text:
Hooke's law
From Wikipedia, the free encyclopedia
Jump to: navigation, search
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 law is a principle of physics that 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 = kX, where k is a constant factor characteristic of the spring: 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 already in 1660.
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, a musician plucking a string of a guitar, and the filling of a party balloon. 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, 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 it is 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
[hide]
1 Formal definition
1.1 For linear springs
1.2 General "scalar" springs
1.3 Vector formulation
1.4 General tensor form
1.5 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 stiffness 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
6.2 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 Homages
9 See also
10 Notes
11 References
12 External links
Formal definition[edit]
For linear springs[edit]
Plot of applied force F vs. elongation X for a helical spring according to Hooke's law (red line) and what the actual plot might look like (dashed line). At bottom, pictures of spring states corresponding to some points of the plot; the middle one is in the relaxed state (no force applied).
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. 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
F = k X {\displaystyle F=kX}
or, equivalently,
X = F k {\displaystyle X={\frac {F}{k}}}
where k is a positive real number, characteristic of the spring. Moreover, the same formula holds when the spring is compressed, with F and X both negative in that case. According to this formula, the graph of the applied force F 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 often stated under the convention that F is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes
F = − k X {\displaystyle F=-kX}
since the direction of the restoring force is opposite to that of the displacement.
General "scalar" springs[edit]
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 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, 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.
The law also applies when a stretched steel wire is twisted by pulling on a lever attached to one end. In this case the stress F can be taken as the force applied to the lever, and X as the distance traveled by it along its circular path. Or, equivalently, one can let F be the torque applied by the lever to the end of the wire, and X be the angle by which that end turns. In either case F is proportional to X (although the constant k is different in each case.)
Vector formulation[edit]
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 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[edit]
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, 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 = 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.
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:
F = [ F 1 F 2 F 3 ] = [ κ 11 κ 12 κ 13 κ 21 κ 22 κ 23 κ 31 κ 32 κ 33 ] [ X 1 X 2 X 3 ] = κ X {\displaystyle \mathbf {F} \,=\,{\begin{bmatrix}F_{1}\\F_{2}\\F_{3}\end{bmatrix}}\,=\,{\begin{bmatrix}\kappa _{11}&\kappa _{12}&\kappa _{13}\\\kappa _{21}&\kappa _{22}&\kappa _{23}\\\kappa _{31}&\kappa _{32}&\kappa _{33}\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}\,=\,{\boldsymbol {\kappa }}\mathbf {X} }
That is,
F i = κ i 1 X 1 + κ i 2 X 2 + κ i 3 X 3 {\displaystyle F_{i}=\kappa _{i1}X_{1}+\kappa _{i2}X_{2}+\kappa _{i3}X_{3}}
for i = 1, 2, 3. Therefore, Hooke's law F = 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[edit]
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.[4]
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
σ = − c ε , {\displaystyle {\boldsymbol {\sigma }}=-\mathbf {c} {\boldsymbol {\varepsilon }},}
where c is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the stiffness tensor or elasticity tensor. One may also write it as
ε = − s σ , {\displaystyle {\boldsymbol {\varepsilon }}=-\mathbf {s} {\boldsymbol {\sigma }},}
where the tensor s, called the compliance tensor, represents the inverse of said linear map.
In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices
ε = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] ; σ = [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] {\displaystyle {\boldsymbol {\varepsilon }}\,=\,{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\,;\qquad {\boldsymbol {\sigma }}\,=\,{\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}
Being a linear mapping between the nine numbers σij and the nine numbers εkl, the stiffness tensor c is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers cijkl. Hooke's law then says that
σ i j = − ∑ k = 1 3 ∑ l = 1 3 c i j k l ε k l {\displaystyle \sigma _{ij}=-\sum _{k=1}^{3}\sum _{l=1}^{3}c_{ijkl}\varepsilon _{kl}}
where i,j = 1,2,3.
All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor ε merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor σ specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor c, on the other hand, is a property of the material, and often depends on physical state variables such as temperature, pressure, and microstructure.
Due to the inherent symmetries of σ, ε, and c, only 21 elastic coefficients of the latter are independent.[5] 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[edit]
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[edit]
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 cijkl are also expressed in units of pressure.
General application to elastic materials[edit]
Stress–strain curve for low-carbon steel. Hooke's law is only valid for the portion of the curve between the origin and the yield point (2).
1: Ultimate strength
2: Yield strength (yield point)
3: Rupture
4: Strain hardening region
5: Necking region
A: Apparent stress (F/A0)
B: 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[edit]
Tensional stiffness of a uniform bar[edit]
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:
σ = E ε {\displaystyle \sigma =E\varepsilon } .
The modulus of elasticity may often be considered constant. In turn,
ε = Δ L L {\displaystyle \varepsilon ={\frac {\Delta L}{L}}}
(that is, the fractional change in length), and since
σ = F A , {\displaystyle \sigma ={\frac {F}{A}}\,,}
it follows that:
ε = σ E = F A E . {\displaystyle \varepsilon ={\frac {\sigma }{E}}={\frac {F}{AE}}\,.}
The change in length may be expressed as
Δ L = ε L = F L A E . {\displaystyle \Delta L=\varepsilon L={\frac {FL}{AE}}\,.}
Spring energy[edit]
The potential energy Uel(x) stored in a spring is given by
U e l ( x ) = 1 2 k x 2 {\displaystyle U_{\mathrm {el} }(x)={\tfrac {1}{2}}kx^{2}}
which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative.
This potential Uel can be visualized as a parabola on the Ux-plane such that Uel(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:
d 2 U e l d x 2 = k . {\displaystyle {\frac {d^{2}U_{\mathrm {el} }}{dx^{2}}}=k\,.}
Note that the change in the change in U is constant even when the displacement and acceleration are zero.
Relaxed force constants (generalized compliance constants)[edit]
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.[6] 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.[7]
Harmonic oscillator[edit]
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 sinusoid 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 π k m {\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[edit]
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 ω 2 r {\displaystyle F_{\mathrm {t} }=kx\,;\qquad F_{\mathrm {c} }=m\omega ^{2}r}
Since Ft = Fc and x = r, then:
k = m ω 2 {\displaystyle k=m\omega ^{2}}
Given that ω = 2πf, this leads to the same frequency equation as above:
f = 1 2 π k m {\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}
Linear elasticity theory for continuous media[edit]
