle c_{pqrs}=l_{pi}l_{qj}l_{rk}l_{sl}c_{ijkl}}$$

 

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

$${\displaystyle [\mathbf {e} _{i}']=[L][\mathbf {e} _{i}]}$$

 

then

$${\displaystyle C_{ij}\varepsilon _{i}\varepsilon _{j}=C_{ij}'\varepsilon '_{i}\varepsilon '_{j}\,.}$$

 

In addition, if the material is symmetric with respect to the transformation [L] then

$${\displaystyle C_{ij}=C'_{ij}\quad \implies \quad C_{ij}(\varepsilon _{i}\varepsilon _{j}-\varepsilon '_{i}\varepsilon '_{j})=0\,.}$$

 

Orthotropic materials

Main article: Orthotropic material

Orthotropic materials have three orthogonal planes of symmetry. If the basis vectors (e₁,e₂,e₃) are normals to the planes of symmetry then the coordinate transformation relations imply that

$${\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}$$

 

The inverse of this relation is commonly written as^{[14][page needed]}

$${\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{zx}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}$$

 

where

• 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

$${\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&0\\0&0&{\frac {1}{G_{xy}}}\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,.}$$

 

The inverse relation is

$${\displaystyle {\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,=\,{\frac {1}{1-\nu _{xy}\nu _{yx}}}{\begin{bmatrix}E_{x}&\nu _{yx}E_{x}&0\\\nu _{xy}E_{y}&E_{y}&0\\0&0&G_{xy}(1-\nu _{xy}\nu _{yx})\end{bmatrix}}{\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,.}$$

 

The transposed form of the above stiffness matrix is also often used.

Transversely isotropic materials

A transversely isotropic material is symmetric with respect to a rotation about an axis of symmetry. For such a material, if e₃ is the axis of symmetry, Hooke's law can be expressed as

$${\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&{\frac {C_{11}-C_{12}}{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}$$

 

More frequently, the x ≡ e₁ axis is taken to be the axis of symmetry and the inverse Hooke's law is written as ^[15]

$${\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{xz}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}$$

 

Universal elastic anisotropy index

To grasp the degree of anisotropy of any class, a universal elastic anisotropy index (AU)^[16] 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

$${\displaystyle \delta W=\delta U}$$

 

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

$${\displaystyle \delta W=\delta W_{\mathrm {s} }+\delta W_{\mathrm {b} }}$$

 

where δW_s is the work done by surface forces while δW_b is the work done by body forces. If δu is a variation of the displacement field u in the body, then the two external work terms can be expressed as

$${\displaystyle \delta W_{\mathrm {s} }=\int _{\partial \Omega }\mathbf {t} \cdot \delta \mathbf {u} \,dS\,;\qquad \delta W_{\mathrm {b} }=\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV}$$

 

where t is the surface traction 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

$${\displaystyle \delta W=\delta U=\int _{\partial \Omega }(\mathbf {n} \cdot {\boldsymbol {\sigma }})\cdot \delta \mathbf {u} \,dS+\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV\,.}$$

 

Converting the surface integral into a volume integral via the divergence theorem gives

$${\displaystyle \delta U=\int _{\Omega }{\big (}\nabla \cdot ({\boldsymbol {\sigma }}\cdot \delta \mathbf {u} )+\mathbf {b} \cdot \delta \mathbf {u} {\big )}\,dV\,.}$$

 

Using the symmetry of the Cauchy stress and the identity

$${\displaystyle \nabla \cdot (\mathbf {a} \cdot \mathbf {b} )=(\nabla \cdot \mathbf {a} )\cdot \mathbf {b} +{\tfrac {1}{2}}\left(\mathbf {a} ^{\mathsf {T}}:\nabla \mathbf {b} +\mathbf {a} :(\nabla \mathbf {b} )^{\mathsf {T}}\right)}$$

 

we have the following

$${\displaystyle \delta U=\int _{\Omega }\left({\boldsymbol {\sigma }}:{\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)+\left(\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} \right)\cdot \delta \mathbf {u} \right)\,dV\,.}$$

 

From the definition of strain and from the equations of equilibrium we have

$${\displaystyle \delta {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)\,;\qquad \nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} =\mathbf {0} \,.}$$

 

Hence we can write

$${\displaystyle \delta U=\int _{\Omega }{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,dV}$$

 

and therefore the variation in the internal energy density is given by

$${\displaystyle \delta U_{0}={\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,.}$$

 

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, U₀ = U₀(ε) and the variation of the internal energy can be expressed as

$${\displaystyle \delta U_{0}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}:\delta {\boldsymbol {\varepsilon }}\,.}$$

 

Since the variation of strain is arbitrary, the stress–strain relation of an elastic material is given by

$${\displaystyle {\boldsymbol {\sigma }}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}\,.}$$

 

For a linear elastic material, the quantity ∂U₀/∂ε is a linear function of ε, and can therefore be expressed as

$${\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}}$$

 

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,

$${\displaystyle {\frac {\partial }{\partial {\boldsymbol {\varepsilon }}}}{\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})={\text{constant}}={\mathsf {c}}\,.}$$

 

In index notation

$${\displaystyle {\frac {\partial \sigma _{ij}}{\partial \varepsilon _{kl}}}={\text{constant}}=c_{ijkl}\,.}$$

 

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
• 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

