# Generalized Maxwell model

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Schematic of Maxwell–Wiechert model

The Generalized Maxwell model also known as the Maxwell–Wiechert model (after James Clerk Maxwell and E Wiechert[1][2]) 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.[3][4]

## General model form

### Solids

Given ${\displaystyle N+1}$ elements with moduli ${\displaystyle E_{i}}$, viscosities ${\displaystyle \eta _{i}}$, and relaxation times ${\displaystyle \tau _{i}={\frac {\eta _{i}}{E_{i}}}}$

The general form for the model for solids is given by[citation needed]:

 General Maxwell Solid Model (1) ${\displaystyle \sigma +}$ ${\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}}$ ${\displaystyle =}$ ${\displaystyle E_{0}\epsilon +}$ ${\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\left({E_{0}+\sum _{j\in \left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod _{k\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{k}}}\right)}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\epsilon }}{\partial {t}^{n}}}}}$

#### Example: standard linear solid model

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

 Standard Linear Solid Model (3) ${\displaystyle \sigma +\tau _{1}{\frac {\partial {\sigma }}{\partial {t}}}=E_{0}\epsilon +\tau _{1}\left({E_{0}+E_{1}}\right){\frac {\partial {\epsilon }}{\partial {t}}}}$

### Liquids

Given ${\displaystyle N+1}$ elements with moduli ${\displaystyle E_{i}}$, viscosities ${\displaystyle \eta _{i}}$, and relaxation times ${\displaystyle \tau _{i}={\frac {\eta _{i}}{E_{i}}}}$

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

 General Maxwell Fluid Model (4) ${\displaystyle \sigma +}$ ${\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}}$ ${\displaystyle =}$ ${\displaystyle \sum _{n=1}^{N}{\left({\eta _{0}+\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\left({\sum _{j\in \left\{{i_{1},...,i_{n}}\right\}}{E_{j}}}\right)\left({\prod _{k\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{k}}}\right)}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\epsilon }}{\partial {t}^{n}}}}}$

#### Example: three parameter fluid

The analogous model to the standard linear solid model is the three parameter fluid, also known as the Jeffrey model:[5]

 Three Parameter Maxwell Fluid Model (6) ${\displaystyle \sigma +\tau _{1}{\frac {\partial {\sigma }}{\partial {t}}}=\left({\eta _{0}+\tau _{1}E_{1}}\right){\frac {\partial {\epsilon }}{\partial {t}}}}$

## References

1. ^ Wiechert, E (1889); "Ueber elastische Nachwirkung", Dissertation, Königsberg University, Germany
2. ^ Wiechert, E (1893); "Gesetze der elastischen Nachwirkung für constante Temperatur", Annalen der Physik, 286, 335–348, 546–570
3. ^ Roylance, David (2001); "Engineering Viscoelasticity", 14-15
4. ^ Tschoegl, Nicholas W. (1989); "The Phenomenological Theory of Linear Viscoelastic Behavior", 119-126
5. ^ Gutierrez-Lemini, Danton (2013). Engineering Viscoelasticity. Springer. p. 88. ISBN 9781461481393.