Note: the Einstein summation convention of summing on repeated indices is used below.
Isotropic materials[edit]
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.[8]:Ch. 10 Thus in index notation:
ε i j = ( 1 3 ε k k δ i j ) + ( ε i j − 1 3 ε k k δ i j ) {\displaystyle \varepsilon _{ij}=\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)}
where δij is the Kronecker delta. In direct tensor notation:
ε = vol  ( ε ) + dev  ( ε ) ; vol  ( ε ) = 1 3 tr  ( ε ) I ; dev  ( ε ) = ε − vol  ( ε ) {\displaystyle {\boldsymbol {\varepsilon }}=\operatorname {vol} ({\boldsymbol {\varepsilon }})+\operatorname {dev} ({\boldsymbol {\varepsilon }})\,;\qquad \operatorname {vol} ({\boldsymbol {\varepsilon }})={\tfrac {1}{3}}\operatorname {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} \,;\qquad \operatorname {dev} ({\boldsymbol {\varepsilon }})={\boldsymbol {\varepsilon }}-\operatorname {vol} ({\boldsymbol {\varepsilon }})}
where I is the second-order identity tensor.
The first term on the right is the constant tensor, also known as the volumetric strain tensor, 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:
σ i j = 3 K ( 1 3 ε k k δ i j ) + 2 G ( ε i j − 1 3 ε k k δ i j ) ; σ = 3 K vol  ( ε ) + 2 G dev  ( ε ) {\displaystyle \sigma _{ij}=3K\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+2G\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)\,;\qquad {\boldsymbol {\sigma }}=3K\operatorname {vol} ({\boldsymbol {\varepsilon }})+2G\operatorname {dev} ({\boldsymbol {\varepsilon }})}
where K is the bulk modulus 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 [9]
σ = λ tr  ( ε ) I + 2 μ ε = c : ε ; c = λ I ⊗ I + 2 μ I {\displaystyle {\boldsymbol {\sigma }}=\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}\,;\qquad {\mathsf {c}}=\lambda \mathbf {I} \otimes \mathbf {I} +2\mu {\mathsf {I}}}
where λ = K − 2/3G and μ = G 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:
σ i j = λ ε k k δ i j + 2 μ ε i j = c i j k l ε k l ; c i j k l = λ δ i j δ k l + μ ( δ i k δ j l + δ i l δ j k ) {\displaystyle \sigma _{ij}=\lambda \varepsilon _{kk}~\delta _{ij}+2\mu \varepsilon _{ij}=c_{ijkl}\varepsilon _{kl}\,;\qquad c_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu \left(\delta _{ik}\delta _{jl}+\delta _{il}~\delta _{jk}\right)}
The inverse relationship is[10]
ε = 1 2 μ σ − λ 2 μ ( 3 λ + 2 μ ) tr  ( σ ) I = 1 2 G σ + ( 1 9 K − 1 6 G ) tr  ( σ ) I {\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2\mu }}{\boldsymbol {\sigma }}-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} ={\frac {1}{2G}}{\boldsymbol {\sigma }}+\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }
Therefore, the compliance tensor in the relation ε = s : σ is
s = − λ 2 μ ( 3 λ + 2 μ ) I ⊗ I + 1 2 μ I = ( 1 9 K − 1 6 G ) I ⊗ I + 1 2 G I {\displaystyle {\mathsf {s}}=-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2\mu }}{\mathsf {I}}=\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2G}}{\mathsf {I}}}
In terms of Young's modulus and Poisson's ratio, Hooke's law for isotropic materials can then be expressed as
ε i j = 1 E ( σ i j − ν ( σ k k δ i j − σ i j ) ) ; ε = 1 E ( σ − ν ( tr  ( σ ) I − σ ) ) = 1 + ν E σ − ν E tr  ( σ ) I {\displaystyle \varepsilon _{ij}={\frac {1}{E}}{\big (}\sigma _{ij}-\nu (\sigma _{kk}\delta _{ij}-\sigma _{ij}){\big )}\,;\qquad {\boldsymbol {\varepsilon }}={\frac {1}{E}}{\big (}{\boldsymbol {\sigma }}-\nu (\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} -{\boldsymbol {\sigma }}){\big )}={\frac {1+\nu }{E}}{\boldsymbol {\sigma }}-{\frac {\nu }{E}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }
This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is
ε 11 = 1 E ( σ 11 − ν ( σ 22 + σ 33 ) ) ε 22 = 1 E ( σ 22 − ν ( σ 11 + σ 33 ) ) ε 33 = 1 E ( σ 33 − ν ( σ 11 + σ 22 ) ) ε 12 = 1 2 G σ 12 ; ε 13 = 1 2 G σ 13 ; ε 23 = 1 2 G σ 23 {\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{E}}{\big (}\sigma _{11}-\nu (\sigma _{22}+\sigma _{33}){\big )}\\\varepsilon _{22}&={\frac {1}{E}}{\big (}\sigma _{22}-\nu (\sigma _{11}+\sigma _{33}){\big )}\\\varepsilon _{33}&={\frac {1}{E}}{\big (}\sigma _{33}-\nu (\sigma _{11}+\sigma _{22}){\big )}\\\varepsilon _{12}&={\frac {1}{2G}}\sigma _{12}\,;\qquad \varepsilon _{13}={\frac {1}{2G}}\sigma _{13}\,;\qquad \varepsilon _{23}={\frac {1}{2G}}\sigma _{23}\end{aligned}}}
where E is Young's modulus and ν is Poisson's ratio. (See 3-D elasticity).
[show]Derivation of Hooke's law in three dimensions
The three-dimensional form of Hooke's law can be derived using Poisson's ratio and the one-dimensional form of Hooke's law as follows.
Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3),
ε 1 ′ = 1 E σ 1 , ε 2 ′ = − ν E σ 1 , ε 3 ′ = − ν E σ 1 , {\displaystyle {\begin{aligned}\varepsilon _{1}'&={\frac {1}{E}}\sigma _{1}\,,\\\varepsilon _{2}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\\\varepsilon _{3}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\end{aligned}}}
where ν is Poisson's ratio and E is Young's modulus.
We get similar equations to the loads in directions 2 and 3,
ε 1 ″ = − ν E σ 2 , ε 2 ″ = 1 E σ 2 , ε 3 ″ = − ν E σ 2 , {\displaystyle {\begin{aligned}\varepsilon _{1}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\\\varepsilon _{2}''&={\frac {1}{E}}\sigma _{2}\,,\\\varepsilon _{3}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\end{aligned}}}
and
ε 1 ‴ = − ν E σ 3 , ε 2 ‴ = − ν E σ 3 , ε 3 ‴ = 1 E σ 3 . {\displaystyle {\begin{aligned}\varepsilon _{1}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{2}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{3}'''&={\frac {1}{E}}\sigma _{3}\,.\end{aligned}}}
Summing the three cases together (εi = εi′ + εi″ + εi‴) we get
ε 1 = 1 E ( σ 1 − ν ( σ 2 + σ 3 ) ) , ε 2 = 1 E ( σ 2 − ν ( σ 1 + σ 3 ) ) , ε 3 = 1 E ( σ 3 − ν ( σ 1 + σ 2 ) ) , {\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}\sigma _{1}-\nu (\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}\sigma _{2}-\nu (\sigma _{1}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}){\big )}\,,\end{aligned}}}
or by adding and subtracting one νσ
ε 1 = 1 E ( ( 1 + ν ) σ 1 − ν ( σ 1 + σ 2 + σ 3 ) ) , ε 2 = 1 E ( ( 1 + ν ) σ 2 − ν ( σ 1 + σ 2 + σ 3 ) ) , ε 3 = 1 E ( ( 1 + ν ) σ 3 − ν ( σ 1 + σ 2 + σ 3 ) ) , {\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{1}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{2}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\end{aligned}}}
and further we get by solving σ1
σ 1 = E 1 + ν ε 1 + ν 1 + ν ( σ 1 + σ 2 + σ 3 ) . {\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {\nu }{1+\nu }}(\sigma _{1}+\sigma _{2}+\sigma _{3})\,.}
Calculating the sum
ε 1 + ε 2 + ε 3 = 1 E ( ( 1 + ν ) ( σ 1 + σ 2 + σ 3 ) − 3 ν ( σ 1 + σ 2 + σ 3 ) ) = 1 − 2 ν E ( σ 1 + σ 2 + σ 3 ) σ 1 + σ 2 + σ 3 = E 1 − 2 ν ( ε 1 + ε 2 + ε 3 ) {\displaystyle {\begin{aligned}\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )(\sigma _{1}+\sigma _{2}+\sigma _{3})-3\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}={\frac {1-2\nu }{E}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\\\sigma _{1}+\sigma _{2}+\sigma _{3}&={\frac {E}{1-2\nu }}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\end{aligned}}}
and substituting it to the equation solved for σ1 gives
σ 1 = E 1 + ν ε 1 + E ν ( 1 + ν ) ( 1 − 2 ν ) ( ε 1 + ε 2 + ε 3 ) = 2 μ ε 1 + λ ( ε 1 + ε 2 + ε 3 ) , {\displaystyle {\begin{aligned}\sigma _{1}&={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {E\nu }{(1+\nu )(1-2\nu )}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\\&=2\mu \varepsilon _{1}+\lambda (\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\,,\end{aligned}}}
where μ and λ are the Lamé parameters.
Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions.
In matrix form, Hooke's law for isotropic materials can be written as
[ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ] = [ ε 11 ε 22 ε 33 γ 23 γ 13 γ 12 ] = 1 E [ 1 − ν − ν 0 0 0 − ν 1 − ν 0 0 0 − ν − ν 1 0 0 0 0 0 0 2 + 2 ν 0 0 0 0 0 0 2 + 2 ν 0 0 0 0 0 0 2 + 2 ν ] [ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] {\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &-\nu &0&0&0\\-\nu &1&-\nu &0&0&0\\-\nu &-\nu &1&0&0&0\\0&0&0&2+2\nu &0&0\\0&0&0&0&2+2\nu &0\\0&0&0&0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}}