1. The anagram was given in alphabetical order, ceiiinosssttuu, 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 978-0674463684.
2. See http://civil.lindahall.org/design.shtml, where one can find also an anagram for catenary.
3. Robert Hooke, De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies, London, 1678.
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. 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.
6. Belen'kii; Salaev (1988). "Deformation effects in layer crystals". Uspekhi Fizicheskikh Nauk. 155 (5): 89. doi:10.3367/UFNr.0155.198805c.0089.
7. 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.
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:2009JChPh.131q4112V. doi:10.1063/1.3259834. PMID 19895003.
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:2012PCCP...14.6787P. doi:10.1039/C2CP40290D. PMID 22461011.
10. Symon, Keith R. (1971). "Chapter 10". Mechanics. Reading, Massachusetts: Addison-Wesley. ISBN 9780201073928.
11. Simo, J. C.; Hughes, T. J. R. (1998). Computational Inelasticity. Springer. ISBN 9780387975207.
12. Milton, Graeme W. (2002). The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press. ISBN 9780521781251.
13. Slaughter, William S. (2001). The Linearized Theory of Elasticity. Birkhäuser. ISBN 978-0817641177.
14. Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. (1993). Advanced Mechanics of Materials (5th ed.). Wiley. ISBN 9780471600091.
15. Tan, S. C. (1994). Stress Concentrations in Laminated Composites. Lancaster, PA: Technomic Publishing Company. ISBN 9781566760775.
16. Ranganathan, S.I.; Ostoja-Starzewski, M. (2008). "Universal Elastic Anisotropy Index". Physical Review Letters. 101 (5): 055504–1–4. Bibcode:2008PhRvL.101e5504R. doi:10.1103/PhysRevLett.101.055504. PMID 18764407. S2CID 6668703.

