Hooke's_law - Wikiparency

# Hooke's law **Hooke's law** states that the force needed to extend or compress an elastic body by some distance is proportional to that distance: `F = k·x`, where k is a constant characteristic of the body. [[Robert_Hooke|Robert Hooke]] announced it in 1676 as the anagram *ceiiinosssttuv* and revealed it two years later as *ut tensio, sic vis* — as the extension, so the force.[^hooke1678] Written for a [[Spring_(device)|spring]] whose restoring force opposes the displacement, the law is usually given as `F = −k·x`, and k is then the spring constant, or spring rate. The law is an approximation, not a fundamental principle: it is the first term in the expansion of any smooth force–displacement relation about an unloaded equilibrium, and it therefore holds for every elastic body provided the deformation is small enough. What counts as small enough differs by material and by geometry. A steel bar obeys it to a few parts in a thousand of strain and then [[Yield_(engineering)|yields]]; a coil spring obeys it until its coils touch; rubber departs from it almost at once. Applied to a continuum rather than to a part, the same proportionality becomes the generalised Hooke's law linking the [[Stress_(mechanics)|stress]] tensor to the [[Strain_(mechanics)|strain]] tensor, which is the foundation of [[Linear_elasticity|linear elasticity]] and of nearly all structural analysis. The framework microsim *Hooke's law: a spring is a line, and so is a bar* draws the force–extension line for either a catalogue coil spring or a metal bar: its slope is the stiffness k, a catalogue number for the spring and `E·A/L` for the bar, the shaded triangle beneath it is the stored energy, and the line stops where the physical object does — the coils go solid, or the metal yields. ## Definition In its scalar form the law relates one force to one displacement measured along the same line, `F = −k·x`, with x measured from the unloaded position. The minus sign records that the force is restoring: a stretched spring pulls back, a compressed one pushes back. The constant k has the units of force per length and is a property of the whole object, not of the material it is made from — geometry and material together set it, which is the distinction between a spring rate and a [[Young's_modulus|modulus]] that the rest of this article turns on. ### Linear springs A helical coil spring is the canonical linear element. Its rate follows from the [[Torsion_(mechanics)|torsion]] of the wire as the coils wind and unwind, and for design purposes it is simply read from a catalogue. The microsim carries three catalogue springs transcribed from the portal's design text, each a candidate for holding 100 N inside 20 mm of travel.[^jensen109] The middle one, S2, has a rate of 9 N/mm, a free length of 40 mm and a solid length of 20 mm: at 100 N it deflects `100/9` = 11.1 mm, so the loaded length is 28.9 mm, and the line ends at `9 × (40 − 20)` = 180 N, the load at which the coils touch.[^simhooke] Springs stack the way resistors do, but the other way around. In series the same force passes through each and the extensions add, so the compliances add and `1/k = Σ 1/kᵢ`; two S2 springs in series make 4.5 N/mm and deflect 22.2 mm under the same 100 N. In parallel the extension is shared and the forces add, so `k = Σ kᵢ`; two S2 springs side by side make 18 N/mm and deflect 5.6 mm. Series stacking is how a suspension gets a soft ride out of stiff parts, and parallel stacking is how a valve gets a fail-safe second spring. *Try: load the catalogue spring S2 to 100 N and read the slope of the force–extension line as the rate, then switch the stacking select from single to series and to parallel and watch the same 100 N land at 22.2, 11.1 and 5.6 mm while the shaded energy triangle grows and shrinks with it.* ### Torsional springs The rotational counterpart replaces force by torque and displacement by angle: `τ = −κ·θ`, with κ the torsional rate in newton-metres per radian. A spiral hairspring in a mechanical watch, a [[Torsion_spring|torsion bar]] in a vehicle [[Car_suspension|suspension]] and the suspension fibre of a torsion balance all obey it, and all of them fail it in the same way — by running out of travel or by yielding. The balance wheel of a watch is a torsional spring and an inertia together, and keeps time for the same reason a mass on a coil spring keeps a period. ### General "scalar" springs Nothing in the law requires a spring. Any system whose configuration is described by a single coordinate, and which sits at a stable equilibrium, obeys a linear force–displacement law for small departures from that equilibrium, with the [[Stiffness|stiffness]] `k = dF/dx` evaluated at the operating point. A diving board, the air in a closed cylinder, a gas strut, a length of taut cable and the [[Bending|bending]] stiffness of a beam at its tip are all "springs" in this sense, and mixing them in one diagram — a