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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}}

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.

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>

==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/>

[[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 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:
* [[hysteresis]] is seen in the [[stress–strain curve]]
* [[stress relaxation]] occurs: step constant strain causes decreasing stress
* [[creep (deformation)|creep]] occurs: step constant stress causes increasing strain
*its stiffness depends on the [[Strain rate|strain rate <math>\dot {\varepsilon} </math>]]<nowiki/>or the stress rate <math>\dot {\sigma}</math>

== Elastic versus viscoelastic behavior ==
[[Image:Elastic v. viscoelastic material.JPG|frame|right|Stress–strain curves for a purely elastic material (a) and a viscoelastic material (b).  The red area is a [[hysteresis]] loop and shows the amount of energy lost (as heat) in a loading and unloading cycle.  It is equal to <math display="inline">\oint \sigma \, d \varepsilon </math>, where <math>\sigma</math> is stress and <math>\varepsilon</math> is strain.<ref name=Meyers/>]]

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==
'''Linear viscoelasticity''' is when the function is [[separable ordinary differential equation|separable]] in both creep response and load. All linear viscoelastic models can be represented by a [[integral equation|Volterra equation]] connecting [[stress (physics)|stress]] and [[Strain (materials science)|strain]]:
<math display="block">\varepsilon(t) = \frac { \sigma(t) }{ E_\text{inst,creep} }+ \int_0^t K(t - t') \dot{\sigma}(t') dt'</math>
or
<math display="block">\sigma(t)= E_\text{inst,relax}\varepsilon(t)+ \int_0^t F(t - t') \dot{\varepsilon}(t') dt'</math>
where 
* {{mvar|t}} is time
* <math>\sigma (t)</math> is [[stress (physics)|stress]]
* <math>\varepsilon (t)</math> is [[Strain (materials science)|strain]]
* <math>E_\text{inst,creep}</math> and <math>E_\text{inst,relax}</math> are instantaneous [[Young's modulus|elastic moduli]] for creep and relaxation
* {{math|''K''(''t'')}} is the [[creep (deformation)|creep]] function
* {{math|''F''(''t'')}} is the relaxation function

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.

When distinguishing between elastic, viscous, and forms of viscoelastic behavior, it is helpful to reference the time scale of the measurement relative to the relaxation times of the material being observed, known as the Deborah number (De) where:<ref name="Macosko 1994"/>
<math display="block">De = \lambda/t</math>
where
* <math>\lambda</math> is the relaxation time of the material
* <math>t</math> is time

== Dynamic modulus ==
{{main|Dynamic modulus}}

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.

A complex [[dynamic modulus]] G can be used to represent the relations between the oscillating stress and strain:
<math display="block">G = G' + iG''</math>
where <math>i^2 = -1</math>; <math>G'</math> is the ''storage modulus'' and <math>G''</math> is the ''loss modulus'':
<math display="block"> G' = \frac {\sigma_0} {\varepsilon_0} \cos \delta </math>
<math display="block"> G'' =  \frac {\sigma_0} {\varepsilon_0} \sin \delta </math>
where <math>\sigma_0</math> and <math>\varepsilon_0</math> are the amplitudes of stress and strain respectively, and <math>\delta</math> is the phase shift between them.

== Constitutive models of linear viscoelasticity ==
[[File:Comparison three four element models.svg|upright=1.8|thumb|right|Comparison of creep and stress relaxation for three and four element models]]
Viscoelastic materials, such as amorphous polymers, semicrystalline polymers, biopolymers and even the living tissue and cells,<ref>{{cite journal |doi=10.1109/TOH.2014.2369416 |title=Multiscale Layered Biomechanical Model of the Pacinian Corpuscle |year=2015 |last1=Biswas |first1=Abhijit |journal=IEEE Transactions on Haptics |volume=8 |issue=1 |pages=31–42 |pmid=25398182 |last2=Manivannan |first2=M. |last3=Srinivasan |first3=Mandyam A.|s2cid=24658742 |url=https://zenodo.org/record/894776 }}</ref> can be modeled in order to determine their stress and strain or force and displacement interactions as well as their temporal dependencies.  These models, which include the [[Maxwell material|Maxwell model]], the [[Kelvin–Voigt material|Kelvin–Voigt model]], the [[standard linear solid model]], and the [[Burgers material|Burgers model]], are used to predict a material's response under different loading conditions.

Viscoelastic behavior has elastic and viscous components modeled as linear combinations of [[spring (device)|springs]] and [[dashpots]], respectively. Each model differs in the arrangement of these elements, and all of these viscoelastic models can be equivalently modeled as electrical circuits.