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

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hooke, physics, empirical, which, states, that, force, needed, extend, compress, spring, some, distance, scales, linearly, with, respect, that, distance, that, where, constant, factor, characteristic, spring, stiffness, small, compared, total, possible, deform. 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 Fs 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 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 Hooke s equation holds to some extent in many other situations where an elastic body is deformed such as wind blowing on a tall building and a musician plucking a string of a guitar An elastic body or material for which this equation can be assumed is said to be linear elastic or Hookean Hooke s law is only a first order linear approximation to the real response of springs and other elastic bodies to applied forces It must eventually fail 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 On the other hand Hooke s law is an accurate approximation for most solid bodies as long as the forces and deformations are small enough For this reason Hooke s law is extensively used in all branches of science and engineering and is the foundation of many disciplines such as seismology molecular mechanics and acoustics It is also the fundamental principle behind the spring scale the manometer the galvanometer and the balance wheel of the mechanical clock 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 Contents 1 Formal definition 1 1 Linear springs 1 2 Torsional springs 1 3 General scalar springs 1 4 Vector formulation 1 5 General tensor form 1 6 Hooke s law for continuous media 2 Analogous laws 3 Units of measurement 4 General application to elastic materials 5 Derived formulae 5 1 Tensional stress of a uniform bar 5 2 Spring energy 5 3 Relaxed force constants generalized compliance constants 5 4 Harmonic oscillator 5 5 Rotation in gravity free space 6 Linear elasticity theory for continuous media 6 1 Isotropic materials 6 1 1 Plane stress 6 1 2 Plane strain 6 2 Anisotropic materials 6 2 1 Matrix representation stiffness tensor 6 2 2 Change of coordinate system 6 2 3 Orthotropic materials 6 2 4 Transversely isotropic materials 6 2 5 Universal elastic anisotropy index 7 Thermodynamic basis 8 See also 9 Notes 10 References 11 External linksFormal definitionLinear springs nbsp 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 Fs 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 thatF s k x displaystyle F s kx nbsp or equivalently x F s k displaystyle x frac F s k nbsp 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 nbsp Graphical derivation According to this formula the graph of the applied force Fs 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 Fs is the restoring force exerted by the spring on whatever is pulling its free end In that case the equation becomesF s k x displaystyle F s kx nbsp 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 t required to rotate an object is directly proportional to the angular displacement 8 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 t k 8 displaystyle tau k theta nbsp Where t 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 8 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 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 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 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 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 between the force and deformation vectors as long as they are small enough Namely there is a function k from vectors to vectors such that F k X and k aX1 bX2 ak X1 bk X2 for any real numbers a b and any displacement vectors X1 X2 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 k connecting them can be represented by a 3 3 matrix k of real coefficients that when multiplied by the displacement vector gives the force vector F F 1 F 2 F 3 k 11 k 12 k 13 k 21 k 22 k 23 k 31 k 32 k 33 X 1 X 2 X 3 k X displaystyle mathbf F begin bmatrix F 1 F 2 F 3 end bmatrix begin bmatrix kappa 11 amp kappa 12 amp kappa 13 kappa 21 amp kappa 22 amp kappa 23 kappa 31 amp kappa 32 amp kappa 33 end bmatrix begin bmatrix X 1 X 2 X 3 end bmatrix boldsymbol kappa mathbf X nbsp That is F i k i 1 X 1 k i 2 X 2 k i 3 X 3 displaystyle F i kappa i1 X 1 kappa i2 X 2 kappa i3 X 3 nbsp for i 1 2 3 Therefore Hooke s law F kX 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 k rather than a single real number k Hooke s law for continuous media Main article linear elasticity nbsp 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 mm 2 0 mm 13 mm 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 mm 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 e in lieu of the displacement X and the stress tensor s replacing the restoring force F The analogue of Hooke s spring law for continuous media is thens c e displaystyle boldsymbol sigma mathbf c boldsymbol varepsilon nbsp 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 e s s displaystyle boldsymbol varepsilon mathbf s boldsymbol sigma nbsp 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 matricese e 11 e 12 e 13 e 21 e 22 e 23 e 31 e 32 e 33 s s 11 s 12 s 13 s 21 s 22 s 23 s 31 s 32 s 33 displaystyle boldsymbol varepsilon begin bmatrix varepsilon 11 amp varepsilon 12 amp varepsilon 13 varepsilon 21 amp varepsilon 22 amp varepsilon 23 varepsilon 31 amp varepsilon 32 amp varepsilon 33 end bmatrix qquad boldsymbol sigma begin bmatrix sigma 11 amp sigma 12 amp sigma 13 sigma 21 amp sigma 22 amp sigma 23 sigma 31 amp sigma 32 amp sigma 33 end bmatrix nbsp Being a linear mapping between the nine numbers sij and the nine numbers ekl the stiffness tensor c is represented by a matrix of 3 3 3 3 81 real numbers cijkl Hooke s law then says that s i j k 1 3 l 1 3 c i j k l e k l displaystyle sigma ij sum k 1 3 sum l 1 3 c ijkl varepsilon kl nbsp 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 e merely specifies the displacement of the medium particles in the neighborhood of the point