HOOKE's Law
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HOOKE´s law
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, specifically when the elastic deformation is proportional to the applied load. This 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 within their range of application, in contrast to metallic material. The 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 thermoplastics, for example, the linear-elastic region is followed, at higher strains, by regions of linear-viscoelastic and non-linear viscoelastic material behaviour (see: elasticity). However, at very low temperatures and for very brittle materials (such as many thermosets), HOOKE's law still applies even for slightly larger deformations. Some polymer materials, such as elastomers above their 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: σij = Eijkl ⋅ εkl, where σij and εkl are the stress and strain tensors respectively, and Eijkl is the 81-component elasticity tensor. Due to the symmetry of the stress and strain tensors, the number of independent components Eijkl is reduced to 36, so that σI = EIJ ⋅ εJ now holds, where σI and εJ are 6-component stress and strain vectors respectively, and EIJ is a 36-component elasticity tensor. The elasticity tensor EIJ 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 modulus of elasticity E and the transverse coefficient of contraction (or Poisson’s ratio) ν.
Application of HOOKE’s law
In the context of polymer testing, the application of HOOKE's law to determine elastic properties under uniaxial loading in tensile and compression tests (e.g. in accordance with ISO 527 and ISO 604) plays a role in determining the modulus of elasticity Et (tension) or Ec (compression) using the secant method according to Et, Ec = Δσ/Δε, where Δε 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 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 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 dynamic mechanical analysis (DMA).
See also
- Deformation
- Elasticity
- Elastic modulus
- Linear-viscoelastic 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: Robert Hooke