In an equivalent electrical circuit, stress is represented by current, and [[strain rate]] by voltage. The elastic modulus of a spring is analogous to the inverse of a circuit's ''inductance'' (it stores energy) and the viscosity of a dashpot to a circuit's ''resistance'' (it dissipates energy).

The elastic components, as previously mentioned, can be modeled as [[spring (device)|springs]] of elastic constant E, given the formula:
<math display="block">\sigma = E \varepsilon</math>
where σ is the stress, E is the elastic modulus of the material, and ε is the strain that occurs under the given stress, similar to [[Hooke's law]].

The viscous components can be modeled as [[dashpots]] such that the stress–strain rate relationship can be given as,
<math display="block">\sigma = \eta \frac{d\varepsilon}{dt}</math>
where σ is the stress, η is the viscosity of the material, and dε/dt is the time derivative of strain.

The relationship between stress and strain can be simplified for specific stress or strain rates. For high stress or strain rates/short time periods, the time derivative components of the stress–strain relationship dominate. In these conditions it can be approximated as a rigid rod capable of sustaining high loads without deforming. Hence, the dashpot can be considered to be a "short-circuit".<ref name="VanVliet">Van Vliet, Krystyn J. (2006). [http://stellar.mit.edu/S/course/3/fa06/3.032/index.html "3.032 Mechanical Behavior of Materials"]</ref><ref name=":0">{{Cite journal|last=Cacopardo|first=Ludovica|date=Jan 2019|title=Engineering hydrogel viscoelasticity|url=https://doi.org/10.1016/j.jmbbm.2018.09.031|journal=Journal of the Mechanical Behavior of Biomedical Materials|volume=89|pages=162–167|doi=10.1016/j.jmbbm.2018.09.031|pmid=30286375|hdl=11568/930491|s2cid=52918639|via=Elsevier|hdl-access=free}}</ref>

Conversely, for low stress states/longer time periods, the time derivative components are negligible and the dashpot can be effectively removed from the system – an "open" circuit.<ref name=":0" /> As a result, only the spring connected in parallel to the dashpot will contribute to the total strain in the system.<ref name=VanVliet/>

===Maxwell model===
{{main|Maxwell material}}

[[Image:Maxwell diagram.svg|right|thumb|upright=1.4|Maxwell model]]
The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. The model can be represented by the following equation:
<math display="block">\sigma + \frac {\eta} {E} \dot {\sigma} = \eta \dot {\varepsilon}</math>

Under this model, if the material is put under a constant strain, the stresses gradually [[Relaxation time|relax]]. When a material is put under a constant stress, the strain has two components. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. One limitation of this model is that it does not predict creep accurately.  The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time.<ref name=McCrum/>

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> 
).

===Kelvin–Voigt model===
{{main|Kelvin–Voigt material}}
[[Image:Kelvin Voigt diagram.svg|right|thumb|upright=0.6|Schematic representation of Kelvin–Voigt model]]

The Kelvin–Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the creep behaviour of polymers.

The constitutive relation is expressed as a linear first-order differential equation:
<math display="block">\sigma = E \varepsilon + \eta \dot {\varepsilon}</math>

This model represents a solid undergoing reversible, viscoelastic strain. Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching the steady-state strain. When the stress is released, the material gradually relaxes to its undeformed state. At constant stress (creep), the model is quite realistic as it predicts strain to tend to σ/E as time continues to infinity.  Similar to the Maxwell model, the Kelvin–Voigt model also has limitations.  The model is extremely good with modelling creep in materials, but with regards to relaxation the model is much less accurate.<ref name=Tanner>{{cite book|last1=Tanner|first1=Roger I.|title=Engineering Rheologu|date=1988|publisher=Oxford University Press|isbn=0-19-856197-0|pages=27}}</ref>

This model can be applied to organic polymers, rubber, and wood when the load is not too high.