where γij = 2εij is the engineering shear strain. The inverse relation may be written as
[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] = E ( 1 + ν ) ( 1 − 2 ν ) [ 1 − ν ν ν 0 0 0 ν 1 − ν ν 0 0 0 ν ν 1 − ν 0 0 0 0 0 0 1 − 2 ν 2 0 0 0 0 0 0 1 − 2 ν 2 0 0 0 0 0 0 1 − 2 ν 2 ] [ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ] {\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &\nu &0&0&0\\\nu &1-\nu &\nu &0&0&0\\\nu &\nu &1-\nu &0&0&0\\0&0&0&{\frac {1-2\nu }{2}}&0&0\\0&0&0&0&{\frac {1-2\nu }{2}}&0\\0&0&0&0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}
which can be simplified thanks to the Lamé constants :
[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] = [ 2 μ + λ λ λ 0 0 0 λ 2 μ + λ λ 0 0 0 λ λ 2 μ + λ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ 0 0 0 0 0 0 μ ] [ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ] {\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}2\mu +\lambda &\lambda &\lambda &0&0&0\\\lambda &2\mu +\lambda &\lambda &0&0&0\\\lambda &\lambda &2\mu +\lambda &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}
In vector notation this becomes
[ σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33 ] = 2 μ [ ε 11 ε 12 ε 13 ε 12 ε 22 ε 23 ε 13 ε 23 ε 33 ] + λ I ( ε 11 + ε 22 + ε 33 ) {\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{12}&\sigma _{22}&\sigma _{23}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}\,=\,2\mu {\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{12}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{13}&\varepsilon _{23}&\varepsilon _{33}\end{bmatrix}}+\lambda \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\right)}
where I is the identity tensor.
Plane stress[edit]
Under plane stress conditions, σ31 = σ13 = σ32 = σ23 = σ33 = 0. In that case Hooke's law takes the form
[ ε 11 ε 22 2 ε 12 ] = 1 E [ 1 − ν 0 − ν 1 0 0 0 2 + 2 ν ] [ σ 11 σ 22 σ 12 ] {\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &0\\-\nu &1&0\\0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}}
The inverse relation is usually written in the reduced form
[ σ 11 σ 22 σ 12 ] = E 1 − ν 2 [ 1 ν 0 ν 1 0 0 0 1 − ν 2 ] [ ε 11 ε 22 2 ε 12 ] {\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&{\frac {1-\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}
In vector notation this becomes
[ σ 11 σ 12 σ 12 σ 22 ] = E 1 − ν 2 ( ( 1 − ν ) [ ε 11 ε 12 ε 12 ε 22 ] + ν I ( ε 11 + ε 22 ) ) {\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{12}&\sigma _{22}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}\left((1-\nu ){\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{12}&\varepsilon _{22}\end{bmatrix}}+\nu \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}\right)\right)}
Anisotropic materials[edit]
The symmetry of the Cauchy stress tensor (σij = σji and the generalized Hooke's laws (σij = cijklεkl) 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), then
σ i j = ∂ U ∂ ε i j ⟹ c i j k l = ∂ 2 U ∂ ε i j ∂ ε k l . {\displaystyle \sigma _{ij}={\frac {\partial U}{\partial \varepsilon _{ij}}}\quad \implies \quad c_{ijkl}={\frac {\partial ^{2}U}{\partial \varepsilon _{ij}\partial \varepsilon _{kl}}}\,.}
The arbitrariness of the order of differentiation implies that cijkl = cklij. These are called the major symmetries of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. The major and minor symmetries indicate that the stiffness tensor has only 21 independent components.
Matrix representation (stiffness tensor)[edit]
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
[ σ ] = [ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] ≡ [ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] ; [ ε ] = [ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ] ≡ [ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] {\displaystyle [{\boldsymbol {\sigma }}]\,=\,{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,;\qquad [{\boldsymbol {\varepsilon }}]\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}
Then the stiffness tensor (c) can be expressed as
[ c ] = [ c 1111 c 1122 c 1133 c 1123 c 1131 c 1112 c 2211 c 2222 c 2233 c 2223 c 2231 c 2212 c 3311 c 3322 c 3333 c 3323 c 3331 c 3312 c 2311 c 2322 c 2333 c 2323 c 2331 c 2312 c 3111 c 3122 c 3133 c 3123 c 3131 c 3112 c 1211 c 1222 c 1233 c 1223 c 1231 c 1212 ] ≡ [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] {\displaystyle [{\mathsf {c}}]\,=\,{\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_{1131}&c_{1112}\\c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}}
and Hooke's law is written as
[ σ ] = [ C ] [ ε ] or σ i = C i j ε j . {\displaystyle [{\boldsymbol {\sigma }}]=[{\mathsf {C}}][{\boldsymbol {\varepsilon }}]\qquad {\text{or}}\qquad \sigma _{i}=C_{ij}\varepsilon _{j}\,.}
Similarly the compliance tensor (s) can be written as
[ s ] = [ s 1111 s 1122 s 1133 2 s 1123 2 s 1131 2 s 1112 s 2211 s 2222 s 2233 2 s 2223 2 s 2231 2 s 2212 s 3311 s 3322 s 3333 2 s 3323 2 s 3331 2 s 3312 2 s 2311 2 s 2322 2 s 2333 4 s 2323 4 s 2331 4 s 2312 2 s 3111 2 s 3122 2 s 3133 4 s 3123 4 s 3131 4 s 3112 2 s 1211 2 s 1222 2 s 1233 4 s 1223 4 s 1231 4 s 1212 ] ≡ [ S 11 S 12 S 13 S 14 S 15 S 16 S 12 S 22 S 23 S 24 S 25 S 26 S 13 S 23 S 33 S 34 S 35 S 36 S 14 S 24 S 34 S 44 S 45 S 46 S 15 S 25 S 35 S 45 S 55 S 56 S 16 S 26 S 36 S 46 S 56 S 66 ] {\displaystyle [{\mathsf {s}}]\,=\,{\begin{bmatrix}s_{1111}&s_{1122}&s_{1133}&2s_{1123}&2s_{1131}&2s_{1112}\\s_{2211}&s_{2222}&s_{2233}&2s_{2223}&2s_{2231}&2s_{2212}\\s_{3311}&s_{3322}&s_{3333}&2s_{3323}&2s_{3331}&2s_{3312}\\2s_{2311}&2s_{2322}&2s_{2333}&4s_{2323}&4s_{2331}&4s_{2312}\\2s_{3111}&2s_{3122}&2s_{3133}&4s_{3123}&4s_{3131}&4s_{3112}\\2s_{1211}&2s_{1222}&2s_{1233}&4s_{1223}&4s_{1231}&4s_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}}
Change of coordinate system[edit]
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[11]
c p q r s = l p i l q j l r k l s l c i j k l {\displaystyle c_{pqrs}=l_{pi}l_{qj}l_{rk}l_{sl}c_{ijkl}}
where lab are the components of an orthogonal rotation matrix [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}]}
then
C i j ε i ε j = C i j ′ ε i ′ ε j ′ . {\displaystyle C_{ij}\varepsilon _{i}\varepsilon _{j}=C_{ij}'\varepsilon '_{i}\varepsilon '_{j}\,.}
In addition, if the material is symmetric with respect to the transformation [L] then
C i j = C i j ′ ⟹ C i j ( ε i ε j − ε i ′ ε j ′ ) = 0 . {\displaystyle C_{ij}=C'_{ij}\quad \implies \quad C_{ij}(\varepsilon _{i}\varepsilon _{j}-\varepsilon '_{i}\varepsilon '_{j})=0\,.}
Orthotropic materials[edit]
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
[ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] = [ C 11 C 12 C 13 0 0 0 C 12 C 22 C 23 0 0 0 C 13 C 23 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66 ] [ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] {\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}
The inverse of this relation is commonly written as[12][page needed]
[ ε x x ε y y ε z z 2 ε y z 2 ε z x 2 ε x y ] = [ 1 E x − ν y x E y − ν z x E z 0 0 0 − ν x y E x 1 E y − ν z y E z 0 0 0 − ν x z E x − ν y z E y 1 E z 0 0 0 0 0 0 1 G y z 0 0 0 0 0 0 1 G z x 0 0 0 0 0 0 1 G x y ] [ σ x x σ y y σ z z σ y z σ z x σ x y ] {\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{zx}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}
where
Ei is the Young's modulus along axis i
Gij is the shear modulus in direction j on the plane whose normal is in direction i
νij 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, σzz = σzx = σyz = 0, Hooke's law for an orthotropic material takes the form
[ ε x x ε y y 2 ε x y ] = [ 1 E x − ν y x E y 0 − ν x y E x 1 E y 0 0 0 1 G x y ] [ σ x x σ y y σ x y ] . {\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&0\\0&0&{\frac {1}{G_{xy}}}\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,.}
The inverse relation is
