Content deleted Content added
No edit summary |
Citation bot (talk | contribs) Add: bibcode, pmid, issue. | Use this bot. Report bugs. | Suggested by Abductive | Category:Rubber properties | #UCB_Category 18/31 |
||
Line 99:
This model can be applied to soft solids: thermoplastic polymers in the vicinity of their melting temperature, fresh concrete (neglecting its aging), and numerous metals at a temperature close to their melting point.
The equation introduced here, however, lacks a consistent derivation from more microscopic model and is not observer independet. The [[Upper-convected Maxwell model]] is its sound formulation in tems of the [[Cauchy stress tensor]] and constitutes the simplest tensorial constitutive model for viscoelasticity (see e.g. <ref>{{Cite book|title=The Structure and Rheology of Complex Fluids (Topics in Chemical Engineering): Larson, Ronald G.: 9780195121971: Amazon.com: Books|isbn=019512197X |last1=Larson |first1=Ronald G. |date=28 January 1999 |publisher=Oup USA }}</ref> or <ref name=":0">{{Cite journal|last1=Winters|first1=A.|last2=Öttinger|first2=H. C.|last3=Vermant|first3=J.|date=2024|title=Comparative analysis of fluctuations in viscoelastic stress: A comparison of the temporary network and dumbbell models|url=https://pubs.aip.org/aip/jcp/article/161/1/014901/3300367/Comparative-analysis-of-fluctuations-in|journal=Journal of Chemical Physics|language=en|volume=161 |issue=1 |pages=014901|doi=10.1063/5.0213660|pmid=38949587 |arxiv=2404.19743|bibcode=2024JChPh.161a4901W }}</ref>
).
Line 225:
The ''strain damping function'' is usually written as:
<math display=block>h(I_1,I_2)=m^*\exp(-n_1 \sqrt{I_1-3})+(1-m^*)\exp(-n_2 \sqrt{I_2-3})</math>
If the value of the strain hardening function is equal to one, then the deformation is small; if it approaches zero, then the deformations are large.<ref>{{cite journal |last1=Wagner |first1=Manfred |journal=Rheologica Acta |date=1976 |volume=15 |pages=136–142|title=Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt|issue=2 |doi=10.1007/BF01517505 |bibcode=1976AcRhe..15..136W |s2cid=96165087 |url=https://www.researchgate.net/publication/226796804}}</ref><ref>{{cite journal |last1=Wagner |first1=Manfred |journal=Rheologica Acta |volume=16 |issue=1977 |pages=43–50|title=Prediction of primary normal stress difference from shear viscosity data usinga single integral constitutive equation|year=1977 |doi=10.1007/BF01516928 |bibcode=1977AcRhe..16...43W |s2cid=98599256 |url=https://www.researchgate.net/publication/226007475}}</ref>
==Prony series==
Line 298:
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}}</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:
Line 306:
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}}</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.
| |||