=== Standard linear solid model ===
{{main|Standard linear solid model}}

The standard linear solid model, also known as the Zener model, consists of two springs and a dashpot. It is the simplest model that describes both the creep and stress relaxation behaviors of a viscoelastic material properly. For this model, the governing constitutive relations are:

{|class="wikitable" style="text-align: center;"
! Maxwell representation
! Kelvin representation
|-----
| [[Image:SLS.svg|200px]]
| [[Image:SLS2.svg|200px]]
|-----
|<math>\sigma + \frac {\eta} {E_2} \dot {\sigma} = E_1 \varepsilon + \frac {\eta (E_1 + E_2)} {E_2} \dot {\varepsilon}</math>
|<math>\sigma + \frac{\eta}{E_1+E_2} \dot \sigma = \frac{E_1E_2}{E_1+E_2}\varepsilon + \frac{E_1\eta}{E_1+E_2} \dot \varepsilon</math>
|}

Under a constant stress, the modeled material will instantaneously deform to some strain, which is the instantaneous elastic portion of the strain. After that it will continue to deform and asymptotically approach a steady-state strain, which is the retarded elastic portion of the strain. Although the standard linear solid model is more accurate than the Maxwell and Kelvin–Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions.

=== Jeffreys model ===
The Jeffreys model like the Zener model is a three element model. It consist of two dashpots and a spring.<ref>{{Cite book |last1=Barnes |first1=Howard A. |url=https://books.google.com/books?id=B1e0uxFg4oYC |title=An Introduction to Rheology |last2=Hutton |first2=John Fletcher |last3=Walters |first3=K. |date=1989 |publisher=Elsevier |isbn=978-0-444-87140-4 |language=en}}</ref>
[[File:Jeffreys rheological model.svg|center|thumb|171x171px|Jeffreys model]]
It was proposed in 1929 by [[Harold Jeffreys]] to study [[Earth's mantle]].<ref>{{Cite book |last=Bird |first=R. Byron |url=https://books.google.com/books?id=posvAQAAIAAJ |title=Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics |date=1987-05-27 |publisher=Wiley |isbn=978-0-471-80245-7 |language=en}}</ref>

=== Burgers model ===
{{main|Burgers material}}
The Burgers model consists of either two Maxwell components in parallel or a Kelvin–Voigt component, a spring and a dashpot in series. For this model, the governing constitutive relations are:

{|class="wikitable" style="text-align: center;"
! Maxwell representation
! Kelvin representation
|-----
| [[Image:Burgers model 2.svg|200px]]
| [[Image:Burgers model.svg|200px]]
|-----
|<math> \sigma + \left( \frac {\eta_1} {E_1} + \frac {\eta_2} {E_2} \right) \dot\sigma + 
\frac {\eta_1 \eta_2} {E_1 E_2} \ddot\sigma = \left( \eta_1 + \eta_2 \right) \dot\varepsilon +
\frac {\eta_1 \eta_2 \left( E_1 + E_2 \right)} {E_1 E_2} \ddot\varepsilon</math>
|<math> \sigma + \left( \frac {\eta_1} {E_1} + \frac {\eta_2} {E_1} + \frac {\eta_2} {E_2} \right) \dot\sigma + 
\frac {\eta_1 \eta_2} {E_1 E_2} \ddot\sigma = \eta_2\dot\varepsilon +
\frac {\eta_1 \eta_2} {E_1} \ddot\varepsilon</math>
|}

This model incorporates viscous flow into the standard linear solid model, giving a linearly increasing asymptote for strain under fixed loading conditions.
{{clear}}

=== Generalized Maxwell model ===
{{main|Generalized Maxwell model}}
[[Image:Weichert.svg|right|thumb|upright=1.4|Schematic of Maxwell-Wiechert Model]]
The generalized Maxwell model, also known as the Wiechert model, is the most general form of the linear model for viscoelasticity. It takes into account that the [[Relaxation time|relaxation]] does not occur at a single time, but at a distribution of times. Due to molecular segments of different lengths with shorter ones contributing less than longer ones, there is a varying time distribution.  The Wiechert model shows this by having as many spring–dashpot Maxwell elements as necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.<ref name=Roylance>Roylance, David (2001); "Engineering Viscoelasticity", 14–15</ref>
Applications: metals and alloys at temperatures lower than one quarter of their absolute melting temperature (expressed in K).
{{clear}}

== 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 ===
{{main|Second-order fluid}}
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 \cdot\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 - (\nabla\mathbf v) ^\mathbf T \cdot \mathbf D - \mathbf D \cdot \nabla \mathbf v</math> and <math>\dot \mathbf D \equiv \frac {\partial}{ \partial t} \mathbf D + \mathbf v \cdot \nabla \mathbf D</math> is the material time derivative of the deformation tensor.<ref name="Macosko 1994"/>

=== Upper-convected Maxwell model ===
{{main|Upper-convected Maxwell model}}
The upper-convected Maxwell model incorporates nonlinear time behavior into the viscoelastic Maxwell model, given by:<ref name="Macosko 1994"/>

<math display=block>\mathbf \tau + \lambda \mathbf \tau ^\triangledown = 2 \eta _0 \mathbf D</math>
where <math>\mathbf \tau</math> denotes the stress tensor.