[ σ x x σ y y σ x y ] = 1 1 − ν x y ν y x [ E x ν y x E x 0 ν x y E y E y 0 0 0 G x y ( 1 − ν x y ν y x ) ] [ ε x x ε y y 2 ε x y ] . {\displaystyle {\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,=\,{\frac {1}{1-\nu _{xy}\nu _{yx}}}{\begin{bmatrix}E_{x}&\nu _{yx}E_{x}&0\\\nu _{xy}E_{y}&E_{y}&0\\0&0&G_{xy}(1-\nu _{xy}\nu _{yx})\end{bmatrix}}{\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,.}
The transposed form of the above stiffness matrix is also often used.
Transversely isotropic materials[edit]
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
[ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] = [ C 11 C 12 C 13 0 0 0 C 12 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 11 − C 12 2 ] [ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] {\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&{\frac {C_{11}-C_{12}}{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}
More frequently, the x ≡ e1 axis is taken to be the axis of symmetry and the inverse Hooke's law is written as [13]
[ ε x x ε y y ε z z 2 ε y z 2 ε z x 2 ε x y ] = [ 1 E x − ν y x E y − ν y x E y 0 0 0 − ν x y E x 1 E y − ν y z E y 0 0 0 − ν x y E x − ν y z E y 1 E y 0 0 0 0 0 0 2 + 2 ν y z E y 0 0 0 0 0 0 1 G x y 0 0 0 0 0 0 1 G x y ] [ σ x x σ y y σ z z σ y z σ z x σ x y ] {\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{yx}}{E_{y}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{yz}}{E_{y}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{y}}}&0&0&0\\0&0&0&{\frac {2+2\nu _{yz}}{E_{y}}}&0&0\\0&0&0&0&{\frac {1}{G_{xy}}}&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[edit]
To grasp the degree of anisotropy of any class, a Universal Elastic Anisotropy Index (AU) [14] was formulated. It replaces the Zener ratio, which is suited for cubic crystals.
Thermodynamic basis[edit]
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
δ W = δ U {\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
δ W = δ W s + δ W b {\displaystyle \delta W=\delta W_{\mathrm {s} }+\delta W_{\mathrm {b} }}
where δWs is the work done by surface forces while δWb 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
δ W s = ∫ ∂ Ω t ⋅ δ u d S ; δ W b = ∫ Ω b ⋅ δ u d V {\displaystyle \delta W_{\mathrm {s} }=\int _{\partial \Omega }\mathbf {t} \cdot \delta \mathbf {u} \,dS\,;\qquad \delta W_{\mathrm {b} }=\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV}
where t is the surface traction 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
δ W = δ U = ∫ ∂ Ω ( n ⋅ σ ) ⋅ δ u d S + ∫ Ω b ⋅ δ u d V . {\displaystyle \delta W=\delta U=\int _{\partial \Omega }(\mathbf {n} \cdot {\boldsymbol {\sigma }})\cdot \delta \mathbf {u} \,dS+\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV\,.}
Converting the surface integral into a volume integral via the divergence theorem gives
δ U = ∫ Ω ( ∇ ⋅ ( σ ⋅ δ u ) + b ⋅ δ u ) d V . {\displaystyle \delta U=\int _{\Omega }{\big (}\nabla \cdot ({\boldsymbol {\sigma }}\cdot \delta \mathbf {u} )+\mathbf {b} \cdot \delta \mathbf {u} {\big )}\,dV\,.}
Using the symmetry of the Cauchy stress and the identity
∇ ⋅ ( a ⋅ b ) = ( ∇ ⋅ a ) ⋅ b + 1 2 ( a T : ∇ b + a : ( ∇ b ) T ) {\displaystyle \nabla \cdot (\mathbf {a} \cdot \mathbf {b} )=(\nabla \cdot \mathbf {a} )\cdot \mathbf {b} +{\tfrac {1}{2}}\left(\mathbf {a} ^{\mathsf {T}}:\nabla \mathbf {b} +\mathbf {a} :(\nabla \mathbf {b} )^{\mathsf {T}}\right)}
we have the following
δ U = ∫ Ω ( σ : 1 2 ( ∇ δ u + ( ∇ δ u ) T ) + ( ∇ ⋅ σ + b ) ⋅ δ u ) d V . {\displaystyle \delta U=\int _{\Omega }\left({\boldsymbol {\sigma }}:{\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)+\left(\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} \right)\cdot \delta \mathbf {u} \right)\,dV\,.}
From the definition of strain and from the equations of equilibrium we have
δ ε = 1 2 ( ∇ δ u + ( ∇ δ u ) T ) ; ∇ ⋅ σ + b = 0 . {\displaystyle \delta {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)\,;\qquad \nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} =\mathbf {0} \,.}
Hence we can write
δ U = ∫ Ω σ : δ ε d V {\displaystyle \delta U=\int _{\Omega }{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,dV}
and therefore the variation in the internal energy density is given by
δ U 0 = σ : δ ε . {\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
δ U 0 = ∂ U 0 ∂ ε : δ ε . {\displaystyle \delta U_{0}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}:\delta {\boldsymbol {\varepsilon }}\,.}
Since the variation of strain is arbitrary, the stress–strain relation of an elastic material is given by
σ = ∂ U 0 ∂ ε . {\displaystyle {\boldsymbol {\sigma }}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}\,.}
For a linear elastic material, the quantity ∂U0/∂ε is a linear function of ε, and can therefore be expressed as
σ = c : ε {\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}}
where c is a fourth-rank tensor of material constants, also called the stiffness tensor. We can see why c must be a fourth-rank tensor by noting that, for a linear elastic material,
∂ ∂ ε σ ( ε ) = constant = c . {\displaystyle {\frac {\partial }{\partial {\boldsymbol {\varepsilon }}}}{\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})={\text{constant}}={\mathsf {c}}\,.}
In index notation
∂ σ i j ∂ ε k l = constant = c i j k l . {\displaystyle {\frac {\partial \sigma _{ij}}{\partial \varepsilon _{kl}}}={\text{constant}}=c_{ijkl}\,.}
The right-hand side constant requires four indices and is a fourth-rank quantity. We can also see that this quantity must be a tensor because it is a linear transformation that takes the strain tensor to the stress tensor. We can also show that the constant obeys the tensor transformation rules for fourth-rank tensors.
Homages[edit]
Ut tensio, sic vis is the motto of the École Polytechnique de Montréal.
See also[edit]
Continuum mechanics
Laws[show]
Conservations
Energy
Mass
Momentum
Inequalities
Clausius–Duhem (entropy)
Solid mechanics[show]
Stress
Deformation
Compatibility
Finite strain
Infinitesimal strain
Elasticity (linear)
Plasticity
Bending
Hooke's law
Material failure theory
Fracture mechanics
Contact mechanics (frictional)
Fluid mechanics[show]
Fluids
Statics · Dynamics
Archimedes' principle · Bernoulli's principle
Navier–Stokes equations
Poiseuille equation · Pascal's law
Viscosity (Newtonian · non-Newtonian)
Buoyancy · Mixing · Pressure
Liquids
Surface tension
Capillary action
Gases
Atmosphere
Boyle's law
Charles's law
Gay-Lussac's law
Combined gas law
Plasma
Rheology[show]
Viscoelasticity
Rheometry
Rheometer
Smart fluids
Magnetorheological
Electrorheological
Ferrofluids
Scientists[show]
Bernoulli
Boyle
Cauchy
Charles
Euler
Gay-Lussac
Hooke
Pascal
Newton
Navier
Stokes
v
t
e
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[edit]
Jump up ^ 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 0674463684.
Jump up ^ See http://civil.lindahall.org/design.shtml, where one can find also an anagram for catenary.
Jump up ^ Robert Hooke, De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies, London, 1678.
Jump up ^ 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.
Jump up ^ Belen'kii; Salaev (1988). "Deformation effects in layer crystals". Uspekhi Fizicheskikh Nauk. 155 (5): 89. doi:10.3367/UFNr.0155.198805c.0089.
Jump up ^ 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.
Jump up ^ 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.
Jump up ^ Keith (1971). Mechanics. Reading, MA: Addison-Wesley. ISBN 0-201-07392-7.
Jump up ^ Simo, J. C.; Hughes, T. J. R. (1998). Computational Inelasticity. Springer. ISBN 9780387975207.
Jump up ^ Milton, Graeme W. (2002). The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press. ISBN 9780521781251.
Jump up ^ Slaughter, William S. (2001). The Linearized Theory of Elasticity. Birkhäuser. ISBN 978-0817641177.
Jump up ^ Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. (1993). Advanced Mechanics of Materials (5th ed.). Wiley. ISBN 9780471600091.
Jump up ^ Tan, S. C. (1994). Stress Concentrations in Laminated Composites. Lancaster, PA: Technomic Publishing Company. ISBN 9781566760775.
