Generalized Maxwell model

Image:Weichert.svg

The generalized Maxwell model also known as the Maxwell–Wiechert model (after James Clerk Maxwell and E WiechertWiechert, E (1889); "Ueber elastische Nachwirkung", Dissertation, Königsberg University, GermanyWiechert, E (1893); "Gesetze der elastischen Nachwirkung für constante Temperatur", Annalen der Physik, Vol. 286, [https://doi.org/10.1002/andp.18932861011 issue 10, p. 335–348] and [https://doi.org/10.1002/andp.18932861110 issue 11, p. 546–570]) is the most general form of the linear model for viscoelasticity. In this model, several Maxwell elements are assembled in parallel. It takes into account that the relaxation does not occur at a single time, but in a set of times. Due to the presence of 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 are necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.Roylance, David (2001); "Engineering Viscoelasticity", 14-15Tschoegl, Nicholas W. (1989); "The Phenomenological Theory of Linear Viscoelastic Behavior", 119-126

The generalized Maxwell model is widely applied to describe how materials deform under mechanical stress when both elastic and viscous effects are present. It assumes linear viscoelastic behavior and is suitable for cases involving small deformations.{{Cite journal |last=Petrie |first=Christopher J. S. |date=1977-05-01 |title=On stretching maxwell models |url=https://linkinghub.elsevier.com/retrieve/pii/0377025777800025 |journal=Journal of Non-Newtonian Fluid Mechanics |volume=2 |issue=3 |pages=221–253 |doi=10.1016/0377-0257(77)80002-5 |issn=0377-0257}} Because of its ability to represent complex time-dependent responses, the model is commonly used in the study of polymers, soft tissues, and other viscoelastic solids.{{Cite journal |last=Song |first=Jake |last2=Holten-Andersen |first2=Niels |last3=McKinley |first3=Gareth H. |date=2023 |title=Non-Maxwellian viscoelastic stress relaxations in soft matter |url=https://xlink.rsc.org/?DOI=D3SM00736G |journal=Soft Matter |language=en |volume=19 |issue=41 |pages=7885–7906 |doi=10.1039/D3SM00736G |issn=1744-683X}} The model can be expressed either in the time domain using a relaxation function or in the frequency domain through a complex modulus, making it adaptable for use in experimental and computational analyses. In engineering practice, it is often implemented using a Prony series to simulate viscoelastic behavior in finite element analysis.{{Cite journal |last=Luk-Cyr |first=Jacques |last2=Crochon |first2=Thibaut |last3=Li |first3=Chun |last4=Lévesque |first4=Martin |date=2012-07-06 |title=Interconversion of linearly viscoelastic material functions expressed as Prony series: a closure |url=https://doi.org/10.1007/s11043-012-9176-y |journal=Mechanics of Time-Dependent Materials |volume=17 |issue=1 |pages=53–82 |doi=10.1007/s11043-012-9176-y |issn=1385-2000}}{{Cite journal |last=Chen |first=T. |title=Determining a Prony Series for a viscoelastic material from time varying strain data |journal=Technical Report, NASA |issue=2000}}

General model form

A common mathematical formulation of the generalized Maxwell model uses a discrete relaxation spectrum, where each Maxwell element contributes a term to the overall stress relaxation behavior. This leads to a Prony series representation of the relaxation modulus:

=<math>G(t) = \sum_{i=1}^{N} G_i \exp\left(-\frac{t}{\tau_i}\right)</math>=

where G_i is the modulus and 𝜏_i is the relaxation time associated with the i^th Maxwell element. This method works well when the number of relaxation times in the material is already known or can be estimated from experiments. A common rule of thumb is to include about one relaxation mode for each decade of time or frequency. More advanced statistical tools can also be used to find the smallest number of modes that still give a good fit, while avoiding overfitting and keeping the model physically realistic.{{Cite journal |last=Renaud |first=Franck |last2=Dion |first2=Jean-Luc |last3=Chevallier |first3=Gaël |last4=Tawfiq |first4=Imad |last5=Lemaire |first5=Rémi |date=2011-04-01 |title=A new identification method of viscoelastic behavior: Application to the generalized Maxwell model |url=https://linkinghub.elsevier.com/retrieve/pii/S0888327010003079 |journal=Mechanical Systems and Signal Processing |volume=25 |issue=3 |pages=991–1010 |doi=10.1016/j.ymssp.2010.09.002 |issn=0888-3270}}

=Solids=

Given $N+1$ elements with moduli $E_i$ , viscosities $\eta_i$ , and relaxation times $\tau_i=\frac{\eta_i}{E_i}$

The general form for the model for solids is given by {{citation needed|date=November 2014}}:

