Хуков закон

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.

У механици, Хуков закон еластичности је апроксимација која казује да је релативна деформација еластичног тела, у одређеним границама, директно пропорционална напону који на њега делује. Закон је назван по Роберту Хуку, енглеском физичару из 17. века, који га је открио и 1675. изразио латинским анаграмом: ceiiinosssttuu. Решење анаграма је објавио 1676. године као: Ut tensio, sic vis (= Колико истезање, толика сила).^[1]^[2]^[3] He published the solution of his anagram in 1678^[4] 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.

У овом првобитном облику, закон се односио пре свега на опруге, тј. чињеницу да је сила коју опруга производи пропорционална њеном истезању или сабијању:

F=-k\Delta x

Где је:

$F$ — сила коју опруга производи, знак „—“ означава супротан смер од помераја. Ако је опруга истегљена, њена сила ће тежити да је скупи и супортно, ако је опруга скупљена, сила опруге ће тежити да је рашири
$k$ — константа еластичности (коефицијент пропорционалности)
$\Delta x$ — означава промену дужине при растезању или скупљању опруге у односу на њен основни, природни положај. Знак $\Delta$ није обавезан, али обично се користи као ознака за промену

Данас је познато да Хуков закон важи за широк спектар еластичних тела, која се називају линеарно-еластичним телима, при деформацијама (истезање, увијање и сл.) које она трпе под утицајем сила. За свако такво тело, закон важи само у одређеним границама карактеристичним за њега — напон не сме прећи тзв. границу еластичности. Линеарни однос између деформације и напона је одређен константом пропорционалности, која се у зависности од типа деформације различито назива, такође карактеристичном за дато тело. Граница еластичности и константа пропорционалности зависе од природе материјала од кога је дато тело начињено и од осталих његових особина.

Формална дефиниција

За линеарне опруге

Consider a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is $F s$ . Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let $x$ be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that

F_{s}=kx

or, equivalently,

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

where $k$ is a positive real number, characteristic of the spring. Moreover, the same formula holds when the spring is compressed, with $F s$ and $x$ both negative in that case. According to this formula, the graph of the applied force $F s$ as a function of the displacement $x$ will be a straight line passing through the origin, whose slope is $k$ .

Hooke's law for a spring is sometimes, but rarely, stated under the convention that $F s$ is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes

F_{s}=-kx

since the direction of the restoring force is opposite to that of the displacement.

General "scalar" springs

Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative.

For example, when a block of rubber attached to two parallel plates is deformed by shearing, rather than stretching or compression, the shearing force $F s$ and the sideways displacement of the plates $x$ obey Hooke's law (for small enough deformations).

Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight $F$ placed at some intermediate point. The displacement $x$ in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.

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 $F s$ 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 $F s$ 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 $F s$ is proportional to $x$ (although the constant $k$ is different in each case.)

Vector formulation

In the case of a helical spring that is stretched or compressed along its axis, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if $F s$ and $x$ are defined as vectors, Hooke's equation still holds and says that the force vector is the elongation vector multiplied by a fixed scalar.

General tensor form

Some elastic bodies will deform in one direction when subjected to a force with a different direction. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. In such cases, the magnitude of the displacement $x$ will be proportional to the magnitude of the force $F s$ , as long as the direction of the latter remains the same (and its value is not too large); so the scalar version of Hooke's law $F s = -kx$ will hold. However, the force and displacement vectors will not be scalar multiples of each other, since they have different directions. Moreover, the ratio $k$ between their magnitudes will depend on the direction of the vector $F s$ .

Yet, in such cases there is often a fixed linear relation between the force and deformation vectors, as long as they are small enough. Namely, there is a function $κ$ from vectors to vectors, such that $F = κ (X)$ , and $κ (α X 1 + β X 2) = α κ (X 1) + β κ (X 2)$ for any real numbers $α$ , $β$ and any displacement vectors $X 1$ , $X 2$ . Such a function is called a (second-order) tensor.

With respect to an arbitrary Cartesian coordinate system, the force and displacement vectors can be represented by 3 × 1 matrices of real numbers. Then the tensor $κ$ connecting them can be represented by a 3 × 3 matrix $κ$ of real coefficients, that, when multiplied by the displacement vector, gives the force vector:

\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}=\kappa _{i1}X_{1}+\kappa _{i2}X_{2}+\kappa _{i3}X_{3}

for $i = 1, 2, 3$ . Therefore, Hooke's law $F = κ X$ can be said to hold also when $X$ and $F$ are vectors with variable directions, except that the stiffness of the object is a tensor $κ$ , rather than a single real number $k$ .

Hooke's law for continuous media

(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.^[5]

The stresses and strains of the material inside a continuous elastic material (such as a block of rubber, the wall of a boiler, or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name.

However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.