Jump up ^ Ranganathan, S.I.; Ostoja-Starzewski, M. (2008). "Universal Elastic Anisotropy Index". Physical Review Latters. APS. 101: 055504–1–4.
References[edit]
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (July 2017) (Learn how and when to remove this template message)
Ugural, A. C.; Fenster, S. K. (2003). Advanced Strength and Applied Elasticity (4th ed.). Prentice-Hall. ISBN 978-0-13-047392-9.
Walter Lewin explains Hooke's law. From Walter Lewin (1 October 1999). Hooke's Law, Simple Harmonic Oscillator. MIT Course 8.01: Classical Mechanics, Lecture 10 (ogg) (videotape). Cambridge, MA USA: MIT OCW. Event occurs at 1:21–10:10. Retrieved 23 December 2010. ...arguably the most important equation in all of Physics.
A test of Hooke's law. From Walter Lewin (1 October 1999). Hooke's Law, Simple Harmonic Oscillator. MIT Course 8.01: Classical Mechanics, Lecture 10 (ogg) (videotape). Cambridge, MA USA: MIT OCW. Event occurs at 10:10–16:33. Retrieved 23 December 2010.
External links[edit]
Java Applet demonstrating Hooke's Law in motion
Hooke's Law for General Stress/Strain Calculations
[hide]
v
t
e
Elastic moduli for homogeneous isotropic materials
Bulk modulus ( K {\displaystyle K} )
Young's modulus ( E {\displaystyle E} )
Lamé's first parameter ( λ {\displaystyle \lambda } )
Shear modulus ( G , μ {\displaystyle G,\mu } )
Poisson's ratio ( ν {\displaystyle \nu } )
P-wave modulus ( M {\displaystyle M} )
[hide]Conversion formulas
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.
K = {\displaystyle K=\,} E = {\displaystyle E=\,} λ = {\displaystyle \lambda =\,} G = {\displaystyle G=\,} ν = {\displaystyle \nu =\,} M = {\displaystyle M=\,} Notes
( K , E ) {\displaystyle (K,\,E)} K {\displaystyle K} E {\displaystyle E} 3 K ( 3 K − E ) 9 K − E {\displaystyle {\tfrac {3K(3K-E)}{9K-E}}} 3 K E 9 K − E {\displaystyle {\tfrac {3KE}{9K-E}}} 3 K − E 6 K {\displaystyle {\tfrac {3K-E}{6K}}} 3 K ( 3 K + E ) 9 K − E {\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}
( K , λ ) {\displaystyle (K,\,\lambda )} K {\displaystyle K} 9 K ( K − λ ) 3 K − λ {\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}} λ {\displaystyle \lambda } 3 ( K − λ ) 2 {\displaystyle {\tfrac {3(K-\lambda )}{2}}} λ 3 K − λ {\displaystyle {\tfrac {\lambda }{3K-\lambda }}} 3 K − 2 λ {\displaystyle 3K-2\lambda \,}
( K , G ) {\displaystyle (K,\,G)} K {\displaystyle K} 9 K G 3 K + G {\displaystyle {\tfrac {9KG}{3K+G}}} K − 2 G 3 {\displaystyle K-{\tfrac {2G}{3}}} G {\displaystyle G} 3 K − 2 G 2 ( 3 K + G ) {\displaystyle {\tfrac {3K-2G}{2(3K+G)}}} K + 4 G 3 {\displaystyle K+{\tfrac {4G}{3}}}
( K , ν ) {\displaystyle (K,\,\nu )} K {\displaystyle K} 3 K ( 1 − 2 ν ) {\displaystyle 3K(1-2\nu )\,} 3 K ν 1 + ν {\displaystyle {\tfrac {3K\nu }{1+\nu }}} 3 K ( 1 − 2 ν ) 2 ( 1 + ν ) {\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}} ν {\displaystyle \nu } 3 K ( 1 − ν ) 1 + ν {\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}
( K , M ) {\displaystyle (K,\,M)} K {\displaystyle K} 9 K ( M − K ) 3 K + M {\displaystyle {\tfrac {9K(M-K)}{3K+M}}} 3 K − M 2 {\displaystyle {\tfrac {3K-M}{2}}} 3 ( M − K ) 4 {\displaystyle {\tfrac {3(M-K)}{4}}} 3 K − M 3 K + M {\displaystyle {\tfrac {3K-M}{3K+M}}} M {\displaystyle M}
( E , λ ) {\displaystyle (E,\,\lambda )} E + 3 λ + R 6 {\displaystyle {\tfrac {E+3\lambda +R}{6}}} E {\displaystyle E} λ {\displaystyle \lambda } E − 3 λ + R 4 {\displaystyle {\tfrac {E-3\lambda +R}{4}}} 2 λ E + λ + R {\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}} E − λ + R 2 {\displaystyle {\tfrac {E-\lambda +R}{2}}} R = E 2 + 9 λ 2 + 2 E λ {\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}
( E , G ) {\displaystyle (E,\,G)} E G 3 ( 3 G − E ) {\displaystyle {\tfrac {EG}{3(3G-E)}}} E {\displaystyle E} G ( E − 2 G ) 3 G − E {\displaystyle {\tfrac {G(E-2G)}{3G-E}}} G {\displaystyle G} E 2 G − 1 {\displaystyle {\tfrac {E}{2G}}-1} G ( 4 G − E ) 3 G − E {\displaystyle {\tfrac {G(4G-E)}{3G-E}}}
( E , ν ) {\displaystyle (E,\,\nu )} E 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {E}{3(1-2\nu )}}} E {\displaystyle E} E ν ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}} E 2 ( 1 + ν ) {\displaystyle {\tfrac {E}{2(1+\nu )}}} ν {\displaystyle \nu } E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}
( E , M ) {\displaystyle (E,\,M)} 3 M − E + S 6 {\displaystyle {\tfrac {3M-E+S}{6}}} E {\displaystyle E} M − E + S 4 {\displaystyle {\tfrac {M-E+S}{4}}} 3 M + E − S 8 {\displaystyle {\tfrac {3M+E-S}{8}}} E − M + S 4 M {\displaystyle {\tfrac {E-M+S}{4M}}} M {\displaystyle M}
S = ± E 2 + 9 M 2 − 10 E M {\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}
There are two valid solutions.
The plus sign leads to ν ≥ 0 {\displaystyle \nu \geq 0} .
The minus sign leads to ν ≤ 0 {\displaystyle \nu \leq 0} .
( λ , G ) {\displaystyle (\lambda ,\,G)} λ + 2 G 3 {\displaystyle \lambda +{\tfrac {2G}{3}}} G ( 3 λ + 2 G ) λ + G {\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}} λ {\displaystyle \lambda } G {\displaystyle G} λ 2 ( λ + G ) {\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}} λ + 2 G {\displaystyle \lambda +2G\,}
( λ , ν ) {\displaystyle (\lambda ,\,\nu )} λ ( 1 + ν ) 3 ν {\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}} λ ( 1 + ν ) ( 1 − 2 ν ) ν {\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}} λ {\displaystyle \lambda } λ ( 1 − 2 ν ) 2 ν {\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}} ν {\displaystyle \nu } λ ( 1 − ν ) ν {\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}} Cannot be used when ν = 0 ⇔ λ = 0 {\displaystyle \nu =0\Leftrightarrow \lambda =0}
( λ , M ) {\displaystyle (\lambda ,\,M)} M + 2 λ 3 {\displaystyle {\tfrac {M+2\lambda }{3}}} ( M − λ ) ( M + 2 λ ) M + λ {\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}} λ {\displaystyle \lambda } M − λ 2 {\displaystyle {\tfrac {M-\lambda }{2}}} λ M + λ {\displaystyle {\tfrac {\lambda }{M+\lambda }}} M {\displaystyle M}
( G , ν ) {\displaystyle (G,\,\nu )} 2 G ( 1 + ν ) 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}} 2 G ( 1 + ν ) {\displaystyle 2G(1+\nu )\,} 2 G ν 1 − 2 ν {\displaystyle {\tfrac {2G\nu }{1-2\nu }}} G {\displaystyle G} ν {\displaystyle \nu } 2 G ( 1 − ν ) 1 − 2 ν {\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}
( G , M ) {\displaystyle (G,\,M)} M − 4 G 3 {\displaystyle M-{\tfrac {4G}{3}}} G ( 3 M − 4 G ) M − G {\displaystyle {\tfrac {G(3M-4G)}{M-G}}} M − 2 G {\displaystyle M-2G\,} G {\displaystyle G} M − 2 G 2 M − 2 G {\displaystyle {\tfrac {M-2G}{2M-2G}}} M {\displaystyle M}
( ν , M ) {\displaystyle (\nu ,\,M)} M ( 1 + ν ) 3 ( 1 − ν ) {\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}} M ( 1 + ν ) ( 1 − 2 ν ) 1 − ν {\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}} M ν 1 − ν {\displaystyle {\tfrac {M\nu }{1-\nu }}} M ( 1 − 2 ν ) 2 ( 1 − ν ) {\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}} ν {\displaystyle \nu } M {\displaystyle M}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hooke%27s_law&oldid=812496433"
Categories:
1660 in science
Springs (mechanical)
Elasticity (physics)
Solid mechanics
Hidden categories:
Use dmy dates from October 2017
Wikipedia articles needing page number citations from November 2016
Articles lacking in-text citations from July 2017
All articles lacking in-text citations
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
In other projects
Wikimedia Commons
Languages
Alemannisch
አማርኛ
العربية
বাংলা
Беларуская
Български
Bosanski
Català
Čeština
Dansk
Deutsch
Eesti
Ελληνικά
Español
Esperanto
فارسی
Français
Gaeilge
Galego
한국어
Հայերեն
हिन्दी
Hrvatski
Bahasa Indonesia
Italiano
עברית
ქართული
Қазақша
Latviešu
Lietuvių
Magyar
Македонски
Bahasa Melayu
Nederlands
日本語
Norsk
Norsk nynorsk
ភាសាខ្មែរ
Piemontèis
Polski
Português
Română
Русский
Simple English
Slovenčina
Slovenščina
Српски / srpski
Srpskohrvatski / српскохрватски
Suomi
Svenska
தமிழ்
Türkçe
Українська
Tiếng Việt
中文
Edit links
This page was last edited on 28 November 2017, at 04:15.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