while the stress tensor s 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 s e 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 lawsSince 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 s ce relating elastic stresses to strains is entirely similar to the equation t me relating the viscous stress tensor t and the strain rate tensor e 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 measurementIn SI units displacements are measured in meters m and forces in newtons N or kg m s2 Therefore the spring constant k and each element of the tensor k is measured in newtons per meter N m or kilograms per second squared kg s2 For continuous media each element of the stress tensor s is a force divided by an area it is therefore measured in units of pressure namely pascals Pa or N m2 or kg m s2 The elements of the strain tensor e are dimensionless displacements divided by distances Therefore the entries of cijkl are also expressed in units of pressure General application to elastic materials nbsp Stress strain curve for low carbon steel showing the relationship between the stress force per unit area and strain 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 Ultimate strengthYield strength yield point RuptureStrain hardening regionNecking region Apparent stress F A0 Actual stress F A 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 formulaeTensional 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 s is linearly proportional to its fractional extension or strain e by the modulus of elasticity E s E e displaystyle sigma E varepsilon nbsp The modulus of elasticity may often be considered constant In turn e D L L displaystyle varepsilon frac Delta L L nbsp that is the fractional change in length and since s F A displaystyle sigma frac F A nbsp it follows that e s E F A E displaystyle varepsilon frac sigma E frac F AE nbsp The change in length may be expressed asD L e L F L A E displaystyle Delta L varepsilon L frac FL AE nbsp Spring energy The potential energy Uel x stored in a spring is given byU e l x 1 2 k x 2 displaystyle U mathrm el x tfrac 1 2 kx 2 nbsp 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 nbsp gives U e l F F 2 2 k displaystyle U mathrm el F frac F 2 2k nbsp This potential Uel can be visualized as a parabola on the Ux plane such that Uel x 1 2 kx2 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 nbsp 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 See also Harmonic oscillator nbsp 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 f 1 2 p k m displaystyle f frac 1 2 pi sqrt frac k m nbsp 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 Ft would supply the required centripetal force Fc F t k x F c m w 2 r displaystyle F mathrm t kx qquad F mathrm c m omega 2 r nbsp Since Ft Fc and x r then k m w 2 displaystyle k m omega 2 nbsp Given that w 2pf this leads to the same frequency equation as above f 1 2 p k m displaystyle f frac 1 2 pi sqrt frac k m nbsp Linear elasticity theory for continuous mediaSee also Elasticity tensor Note the Einstein summation convention of summing on repeated indices is used below 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 e i j 1 3 e k k d i j e i j 1 3 e k k d 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 nbsp where dij is the Kronecker delta In direct tensor notation e vol e dev e vol e 1 3 tr e I dev e e vol e 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 nbsp 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 s i j 3 K 1 3 e k k d i j 2 G e i j 1 3 e k k d i j s 3 K vol e 2 G dev e 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 nbsp 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 s l tr e I 2 m e c e c l I I 2 m 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 nbsp where l K 2 3 G c1111 2c1212 and m G c1212 are the Lame constants I is the second rank identity tensor and I is the symmetric part of the fourth rank identity tensor In index notation s i j l e k k d i j 2 m e i j c i j k l e k l c i j k l l d i j d k l m d i k d j l d i l d 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 nbsp The inverse relationship is 12 e 1 2 m s l 2 m 3 l 2 m tr s I 1 2 G s 1 9 K 1 6 G tr s 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 nbsp Therefore the compliance tensor in the relation e s s is s l 2 m 3 l 2 m I I 1 2 m 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 nbsp In terms of Young s modulus and Poisson s ratio Hooke s law for isotropic materials can then be expressed as e i j 1 E s i j n s k k d i j s i j e 1 E s n tr s I s 1 n E s n E tr s 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 nbsp This is the form in which the strain is expressed in terms of the stress tensor in engineering The expression in expanded form is e 11 1 E s 11 n s 22 s 33 e 22 1 E s 22 n s 11 s 33 e 33 1 E s 33 n s 11 s 22 e 12 1 2 G s 12 e 13 1 2 G s 13 e 23 1 2 G s 23 displaystyle begin aligned varepsilon 11 amp frac 1 E big sigma 11 nu sigma 22 sigma 33 big varepsilon 22 amp frac 1 E big sigma 22 nu sigma 11 sigma 33 big varepsilon 33 amp frac 1 E big sigma 33 nu sigma 11 sigma 22 big varepsilon 12 amp frac 1 2G sigma 12 qquad varepsilon 13 frac 1 2G sigma 13 qquad varepsilon 23 frac 1 2G sigma 23 end aligned nbsp where E is Young s modulus and n 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 e 1 1 E s 1 e 2 n E s 1 e 3 n E s 1 displaystyle begin aligned varepsilon 1 amp frac 1 E sigma 1 varepsilon 2 amp frac nu E sigma 1 varepsilon 3 amp frac nu E sigma 1 end aligned nbsp where n is Poisson s ratio and E is Young s modulus We get similar equations to the loads in directions 2 and 3 e 1 n E s 2 e 2 1 E s 2 e 3 n E s 2 displaystyle begin aligned varepsilon 1 amp frac nu E sigma 2 varepsilon 2 amp frac 1 E sigma 2 varepsilon 3 amp frac nu E sigma 2 end aligned nbsp and e 1 n E s 3 e 2 n E s 3 e 3 1 E s 3 displaystyle begin aligned varepsilon 1 amp frac nu E sigma 3 varepsilon 2 amp frac nu E sigma 3 varepsilon 3 amp frac 1 E sigma 3 end aligned nbsp Summing the three cases together ei ei ei ei we gete 1 1 E s 1 n s 2 s 3 e 2 1 E s 2 n s 1 s 3 e 3 1 E s 3 n s 