Editing Deborah number (section)

== Definition ==
The Deborah number is the ratio of fundamentally different characteristic times. The Deborah number is defined as the ratio of the time it takes for a material to adjust to applied stresses or deformations<!-- characterizing the intrinsic fluidity of a material: THIS WILL BE A CIRCULAR EXPLANATION-->, and the characteristic time scale of an experiment (or a computer simulation) probing the response of the material:

:<math alt="De equals tc divided by tp">
\mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}},</math>

where {{math|''t''<sub>c</sub>}} stands for the relaxation time and {{math|''t''<sub>p</sub>}} for the "time of observation", typically taken to be the time scale of the process.<ref name="Poole"/>

The numerator, [[relaxation time]], is the time needed for a reference amount of deformation to occur under a suddenly applied reference load (a more fluid-like material will therefore require less time to flow, giving a lower Deborah number relative to a solid subjected to the same loading rate).

The denominator, material time,<ref name="TAImaterialProperties">{{cite web |last1=Franck |first1=A. |title=Viscoelasticity and dynamic mechanical testing |url=http://tainstruments.com/pdf/literature/AAN004_Viscoelasticity_and_DMA.pdf |website=TA Instruments |publisher=TA Instruments Germany |access-date=26 March 2019}}</ref> is the amount of time required to reach a given reference strain (a faster loading rate will therefore reach the reference strain sooner, giving a higher Deborah number).

Equivalently, the relaxation time is the time required for the stress induced, by a suddenly applied reference strain, to reduce by a certain reference amount. The relaxation time is actually based on the rate of relaxation that exists at the moment of the suddenly applied load.

This incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the material behaves in a more fluidlike manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior enters the non-Newtonian regime, increasingly dominated by elasticity and demonstrating solidlike behavior.<ref name="Reiner1964">{{citation|first=M. |last=Reiner |year=1964|journal=Physics Today|volume =17|issue= 1| page= 62 |title=The Deborah Number|doi=10.1063/1.3051374|bibcode = 1964PhT....17a..62R }}</ref><ref>[http://rrc.engr.wisc.edu/deborah.html The Deborah Number] {{webarchive|url=https://web.archive.org/web/20110413144406/http://rrc.engr.wisc.edu/deborah.html |date=2011-04-13 }}</ref><!--Note that the Deborah number is relevant for materials that flow on long time scales (like a [[Maxwell material|Maxwell fluid]]) but ''not'' for the reverse kind of materials (like the Voigt or [[Kelvin material|Kelvin model]]) that are viscous on short time scales but solid on the long term.-->

For example, for a Hookean elastic solid, the relaxation time {{math|''t''<sub>c</sub>}} will be infinite and it will vanish for a Newtonian viscous fluid. For liquid water, {{math|''t''<sub>c</sub>}} is typically 10<sup>−12</sup> s, for lubricating oils passing through gear teeth at high pressure it is of the order of 10<sup>−6</sup> s and for polymers undergoing plastics processing, the relaxation time will be of the order of a few seconds. Therefore, depending on the situation, these liquids may exhibit elastic properties, departing from purely viscous behavior.<ref>{{cite book|last1=Barnes|first1=H.A.|last2=Hutton|first2=J.F.|last3=Walters|first3=K.|title=An introduction to rheology|url=https://archive.org/details/introductiontorh00barn|url-access=limited|date=1989|publisher=Elsevier|location=Amsterdam|isbn=978-0-444-87140-4|pages=[https://archive.org/details/introductiontorh00barn/page/n12 5]–6|edition=5. impr.}}</ref>

While {{math|De}} is similar to the [[Weissenberg number]] and is often confused with it in technical literature, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flows with a constant stretch history, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released.<ref name="Poole">{{Cite journal|last=Poole|first=R J|year=2012|title=The Deborah and Weissenberg numbers|url=http://pcwww.liv.ac.uk/~robpoole/PAPERS/POOLE_45.pdf|journal=Rheology Bulletin|volume=53 |issue=2|pages=32–39}}</ref>