Generalized Maxwell 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]

Schematic of Maxwell–Wiechert model

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)

${\displaystyle \sigma +}$ ${\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}}$

${\displaystyle =}$

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

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

General Maxwell Solid Model (2)

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

${\displaystyle \left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}$ ${\displaystyle +...+}$ ${\displaystyle \left({\prod _{i=1}^{N}{\tau _{i}}}\right){\frac {\partial ^{N}{\sigma }}{\partial {t}^{N}}}}$

${\displaystyle =}$

${\displaystyle E_{0}\epsilon +}$ ${\displaystyle {\left({\sum _{i=1}^{N}{\left({E_{0}+E_{i}}\right)\tau _{i}}}\right)}{\frac {\partial {\epsilon }}{\partial {t}}}+}$ ${\displaystyle {\left({\sum _{i=1}^{N-1}{\left({\sum _{j=i+1}^{N}{\left({E_{0}+E_{i}+E_{j}}\right)\tau _{i}\tau _{j}}}\right)}}\right)}{\frac {\partial ^{2}{\epsilon }}{\partial {t}^{2}}}}$ ${\displaystyle +...+}$

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

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)

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

Fluids

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 fluids is given by:

General Maxwell Fluid Model (4)

${\displaystyle \sigma +}$ ${\displaystyle \sum _{n=1}^{N}{\left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}}$

${\displaystyle =}$

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

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

General Maxwell Fluid Model (5)

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

${\displaystyle \left({\sum _{i_{1}=1}^{N-n+1}{...\left({\sum _{i_{a}=i_{a-1}+1}^{N-\left({n-a}\right)+1}{...\left({\sum _{i_{n}=i_{n-1}+1}^{N}{\left({\prod _{j\in \left\{{i_{1},...,i_{n}}\right\}}{\tau _{j}}}\right)}}\right)...}}\right)...}}\right){\frac {\partial ^{n}{\sigma }}{\partial {t}^{n}}}}$ ${\displaystyle +...+}$ ${\displaystyle \left({\prod _{i=1}^{N}{\tau _{i}}}\right){\frac {\partial ^{N}{\sigma }}{\partial {t}^{N}}}}$

${\displaystyle =}$

${\displaystyle {\left({\eta _{0}+\sum _{i=1}^{N}{E_{i}\tau _{i}}}\right)}{\frac {\partial {\epsilon }}{\partial {t}}}+}$ ${\displaystyle {\left({\eta _{0}+\sum _{i=1}^{N-1}{\left({\sum _{j=i+1}^{N}{\left({E_{i}+E_{j}}\right)\tau _{i}\tau _{j}}}\right)}}\right)}{\frac {\partial ^{2}{\epsilon }}{\partial {t}^{2}}}}$ ${\displaystyle +...+}$

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

Example: three parameter fluid

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

Three Parameter Maxwell Fluid Model (6)

${\displaystyle \sigma +\tau _{1}{\frac {\partial {\sigma }}{\partial {t}}}=\left({\eta _{0}+\tau _{1}E_{1}{\frac {\partial }{\partial t}}}\right){\frac {\partial {\epsilon }}{\partial {t}}}}$