Viscoelasticity: Difference between revisions

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{{short description|Property of materials with both viscous and elastic characteristics under deformation}}

{{Lead too short|date=June 2025}}

{{Continuum mechanics|cTopic=rheology}}

'''Viscoelasticity''' is a material property that combines both viscous and elastic characteristics. Many materials have such viscoelastic properties. Especially materials that consist of large molecules show viscoelastic properties. [[Polymer|Polymers]] are viscoelastic because their [[Macromolecule|macromolecules]] can make temporary [https://www.stevenabbott.co.uk/practical-solubility/polymer-entanglement.php entanglements] with neighbouring molecules which causes elastic properties<ref>{{Cite book |last=Doi |first=M |title=The theory of polymer dynamics |publisher=Oxford University Press |year=1986 |isbn=0198520336}}</ref>. After some time these entanglements will disappear again and the macromolecules will flow into other positions where new entanglements will be made (viscous properties).

In [[materials science]] and [[continuum mechanics]], '''viscoelasticity''' is the property of [[materials]] that exhibit both [[Viscosity|viscous]] and [[Elasticity (physics)|elastic]] characteristics when undergoing [[deformation (engineering)|deformation]]. Viscous materials, like water, resist both [[shear flow]] and [[Strain (materials science)|strain]] linearly with time when a [[Stress (physics)|stress]] is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

A viscoelastic material will show elastic properties on short time scales and viscous properties on long time scales. These materials exhibit behavior that depends on the time and rate of applied forces, allowing them to both store and dissipate energy.

Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Whereas elasticity is usually the result of [[chemical bond|bond]] stretching along [[crystallographic plane]]s in an ordered solid, viscosity is the result of the diffusion of atoms or molecules inside an [[amorphous]] material.<ref name=Meyers>Meyers and Chawla (1999): "Mechanical Behavior of Materials", 98-103.</ref>

Viscoelasticity has been studied since the nineteenth century by researchers such as [[James Clerk Maxwell]], [[Ludwig Boltzmann]], and [[William Thomson, 1st Baron Kelvin|Lord Kelvin]].

Several models are available for the mathematical description of the viscoelastic properties of a substance:

* [[Constitutive equation|Constitutive models]] of linear viscoelasticity assume a linear relationship between [[Stress (mechanics)|stress]] and [[Strain (mechanics)|strain]]. These models are valid for relatively small [[Deformation (physics)|deformations]] only.

* Constitutive models of non-linear viscoelasticity are based on a more realistic non-linear relationship between stress and strain. These models are valid for relatively large deformations.

The viscoelastic properties of polymers are highly temperature dependent. From low to high temperature the material can be in the glass phase, rubber phase or the melt phase. These [[Phase (matter)|phases]] have a very strong effect on the mechanical and viscous properties of the polymers.

Typical viscoelastic properties are:

* A time dependant stress in the polymer under constant deformation (strain).

* A time dependant strain in the polymer under constant stress.

* A time and temperature dependant [[stiffness]] of the polymer.

* Viscous energy loss during deformation of the polymer in the glass or rubber phase ([[hysteresis]]).

* A [[strain rate]] dependant [[viscosity]] of the molten polymer.

* An ongoing deformation of a polymer in the glass phase at constant load ([[Creep (deformation)|creep]]).

The viscoelasticity properties are measured with various techniques, such as [[tensile testing]], [[dynamic mechanical analysis]], shear rheometry and extensional rheometry.

==Background==

In the nineteenth century, physicists such as [[James Clerk Maxwell]], [[Ludwig Boltzmann]], and [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] researched and experimented with [[Creep (deformation)|creep]] and recovery of [[glass]]es, [[metal]]s, and [[rubber]]s. Viscoelasticity was further examined in the late twentieth century when [[synthetic polymer]]s were engineered and used in a variety of applications.<ref name=McCrum>McCrum, Buckley, and Bucknell (2003): "Principles of Polymer Engineering," 117-176.</ref>

Viscoelasticity calculations depend heavily on the [[viscosity]] variable, ''η''. The inverse of ''η'' is also known as [[Viscosity#Fluidity|fluidity]], ''φ''. The value of either can be derived as a [[Temperature dependence of liquid viscosity|function of temperature]] or as a given value (i.e. for a [[dashpot]]).<ref name="Meyers">Meyers and Chawla (1999): "Mechanical Behavior of Materials", 98-103.</ref>

