Viscoelasticity: Difference between revisions

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In the nineteenth century, physicists such as [[James Clerk Maxwell|Maxwell]], [[Ludwig Boltzmann|Boltzmann]], and [[William Thomson, 1st Baron Kelvin|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|(d<math>d\varepsilon/dt</math>/d''t'')}}]]

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 include 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 both on the rate of the change of their length as well as on 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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== 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/>

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== 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]]:

{{block indent|<math display="block">\varepsilon(t) = \frac { \sigma(t) }{ E_\text{inst,creep} }+ \int_0^t K(t - t^\prime') \dot{\sigma}(t^\prime') d t^\primedt'</math>}}

or

{{block indent|<math display="block">\sigma(t)= E_\text{inst,relax}\varepsilon(t)+ \int_0^t F(t - t^\prime') \dot{\varepsilon}(t^\prime') d t^\primedt'</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.

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A complex [[dynamic modulus]] G can be used to represent the relations between the oscillating stress and strain:

{{block indent|<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'':

{{block indent|<math display="block"> G' = \frac {\sigma_0} {\varepsilon_0} \cos \delta </math>}}

{{block indent|<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.

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The elastic components, as previously mentioned, can be modeled as [[spring (device)|springs]] of elastic constant E, given the formula:

{{block indent|<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,

{{block indent|<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.

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[[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:

{{block indent|<math display="block">\sigma + \frac {\eta} {E} \dot {\sigma} = \eta \dot {\varepsilon}</math>}}

{{block indent|<math>\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/>

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The constitutive relation is expressed as a linear first-order differential equation:

{{block indent|<math display="block">\sigma = E \varepsilon + \eta \dot {\varepsilon}</math>}}

{{block indent|<math>\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>

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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>\boldmathbf T = -p\boldmathbf I + 2 \eta_0\boldmathbf D - \psi_1 \boldmathbf D^\triangledown + 4\psi _2 \boldmathbf D \bullet\boldmathbf D</math>

where:

* <math>\boldmathbf DI</math> is the deformationidentity tensor

* <math>\boldmathbf ID</math> is the identitydeformation tensor

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

<math display=block>\bold D ^\triangledown \equiv \dot \bold D - (\bigtriangledown\bold v) ^\bold T \bullet \bold D - \bold D \bullet \bigtriangledown \bold v</math> and <math>\dot \bold D \equiv \frac {\partial}{ \partial t} \bold D + \bold v \bullet \bigtriangledown \bold D</math>is the material time derivative of the deformation tensor.<ref name="Macosko 1994"/>

=== Upper-convected Maxwell model ===

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

<math display=block>\boldmathbf \tau + \lambda \boldmathbf \tau ^\triangledown = 2 \eta _0 \boldmathbf D</math>

*where <math>\boldmathbf \tau</math> denotes the stress tensor.

where:

* <math>\bold \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:

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* <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"/>

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=== Wagner model ===

{{main|Wagner model}}

Wagner model is might be considered as a simplified practical form of the [[Bernstein-Kearsley-Zapas]]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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*''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 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>.

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{{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

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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</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>

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>

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<math display=block>

G(t) = G_0 - \sum_{i=1}^{N} G_i \left[1-\exp(e^{-t / \tau_i)}\right]

</math>

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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=free}}</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 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 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.

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]]

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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 |last=Bach |first=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 |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>

<math>\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>

<math>\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 |issn=1435-1528}}</ref> It then calculates the sample viscosity using the well known equation:

<math display="block">\sigma = \eta \dot{\epsilon}</math>

<math>\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.]]

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* [[Visco-elastic jets]]

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{{clear}}

== References ==

{{Reflist}}