{{Equation box 1 |title = General Maxwell Solid Model ({{EquationRef|1}}) |equation =

$\sigma+$

$\sum^{N}_{n=1}{
\left({
\sum^{N-n+1}_{i_1=1}{
...
\left({
\sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{
...
\left({
\sum^{N}_{i_n=i_{n-1}+1}{
\left({
\prod_{j\in\left\{{i_1,...,i_n}\right\}}{
\tau_j
}
}\right)
}
}\right)
...
}
}\right)
...
}
}\right)
\frac{\partial^{n}{\sigma}}{\partial{t}^{n}}
}$

$=$

$E_0\epsilon+$

$\sum^{N}_{n=1}{
\left({
\sum^{N-n+1}_{i_1=1}{
...
\left({
\sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{
...
\left({
\sum^{N}_{i_n=i_{n-1}+1}{
\left({
\left({
E_0+\sum_{j\in\left\{{i_1,...,i_n}\right\}}{
E_j
}
}\right)
\left({
\prod_{k\in\left\{{i_1,...,i_n}\right\}}{
\tau_k
}
}\right)
}\right)
}
}\right)
...
}
}\right)
...
}
}\right)
\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}
}$

|cellpadding = 6 |border = 1 |border colour = black |background colour = white}}

{{divhide|This may be more easily understood by showing the model in a slightly more expanded form:}}

{{Equation box 1 |title = General Maxwell Solid Model ({{EquationRef|2}}) |equation =

$\sigma+$

${\left({\sum^{N}_{i=1}{\tau_i}}\right)}
\frac{\partial{\sigma}}{\partial{t}}
+$

${\left({\sum^{N-1}_{i=1}{
\left({\sum^{N}_{j=i+1}{
\tau_i\tau_j
}}\right)
}}\right)}
\frac{\partial^{2}{\sigma}}{\partial{t}^{2}}$

$+...+$

$\left({
\sum^{N-n+1}_{i_1=1}{
...
\left({
\sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{
...
\left({
\sum^{N}_{i_n=i_{n-1}+1}{
\left({
\prod_{j\in\left\{{i_1,...,i_n}\right\}}{
\tau_j
}
}\right)
}
}\right)
...
}
}\right)
...
}
}\right)
\frac{\partial^{n}{\sigma}}{\partial{t}^{n}}$

$+...+$

$\left({
\prod^{N}_{i=1}{
\tau_i
}
}\right)
\frac{\partial^{N}{\sigma}}{\partial{t}^{N}}$

$=$

$E_0\epsilon+$

${\left({\sum^{N}_{i=1}{\left({E_0+E_i}\right)\tau_i}}\right)}
\frac{\partial{\epsilon}}{\partial{t}}
+$

${\left({\sum^{N-1}_{i=1}{
\left({\sum^{N}_{j=i+1}{
\left({E_0+E_i+E_j}\right)
\tau_i\tau_j
}}\right)
}}\right)}
\frac{\partial^{2}{\epsilon}}{\partial{t}^{2}}$

$+...+$

$\left({
\sum^{N-n+1}_{i_1=1}{
...
\left({
\sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{
...
\left({
\sum^{N}_{i_n=i_{n-1}+1}{
\left({
\left({
E_0+\sum_{j\in\left\{{i_1,...,i_n}\right\}}{
E_j
}
}\right)
\left({
\prod_{k\in\left\{{i_1,...,i_n}\right\}}{
\tau_k
}
}\right)
}\right)
}
}\right)
...
}
}\right)
...
}
}\right)
\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}$

$+...+$

$\left({
E_0+\sum_{j=1}^{N} E_j
}\right)
\left({
\prod^{N}_{i=1}{
\tau_i
}
}\right)
\frac{\partial^{N}{\epsilon}}{\partial{t}^{N}}$

|cellpadding = 6 |border = 1 |border colour = black |background colour = white}}

==Example: [[standard linear solid model]]==

Following the above model with $N+1=2$ elements yields the standard linear solid model:

{{Equation box 1 |title = Standard Linear Solid Model ({{EquationRef|3}}) |equation =

$\sigma+\tau_1\frac{\partial{\sigma}}{\partial{t}}=E_0\epsilon+\tau_1\left({E_0+E_1}\right)\frac{\partial{\epsilon}}{\partial{t}}$

|cellpadding = 6 |border = 1 |border colour = black |background colour = white}}

=Fluids=

Given $N+1$ elements with moduli $E_i$ , viscosities $\eta_i$ , and relaxation times $\tau_i=\frac{\eta_i}{E_i}$

The general form for the model for fluids is given by:

{{Equation box 1 |title = General Maxwell Fluid Model ({{EquationRef|4}}) |equation =