[[Image:Non-Newtonian fluid.svg|thumb|350px| Different types of responses {{nowrap|(<math>\sigma</math>)}} to a change in strain rate {{nowrap|(<math>d\varepsilon/dt</math>)}}]]

Depending on the change of strain rate versus stress inside a material, the viscosity can be categorized as having a linear, non-linear, or plastic response.:

* When a material exhibits a linear response it is categorized as a [[Newtonian material]]. In this case the stress is linearly proportional to the strain rate.
* If the material exhibits a non-linear response to the strain rate, it is categorized as [[non-Newtonian fluid]].
* There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. A material which exhibits this type of behavior is known as [[thixotropy|thixotropic]].
* In addition, when the stress is independent of this strain rate, the material exhibits plastic deformation.<ref name="Meyers" /> Many viscoelastic materials exhibit [[rubber]] like behavior explained by the thermodynamic theory of polymer elasticity.

Some examples of viscoelastic materials are amorphous polymers, semicrystalline polymers, biopolymers, metals at very high temperatures, and bitumen materials. Cracking occurs when the strain is applied quickly and outside of the elastic limit. [[Ligament]]s and [[tendon]]s in the human body are viscoelastic, so the extent of the potential damage to them depends on both the rate of the change of their length and the force applied.{{Citation needed|reason=maybe https://doi.org/10.1114/1.1408926| date=February 2017}}

A viscoelastic material has the following properties:

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Unlike purely elastic substances, a viscoelastic substance has an elastic component and a viscous component. The [[viscosity]] of a viscoelastic substance gives the substance a strain rate dependence on time. Purely elastic materials do not dissipate energy (heat) when a load is applied, then removed. However, a viscoelastic substance dissipates energy when a load is applied, then removed. [[Hysteresis]] is observed in the stress–strain curve, with the area of the loop being equal to the energy lost during the loading cycle. Since viscosity is the resistance to thermally activated plastic deformation, a viscous material will lose energy through a loading cycle. Plastic deformation results in lost energy, which is uncharacteristic of a purely elastic material's reaction to a loading cycle.<ref name=Meyers/>

Specifically, viscoelasticity is a molecular rearrangement. When a stress is applied to a viscoelastic material such as a [[polymer]], parts of the long polymer chain change positions. This movement or rearrangement is called [[Creep (deformation)|creep]]. Polymers remain a solid material even when these parts of their chains are rearranging in order to accommodate the stress, and as this occurs, it creates a back stress in the material. When the back stress is the same magnitude as the applied stress, the material no longer creeps. When the original stress is taken away, the accumulated back stresses will cause the polymer to return to its original form. The material creeps, which gives the prefix visco-, and the material fully recovers, which gives the suffix -elasticity.<ref name=McCrum/>

== Linear viscoelasticity and nonlinear viscoelasticity==

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Linear viscoelasticity is usually applicable only for small [[deformation (engineering)|deformation]]s.

'''Nonlinear viscoelasticity''' is when the function is not separable. It usually happens when the [[deformation (engineering)|deformation]]s are large or if the material changes its properties under deformations. Nonlinear viscoelasticity also elucidates observed phenomena such as normal stresses, shear thinning, and extensional thickening in viscoelastic fluids.<ref name="Macosko 1994">{{Cite book|last=Macosko|first=Christopher W.|url=https://www.worldcat.org/oclc/232602530| title=Rheology : principles, measurements, and applications|date=1994|publisher=VCH|isbn=978-1-60119-575-3|___location=New York| oclc=232602530}}</ref>

An '''anelastic''' material is a special case of a viscoelastic material: an anelastic material will fully recover to its original state on the removal of load.

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Viscoelasticity is studied using [[dynamic mechanical analysis]], applying a small oscillatory stress and measuring the resulting strain.

* Purely elastic materials have stress and strain in phase, so that the response of one caused by the other is immediate.

* In purely viscous materials, strain lags stress by a 90 -degree phase.

* Viscoelastic materials exhibit behavior somewhere in the middle of these two types of material, exhibiting some lag in strain.

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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 independetindependent. The [[Upper-convected Maxwell model]] is its sound formulation in temsterms 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 |hdl=20.500.11850/682077 }}</ref>

).