=== Oldroyd-B model ===
{{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| pages=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:
<math display=block> \mathbf{T} + \lambda_1 \stackrel{\nabla}{\mathbf{T}} = 2\eta_0 (\mathbf{D} + \lambda_2 \stackrel{\nabla}{\mathbf{D}}) </math>
where:
* <math>\mathbf{T}</math> is the [[Stress (physics)|stress]] [[tensor]];
* <math>\lambda_1</math> is the relaxation time;
* <math>\lambda_2</math> is the retardation time = <math> \frac{\eta_s}{\eta_0}\lambda_1 </math>;
* <math> \stackrel{\nabla}{\mathbf{T}} </math> is the [[upper convected time derivative]] of stress tensor:<math display=block> \stackrel{\nabla}{\mathbf{T}} = \frac{\partial}{\partial t} \mathbf{T} + \mathbf{v} \cdot \nabla \mathbf{T} -( (\nabla \mathbf{v})^T \cdot \mathbf{T} + \mathbf{T} \cdot (\nabla \mathbf{v})); </math>
*<math>\mathbf{v}</math> is the fluid velocity;
*<math>\eta_0</math> is the total [[viscosity]] composed of solvent and polymer components, <math> \eta_0 = \eta_s + \eta_p </math>;
*<math>\mathbf {D}</math> is the deformation rate tensor or rate of strain tensor, <math>\mathbf{D} = \frac{1}{2} \left[\boldsymbol\nabla \mathbf{v} + (\boldsymbol\nabla \mathbf{v})^T\right]</math>.

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:
<math display=block>\mathbf{\sigma}(t) = -p \mathbf{I} + \int_{-\infty}^{t} M(t-t')h(I_1,I_2)\mathbf{B}(t')\, dt'</math>

where:
*<math>\mathbf{\sigma}(t)</math> is the [[Cauchy stress tensor]] as function of time ''t'',
*''p'' is the pressure
*<math>\mathbf{I}</math> is the unity tensor
*''M'' is the memory function showing, usually expressed as a sum of exponential terms for each mode of [[Relaxation (physics)|relaxation]]: <math display=block>M(x)=\sum_{k=1}^m \frac{g_i}{\theta_i}\exp \left(\frac{-x}{\theta_i}\right),</math> where for each mode of the relaxation, <math>g_i</math> is the relaxation modulus and <math>\theta_i</math> is the relaxation time;
*<math>h(I_1,I_2)</math> is the ''strain damping'' function that depends upon the first and second [[Invariants of tensors|invariants]] of [[Finite strain theory#The Finger deformation tensor|Finger tensor]] <math>\mathbf{B}</math>.

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>
G(t) = G_\infty + \sum_{i=1}^{N} G_i \exp(-t/\tau_i)
</math>

where <math>G_\infty</math> is the long term modulus once the material is totally relaxed, <math>\tau_i</math> are the relaxation times (not to be confused with <math>\tau_i</math> in the diagram); the higher their values, the longer it takes for the stress to relax. The data is fitted with the equation by using a minimization algorithm that adjust the parameters (<math>G_\infty, G_i, \tau_i</math>) to minimize the error between the predicted and data values.<ref name=ETT2011>E. J. Barbero. "Time-temperature-age Superposition Principle for Predicting Long-term Response of Linear Viscoelastic Materials", chapter 2 in [https://www.amazon.com/exec/obidos/ASIN/1439852367/booksoncomposite ''Creep and fatigue in polymer matrix composites'']. Woodhead, 2011.</ref>

An alternative form is obtained noting that the elastic modulus is related to the long term modulus by

<math display=block>
G(t=0) = G_0 = G_\infty+\sum_{i=1}^{N} G_i
</math>

Therefore,

<math display=block>
G(t) = G_0 - \sum_{i=1}^{N} G_i \left[1-e^{-t / \tau_i}\right]
</math>

This form is convenient when the elastic shear modulus <math>G_0</math> is obtained from data independent from the relaxation data, and/or for computer implementation, when it is desired to specify the elastic properties separately from the viscous properties, as in Simulia (2010).<ref name=Abaqus_Analysis_19.7.1>Simulia. ''Abaqus Analysis User's Manual'', 19.7.1 "Time domain vicoelasticity", 6.10 edition, 2010</ref>

