View source for HOOKE's Law

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<span style="font-size:1.2em;font-weight:bold;">HOOKE´s law</span>
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==General principles==

HOOKE's law (named after Robert Hooke (1635–1703)) describes the elastic behaviour of solids as a special linear case of the law of [[Elasticity|elasticity]], specifically when the elastic [[Deformation|deformation]] is proportional to the applied load. This [[Material & Werkstoff|material]] behaviour, also referred to as linear-elastic behaviour, is often only valid for very small deformations (generally for strains of up to about 0.2 %) for most [[Plastics|plastics]] within their range of application, in contrast to metallic [[Material & Werkstoff|material]]. The [[Deformation Mechanisms|deformation mechanisms]] are rooted at the molecular level in changes to the distances between atoms in the main chain of the macromolecules, as well as in the bond angles between these atoms. As is typical for [[Thermoplastic Material|thermoplastics]], for example, the linear-elastic region is followed, at higher strains, by regions of [[Linear-viscoelastic Behaviour|linear-viscoelastic]] and non-linear viscoelastic material behaviour (see: [[Elasticity|elasticity]]). However, at very low temperatures and for very brittle materials (such as many [[Thermosets|thermosets]]), HOOKE's law still applies even for slightly larger deformations. Some [[Polymer|polymer]] [[Material & Werkstoff|materials]], such as [[Elastomers|elastomers]] above their [[Glass Transition Temperature|glass transition temperature]], exhibit exclusively non-linear elastic behaviour, however.

==Three-dimensional stress==

In the general case of an anisotropic solid subjected to three-dimensional loading, HOOKE's law is expressed by a fourth-order linear tensor equation: ''σ''<sub>ij</sub> = ''E''<sub>ijkl</sub> ⋅ ''ε''<sub>kl</sub>, where ''σ''<sub>ij</sub> and ''ε''<sub>kl</sub> are the stress and strain tensors respectively, and ''E''<sub>ijkl</sub> is the 81-component elasticity tensor. Due to the symmetry of the stress and strain tensors, the number of independent components ''E''<sub>ijkl</sub> is reduced to 36, so that ''σ''<sub>I</sub> = ''E''<sub>IJ</sub> ⋅ ''ε''<sub>J</sub>  now holds, where ''σ''<sub>I</sub> and ''ε''<sub>J</sub> are 6-component stress and strain vectors respectively, and ''E''<sub>IJ</sub> is a 36-component elasticity tensor. The elasticity tensor ''E''<sub>IJ</sub> is also symmetric, so that the number of independent elastic constants is further reduced to a maximum of 21. In the case of an isotropic solid, only two of these constants remain: the [[Elastic Modulus|modulus of elasticity]] ''E'' and the transverse coefficient of contraction (or [[Poisson's Ratio|Poisson’s ratio]]) ''&nu;''.

==Application of HOOKE’s law==

In the context of [[Polymer Testing|polymer testing]], the application of HOOKE's law to determine elastic properties under [[Uniaxial Stress State|uniaxial loading]] in [[Tensile Test|tensile]] and [[Compression Test|compression]] tests (e.g. in accordance with ISO 527 and ISO 604) plays a role in determining the [[Elastic Modulus|modulus of elasticity]] ''E''<sub>t</sub> (tension) or ''E''<sub>c</sub> (compression) using the secant method according to ''E''<sub>t</sub>, ''E''<sub>c</sub> = Δ''σ''/Δ''ε'', where Δ''ε'' is the difference in strain or compression and Δσ is the difference in the corresponding tensile or compressive stresses. Strictly taken, since this takes into account strains or compressions of between 0.05 % and 0.25 %, it covers not only the linear-elastic range but also part of the [[Linear-viscoelastic Behaviour|linear-viscoelastic range]].

Another important, fundamental application of HOOKE's law, extending beyond its original validity range for describing linear-elastic behaviour, is made possible for polymeric materials through the formal inclusion of the time dependence of stresses and strains: ''σ''(''t'') = ''E*'' ⋅ ''ε''(''t''). This simple extension of HOOKE's law provides the basis for describing the [[Linear-viscoelastic Behaviour|linear-viscoelastic behaviour]] of materials using the complex modulus of elasticity ''E*'' (specifically, its real part ''E′'' – the storage modulus – and its imaginary part ''E′′'' – the loss modulus), a concept which is utilised experimentally in applications such as [[Elastic Modulus#Dynamic-mechanical analysis (DMA)|dynamic mechanical analysis (DMA)]].

==See also==

* [[Deformation]]
* [[Elasticity]]
* [[Elastic Modulus|Elastic modulus]]
* [[Linear-viscoelastic Behaviour|Linear-viscoelastic behaviour]]
* [[Viscoelastic Material Behaviour|Viscoelastic material behaviour]]

==References==

* Gross, D., Hauger, W., Schröder, J., Wall, W. A.: Technische Mechanik 2 – Elastostatik. 12th aktualisierte Aufl., Springer Verlag, Berlin Heidelberg (2014) (ISBN 978-3-642-40966-0)
* ISO 527-1 (2019-07): Plastics – Determination of Tensile Properties – Part 1: General Principles
* ISO 604 (2002-03): Plastics – Determination of Compressive Properties

==Weblink==

* Wikipedia – Die freie Enzyklopädie: [https://en.wikipedia.org/wiki/Robert_Hooke Robert Hooke]

[[Category:Deformation]]