Hooke's law: the force is proportional to the extensionBourdon tubes are based on Hooke's law. The force created by gaspressure inside the coiled metal tube above unwinds it by an amount proportional to the pressure.Thebalance 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.
Inphysics,Hooke's law is anempirical law which states that theforce (F) needed to extend or compress aspring by some distance (x)scales linearly with respect to that distance—that is,Fs =kx, wherek is a constant factor characteristic of the spring (i.e., itsstiffness), andx is small compared to the total possible deformation of the spring.
The law is named after 17th-century British physicistRobert Hooke. He first stated the law in 1676 as a Latinanagram.[1][2] He published the solution of his anagram in 1678[3] as:ut tensio, sic vis ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. It is the fundamental principle behind thespring scale, themanometer, thegalvanometer, and thebalance wheel of themechanical clock.
The equation holds in many situations where anelastic body isdeformed. An elastic body or material for which this equation can be assumed is said to belinear-elastic orHookean. Hooke's law is afirst-order linear approximation to the real response of springs and other elastic bodies to applied forces. It fails once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before thoseelastic limits are reached.
Definition
The moderntheory of elasticity generalizes Hooke's law to say that thestrain (deformation) of an elastic object or material is proportional to thestress 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 alinear map (atensor) that can be represented by amatrix 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 ahomogeneous rod with uniformcross section will behave like a simple spring when stretched, with a stiffnessk directly proportional to its cross-section area and inversely proportional to its length.
Linear springs
Elongation and compression of a spring
Consider a simplehelical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude isFs. Suppose that the spring has reached a state ofequilibrium, where its length is not changing anymore. Letx be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that or, equivalently,wherek 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, withFs andx both negative in that case.[4]
Graphical derivation
According to this formula, thegraph of the applied forceFs as a function of the displacementx will be a straight line passing through theorigin, whoseslope isk.
Hooke's law for a spring is also stated under the convention thatFs is therestoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes since the direction of the restoring force is opposite to that of the displacement.
Torsional springs
Thetorsional analog of Hooke's law applies totorsional 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 angulardeformation due to torsion. Mathematically, it can be expressed as:
k is thetorsional constant (measured in N·m/radian), which characterizes the stiffness of the torsional spring or the resistance to angular displacement.
θ is theangular 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 byshearing, rather than stretching or compression, the shearing forceFs and the sideways displacement of the platesx 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 weightF placed at some intermediate point. The displacementx 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 itsaxis, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, ifFs andx are defined asvectors, Hooke'sequation still holds and says that the force vector is theelongation vector multiplied by a fixedscalar.
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, themagnitude of the displacementx will be proportional to the magnitude of the forceFs, 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 lawFs = −kx will hold. However, the force and displacementvectors will not be scalar multiples of each other, since they have different directions. Moreover, the ratiok between their magnitudes will depend on the direction of the vectorFs.
Yet, in such cases there is often a fixedlinear relation between the force and deformation vectors, as long as they are small enough. Namely, there is afunctionκ from vectors to vectors, such thatF =κ(X), andκ(αX1 +βX2) =ακ(X1) +βκ(X2) for any real numbersα,β and any displacement vectorsX1,X2. Such a function is called a (second-order)tensor.
With respect to an arbitraryCartesian coordinate system, the force and displacement vectors can be represented by 3 × 1matrices of real numbers. Then the tensorκ connecting them can be represented by a 3 × 3 matrixκ of real coefficients, that, whenmultiplied by the displacement vector, gives the force vector:
That is, fori = 1, 2, 3. Therefore, Hooke's lawF =κX can be said to hold also whenX andF are vectors with variable directions, except that the stiffness of the object is a tensorκ, rather than a single real numberk.
(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 acontinuous elastic material (such as a block of rubber, the wall of aboiler, 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, thestrain tensorε (in lieu of the displacementX) and thestress tensorσ (replacing the restoring forceF). The analogue of Hooke's spring law for continuous media is then wherec is a fourth-order tensor (that is, a linear map between second-order tensors) usually called thestiffness tensor orelasticity tensor. One may also write it as where the tensors, called thecompliance tensor, represents the inverse of said linear map.