tyre in series with a coil, a mount in parallel with a bush — is ordinary practice in [[Vibration_isolation|vibration isolation]]. ### Vector formulation For a body that can move in more than one direction, displacement and force are vectors and the law reads `F = −k·x` with k a scalar only when the spring is isotropic — that is, when it is equally stiff in every direction, so that the restoring force always points back along the displacement. Real mounts are not isotropic, and the general linear relation is `F = −K·x` with K a stiffness matrix; the force then need not be parallel to the displacement, and the directions in which it is are the eigenvectors of K. ### General tensor form Inside a material the same idea is written between two tensors. The generalised Hooke's law is `σij = cijkl·εkl`, with c the fourth-order stiffness tensor. Its 81 components reduce to 36 by the symmetry of the stress and strain tensors, to 21 by the existence of a strain energy function, and to 2 for an isotropic material — the numbers usually quoted as [[Young's_modulus|E]] and [[Poisson's_ratio|ν]]. ### Hooke's law for continuous media Written for a continuum, the law loses all reference to a particular object: it is a statement about the material at a point, and the object's stiffness must be recovered by integrating it over the geometry. That is why a single figure for steel, `E` ≈ 207 GPa, serves for a bar, a beam, a shaft and a shell, while each of those has its own k.[^callister] ## Analogous laws Hooke's law is one member of a family of linear constitutive laws, all of which assert that a flux or response is proportional to the driving quantity for small departures from equilibrium. [[Ohm's_law|Ohm's law]], `V = I·R`, plays exactly the same role in circuits that Hooke's law plays in structures, and the analogy is exploited directly whenever a mechanical system is analysed as an electrical one. Fourier's law of [[Thermal_conduction|heat conduction]], Fick's law of [[Diffusion|diffusion]], Darcy's law for flow through porous media and Newton's law of [[Viscosity|viscosity]] have the same shape. The family resemblance is not a coincidence. Each is the leading term of a Taylor expansion about a state of equilibrium, so each is exact in the limit of small disturbance and each acquires corrections of the same kind — nonlinearity at large amplitude, hysteresis, and rate dependence. The mechanical analogue of an electrical circuit's resistance is the dashpot, which contributes a force proportional to velocity instead of to displacement, and combining springs and dashpots gives the standard [[Viscoelasticity|viscoelastic]] models used for polymers and for damping materials. ## Units of measurement The spring constant k has the units of force per length. The SI unit is the [[Newton_(unit)|newton]] per metre, but design practice works in newtons per millimetre because the numbers are more convenient: the catalogue spring above is a 9 N/mm spring, not a 9,000 N/m one, though the two are the same. United States catalogues quote pounds-force per inch, and the conversion is `1 N/mm = 5.710 lbf/in`, so the 9 N/mm spring is a 51.4 lbf/in spring. The torsional rate κ is in newton-metres per radian; care is needed because some catalogues quote it per degree or per turn, a factor of 57.3 or 6.28 either way. The components of the stiffness tensor, and the moduli derived from them, have the units of stress — [[Pascal_(unit)|pascals]], or in practice gigapascals — because strain is dimensionless. Compliance, the reciprocal of stiffness, is quoted in metres per newton or in reciprocal pascals, and appears whenever springs in series or materials under a fixed load are being added up. ## General application to elastic materials Hooke's law describes a region, not a material. Every real force–extension curve is straight near the origin and stops being straight somewhere, and the useful question is always where. For metals the departure is the onset of plastic flow: the [[Stress–strain_curve|stress–strain curve]] is linear to the proportional limit, and beyond the yield strength the material keeps part of its deformation permanently. Loading a bar of 6061-T6 aluminium of 5 mm² section with 2,000 N puts it at 400 MPa, well past the yield strength of that alloy, and the microsim marks the point where the straight line stops meaning anything.[^callister][^simhooke] For a coil spring the limit is usually geometric rather than material: the coils touch at the solid length and the spring becomes, abruptly, a short steel tube. Materials fail the law in other ways too. Rubber and soft biological tissue are nonlinear from the start and are treated as [[Hyperelastic_material|hyperelastic]] rather than linear. Concrete and cast iron are much stiffer and stronger in compression than in tension. Polymers and wood creep under sustained load, so their apparent stiffness depends on how long the load has been there, and many materials show hysteresis — a loading curve that does not retrace itself on unloading, the area between the two being the energy lost per cycle. *Try: open the Spring (device) variant on catalogue spring S2 at its 100 N duty load, then raise the load past 180 N and watch the operating point stop at the vertical wall labelled "coils touch": the line does not continue, because the spring has become a solid stack 20 mm high.