1 s 2 displaystyle begin aligned varepsilon 1 amp frac 1 E big sigma 1 nu sigma 2 sigma 3 big varepsilon 2 amp frac 1 E big sigma 2 nu sigma 1 sigma 3 big varepsilon 3 amp frac 1 E big sigma 3 nu sigma 1 sigma 2 big end aligned nbsp or by adding and subtracting one ns e 1 1 E 1 n s 1 n s 1 s 2 s 3 e 2 1 E 1 n s 2 n s 1 s 2 s 3 e 3 1 E 1 n s 3 n s 1 s 2 s 3 displaystyle begin aligned varepsilon 1 amp frac 1 E big 1 nu sigma 1 nu sigma 1 sigma 2 sigma 3 big varepsilon 2 amp frac 1 E big 1 nu sigma 2 nu sigma 1 sigma 2 sigma 3 big varepsilon 3 amp frac 1 E big 1 nu sigma 3 nu sigma 1 sigma 2 sigma 3 big end aligned nbsp and further we get by solving s1 s 1 E 1 n e 1 n 1 n s 1 s 2 s 3 displaystyle sigma 1 frac E 1 nu varepsilon 1 frac nu 1 nu sigma 1 sigma 2 sigma 3 nbsp Calculating the sume 1 e 2 e 3 1 E 1 n s 1 s 2 s 3 3 n s 1 s 2 s 3 1 2 n E s 1 s 2 s 3 s 1 s 2 s 3 E 1 2 n e 1 e 2 e 3 displaystyle begin aligned varepsilon 1 varepsilon 2 varepsilon 3 amp 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 amp frac E 1 2 nu varepsilon 1 varepsilon 2 varepsilon 3 end aligned nbsp and substituting it to the equation solved for s1 gives s 1 E 1 n e 1 E n 1 n 1 2 n e 1 e 2 e 3 2 m e 1 l e 1 e 2 e 3 displaystyle begin aligned sigma 1 amp frac E 1 nu varepsilon 1 frac E nu 1 nu 1 2 nu varepsilon 1 varepsilon 2 varepsilon 3 amp 2 mu varepsilon 1 lambda varepsilon 1 varepsilon 2 varepsilon 3 end aligned nbsp where m and l are the Lame 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 e 11 e 22 e 33 2 e 23 2 e 13 2 e 12 e 11 e 22 e 33 g 23 g 13 g 12 1 E 1 n n 0 0 0 n 1 n 0 0 0 n n 1 0 0 0 0 0 0 2 2 n 0 0 0 0 0 0 2 2 n 0 0 0 0 0 0 2 2 n s 11 s 22 s 33 s 23 s 13 s 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 amp nu amp nu amp 0 amp 0 amp 0 nu amp 1 amp nu amp 0 amp 0 amp 0 nu amp nu amp 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 2 2 nu amp 0 amp 0 0 amp 0 amp 0 amp 0 amp 2 2 nu amp 0 0 amp 0 amp 0 amp 0 amp 0 amp 2 2 nu end bmatrix begin bmatrix sigma 11 sigma 22 sigma 33 sigma 23 sigma 13 sigma 12 end bmatrix nbsp where gij 2eij is the engineering shear strain The inverse relation may be written as s 11 s 22 s 33 s 23 s 13 s 12 E 1 n 1 2 n 1 n n n 0 0 0 n 1 n n 0 0 0 n n 1 n 0 0 0 0 0 0 1 2 n 2 0 0 0 0 0 0 1 2 n 2 0 0 0 0 0 0 1 2 n 2 e 11 e 22 e 33 2 e 23 2 e 13 2 e 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 amp nu amp nu amp 0 amp 0 amp 0 nu amp 1 nu amp nu amp 0 amp 0 amp 0 nu amp nu amp 1 nu amp 0 amp 0 amp 0 0 amp 0 amp 0 amp frac 1 2 nu 2 amp 0 amp 0 0 amp 0 amp 0 amp 0 amp frac 1 2 nu 2 amp 0 0 amp 0 amp 0 amp 0 amp 0 amp 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 nbsp which can be simplified thanks to the Lame constants s 11 s 22 s 33 s 23 s 13 s 12 2 m l l l 0 0 0 l 2 m l l 0 0 0 l l 2 m l 0 0 0 0 0 0 m 0 0 0 0 0 0 m 0 0 0 0 0 0 m e 11 e 22 e 33 2 e 23 2 e 13 2 e 12 displaystyle begin bmatrix sigma 11 sigma 22 sigma 33 sigma 23 sigma 13 sigma 12 end bmatrix begin bmatrix 2 mu lambda amp lambda amp lambda amp 0 amp 0 amp 0 lambda amp 2 mu lambda amp lambda amp 0 amp 0 amp 0 lambda amp lambda amp 2 mu lambda amp 0 amp 0 amp 0 0 amp 0 amp 0 amp mu amp 0 amp 0 0 amp 0 amp 0 amp 0 amp mu amp 0 0 amp 0 amp 0 amp 0 amp 0 amp mu end bmatrix begin bmatrix varepsilon 11 varepsilon 22 varepsilon 33 2 varepsilon 23 2 varepsilon 13 2 varepsilon 12 end bmatrix nbsp In vector notation this becomes s 11 s 12 s 13 s 12 s 22 s 23 s 13 s 23 s 33 2 m e 11 e 12 e 13 e 12 e 22 e 23 e 13 e 23 e 33 l I e 11 e 22 e 33 displaystyle begin bmatrix sigma 11 amp sigma 12 amp sigma 13 sigma 12 amp sigma 22 amp sigma 23 sigma 13 amp sigma 23 amp sigma 33 end bmatrix 2 mu begin bmatrix varepsilon 11 amp varepsilon 12 amp varepsilon 13 varepsilon 12 amp varepsilon 22 amp varepsilon 23 varepsilon 13 amp varepsilon 23 amp varepsilon 33 end bmatrix lambda mathbf I left varepsilon 11 varepsilon 22 varepsilon 33 right nbsp where I is the identity tensor Plane stress Under plane stress conditions s31 s13 s32 s23 s33 0 In that case Hooke s law takes the form s 11 s 22 s 12 E 1 n 2 1 n 0 n 1 0 0 0 1 n 2 e 11 e 22 2 e 12 displaystyle begin bmatrix sigma 11 sigma 22 sigma 12 end bmatrix frac E 1 nu 2 begin bmatrix 1 amp nu amp 0 nu amp 1 amp 0 0 amp 0 amp frac 1 nu 2 end bmatrix begin bmatrix varepsilon 11 varepsilon 22 2 varepsilon 12 end bmatrix nbsp In vector notation this becomes s 11 s 12 s 12 s 22 E 1 n 2 1 n e 11 e 12 e 12 e 22 n I e 11 e 22 displaystyle begin bmatrix sigma 11 amp sigma 12 sigma 12 amp sigma 22 end bmatrix frac E 1 nu 2 left 1 nu begin bmatrix varepsilon 11 amp varepsilon 12 varepsilon 12 amp varepsilon 22 end bmatrix nu mathbf I left varepsilon 11 varepsilon 22 right right nbsp The inverse relation is usually written in the reduced form e 11 e 22 2 e 12 1 E 1 n 0 n 1 0 0 0 2 2 n s 11 s 22 s 12 displaystyle begin bmatrix varepsilon 11 varepsilon 22 2 varepsilon 12 end bmatrix frac 1 E begin bmatrix 1 amp nu amp 0 nu amp 1 amp 0 0 amp 0 amp 2 2 nu end bmatrix begin bmatrix sigma 11 sigma 22 sigma 12 end bmatrix nbsp Plane strain Under plane strain conditions e31 e13 e32 e23 e33 0 In this case Hooke s law takes the form s 11 s 22 s 12 E 1 n 1 2 n 1 n n 0 n 1 n 0 0 0 1 2 n 2 e 11 e 22 2 e 12 displaystyle begin bmatrix sigma 11 sigma 22 sigma 12 end bmatrix frac E 1 nu 1 2 nu begin bmatrix 1 nu amp nu amp 0 nu amp 1 nu amp 0 0 amp 0 amp frac 1 2 nu 2 end bmatrix begin bmatrix varepsilon 11 varepsilon 22 2 varepsilon 12 end bmatrix nbsp Anisotropic materials The symmetry of the Cauchy stress tensor sij sji and the generalized Hooke s laws sij cijklekl implies that cijkl cjikl Similarly the symmetry of the infinitesimal strain tensor 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 thens i j U e i j c i j k l 2 U e i j e 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 nbsp 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 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 e1 e2 e3 as s s 11 s 22 s 33 s 23 s 13 s 12 s 1 s 2 s 3 s 4 s 5 s 6 e e 11 e 22 e 33 2 e 23 2 e 13 2 e 12 e 1 e 2 e 3 e 4 e 5 e 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 nbsp 