In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the strain tensor $ε$ (in lieu of the displacement $X$ ) and the stress tensor $σ$ (replacing the restoring force $F$ ). The analogue of Hooke's spring law for continuous media is then

{\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

{\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

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

Being a linear mapping between the nine numbers $σ ij$ and the nine numbers $ε kl$ , the stiffness tensor $c$ is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers $c ijkl$ . Hooke's law then says that

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

where $i, j = 1,2,3$ .

All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor $ε$ merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor $σ$ specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor $c$ , on the other hand, is a property of the material, and often depends on physical state variables such as temperature, pressure, and microstructure.

Due to the inherent symmetries of $σ$ , $ε$ , and $c$ , only 21 elastic coefficients of the latter are independent.^[6] This number can be further reduced by the symmetry of the material: 9 for an orthorhombic crystal, 5 for an hexagonal structure, and 3 for a cubic symmetry.^[7] For isotropic media (which have the same physical properties in any direction), $c$ can be reduced to only two independent numbers, the bulk modulus $K$ and the shear modulus $G$ , that quantify the material's resistance to changes in volume and to shearing deformations, respectively.

Референце

^ Elert, Glenn. „Springs”. The Physics Hypertextbook (на језику: (језик: енглески)). Приступљено 18. 7. 2010.
^ 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. стр. 11. ISBN 978-0674463684.
^ See http://civil.lindahall.org/design.shtml, where one can find also an anagram for catenary.
^ Robert Hooke, De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies, London, 1678.
^ 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. PMC 4661696 . PMID 26612544. doi:10.1038/srep17152.
^ Belen'kii; Salaev (1988). „Deformation effects in layer crystals”. Uspekhi Fizicheskikh Nauk. 155 (5): 89. doi:10.3367/UFNr.0155.198805c.0089.
^ Mouhat, Félix; Coudert, François-Xavier (2014-12-05). „Necessary and sufficient elastic stability conditions in various crystal systems”. Physical Review B (на језику: енглески). 90 (22): 224104. Bibcode:2014PhRvB..90v4104M. ISSN 1098-0121. S2CID 54058316. arXiv:1410.0065 . doi:10.1103/PhysRevB.90.224104.

Литература

Timshenko, S.; Goodier, J. N. (1951). Theory of Elasticity (на језику: (језик: енглески)) (2. изд.). New York: McGraw-Hill Book Company.
Krpić, Dragomir (2005) [1996]. Fizička mehanika (3. изд.). Beograd: Fizički fakultet. ISBN 978-86-81019-06-1 (prvo izdanje).
Ugural, A. C.; Fenster, S. K. (2003). Advanced Strength and Applied Elasticity (4th изд.). Prentice-Hall. ISBN 978-0-13-047392-9.
Walter Lewin explains Hooke's law. From Walter Lewin (1. 10. 1999). Hooke's Law, Simple Harmonic Oscillator. MIT Course 8.01: Classical Mechanics, Lecture 10. (videotape). Cambridge, MA USA: MIT OCW. Корисна информација се налази на: 1:21–10:10. Архивирано из оригинала (ogg) 29. 6. 2011. г. Приступљено 23. 12. 2010. „...arguably the most important equation in all of Physics.”
A test of Hooke's law. From Walter Lewin (1. 10. 1999). Hooke's Law, Simple Harmonic Oscillator. MIT Course 8.01: Classical Mechanics, Lecture 10. (videotape). Cambridge, MA USA: MIT OCW. Корисна информација се налази на: 10:10–16:33. Архивирано из оригинала (ogg) 29. 6. 2011. г. Приступљено 23. 12. 2010.

Спољашње везе

[anagram-1] Elert, Glenn. „Springs”. The Physics Hypertextbook (на језику: (језик: енглески)). Приступљено 18. 7. 2010.

[2] 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. стр. 11. ISBN 978-0674463684.

[3] See http://civil.lindahall.org/design.shtml, where one can find also an anagram for catenary.

[4] Robert Hooke, De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies, London, 1678.

[5] Ushiba, Shota; Masui, Kyoko; Taguchi, Natsuo; Hamano, Tomoki; Kawata, Satoshi; Shoji, Satoru (2015). „Size dependent nanomechanics of coil spring shaped polymer nanowires”. Scientific Reports. 5: 17152. Bibcode:2015NatSR...517152U. PMC 4661696 . PMID 26612544. doi:10.1038/srep17152.

[6] Belen'kii; Salaev (1988). „Deformation effects in layer crystals”. Uspekhi Fizicheskikh Nauk. 155 (5): 89. doi:10.3367/UFNr.0155.198805c.0089.

[7] Mouhat, Félix; Coudert, François-Xavier (2014-12-05). „Necessary and sufficient elastic stability conditions in various crystal systems”. Physical Review B (на језику: енглески). 90 (22): 224104. Bibcode:2014PhRvB..90v4104M. ISSN 1098-0121. S2CID 54058316. arXiv:1410.0065 . doi:10.1103/PhysRevB.90.224104.

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