* ## Derived formulae ### Tensional stress of a uniform bar A prismatic bar of length L and cross-sectional area A, pulled along its axis, is a spring whose rate follows from the material's [[Young's_modulus|Young's modulus]]: combining `σ = F/A`, `ε = x/L` and `σ = E·ε` gives `k = E·A/L`, so stiffness rises with the modulus and the area and falls with the length.[^up12-3] A steel bar 1 m long of 50 mm² section, with `E` = 207 GPa, has `k` = 10.35 kN/mm — more than a thousand times the catalogue coil spring, from an object that looks nothing like a spring. Under 1,000 N it carries a stress of 20 MPa, a strain of 9.66 × 10⁻⁵, and stretches 0.0966 mm.[^simhooke] The elastic stretch of metal parts is almost always this small, which is why the sim draws a bar's stretch two hundred times life size and says so on the picture: the exaggeration is ILLUSTRATIVE, and the readout, not the drawing, carries the number. Where the drawn stretch would still run off the picture the sim clamps it and labels it clamped.[^simhooke] *Try: open the Young's modulus variant on the 1020 steel bar, 1 m long and 50 mm² in section, at 1,000 N, then switch the material through the six-metal table and watch the slope of the line — and the stretch readout — change by a factor of three between steel and aluminium while the bar's dimensions never move.* ### Spring energy The work done in stretching a linear spring is the area under its force–extension line, a triangle of base x and height kx, so the stored [[Potential_energy|potential energy]] is `U = ½·k·x²`. It grows with the square of the deflection, so the last millimetre of travel stores far more than the first. The catalogue spring S2 at 100 N holds `½ × 9,000 × 0.0111²` = 0.56 J.[^simhooke] That is a modest number, and it is why coil springs are poor [[Energy_storage|energy stores]] by mass compared with a [[Flywheel|flywheel]] or a battery, while being excellent at returning a known force over a known travel. The same expression, integrated over a body as `½·σ:ε` per unit volume, is the strain energy density that underlies [[Castigliano's_method|Castigliano's method]] and the energy formulations of the [[Finite_element_method|finite element method]]. ### Relaxed force constants (generalized compliance constants) Inverting the stiffness matrix of a multi-coordinate system gives its compliance matrix, and the diagonal elements of that inverse are the relaxed force constants. The distinction matters whenever several coordinates are coupled: a stiffness element answers the question "what force is needed to move this coordinate while holding all the others fixed", whereas a compliance element answers "how far does this coordinate move under a unit force when all the others are free to relax". The second question is usually the physically meaningful one, and in molecular systems the relaxed constants are used for exactly that reason, as a measure of the strength of an individual [[Chemical_bond|chemical bond]] that does not depend on how the remaining internal coordinates were chosen. ### Harmonic oscillator A mass m on a spring of rate k obeys `m·x'' + k·x = 0`, whose solution is [[Simple_harmonic_motion|simple harmonic motion]] at the angular frequency `ω = √(k/m)`, period `T = 2π·√(m/k)`. A 1 kg mass on the catalogue spring S2 oscillates at 94.9 rad/s, or 15.1 Hz. The frequency depends on the ratio of stiffness to mass and not at all on the amplitude, which is what makes a spring a timekeeper and what makes a structure's natural frequencies a property of the structure. Adding a dashpot gives the [[Damping|damped]] oscillator and, with a driving force, [[Resonance|resonance]] — the reason [[Vibration_isolation|isolation mounts]] are chosen so that the disturbing frequency lands well above `√(k/m)`, not near it. ### Rotation in gravity-free space A mass on a spring swung about the spring's fixed end, far from gravity, shows the law and its limits in one picture. The spring must supply the [[Centripetal_force|centripetal force]], so `k·x = m·ω²·(L₀ + x)` for a natural length L₀, giving `x = m·ω²·L₀ / (k − m·ω²)`. The extension grows without bound as ω approaches `√(k/m)`, the same natural frequency as before: at the resonant spin rate no finite extension can supply the force required. Taking the S2 spring with a 1 kg mass and its 40 mm free length, a spin of 50 rad/s stretches it 15.4 mm, and by about 55 rad/s the extension has reached 20 mm and the coils are solid — the real spring runs out of geometry long before the idealised one runs out of