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 amp c 1122 amp c 1133 amp c 1123 amp c 1131 amp c 1112 c 2211 amp c 2222 amp c 2233 amp c 2223 amp c 2231 amp c 2212 c 3311 amp c 3322 amp c 3333 amp c 3323 amp c 3331 amp c 3312 c 2311 amp c 2322 amp c 2333 amp c 2323 amp c 2331 amp c 2312 c 3111 amp c 3122 amp c 3133 amp c 3123 amp c 3131 amp c 3112 c 1211 amp c 1222 amp c 1233 amp c 1223 amp c 1231 amp c 1212 end bmatrix equiv begin bmatrix C 11 amp C 12 amp C 13 amp C 14 amp C 15 amp C 16 C 12 amp C 22 amp C 23 amp C 24 amp C 25 amp C 26 C 13 amp C 23 amp C 33 amp C 34 amp C 35 amp C 36 C 14 amp C 24 amp C 34 amp C 44 amp C 45 amp C 46 C 15 amp C 25 amp C 35 amp C 45 amp C 55 amp C 56 C 16 amp C 26 amp C 36 amp C 46 amp C 56 amp C 66 end bmatrix nbsp and Hooke s law is written as s C e or s i C i j e j displaystyle boldsymbol sigma mathsf C boldsymbol varepsilon qquad text or qquad sigma i C ij varepsilon j nbsp 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 amp s 1122 amp s 1133 amp 2s 1123 amp 2s 1131 amp 2s 1112 s 2211 amp s 2222 amp s 2233 amp 2s 2223 amp 2s 2231 amp 2s 2212 s 3311 amp s 3322 amp s 3333 amp 2s 3323 amp 2s 3331 amp 2s 3312 2s 2311 amp 2s 2322 amp 2s 2333 amp 4s 2323 amp 4s 2331 amp 4s 2312 2s 3111 amp 2s 3122 amp 2s 3133 amp 4s 3123 amp 4s 3131 amp 4s 3112 2s 1211 amp 2s 1222 amp 2s 1233 amp 4s 1223 amp 4s 1231 amp 4s 1212 end bmatrix equiv begin bmatrix S 11 amp S 12 amp S 13 amp S 14 amp S 15 amp S 16 S 12 amp S 22 amp S 23 amp S 24 amp S 25 amp S 26 S 13 amp S 23 amp S 33 amp S 34 amp S 35 amp S 36 S 14 amp S 24 amp S 34 amp S 44 amp S 45 amp S 46 S 15 amp S 25 amp S 35 amp S 45 amp S 55 amp S 56 S 16 amp S 26 amp S 36 amp S 46 amp S 56 amp S 66 end bmatrix nbsp 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 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 nbsp where lab are the components of an orthogonal rotation matrix L The same relation also holds for inversions In matrix notation if the transformed basis rotated or inverted is related to the reference basis by e i L e i displaystyle mathbf e i L mathbf e i nbsp then C i j e i e j C i j e i e j displaystyle C ij varepsilon i varepsilon j C ij varepsilon i varepsilon j nbsp In addition if the material is symmetric with respect to the transformation L then C i j C i j C i j e i e j e i e j 0 displaystyle C ij C ij quad implies quad C ij varepsilon i varepsilon j varepsilon i varepsilon j 0 nbsp Orthotropic materials Main article Orthotropic material Orthotropic materials have three orthogonal planes of symmetry If the basis vectors e1 e2 e3 are normals to the planes of symmetry then the coordinate transformation relations imply that s 1 s 2 s 3 s 4 s 5 s 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 e 1 e 2 e 3 e 4 e 5 e 6 displaystyle begin bmatrix sigma 1 sigma 2 sigma 3 sigma 4 sigma 5 sigma 6 end bmatrix begin bmatrix C 11 amp C 12 amp C 13 amp 0 amp 0 amp 0 C 12 amp C 22 amp C 23 amp 0 amp 0 amp 0 C 13 amp C 23 amp C 33 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp C 44 amp 0 amp 0 0 amp 0 amp 0 amp 0 amp C 55 amp 0 0 amp 0 amp 0 amp 0 amp 0 amp C 66 end bmatrix begin bmatrix varepsilon 1 varepsilon 2 varepsilon 3 varepsilon 4 varepsilon 5 varepsilon 6 end bmatrix nbsp The inverse of this relation is commonly written as 14 page needed e x x e y y e z z 2 e y z 2 e z x 2 e x y 1 E x n y x E y n z x E z 0 0 0 n x y E x 1 E y n z y E z 0 0 0 n x z E x n 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 s x x s y y s z z s y z s z x s 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 amp frac nu yx E y amp frac nu zx E z amp 0 amp 0 amp 0 frac nu xy E x amp frac 1 E y amp frac nu zy E z amp 0 amp 0 amp 0 frac nu xz E x amp frac nu yz E y amp frac 1 E z amp 0 amp 0 amp 0 0 amp 0 amp 0 amp frac 1 G yz amp 0 amp 0 0 amp 0 amp 0 amp 0 amp frac 1 G zx amp 0 0 amp 0 amp 0 amp 0 amp 0 amp frac 1 G xy end bmatrix begin bmatrix sigma xx sigma yy sigma zz sigma yz sigma zx sigma xy end bmatrix nbsp where Ei is the Young s modulus along axis i Gij is the shear modulus in direction j on the plane whose normal is in direction i nij 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 szz szx syz 0 Hooke s law for an orthotropic material takes the form e x x e y y 2 e x y 1 E x n y x E y 0 n x y E x 1 E y 0 0 0 1 G x y s x x s y y s x y displaystyle begin bmatrix varepsilon xx varepsilon yy 2 varepsilon xy end bmatrix begin bmatrix frac 1 E x amp frac nu yx E y amp 0 frac nu xy E x amp frac 1 E y amp 0 0 amp 0 amp frac 1 G xy end bmatrix begin bmatrix sigma xx sigma yy sigma xy end bmatrix nbsp The inverse relation is s x x s y y s x y 1 1 n x y n y x E x n y x E x 0 n x y E y E y 0 0 0 G x y 1 n x y n y x e x x e y y 2 e x y displaystyle begin bmatrix sigma xx sigma yy sigma xy end bmatrix frac 1 1 nu xy nu yx begin bmatrix E x amp nu yx E x amp 0 nu xy E y amp E y amp 0 0 amp 0 amp G xy 1 nu xy nu yx end bmatrix begin bmatrix varepsilon xx varepsilon yy 2 varepsilon xy end bmatrix nbsp 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 e3 is the axis of symmetry Hooke s law can be expressed as s 1 s 2 s 3 s 4 s 5 s 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 e 1 e 2 e 3 e 4 e 5 e 6 displaystyle begin bmatrix sigma 1 sigma 2 sigma 3 sigma 4 sigma 5 sigma 6 end bmatrix begin bmatrix C 11 amp C 12 amp C 13 amp 0 amp 0 amp 0 C 12 amp C 11 amp C 13 amp 0 amp 0 amp 0 C 13 amp C 13 amp C 33 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp C 44 amp 0 amp 0 0 amp 0 amp 0 amp 0 amp C 44 amp 0 0 amp 0 amp 0 amp 0 amp 0 amp frac C 11 C 12 2 end bmatrix begin bmatrix varepsilon 1 varepsilon 2 varepsilon 3 varepsilon 4 varepsilon 5 varepsilon 6 end bmatrix nbsp More frequently the x e1 axis is taken to be the axis of symmetry and the inverse Hooke s law is written as 15 e x x e y y e z z 2 e y z 2 e z x 2 e x y 1 E x n y x E y n z x E z 0 0 0 n x y E x 1 E y n z y E z 0 0 0 n x z E x n 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 s x x s y y s z z s y z s z x s 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 amp frac nu yx E y amp frac nu zx E z amp 0 amp 0 amp 