$\sigma+$

$=$

$\sum^{N}_{n=1}{
\left({
\eta_0+\sum^{N-n+1}_{i_1=1}{
...
\left({
\sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{
...
\left({
\sum^{N}_{i_n=i_{n-1}+1}{
\left({
\left({
\sum_{j\in\left\{{i_1,...,i_n}\right\}}{
E_j
}
}\right)
\left({
\prod_{k\in\left\{{i_1,...,i_n}\right\}}{
\tau_k
}
}\right)
}\right)
}
}\right)
...
}
}\right)
...
}
}\right)
\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}
}$

|cellpadding = 6 |border = 1 |border colour = black |background colour = white}}

{{divhide|This may be more easily understood by showing the model in a slightly more expanded form:}}

{{Equation box 1 |title = General Maxwell Fluid Model ({{EquationRef|5}}) |equation =

$\sigma+$

${\left({\sum^{N}_{i=1}{\tau_i}}\right)}
\frac{\partial{\sigma}}{\partial{t}}
+$

${\left({\sum^{N-1}_{i=1}{
\left({\sum^{N}_{j=i+1}{
\tau_i\tau_j
}}\right)
}}\right)}
\frac{\partial^{2}{\sigma}}{\partial{t}^{2}}$

$+...+$

$\left({
\prod^{N}_{i=1}{
\tau_i
}
}\right)
\frac{\partial^{N}{\sigma}}{\partial{t}^{N}}$

$=$

${\left({\eta_0+\sum^{N}_{i=1}{E_i\tau_i}}\right)}
\frac{\partial{\epsilon}}{\partial{t}}
+$

${\left({\eta_0+\sum^{N-1}_{i=1}{
\left({\sum^{N}_{j=i+1}{
\left({E_i+E_j}\right)
\tau_i\tau_j
}}\right)
}}\right)}
\frac{\partial^{2}{\epsilon}}{\partial{t}^{2}}$

$+...+$

$\left({
\eta_0+
\sum^{N-n+1}_{i_1=1}{
...
\left({
\sum^{N-\left({n-a}\right)+1}_{i_a=i_{a-1}+1}{
...
\left({
\sum^{N}_{i_n=i_{n-1}+1}{
\left({
\left({
\sum_{j\in\left\{{i_1,...,i_n}\right\}}{
E_j
}
}\right)
\left({
\prod_{k\in\left\{{i_1,...,i_n}\right\}}{
\tau_k
}
}\right)
}\right)
}
}\right)
...
}
}\right)
...
}
}\right)
\frac{\partial^{n}{\epsilon}}{\partial{t}^{n}}$

$+...+$

$\left({
\eta_0+
\left({
\sum_{j=1}^{N} E_j
}\right)
\left({
\prod^{N}_{i=1}{
\tau_i
}
}\right)
}\right)
\frac{\partial^{N}{\epsilon}}{\partial{t}^{N}}$

|cellpadding = 6 |border = 1 |border colour = black |background colour = white}}

==Example: three parameter fluid==

The analogous model to the standard linear solid model is the three parameter fluid, also known as the Jeffreys model:{{cite book

| last = Gutierrez-Lemini

| first = Danton

| authorlink =

| title = Engineering Viscoelasticity

| publisher = Springer

| series =

| volume =

| edition =

| year = 2013

| location =

| pages = 88

| language =

| url = https://books.google.com/books?id=MbjBAAAAQBAJ&q=three+parameter+fluid+viscoelasticity&pg=PA88

| doi =

| id =

| isbn = 9781461481393

| mr =

| zbl =

| jfm =

}}

{{Equation box 1 |title = Three Parameter Maxwell Fluid Model ({{EquationRef|6}}) |equation =

$\sigma+\tau_1\frac{\partial{\sigma}}{\partial{t}}=\left({\eta_0+\tau_1 E_1\frac{\partial}{\partial t}}\right)\frac{\partial{\epsilon}}{\partial{t}}$

|cellpadding = 6 |border = 1 |border colour = black |background colour = white}}

Comparison of Linear Viscoelastic Models

class="wikitable"

!Model

!Configuration

!Best for

!Limitation

Maxwell model

|Spring and dashpot in series

|Describing stress relaxation

|Fails to model creep accurately; predicts unbounded strain under constant stress

Kelvin-Voigt model

|Spring and dashpot in parallel

|Describing creep (delayed strain under constant stress)

|Cannot describe stress relaxation behavior

Generalized Maxwell model

|Multiple Maxwell elements in parallel

|Modeling realistic stress relaxation and frequency-dependent behavior

|Requires fitting many parameters for accurate behavior

References

Category:Materials science

Category:Non-Newtonian fluids

Category:James Clerk Maxwell