values: mw-page-base noprint

values: mw-head-base noprint

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

value: img

attrs: alt src width height class srcset data-file-width data-file-height

attrs: href class

values:
/wiki/Bourdon_tube
mw-redirect
Bourdon tube

values:
/wiki/Spring_(mechanics)
mw-redirect
Spring (mechanics)

values:
/wiki/Party_balloon
mw-redirect
Party balloon

values:
/wiki/Elastic_limit
mw-redirect
Elastic limit

values:
/wiki/Mechanical_clock
mw-redirect
Mechanical clock

values:
/wiki/Theory_of_elasticity
mw-redirect
Theory of elasticity

values: vertical-align: -0.338ex; width:8.072ex; height:2.176ex;

values: vertical-align: -2.005ex; width:7.687ex; height:5.509ex;

values:
/wiki/Cartesian_coordinates
mw-redirect
Cartesian coordinates

values: vertical-align: -0.338ex; width:9.891ex; height:2.176ex;

values:
/wiki/Axis_(mathematics)
mw-redirect
Axis (mathematics)

values:
/wiki/Vector_(mathematics)
mw-redirect
Vector (mathematics)

values:
/wiki/Matrix_product
mw-redirect
Matrix product

values: vertical-align: -4.005ex; width:48.484ex; height:9.176ex;

values: vertical-align: -0.671ex; width:29.069ex; height:2.509ex;

values: Edit section: Hooke's law for continuous media

values:
/wiki/Strain_tensor
mw-redirect
Strain tensor

values: vertical-align: -0.671ex; width:9.628ex; height:2.009ex;

values:
/wiki/Stiffness_tensor
mw-redirect
Stiffness tensor

values:
/wiki/Elasticity_tensor
mw-redirect
Elasticity tensor

values: vertical-align: -0.671ex; width:9.496ex; height:2.009ex;

values: vertical-align: -4.005ex; width:52.366ex; height:9.176ex;

values: vertical-align: -3.171ex; width:22.208ex; height:7.509ex;

values:
/wiki/Ionic_polarization
mw-redirect
Ionic polarization

values:
/wiki/Dynamics_(physics)
mw-redirect
Dynamics (physics)

values: Edit section: General application to elastic materials

values:
/wiki/Strain_hardening
mw-redirect
Strain hardening

values:
/wiki/Proportional_limit
mw-redirect
Proportional limit

values:
/wiki/Mooney-Rivlin_solid
mw-redirect
Mooney-Rivlin solid

values: Edit section: Tensional stiffness of a uniform bar

values:
/wiki/Tensile_stress
mw-redirect
Tensile stress

values:
/wiki/Modulus_of_elasticity
mw-redirect
Modulus of elasticity

values: vertical-align: -0.338ex; width:7.329ex; height:2.176ex;

values: vertical-align: -2.005ex; width:8.579ex; height:5.509ex;

values: vertical-align: -2.005ex; width:8.083ex; height:5.509ex;

values: {\displaystyle \sigma ={\frac {F}{A}}\,,}

values: vertical-align: -2.005ex; width:15.354ex; height:5.509ex;

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

values: vertical-align: -2.005ex; width:17.865ex; height:5.509ex;

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

values: vertical-align: -1.171ex; width:14.601ex; height:3.509ex;

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

values: display:block; line-height:1em; margin:0 0.1em;

values: display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;

values: display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;

values: vertical-align: -2.171ex; width:11.534ex; height:6.009ex;

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

values: Edit section: Relaxed force constants (generalized compliance constants)

values: Compliance constant (page does not exist)

values: vertical-align: -2.338ex; width:12.971ex; height:6.343ex;

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

values: Edit section: Rotation in gravity-free space

values: vertical-align: -0.671ex; width:25.343ex; height:3.009ex;

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

values: vertical-align: -0.338ex; width:8.899ex; height:2.676ex;

values: Edit section: Linear elasticity theory for continuous media

values:
/wiki/Einstein_summation_convention
mw-redirect
Einstein summation convention

values: vertical-align: -1.838ex; width:34.083ex; height:4.843ex;

values: vertical-align: -1.338ex; width:69.952ex; height:3.676ex;

values: vertical-align: -1.838ex; width:75.216ex; height:4.843ex;

values: vertical-align: -0.838ex; width:49.001ex; height:2.843ex;

values:
/wiki/Lam%C3%A9_constants
mw-redirect
Lamé constants

values: vertical-align: -1.005ex; width:71.01ex; height:3.009ex;

values: vertical-align: -2.671ex; width:64.455ex; height:6.343ex;

values: vertical-align: -2.671ex; width:60.858ex; height:6.343ex;

values: vertical-align: -2.005ex; width:87.187ex; height:5.343ex;

attrs: columnalign rowspacing columnspacing displaystyle

values: right left right left right left right left right left right left

values: 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em

values: vertical-align: -10.338ex; width:52.411ex; height:21.843ex;

values:
/wiki/3-D_elasticity
mw-redirect
3-D elasticity

values: vertical-align: -6.814ex; margin-bottom: -0.19ex; width:13.901ex; height:15.176ex;

values: vertical-align: -6.814ex; margin-bottom: -0.19ex; width:13.991ex; height:15.176ex;

values: vertical-align: -6.814ex; margin-bottom: -0.19ex; width:14.451ex; height:15.176ex;

values: vertical-align: -7.334ex; margin-bottom: -0.337ex; width:27.806ex; height:16.343ex;

values: vertical-align: -7.334ex; margin-bottom: -0.337ex; width:40.153ex; height:16.343ex;

values: vertical-align: -2.005ex; width:38.494ex; height:5.509ex;

values: vertical-align: -4.582ex; margin-bottom: -0.256ex; width:88.471ex; height:10.843ex;

values: vertical-align: -3.793ex; margin-bottom: -0.212ex; width:47.569ex; height:9.176ex;

values: vertical-align: -9.005ex; width:76.487ex; height:19.176ex;

values: vertical-align: -10.171ex; width:80.481ex; height:21.509ex;

values: vertical-align: -9.171ex; width:59.143ex; height:19.509ex;

values: vertical-align: -4.005ex; width:64.26ex; height:9.176ex;

values: vertical-align: -4.005ex; width:42.718ex; height:9.176ex;

values: vertical-align: -4.505ex; width:41.039ex; height:10.176ex;

values: vertical-align: -2.505ex; width:61.681ex; height:6.176ex;

values:
/wiki/Stress_(physics)
mw-redirect
Stress (physics)

values: vertical-align: -2.671ex; width:38.522ex; height:6.509ex;

values: Edit section: Matrix representation (stiffness tensor)

values: vertical-align: -9.005ex; width:54.517ex; height:19.176ex;

values: vertical-align: -9.005ex; width:89.877ex; height:19.176ex;

values: vertical-align: -1.005ex; width:34.252ex; height:3.009ex;

values: vertical-align: -9.005ex; width:95.853ex; height:19.176ex;

values: Edit section: Change of coordinate system

values: vertical-align: -1.005ex; width:20.845ex; height:2.843ex;

values: vertical-align: -1.005ex; width:12.731ex; height:3.009ex;

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

values: vertical-align: -1.338ex; width:18.306ex; height:3.343ex;

values: vertical-align: -1.338ex; width:41.475ex; height:3.343ex;

values:
/wiki/Plane_of_symmetry
mw-redirect
Plane of symmetry

values: vertical-align: -9.005ex; width:53.05ex; height:19.176ex;

values: This citation requires a reference to the specific page or range of pages in which the material appears. (November 2016)

values: vertical-align: -12.987ex; margin-bottom: -0.184ex; width:61.081ex; height:27.509ex;

values: vertical-align: -6.062ex; margin-bottom: -0.276ex; width:42.821ex; height:13.843ex;

values: vertical-align: -4.188ex; margin-bottom: -0.317ex; width:68.151ex; height:10.176ex;

values: Edit section: Transversely isotropic materials

values:
/wiki/Transversely_isotropic
mw-redirect
Transversely isotropic

values:
/wiki/Axis_of_symmetry
mw-redirect
Axis of symmetry

values: vertical-align: -9.505ex; width:56.995ex; height:20.176ex;

values: vertical-align: -13.671ex; width:63.624ex; height:28.509ex;

values: Edit section: Universal Elastic Anisotropy Index

values: vertical-align: -0.338ex; width:9.466ex; height:2.343ex;

values: vertical-align: -0.671ex; width:18.032ex; height:2.676ex;