In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices
Being a linear mapping between the nine numbersσij and the nine numbersεkl, the stiffness tensorc is represented by a matrix of3 × 3 × 3 × 3 = 81 real numberscijkl. Hooke's law then says thatwherei,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 tensorε merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensorσ 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 tensorc, on the other hand, is a property of the material, and often depends on physical state variables such as temperature,pressure, andmicrostructure.
Due to the inherent symmetries ofσ,ε, andc, only 21 elastic coefficients of the latter are independent.[6] This number can be further reduced by the symmetry of the material: 9 for anorthorhombic crystal, 5 for anhexagonal structure, and 3 for acubic symmetry.[7] Forisotropic media (which have the same physical properties in any direction),c can be reduced to only two independent numbers, thebulk modulusK and theshear modulusG, 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 offluids, or thepolarization of adielectric by anelectric field.
In particular, the tensor equationσ =cε relating elastic stresses to strains is entirely similar to the equationτ =με̇ relating theviscous stress tensorτ and thestrain rate tensorε̇ in flows ofviscous fluids; although the former pertains tostatic stresses (related toamount of deformation) while the latter pertains todynamical stresses (related to therate of deformation).
Units of measurement
InSI units, displacements are measured in meters (m), and forces innewtons (N or kg·m/s2). Therefore, the spring constantk, and each element of the tensorκ, is measured in newtons per meter (N/m), or kilograms per second squared (kg/s2).
For continuous media, each element of the stress tensorσ is a force divided by an area; it is therefore measured in units of pressure, namelypascals (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensorε aredimensionless (displacements divided by distances). Therefore, the entries ofcijkl are also expressed in units of pressure.
General application to elastic materials
Stress–strain curve for low-carbon steel, showing the relationship between thestress (force per unit area) andstrain (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).
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 itselastic range (i.e., for stresses below theyield strength). For some other materials, such as aluminium, Hooke's law is only valid for a portion of the elastic range. For these materials aproportional 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.
A rod of anyelastic material may be viewed as a linearspring. The rod has lengthL and cross-sectional areaA. Itstensile stressσ is linearly proportional to its fractional extension or strainε by themodulus of elasticityE:
The modulus of elasticity may often be considered constant. In turn,(that is, the fractional change in length), and sinceit follows that:
The change in length may be expressed as
Spring energy
The potential energyUel(x) stored in a spring is given by which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative. Substituting gives
This potentialUel can be visualized as aparabola on theUx-plane such thatUel(x) =1/2kx2. As the spring is stretched in the positivex-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate:Note that the change in the change inU is constant even when the displacement and acceleration are zero.
Relaxed force constants (generalized compliance constants)
Relaxed force constants (the inverse of generalizedcompliance 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 forreactants,transition states, and products of achemical reaction. Just as thepotential 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 termedcompliance 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) ascovalent bond strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.[9]
A mass suspended by a spring is the classical example of a harmonic oscillator
A massm attached to the end of a spring is a classic example of aharmonic oscillator. By pulling slightly on the mass and then releasing it, the system will be set insinusoidal oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglectfriction and the mass of the spring, the amplitude of the oscillation will remain constant; and itsfrequencyf will be independent of its amplitude, determined only by the mass and the stiffness of the spring:This phenomenon made possible the construction of accuratemechanical clocks and watches that could be carried on ships and people's pockets.
Rotation in gravity-free space
If the massm were attached to a spring with force constantk and rotating in free space, the spring tension (Ft) would supply the requiredcentripetal force (Fc):
SinceFt =Fc andx =r, then:Given thatω = 2πf, this leads to the same frequency equation as above:
For an analogous development for viscous fluids, seeViscosity.
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 thetrace 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 inindex notation:
The first term on the right is the constant tensor, also known as thevolumetric strain tensor, and the second term is the traceless symmetric tensor, also known as thedeviatoric 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:
Using the relationships between theelastic 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]
whereλ =K −2/3G =c1111 − 2c1212 andμ =G =c1212 are theLamé constants,I is the second-rank identity tensor, andI is the symmetric part of the fourth-rank identity tensor. In index notation:
This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form iswhereE isYoung's modulus andν isPoisson's ratio. (See3-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),whereν is Poisson's ratio andE is Young's modulus.