algebra. ## Linear elasticity theory for continuous media For a continuum the law is written between tensors, `σ = c : ε`, and the whole of [[Linear_elasticity|linear elasticity]] follows from it together with the equilibrium equations and the strain–displacement relations. The stiffness tensor's symmetry, and hence how many independent constants a material has, is decided by the symmetry of the material itself. ### Isotropic materials An isotropic material has the same properties in every direction, and just two independent elastic constants. The pair usually quoted is Young's modulus E and [[Poisson's_ratio|Poisson's ratio]] ν, from which the [[Shear_modulus|shear modulus]] and the [[Bulk_modulus|bulk modulus]] follow: `G = E / (2(1 + ν))`, `K = E / (3(1 − 2ν))`. For 6061-T6 aluminium, with `E` = 68.9 GPa and `ν` = 0.33, the shear modulus is 25.9 GPa.[^callister][^simhooke] Poisson's ratio is the lateral contraction per unit axial extension: a bar pulled along its length gets thinner across it, and the effect is why a stretched rubber band narrows visibly and why a bar in tension loses a little volume less than one would guess. Thermodynamic stability requires ν between −1 and 0.5, and the upper bound is the incompressible limit — rubber sits close to it at about 0.4999, which is why rubber blocks are stiff in compression when confined and soft when free to bulge. Two two-dimensional reductions dominate practice. Plane stress assumes the stress through the thickness vanishes, which suits a thin plate loaded in its own plane; plane strain assumes the strain along one axis vanishes, which suits a long body such as a dam, a tunnel lining or a rolled section. They are different problems with different effective stiffnesses, and choosing the wrong one is a common source of error in [[Finite_element_method|finite element]] modelling. *Try: open the Poisson's ratio variant on the aluminium bar, 20 mm² at 1,500 N, and drag nu from 0 to 0.5: the drawn bar thins against its unloaded outline — the lateral change is drawn five hundred times life size, ILLUSTRATIVE like the axial stretch — while the shear modulus readout falls from 34.5 to 23.0 GPa.* ### Anisotropic materials Materials with directional structure need more constants. In Voigt notation the stiffness tensor becomes a symmetric 6 × 6 matrix, with 21 independent entries in the fully anisotropic case, called triclinic. An [[Orthotropic_material|orthotropic material]] — one with three mutually perpendicular planes of symmetry — needs 9: [[Wood|wood]], rolled sheet metal and a woven composite ply are orthotropic to a good approximation. A transversely isotropic material, such as a unidirectional fibre composite or many sedimentary rocks, needs 5, and a cubic single crystal needs 3. Single crystals of metals are strongly anisotropic even in the cubic case, and a dimensionless index constructed from the bounds on the aggregate moduli is used to say by how much. Anisotropy is a design variable, not only a nuisance. A [[Composite_material|composite]] laminate is stacked ply by ply so that the stiffness matrix of the whole has the directional properties the part needs, and the laminate's behaviour under combined loads is computed by summing the plies' contributions with each ply's stiffness rotated into the laminate frame. ## Thermodynamic basis Hooke's law can be derived rather than assumed. The elastic energy of a body is a function of its state of strain, and expanding that function about the unstressed, unstrained configuration gives a constant term, no linear term — because the unstressed state is an equilibrium, where the first derivative vanishes — and a quadratic term whose coefficients are the components of the stiffness tensor. Differentiating the quadratic gives a stress linear in strain, which is Hooke's law; the higher terms of the expansion are the nonlinear elastic corrections that matter at large strain. The symmetry `cijkl = cklij`, which cuts the independent constants from 36 to 21, is exactly the statement that the mixed second derivatives of the energy function commute. Because the expansion is of a thermodynamic potential, the moduli depend on what is held fixed during the deformation. The isothermal moduli, measured slowly enough for heat to flow, differ slightly from the adiabatic moduli that govern fast deformations such as the passage of a sound wave; for metals the difference is under a per cent, and for gases it is the factor that separates the isothermal from the adiabatic bulk modulus. The deeper distinction is what stores the energy. In metals and ceramics, stretching raises the internal energy by displacing atoms in their potential wells — energy elasticity. In [[Natural_rubber|rubber]], stretching mainly straightens coiled molecules and so lowers the [[Entropy|entropy]], which is why a stretched rubber band warms as it is stretched and contracts when heated, the reverse of ordinary [[Thermal_expansion|thermal expansion]]; the observation is known as the Gough–Joule effect.