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

values: vertical-align: -2.338ex; width:45.603ex; height:5.843ex;

values: vertical-align: -2.338ex; width:46.394ex; height:5.843ex;

values: vertical-align: -2.338ex; width:35.59ex; height:5.843ex;

values: vertical-align: -1.171ex; width:50.884ex; height:3.509ex;

values: vertical-align: -2.338ex; width:58.027ex; height:5.843ex;

values: vertical-align: -1.171ex; width:46.205ex; height:3.509ex;

values: vertical-align: -2.338ex; width:18.34ex; height:5.843ex;

values: vertical-align: -0.671ex; width:13.724ex; height:2.676ex;

values: vertical-align: -2.005ex; width:16.943ex; height:5.509ex;

values: vertical-align: -2.005ex; width:10.582ex; height:5.509ex;

values: vertical-align: -0.338ex; width:8.945ex; height:1.676ex;

values: vertical-align: -2.005ex; width:25.137ex; height:5.509ex;

values: vertical-align: -2.338ex; width:24.879ex; height:6.176ex;

values:
/wiki/List_of_Latin_mottos
mw-redirect
List of Latin mottos

values: float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none

values: font-size: 105%; padding: 0.2em 0px 0.4em; text-align: center; display: none;

values:
/wiki/Fluid_statics
mw-redirect
Fluid statics

values:
/wiki/Elastic_potential_energy
mw-redirect
Elastic potential energy

values: List of scientific laws named after people

values:
nofollow
external free
http://civil.lindahall.org/design.shtml

values:
nofollow
external text
//doi.org/10.1038%2Fsrep17152

values:
/wiki/PubMed_Identifier
mw-redirect
PubMed Identifier

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/26612544

values:
nofollow
external text
//doi.org/10.1063%2F1.3259834

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/19895003

values:
nofollow
external text
//doi.org/10.1039%2FC2CP40290D

values: -moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;

values: Wikipedia:WikiProject Fact and Reference Check

values: plainlinks metadata ambox ambox-style ambox-More_footnotes

values: vertical-align: -0.338ex; width:2.076ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.786ex; height:2.176ex;

values: vertical-align: -0.338ex; width:1.366ex; height:2.176ex;

values: vertical-align: -0.838ex; width:4.294ex; height:2.676ex;

values: vertical-align: -0.338ex; width:1.243ex; height:1.676ex;

values: vertical-align: -0.338ex; width:2.453ex; height:2.176ex;

values: nowraplinks collapsible show navbox-inner

values: vertical-align: -0.338ex; width:4.927ex; height:2.176ex;

values: vertical-align: -0.338ex; width:4.637ex; height:2.176ex;

values: vertical-align: -0.338ex; width:4.217ex; height:2.176ex;

values: vertical-align: -0.338ex; width:4.688ex; height:2.176ex;

values: vertical-align: -0.338ex; width:4.093ex; height:1.676ex;

values: vertical-align: -0.338ex; width:5.304ex; height:2.176ex;

values: vertical-align: -0.838ex; width:7.124ex; height:2.843ex;

values: vertical-align: -1.338ex; width:9.274ex; height:4.343ex;

values: vertical-align: -1.338ex; width:5.683ex; height:3.843ex;

values: vertical-align: -0.838ex; width:6.704ex; height:2.843ex;

values: vertical-align: -1.338ex; width:8.148ex; height:4.343ex;

values: vertical-align: -1.171ex; width:6.68ex; height:4.176ex;

values: vertical-align: -1.338ex; width:5.385ex; height:3.843ex;

values: vertical-align: -0.338ex; width:9.026ex; height:2.176ex;

values: vertical-align: -0.838ex; width:7.175ex; height:2.843ex;

values: vertical-align: -1.505ex; width:5.719ex; height:4.009ex;

values: vertical-align: -1.338ex; width:7.892ex; height:3.843ex;

values: vertical-align: -0.338ex; width:1.837ex; height:2.176ex;

values: vertical-align: -1.838ex; width:7.842ex; height:4.343ex;

values: vertical-align: -0.838ex; width:6.581ex; height:2.843ex;

values: vertical-align: -0.838ex; width:11.906ex; height:2.843ex;

values: vertical-align: -1.338ex; width:4.012ex; height:3.843ex;

values: vertical-align: -1.838ex; width:8.251ex; height:4.843ex;

values: vertical-align: -1.338ex; width:7.422ex; height:4.343ex;

values: vertical-align: -0.838ex; width:7.791ex; height:2.843ex;

values: vertical-align: -1.338ex; width:8.916ex; height:4.343ex;

values: vertical-align: -1.171ex; width:6.154ex; height:3.676ex;

values: vertical-align: -1.338ex; width:7.448ex; height:4.343ex;

values: vertical-align: -1.338ex; width:6.154ex; height:3.843ex;

values: vertical-align: -0.838ex; width:6.413ex; height:2.843ex;

values: vertical-align: -1.338ex; width:7.721ex; height:4.009ex;

values: vertical-align: -1.505ex; width:6.891ex; height:4.009ex;

values: vertical-align: -1.171ex; width:6.891ex; height:3.843ex;

values: vertical-align: -0.838ex; width:23.71ex; height:3.509ex;

values: vertical-align: -0.838ex; width:6.885ex; height:2.843ex;

values: vertical-align: -1.838ex; width:7.637ex; height:4.343ex;

values: vertical-align: -1.338ex; width:8.107ex; height:4.343ex;

values: vertical-align: -1.338ex; width:6.988ex; height:3.843ex;

values: vertical-align: -0.838ex; width:6.29ex; height:2.843ex;

values: vertical-align: -1.838ex; width:6.783ex; height:4.343ex;

values: vertical-align: -1.838ex; width:10.242ex; height:4.343ex;

values: vertical-align: -1.838ex; width:5.954ex; height:4.343ex;

values: vertical-align: -1.838ex; width:10.242ex; height:4.843ex;

values: vertical-align: -0.838ex; width:7.5ex; height:2.843ex;

values: vertical-align: -1.338ex; width:8.302ex; height:4.009ex;

values: vertical-align: -1.338ex; width:7.473ex; height:4.009ex;

values: vertical-align: -0.838ex; width:28.667ex; height:3.509ex;

values: vertical-align: -0.671ex; width:5.524ex; height:2.509ex;

values: vertical-align: -0.838ex; width:6.465ex; height:2.843ex;

values: vertical-align: -1.338ex; width:7.181ex; height:3.843ex;

values: vertical-align: -1.505ex; width:8.639ex; height:4.509ex;

values: vertical-align: -1.838ex; width:6.51ex; height:4.343ex;

values: vertical-align: -0.505ex; width:7.614ex; height:2.343ex;

values: vertical-align: -0.838ex; width:5.87ex; height:2.843ex;

values: vertical-align: -1.338ex; width:6.09ex; height:4.343ex;

values: vertical-align: -1.005ex; width:11.207ex; height:4.009ex;

values: vertical-align: -1.338ex; width:6.919ex; height:4.343ex;

values: vertical-align: -1.005ex; width:6.09ex; height:4.009ex;

values: vertical-align: -0.338ex; width:14.796ex; height:2.176ex;

values: vertical-align: -0.838ex; width:7.08ex; height:2.843ex;

values: vertical-align: -1.338ex; width:5.651ex; height:4.009ex;

values: vertical-align: -1.505ex; width:12.226ex; height:4.509ex;

values: {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}

values: vertical-align: -1.171ex; width:4.822ex; height:3.676ex;

values: vertical-align: -1.505ex; width:4.822ex; height:4.009ex;

values: vertical-align: -0.838ex; width:6.341ex; height:2.843ex;

values: vertical-align: -1.838ex; width:7.253ex; height:4.843ex;

values: vertical-align: -0.838ex; width:10.494ex; height:2.843ex;

values: vertical-align: -1.338ex; width:4.659ex; height:3.843ex;

values: vertical-align: -1.338ex; width:7.253ex; height:4.343ex;

values: vertical-align: -0.838ex; width:7.552ex; height:2.843ex;

values: vertical-align: -1.338ex; width:8.268ex; height:3.843ex;

values: vertical-align: -1.338ex; width:9.407ex; height:4.343ex;

values: vertical-align: -0.338ex; width:8.701ex; height:2.176ex;

values: vertical-align: -1.338ex; width:6.814ex; height:3.843ex;

values: vertical-align: -0.838ex; width:6.957ex; height:2.843ex;

values: vertical-align: -1.838ex; width:6.859ex; height:4.843ex;

values: vertical-align: -1.338ex; width:11.976ex; height:4.343ex;

values: vertical-align: -1.338ex; width:3.83ex; height:3.843ex;

values: vertical-align: -1.838ex; width:7.688ex; height:4.843ex;

values: Category:Use dmy dates from October 2017

values: Category:Wikipedia articles needing page number citations from November 2016

values: Category:Articles lacking in-text citations from July 2017

values: Category:All articles lacking in-text citations

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/Hooke%27s_law
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values: p-views navigation vectorTabs p-views-label