A creep experiment is usually easier to perform than a relaxation one, so most data is available as (creep) compliance vs. time.<ref name=campusplastics>[http://www.campusplastics.com/ Computer Aided Material Preselection by Uniform Standards]</ref>  Unfortunately, there is no known closed form for the (creep) compliance in terms of the coefficient of the Prony
series. So, if one has creep data, it is not easy to get the coefficients of the (relaxation) Prony series, which are needed for example in.<ref name=Abaqus_Analysis_19.7.1 /> An expedient way to obtain these coefficients is the following. First, fit the creep data with a model that has closed form solutions in both compliance and relaxation; for example the Maxwell-Kelvin model
(eq. 7.18-7.19) in Barbero (2007)<ref name=Barbero2007>E. J. Barbero. [https://www.amazon.com/exec/obidos/ASIN/1420054333/booksoncomposite ''Finite Element Analysis of Composite Materials'']. CRC Press, Boca Raton, Florida, 2007.</ref> or the Standard Solid Model (eq. 7.20-7.21) in Barbero (2007)<ref name=Barbero2007 /> (section 7.1.3). Once the parameters of the creep model are known, produce relaxation pseudo-data with the conjugate relaxation model for the same
times of the original data. Finally, fit the pseudo data with the Prony series.

==Effect of temperature==
{{main|Time–temperature superposition}}

The secondary bonds of a polymer constantly break and reform due to thermal motion.  Application of a stress favors some conformations over others, so the molecules of the polymer will gradually "flow" into the favored conformations over time.<ref>S.A. Baeurle, A. Hotta, A.A. Gusev, Polymer '''47''', 6243-6253 (2006).</ref>  Because thermal motion is one factor contributing to the deformation of polymers, viscoelastic properties change with increasing or decreasing temperature.  In most cases, the creep modulus, defined as the ratio of applied stress to the time-dependent strain, decreases with increasing temperature. Generally speaking, an increase in temperature correlates to a logarithmic decrease in the time required to impart equal strain under a constant stress. In other words, it takes less work to stretch a viscoelastic material an equal distance at a higher temperature than it does at a lower temperature.

More detailed effect of temperature on the viscoelastic behavior of polymer can be plotted as shown.

There are mainly five regions (some denoted four, which combines IV and V together) included in the typical polymers.<ref>{{Cite journal|last=Aklonis.|first=J.J.|date=1981|title=Mechanical properties of polymer|journal=J Chem Educ|volume=58 | issue=11 | page=892 |doi=10.1021/ed058p892|bibcode=1981JChEd..58..892A|doi-access=}}</ref>

*Region I: Glassy state of the polymer is presented in this region. The temperature in this region for a given polymer is too low to endow molecular motion. Hence the motion of the molecules is frozen in this area. The mechanical property is hard and brittle in this region.<ref>{{Cite journal|last=I. M.|first=Kalogeras|date=2012|title=The nature of the glassy state: structure and glass transitions.|journal=Journal of Materials Education|volume=34 |issue=3|page=69}}</ref>
*Region II: Polymer passes glass transition temperature in this region. Beyond Tg, the thermal energy provided by the environment is enough to unfreeze the motion of molecules. The molecules are allowed to have local motion in this region hence leading to a sharp drop in stiffness compared to Region I.
*Region III: Rubbery plateau region. Materials lie in this region would exist long-range elasticity driven by entropy. For instance, a rubber band is disordered in the initial state of this region. When stretching the rubber band, you also align the structure to be more ordered. Therefore, when releasing the rubber band, it will spontaneously seek higher entropy state hence goes back to its initial state. This is what we called entropy-driven elasticity shape recovery.
*Region IV: The behavior in the rubbery flow region is highly time-dependent. Polymers in this region would need to use a time-temperature superposition to get more detailed information to cautiously decide how to use the materials. For instance, if the material is used to cope with short interaction time purpose, it could present as 'hard' material. While using for long interaction time purposes, it would act as 'soft' material.<ref>{{Cite book|last=I|first=Emri|title=Time-dependent behavior of solid polymers|year=2010}}</ref>
*Region V: Viscous polymer flows easily in this region. Another significant drop in stiffness.
[[File:Visco.jpg|thumb|Temperature dependence of modulus]]

Extreme cold temperatures can cause viscoelastic materials to change to the [[Glass transition|glass]] phase and become [[brittle]]. For example, exposure of [[pressure sensitive adhesive]]s to extreme cold ([[dry ice]], [[freeze spray]], etc.) causes them to lose their tack, resulting in debonding.