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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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The second-order fluid is typically considered the simplest nonlinear viscoelastic model, and typically occurs in a narrow region of materials behavior occurring at high strain amplitudes and Deborah number between Newtonian fluids and other more complicated nonlinear viscoelastic fluids.<ref name="Macosko 1994"/> The second-order fluid constitutive equation is given by:

<math display=block>\mathbf T = -p\mathbf I + 2 \eta_0\mathbf D - \psi_1 \mathbf D^\triangledown + 4\psi _2 \mathbf D \bulletcdot\mathbf D</math>

where:

* <math>\mathbf I</math> is the identity tensor

* <math>\mathbf D</math> is the deformation tensor

* <math>\eta_0 , \psi_1 , \psi_2</math> denote viscosity, and first and second normal stress coefficients, respectively

* <math>\mathbf D ^\triangledown</math> denotes the upper-convected derivative of the deformation tensor where <math display=block>\mathbf D ^\triangledown \equiv \dot \mathbf D - (\bigtriangledownnabla\mathbf v) ^\mathbf T \bulletcdot \mathbf D - \mathbf D \bulletcdot \bigtriangledownnabla \mathbf v</math> and <math>\dot \mathbf D \equiv \frac {\partial}{ \partial t} \mathbf D + \mathbf v \bulletcdot \bigtriangledownnabla \mathbf D</math> is the material time derivative of the deformation tensor.<ref name="Macosko 1994"/>

=== Upper-convected Maxwell model ===

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{{main|Oldroyd-B model}}

The Oldroyd-B model is an extension of the [[Upper Convected Maxwell model]] and is interpreted as a solvent filled with elastic bead and spring dumbbells.

The model is named after its creator [[James G. Oldroyd]].<ref name="a">{{cite journal|last=Oldroyd|first=James|title=On the Formulation of Rheological Equations of State|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|date=February 1950|volume=200|issue=1063|pages=523–541|bibcode=1950RSPSA.200..523O| doi=10.1098/rspa.1950.0035| s2cid=123239889}}</ref><ref name="b">{{cite book | last1=Owens |first1=R. G. |last2=Phillips |first2=T. N.| title=Computational Rheology| publisher=Imperial College Press | year=2002 | isbn=978-1-86094-186-3}}</ref><ref name="c">{{cite journal| last=Poole|first=Rob| journal=Physical Review Letters |title=Purely elastic flow asymmetries|date=October 2007|volume=99| number=16| pagesarticle-number=164503|doi=10.1103/PhysRevLett.99.164503|pmid=17995258|bibcode=2007PhRvL..99p4503P|hdl=10400.6/634|hdl-access=free}}</ref>

The model can be written as:

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Whilst the model gives good approximations of viscoelastic fluids in shear flow, it has an unphysical singularity in extensional flow, where the dumbbells are infinitely stretched. This is, however, specific to idealised flow; in the case of a cross-slot geometry the extensional flow is not ideal, so the stress, although singular, remains integrable, although the stress is infinite in a correspondingly infinitely small region.<ref name="c"/>

If the solvent viscosity is zero, the Oldroyd-B becomes the [[Upper Convected Maxwell model|upper convected Maxwell model]].

=== Wagner model ===

{{main|Wagner model}}

Wagner model is might be considered as a simplified practical form of the Bernstein–Kearsley–Zapas model. The model was developed by German rheologist [[Manfred Wagner]].

For the [[Isothermal process|isothermal]] conditions the model can be written as:

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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 ===

{{main|Prony's method}}

In a one-dimensional relaxation test, the material is subjected to a sudden strain that is kept constant over the duration of the test, and the stress is measured over time. The initial stress is due to the elastic response of the material. Then, the stress relaxes over time due to the viscous effects in the material. Typically, either a tensile, compressive, bulk compression, or shear strain is applied. The resulting stress vs. time data can be fitted with a number of equations, called models. Only the notation changes depending on the type of strain applied: tensile-compressive relaxation is denoted <math>E</math>, shear is denoted <math>G</math>, bulk is denoted <math>K</math>. The Prony series for the shear relaxation is

<math display=block>

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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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{{Non-Newtonian fluids}}

[[Category:Materials science]]

[[Category:Elasticity (physics)]]