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== Constitutive models for nonlinear viscoelasticity ==
Non-linear viscoelastic constitutive equations are needed to quantitatively account for phenomena in fluids like differences in normal stresses, shear thinning, and extensional thickening.<ref name="Macosko 1994"/> Necessarily, the history experienced by the material is needed to account for time-dependent behavior, and is typically included in models as a history kernel '''K'''.<ref>{{Cite journal|last1=Drapaca|first1=C.S.|last2=Sivaloganathan|first2=S.|last3=Tenti|first3=G.|date=2007-10-01|title=Nonlinear Constitutive Laws in Viscoelasticity|url=https://doi.org/10.1177/1081286506062450|journal=Mathematics and Mechanics of Solids|language=en|volume=12|issue=5|pages=475–501|doi=10.1177/1081286506062450|s2cid=121260529|issn=1081-2865|url-access=subscription}}</ref>
=== Second-order fluid ===
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=== Shear rheometry ===
Shear rheometers are based on the idea of putting the material to be measured between two plates, one or both of which move in a shear direction to induce stresses and strains in the material. The testing can be done at constant strain rate, stress, or in an oscillatory fashion (a form of [[dynamic mechanical analysis]]).<ref>{{Cite journal|date=1987-01-01|title=Shear rheometry of fluids with a yield stress|url=https://www.sciencedirect.com/science/article/abs/pii/0377025787800125|journal=Journal of Non-Newtonian Fluid Mechanics|language=en|volume=23|pages=91–106|doi=10.1016/0377-0257(87)80012-5|issn=0377-0257|last1=Magnin|first1=A.|last2=Piau|first2=J.M.|bibcode=1987JNNFM..23...91M |url-access=subscription}}</ref> Shear rheometers are typically limited by edge effects where the material may leak out from between the two plates and slipping at the material/plate interface.
=== Extensional rheometry ===
Extensional rheometers, also known as extensiometers, measure viscoelastic properties by pulling a viscoelastic fluid, typically uniaxially.<ref name="sciencedirect.com">{{Cite journal|date=1978-01-01|title=Extensional Rheometers for molten polymers; a review|url=https://www.sciencedirect.com/science/article/abs/pii/0377025778850034|journal=Journal of Non-Newtonian Fluid Mechanics|language=en|volume=4|issue=1–2|pages=9–21|doi=10.1016/0377-0257(78)85003-4|issn=0377-0257|last1=Dealy|first1=J.M.|bibcode=1978JNNFM...4....9D |url-access=subscription}}</ref> Because this typically makes use of capillary forces and confines the fluid to a narrow geometry, the technique is often limited to fluids with relatively low viscosity like dilute polymer solutions or some molten polymers.<ref name="sciencedirect.com"/> Extensional rheometers are also limited by edge effects at the ends of the extensiometer and pressure differences between inside and outside the capillary.<ref name="Macosko 1994"/>
Despite the apparent limitations mentioned above, extensional rheometry can also be performed on high viscosity fluids. Although this requires the use of different instruments, these techniques and apparatuses allow for the study of the extensional viscoelastic properties of materials such as polymer melts. Three of the most common extensional rheometry instruments developed within the last 50 years are the Meissner-type rheometer, the filament stretching rheometer (FiSER), and the Sentmanat Extensional Rheometer (SER).
The Meissner-type rheometer, developed by Meissner and Hostettler in 1996, uses two sets of counter-rotating rollers to strain a sample uniaxially.<ref>{{Cite journal |last1=Meissner |first1=J. |last2=Hostettler |first2=J. |date=1994-01-01 |title=A new elongational rheometer for polymer melts and other highly viscoelastic liquids |url=https://doi.org/10.1007/BF00453459 |journal=Rheologica Acta |language=en |volume=33 |issue=1 |pages=1–21 |doi=10.1007/BF00453459 |bibcode=1994AcRhe..33....1M |s2cid=93395453 |issn=1435-1528|url-access=subscription }}</ref> This method uses a constant sample length throughout the experiment, and supports the sample in between the rollers via an air cushion to eliminate sample sagging effects. It does suffer from a few issues – for one, the fluid may slip at the belts which leads to lower strain rates than one would expect. Additionally, this equipment is challenging to operate and costly to purchase and maintain.
The FiSER rheometer simply contains fluid in between two plates. During an experiment, the top plate is held steady and a force is applied to the bottom plate, moving it away from the top one.<ref>{{Cite journal |last1=Bach |first1=Anders |last2=Rasmussen |first2=Henrik Koblitz |last3=Hassager |first3=Ole |date=March 2003 |title=Extensional viscosity for polymer melts measured in the filament stretching rheometer |url=http://sor.scitation.org/doi/10.1122/1.1545072 |journal=Journal of Rheology |language=en |volume=47 |issue=2 |pages=429–441 |doi=10.1122/1.1545072 |bibcode=2003JRheo..47..429B |s2cid=44889615 |issn=0148-6055}}</ref> The strain rate is measured by the rate of change of the sample radius at its middle. It is calculated using the following equation:
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where <math>\eta</math> is the sample viscosity, and <math>F</math> is the force applied to the sample to pull it apart.
Much like the Meissner-type rheometer, the SER rheometer uses a set of two rollers to strain a sample at a given rate.<ref>{{Cite journal |last=Sentmanat |first=Martin L. |date=2004-12-01 |title=Miniature universal testing platform: from extensional melt rheology to solid-state deformation behavior |url=https://doi.org/10.1007/s00397-004-0405-4 |journal=Rheologica Acta |language=en |volume=43 |issue=6 |pages=657–669 |doi=10.1007/s00397-004-0405-4 |bibcode=2004AcRhe..43..657S |s2cid=73671672 |issn=1435-1528|url-access=subscription }}</ref> It then calculates the sample viscosity using the well known equation:
<math display="block">\sigma = \eta \dot{\epsilon}</math>
where <math>\sigma</math> is the stress, <math>\eta</math> is the viscosity and <math>\dot{\epsilon}</math> is the strain rate. The stress in this case is determined via torque transducers present in the instrument. The small size of this instrument makes it easy to use and eliminates sample sagging between the rollers. A schematic detailing the operation of the SER extensional rheometer can be found on the right.
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