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: Guidance on how to use and edit Wikipedia

values: About the project, what you can do, where to find things

values: A list of recent changes in the wiki [ctrl-alt-r]

values: portal navigation p-wikibase-otherprojects p-wikibase-otherprojects-label

values:
uls-settings-trigger
Language settings
0
button
true

values: قانون هوك – Arabic

values: হুকের সূত্র – Bangla

values: Закон Гука – Belarusian

values: Закон на Хук – Bulgarian

values: Νόμος του Χουκ – Greek

values: قانون هوک – Persian

values: Հուկի օրենք – Armenian

values: हुक का नियम – Hindi

values: חוק הוק – Hebrew

values: ჰუკის კანონი – Georgian

values: Гук заңы – Kazakh

values: Хуков закон – Macedonian

values: ច្បាប់ហ៊ូក – Khmer

values: Закон Гука – Russian

values: Хуков закон – Serbian

values: ஊக்கின் விதி – Tamil

values: Закон Гука – Ukrainian

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Hooke_s_law rootpage-Hooke_s_law skin-vector action-view


text:
There's a sucker born every minute
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"There's a sucker born every minute" is a phrase closely associated with P. T. Barnum, an American showman of the mid-19th century, although there is no evidence that he said it. Early examples of its use are found instead among gamblers and confidence men.
Contents
[hide]
1 Attribution to Barnum
2 History
3 See also
4 References
5 External links
Attribution to Barnum[edit]
When Barnum's biographer, Arthur H. Saxon, tried to track down when Barnum had uttered this phrase, he was unable to verify it. According to Saxon, "There's no contemporary account of it, or even any suggestion that the word 'sucker' was used in the derogatory sense in his day. Barnum was just not the type to disparage his patrons."[1]
Photo of P. T. Barnum by Charles Eisenmann
Some sources claim the quote is most likely from famous con-man Joseph ("Paper Collar Joe") Bessimer,[2] and other sources say it was actually uttered by David Hannum, spoken in reference to Barnum's part in the Cardiff Giant hoax.[citation needed] Hannum, who was exhibiting the "original" giant and had unsuccessfully sued Barnum for exhibiting a copy and claiming it was the original, was referring to the crowds continuing to pay to see Barnum's exhibit even after both it and the original had been proven to be fakes.
Another source credits late 1860s Chicago "bounty broker, saloon and gambling-house keeper, eminent politician, and dispenser of cheating privileges..." Michael Cassius McDonald as the originator of the aphorism. According to the book Gem of the Prairie: Chicago Underworld (1940) by Herbert Asbury, when McDonald was equipping his gambling house known as The Store (at Clark and Monroe Streets in Chicago) his partner Harry Lawrence expressed concern over the large number of roulette wheels and faro tables being installed and their ability to get enough players to play the games. McDonald then allegedly said, "Don't worry about that, there's a sucker born every minute."[page needed]
History[edit]
Early uses of the phrase refer to it as a catch-phrase among gamblers. In an 1879 discussion of gambling in Chicago, an "old-timer" is quoted as saying, “[G]oodness knows how they live, it’s mighty hard times with the most of them; in the season they make a bit on base ball, or on the races, and then, you know, ‘there’s a sucker born every minute,’ and rigid city legislation drives the hard-up gambler, who would be a decent one of the kind, to turn skin-dealer and sure-thing player.”[3] The use of quotation marks indicates that it must already have been an established catch-phrase.
The phrase appears in print in the 1885 biography of confidence man Hungry Joe, The Life of Hungry Joe, King of the Bunco Men.[4][5]
In a slightly different form, the phrase shows up in the January 1806 European Magazine: "It was the observation of one of the tribe of Levi, to whom some person had expressed his astonishment at his being able to sell his damaged and worthless commodities, 'That there vash von fool born every minute.'"[6]
According to David W. Maurer, writing in The Big Con (1940),[7] there was a similar saying amongst con men: "There's a mark born every minute, and one to trim 'em and one to knock 'em." Here "trim" means to steal from, and "knock" means to persuade away from a scam. The meaning is that there is no shortage of new victims, nor of con men, nor of honest men.
In the 1930 John Dos Passos novel The 42nd Parallel, the quotation is attributed to Mark Twain.
See also[edit]
Forer effect or the Barnum effect
References[edit]
Jump up ^ Brooks, Andree (October 3, 1982). "Debunking the Myth of P. T. Barnum". The New York Times. Retrieved April 27, 2014.
Jump up ^ Saxon, A. H. (1989). P. T. Barnum: the Legend and the Man. Columbia University Press.
Jump up ^ "Gambling in Chicago". The Inter-Ocean. January 2, 1879. p. 5. Retrieved April 18, 2016 – via Newspapers.com.
Jump up ^ Jay, Ricky (February 2011). "Grifters, Bunco Artists & Flimflam Men". Wired (19.02): 90.
Jump up ^ The Life of Hungry Joe, King of the Bunco Men. New York: Frank Tousey. 1885. p. 22. OCLC 78310948.
Jump up ^ "Essay on False Genius". The European Magazine And London Review. January 1806.
Jump up ^ Maurer, David W. (1999) [1940]. The Big Con: The Story of the Confidence Man and the Confidence Game (reprinted). New York: Bobbs Merrill / Anchor Books. ISBN 0-385-49538-2.
External links[edit]
Quotations related to P. T. Barnum at Wikiquote
[hide]
v
t
e
P. T. Barnum
Acts and exhibits
Ringling Bros. and Barnum & Bailey Circus
Anna Haining Bates
Eli Bowen
Chang and Eng Bunker
Cardiff Giant
Circassian beauties
Henry Alexander Cooper
Captain George Costentenus
Nala Damajanti
John Diamond (dancer)
Admiral Dot
Colonel Routh Goshen
Fiji mermaid
Joice Heth
Fedor Jeftichew
Annie Jones (bearded woman)
Jumbo
William S. Hutchings
Nellie Keeler
Jenny Lind
Count Primo Magri
Master Juba
Maximo and Bartola
Millie and Christine McKoy
Edmund Newell
Commodore Nutt
Isaac W. Sprague
General Tom Thumb
Charles B. Tripp
Lavinia Warren
Minnie Warren
Wild Men of Borneo
Zip the Pinhead
Related articles
James "Grizzly" Adams
Barnum (musical)
Barnum's American Museum
Barnum's Aquarial Gardens
Barnum Museum
Bridgeport half dollar
William C. Coup
The Herald of Freedom
Iranistan
The Mighty Barnum
Moses Kimball
"There's a sucker born every minute"
The Greatest Showman
The Greatest Show on Earth
Retrieved from "https://en.wikipedia.org/w/index.php?title=There%27s_a_sucker_born_every_minute&oldid=796181064"
Categories:
English phrases
Hidden categories:
All articles with unsourced statements
Articles with unsourced statements from April 2016
Wikipedia articles needing page number citations from April 2014
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
Languages
Русский
Edit links
This page was last edited on 19 August 2017, at 01:47.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

key: html

type: local

value: div

attrs: id class

values: mw-page-base noprint

values: mw-head-base noprint

value: a

attrs: id class lang

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

value: b

attrs: href title

value: p

attrs: role tabindex class

values: button 0 togglelink

value: span

value: li

values: This claim needs references to reliable sources. (April 2016)

values:
noprint Inline-Template Template-Fact
white-space:nowrap;

values: This citation requires a reference to the specific page or range of pages in which the material appears. (April 2014)

values:
/wiki/The_42nd_Parallel
mw-redirect
The 42nd Parallel

values:
/wiki/Forer_effect
mw-redirect
Forer effect

attrs: alt src width height srcset data-file-width data-file-height

values:
nofollow
external text
//www.worldcat.org/oclc/78310948

values:
reflist mw-references-wrap
list-style-type: decimal;

values:
/wiki/James_%22Grizzly%22_Adams
mw-redirect
James "Grizzly" Adams

values:
/wiki/Bridgeport_half_dollar
mw-redirect
Bridgeport half dollar

values: nowraplinks collapsible autocollapse navbox-inner

values: Category:All articles with unsourced statements

values: Category:Articles with unsourced statements from April 2016

values: Category:Wikipedia articles needing page number citations from April 2014

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: portal navigation p-interaction p-interaction-label

values: List of all English Wikipedia pages containing links to this page [ctrl-alt-j]

values: Recent changes in pages linked from this page [ctrl-alt-k]

values:
/wiki/Wikipedia:File_Upload_Wizard
Upload files [ctrl-alt-u]
u

values: A list of all special pages [ctrl-alt-q]

values: Permanent link to this revision of the page

values: Link to connected data repository item [ctrl-alt-g]

values: Printable version of this page [ctrl-alt-p]

values: Каждую минуту рождается лох – Russian

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-There_s_a_sucker_born_every_minute rootpage-There_s_a_sucker_born_every_minute vector-nav-directionality skin-vector action-view