[^treloar] ## Minnesota *This section is specific to Wikitube.* The most consequential application of Hooke's law in [[Minnesota]] is one nobody sees: the space left for a structure to move. A bar restrained against a temperature change carries the stress the change would have produced as free strain, `σ = E·α·ΔT`, with no load applied at all. Minnesota's recorded extreme low is −51 °C (−60 °F), measured at Tower on February 2, 1996,[^mndnr] and summer surfaces in the state run far above air temperature, so a member fully restrained across a 100 K swing would pick up about 248 MPa in steel, taking `E` = 207 GPa and a [[Thermal_expansion|coefficient of expansion]] of 12 × 10⁻⁶ K⁻¹ — comparable with the yield strength of ordinary structural steel, from the weather alone. The answer is not to make the structure stronger but to let it move. Expansion joints in bridge decks and pavements, sliding and elastomeric bearings under girders, and the loops built into long pipe runs all exist so that `α·ΔT` is taken up as displacement rather than as strain, and the small residual stiffness of the bearing, rather than `E·A/L` of the whole member, sets the force that remains. Elastomeric bearings are springs in the sense of this article, chosen from a catalogue by rate exactly as the microsim's coil springs are; the difference between a joint that works through a Minnesota winter and one that does not is usually a stiffness, not a strength. ## See also - [[Elasticity_(physics)]] - [[Young's_modulus]] - [[Poisson's_ratio]] - [[Spring_(device)]] - [[Stress_(mechanics)]] - [[Strain_(mechanics)]] - [[Stress–strain_curve]] - [[Linear_elasticity]] - [[Simple_harmonic_motion]] - [[Shear_modulus]] - [[Bulk_modulus]] - [[Viscoelasticity]] ## References [^hooke1678]: Hooke, Robert (1678). *Lectures de Potentia Restitutiva, or of Spring, Explaining the Power of Springing Bodies*. London: John Martyn. (Publisher as recalled; the 1676 anagram *ceiiinosssttuv*, its 1678 solution *ut tensio, sic vis*, and the year are standard in histories of mechanics.) [^up12-3]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. Chapter 12 "Static Equilibrium and Elasticity," §12.3 "Stress, Strain, and Elastic Modulus," pp. 588–591 (stress and strain, Young's modulus, the tensile bar). https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077). [^jensen109]: Jensen, Alfred (and later editors). *Introduction to Mechanical Design and Manufacturing*. Portal Books 109, pp. 173–175: helical springs, the spring rate, and the worked catalogue selection of a spring to hold a 100 N load inside 20 mm of travel (options S1–S3 tabulated on p. 175). The microsim's three catalogue springs are transcribed from that table. [^simhooke]: Portal engineering pack, `design.spring` — `CATALOG` (the book 109 p. 175 table: rating, outside diameter, free and solid length, rate), `length` for the loaded length clamped at the solid height, `solidForce`, `series` and `parallel` for the stacking, and `barStiffness` for `E·A/L` — with `design.stress.axial` and `strain`, `solid.mech.MATERIALS` for E and the yield strengths, and `mech.dynamics.springEnergy`; sim spec `specs/sims/Hooke's_law.json`. Hand-checked in the run report of 2026-09-18 (S2 at 100 N: x = 11.1 mm, loaded length 28.89 mm, solid at 180 N, U = 0.56 J; steel bar 1 m × 50 mm²: k = 10.35 kN/mm, σ = 20 MPa, ε = 9.66 × 10⁻⁵, x = 0.0966 mm). ILLUSTRATIVE, and labelled as such on the sim's picture sheet: a bar's stretch is drawn ×200 and clamped when it would still overflow, and the springs are drawn to a per-object scale rather than to a common one. [^callister]: Callister, William D.; Rethwisch, David G. *Materials Science and Engineering: An Introduction*, Appendix B (properties of selected engineering materials): the modulus, Poisson's ratio and yield strength values used by the microsim's material table, including 1020 steel (E ≈ 207 GPa) and 6061-T6 aluminium (E ≈ 68.9 GPa, ν ≈ 0.33, yield ≈ 276 MPa). Edition and page not re-checked for this article; Appendix B carries these tables in every edition. [^treloar]: Treloar, L. R. G. (1975). *The Physics of Rubber Elasticity*, 3rd edition. Oxford: Clarendon Press. The entropic origin of rubber elasticity and the Gough–Joule effect. (Chapter and pages not re-checked for this article.) [^mndnr]: Minnesota Department of Natural Resources, State Climatology Office. "Minnesota's record low temperature": −60 °F at Tower, February 2, 1996. https://www.dnr.state.mn.us/climate/index.html  **Microsim — three.js (Wikitube framework):** *Hooke's law* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Hooke's_law.html" data-title="Hooke's law"></div> *Built from `MICROSIM_GUIDE/specs/sims/Hooke's_law.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Hooke's_law) : [Wikitube](https://en.wikitube.io/wiki/Hooke's_law) - skeleton pinned to revision 1369639884 (2026-09-18).