0 frac nu xy E x amp frac 1 E y amp frac nu zy E z amp 0 amp 0 amp 0 frac nu xz E x amp frac nu yz E y amp frac 1 E z amp 0 amp 0 amp 0 0 amp 0 amp 0 amp frac 1 G yz amp 0 amp 0 0 amp 0 amp 0 amp 0 amp frac 1 G xz amp 0 0 amp 0 amp 0 amp 0 amp 0 amp frac 1 G xy end bmatrix begin bmatrix sigma xx sigma yy sigma zz sigma yz sigma zx sigma xy end bmatrix nbsp 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 basisLinear 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 asd W d U displaystyle delta W delta U nbsp where dU is the increase in internal energy and dW is the work done by external forces The work can be split into two terms d W d W s d W b displaystyle delta W delta W mathrm s delta W mathrm b nbsp where dWs is the work done by surface forces while dWb is the work done by body forces If du is a variation of the displacement field u in the body then the two external work terms can be expressed as d W s W t d u d S d W b W b d 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 nbsp where t is the surface traction vector b is the body force vector W represents the body and W represents its surface Using the relation between the Cauchy stress and the surface traction t n s where n is the unit outward normal to W we have d W d U W n s d u d S W b d 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 nbsp Converting the surface integral into a volume integral via the divergence theorem gives d U W s d u b d 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 nbsp 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 nbsp we have the following d U W s 1 2 d u d u T s b d 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 nbsp From the definition of strain and from the equations of equilibrium we have d e 1 2 d u d u T s 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 nbsp Hence we can write d U W s d e d V displaystyle delta U int Omega boldsymbol sigma delta boldsymbol varepsilon dV nbsp and therefore the variation in the internal energy density is given by d U 0 s d e displaystyle delta U 0 boldsymbol sigma delta boldsymbol varepsilon nbsp 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 U0 U0 e and the variation of the internal energy can be expressed as d U 0 U 0 e d e displaystyle delta U 0 frac partial U 0 partial boldsymbol varepsilon delta boldsymbol varepsilon nbsp Since the variation of strain is arbitrary the stress strain relation of an elastic material is given by s U 0 e displaystyle boldsymbol sigma frac partial U 0 partial boldsymbol varepsilon nbsp For a linear elastic material the quantity U0 e is a linear function of e and can therefore be expressed as s c e displaystyle boldsymbol sigma mathsf c boldsymbol varepsilon nbsp 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 e s e constant c displaystyle frac partial partial boldsymbol varepsilon boldsymbol sigma boldsymbol varepsilon text constant mathsf c nbsp In index notation s i j e k l constant c i j k l displaystyle frac partial sigma ij partial varepsilon kl text constant c ijkl nbsp 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 alsoAcoustoelastic 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 pendulumNotes The anagram was given in alphabetical order ceiiinosssttuu 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 0674463684 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 Felix Coudert Francois 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 14 19 6787 6795 Bibcode 2012PCCP 14 6787P doi 10 1039 C2CP40290D PMID 22461011 Symon Keith R 1971 Chapter 10 Mechanics Reading Massachusetts Addison Wesley ISBN 9780201073928 Simo J C Hughes T J R 1998 Computational Inelasticity Springer ISBN 9780387975207 Milton Graeme W 2002 The Theory of Composites Cambridge Monographs on Applied and Computational Mathematics Cambridge University Press ISBN 9780521781251 Slaughter William S 2001 The Linearized Theory of Elasticity Birkhauser ISBN 978 0817641177 Boresi A P Schmidt R J Sidebottom O M 1993 Advanced Mechanics of Materials 5th ed Wiley ISBN 9780471600091 Tan S C 1994 Stress Concentrations in Laminated Composites Lancaster PA Technomic Publishing Company ISBN 9781566760775 Ranganathan S I Ostoja Starzewski M 2008 Universal Elastic Anisotropy Index Physical Review Letters 101 5 055504 1 4 Bibcode 2008PhRvL 101e5504R doi 10 1103 PhysRevLett 101 055504 PMID 18764407 S2CID 6668703 ReferencesThis article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations July 2017 Learn how and when to remove this message Hooke s law The Feynman Lectures on Physics Hooke s Law Classical Mechanics Physics MIT OpenCourseWareExternal linksJavaScript Applet demonstrating Springs and Hooke s law JavaScript Applet demonstrating Spring Force Conversion formulae Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli 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 K displaystyle K nbsp E displaystyle E nbsp l displaystyle lambda nbsp G displaystyle G nbsp n displaystyle nu nbsp M displaystyle M nbsp Notes K E displaystyle K E nbsp 3 K 3 K E 9 K E displaystyle tfrac 3K 3K E 9K E nbsp 3 K E 9 K E displaystyle tfrac 3KE 9K E nbsp 3 K E 6 K displaystyle tfrac 3K E 6K nbsp 3 K 3 K E 9 K E displaystyle tfrac 3K 3K E 9K E nbsp K l displaystyle K lambda nbsp 9 K K l 3 K l displaystyle tfrac 9K K lambda 3K lambda nbsp 3 K l 2 displaystyle tfrac 3 K lambda 2 nbsp l 3 K l displaystyle tfrac lambda 3K lambda nbsp 3 K 2 l displaystyle 3K 2 lambda nbsp K G displaystyle K G nbsp 9 K G 3 K G displaystyle tfrac 9KG 3K G nbsp K 2 G 3 displaystyle K tfrac 2G 3 nbsp 3 K 2 G 2 3 K G displaystyle tfrac 3K 2G 2 3K G nbsp K 4 G 3 displaystyle K tfrac 4G 3 nbsp K n displaystyle K nu nbsp 3 K 1 2 n displaystyle 3K 1 2 nu nbsp 3 K n 1 n displaystyle tfrac 3K nu 1 nu nbsp 3 K 1 2 n 2 1 n displaystyle tfrac 3K 1 2 nu 2 1 nu nbsp 3 K 1 n 1 n displaystyle tfrac 3K 1 nu 1 nu nbsp K M displaystyle K M nbsp 9 K M K 3 K M displaystyle tfrac 9K M K 3K M nbsp 3 K M 2 displaystyle tfrac 3K M 2 nbsp 3 M K 4 displaystyle tfrac 3 M K 4 nbsp 3 K M 3 K M displaystyle tfrac 3K M 3K M nbsp td, wikipedia, wiki, book, books, library,