We get similar equations to the loads in directions 2 and 3,and
Summing the three cases together (εi =εi′ +εi″ +εi‴) we getor by adding and subtracting oneνσand further we get by solvingσ1
Calculating the sumand substituting it to the equation solved forσ1 giveswhereμ andλ are theLamé 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 aswhereγij = 2εij is theengineering shear strain. The inverse relation may be written aswhich can be simplified thanks to the Lamé constants:In vector notation this becomeswhereI is the identity tensor.
Plane stress
Underplane stress conditions,σ31 =σ13 =σ32 =σ23 =σ33 = 0. In that case Hooke's law takes the form
In vector notation this becomes
The inverse relation is usually written in the reduced form
Plane strain
Underplane strain conditions,ε31 =ε13 =ε32 =ε23 =ε33 = 0. In this case Hooke's law takes the form
Anisotropic materials
The symmetry of theCauchy stress tensor (σij =σji) and the generalized Hooke's laws (σij =cijklεkl) implies thatcijkl =cjikl. Similarly, the symmetry of theinfinitesimal strain tensor implies thatcijkl =cijlk. These symmetries are called theminor symmetries of the stiffness tensorc. 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), thenThe arbitrariness of the order of differentiation implies thatcijkl =cklij. These are called themajor 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 calledVoigt 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) asThen the stiffness tensor (c) can be expressed as
and Hooke's law is written as
Similarly the compliance tensor (s) can be written as
Change of coordinate system
If a linear elastic material is rotated from a reference configuration to another, then the material is symmetric with respect to the rotation if the components of the stiffness tensor in the rotated configuration are related to the components in the reference configuration by the relation[13]
wherelab are the components of anorthogonal rotation matrix[L]. The same relation also holds for inversions.
In matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by
then
In addition, if the material is symmetric with respect to the transformation[L] then
Gij is theshear modulus in directionj on the plane whose normal is in directioni
νij is thePoisson's ratio that corresponds to a contraction in directionj when an extension is applied in directioni.
Underplane stress conditions,σzz =σzx =σyz = 0, Hooke's law for an orthotropic material takes the formThe inverse relation isThe transposed form of the above stiffness matrix is also often used.
Transversely isotropic materials
Atransversely isotropic material is symmetric with respect to a rotation about anaxis of symmetry. For such a material, ife3 is the axis of symmetry, Hooke's law can be expressed as
More frequently, thex ≡e1 axis is taken to be the axis of symmetry and the inverse Hooke's law is written as[15]
Universal elastic anisotropy index
To grasp the degree of anisotropy of any class, auniversal elastic anisotropy index (AU)[16] was formulated. It replaces theZener ratio, which is suited forcubic crystals.
Thermodynamic basis
Linear deformations of elastic materials can be approximated asadiabatic. Under these conditions and for quasistatic processes thefirst law of thermodynamics for a deformed body can be expressed aswhereδU is the increase ininternal energy andδW is thework done by external forces. The work can be split into two termswhereδWs is the work done bysurface forces whileδWb is the work done bybody forces. Ifδu is avariation of the displacement fieldu in the body, then the two external work terms can be expressed aswheret is the surfacetraction vector,b is the body force vector,Ω represents the body and∂Ω represents its surface. Using the relation between theCauchy stress and the surface traction,t =n ·σ (wheren is the unit outward normal to∂Ω), we haveConverting thesurface integral into avolume integral via thedivergence theorem givesUsing the symmetry of the Cauchy stress and the identitywe have the following
From the definition ofstrain and from the equations ofequilibrium we haveHence we can writeand therefore the variation in theinternal energy density is given byAnelastic material is defined as one in which the total internal energy is equal to thepotential energy of the internal forces (also called theelastic 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 asSince the variation of strain is arbitrary, the stress–strain relation of an elastic material is given byFor a linear elastic material, the quantity∂U0/∂ε is a linear function ofε, and can therefore be expressed aswherec is a fourth-rank tensor of material constants, also called thestiffness tensor. We can see whyc must be a fourth-rank tensor by noting that, for a linear elastic material,In index notation
The right-hand side constant requires four indices and is a fourth-rank quantity. We can also see that this quantity must be a tensor because it is a linear transformation that takes the strain tensor to the stress tensor. We can also show that the constant obeys the tensor transformation rules for fourth-rank tensors.
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^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.PMID22461011.
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Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two quantities among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part).
