Viscoelasticity: Difference between revisions

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{{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 stress and 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>)}}]]

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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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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 independent. The [[Upper-convected Maxwell model]] is its sound formulation in terms 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>{{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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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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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>