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Conversion formulae
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli 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	${\displaystyle K=\,}$	${\displaystyle E=\,}$	${\displaystyle \lambda =\,}$	${\displaystyle G=\,}$	${\displaystyle \nu =\,}$	${\displaystyle M=\,}$	Notes
${\displaystyle (K,\,E)}$			${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$	${\displaystyle {\tfrac {3KE}{9K-E}}}$	${\displaystyle {\tfrac {3K-E}{6K}}}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$
${\displaystyle (K,\,\lambda )}$		${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$		${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$	${\displaystyle 3K-2\lambda \,}$
${\displaystyle (K,\,G)}$		${\displaystyle {\tfrac {9KG}{3K+G}}}$	${\displaystyle K-{\tfrac {2G}{3}}}$		${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$	${\displaystyle K+{\tfrac {4G}{3}}}$
${\displaystyle (K,\,\nu )}$		${\displaystyle 3K(1-2\nu )\,}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$		${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$
${\displaystyle (K,\,M)}$		${\displaystyle {\tfrac {9K(M-K)}{3K+M}}}$	${\displaystyle {\tfrac {3K-M}{2}}}$	${\displaystyle {\tfrac {3(M-K)}{4}}}$	${\displaystyle {\tfrac {3K-M}{3K+M}}}$