==Viscoelastic creep==
{{main|Creep (deformation)}}

[[Image:Creep.svg|right|thumb|250px| a) Applied stress and b) induced strain as functions of time over a short period for a viscoelastic material.]]

When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain.  This phenomenon is known as viscoelastic creep.

At time <math>t_0</math>, a viscoelastic material is loaded with a constant stress that is maintained for a sufficiently long time period.  The material responds to the stress with a strain that increases until the material ultimately fails, if it is a viscoelastic liquid. If, on the other hand, it is a viscoelastic solid, it may or may not fail depending on the applied stress versus the material's ultimate resistance.  When the stress is maintained for a shorter time period, the material undergoes an initial strain until a time <math>t_1</math>, after which the strain immediately decreases (discontinuity) then gradually decreases at times <math>t > t_1</math> to a residual strain.

Viscoelastic creep data can be presented by plotting the creep modulus (constant applied stress divided by total strain at a particular time) as a function of time.<ref name=Rosato>Rosato, et al. (2001): "Plastics Design Handbook", 63-64.</ref>  Below its critical stress, the viscoelastic creep modulus is independent of stress applied.  A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep modulus versus time curve if the applied stresses are below the material's critical stress value.

Viscoelastic creep is important when considering long-term structural design.  Given loading and temperature conditions, designers can choose materials that best suit component lifetimes.

== Measurement ==

=== 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:
<math display="block">\dot{\epsilon} = -\frac{2}{R}{dR \over dt}</math>
where <math>R</math> is the mid-radius value and <math>\dot{\epsilon}</math> is the strain rate. The viscosity of the sample is then calculated using the following equation:
<math display="block">\eta = \frac{F}{\pi R^2 \dot{\epsilon}}</math>
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.
[[File:Ser_extensional_rheometer.png|thumb|Schematic of the SER extensional rheometer. The sample (brown) is held to two cylinders (grey) which are then counterrotated at varying strain rates. The torque required to strain the sample at these rates is calculated via a set of torque transducers present in the instrument. These torque values are then converted to stress values, and the stresses and strain rates are then used to determine the viscosity of the sample.]]

=== Other methods ===
Though there are many instruments that test the mechanical and viscoelastic response of materials, [[broadband viscoelastic spectroscopy]] (BVS) and [[resonant ultrasound spectroscopy]] (RUS) are more commonly used to test viscoelastic behavior because they can be used above and below ambient temperatures and are more specific to testing viscoelasticity. These two instruments employ a damping mechanism at various frequencies and time ranges with no appeal to [[time–temperature superposition]]. Using BVS and RUS to study the mechanical properties of materials is important to understanding how a material exhibiting viscoelasticity will perform.<ref name=Lakes>{{cite book |author=Rod Lakes |year=1998 |title= Viscoelastic solids |publisher = CRC Press |isbn= 0-8493-9658-1}}</ref>

== See also ==
{{div col|colwidth=22em}}
* [[Bingham plastic]]
* [[Biomaterial]]
* [[Biomechanics]]
* [[Blood viscoelasticity]]
* [[Constant viscosity elastic fluid]]s
* [[Deformation index]]
* [[Glass transition]]
*[[Pressure-sensitive adhesive]]
* [[Rheology]]
* [[Rubber elasticity]]
* [[Silly Putty]]
* [[Viscoelasticity of bone]]
* [[Viscoplasticity]]
* [[Visco-elastic jets]]
{{div col end}}
{{clear}}

== References ==
{{Reflist}}
{{refbegin}}
* Silbey and Alberty (2001): ''Physical Chemistry'', 857. John Wiley & Sons, Inc.
* Alan S. Wineman and K. R. Rajagopal (2000): ''Mechanical Response of Polymers: An Introduction'' 
* Allen and Thomas (1999): ''The Structure of Materials'', 51.
* Crandal et al. (1999): ''An Introduction to the Mechanics of Solids'' 348
* J. Lemaitre and J. L. Chaboche (1994) ''Mechanics of solid materials''
* Yu. Dimitrienko (2011) ''Nonlinear continuum mechanics and Large Inelastic Deformations'', Springer, 772p
{{refend}}
{{Authority control}}
{{Non-Newtonian fluids}}
[[Category:Materials science]]
[[Category:Elasticity (physics)]]
[[Category:Non-Newtonian fluids]]
[[Category:Continuum mechanics]]
[[Category:Rubber properties]]
[[Category:Hysteresis]]