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

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.^[5] 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.^[6] 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.^[7][8]

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.^[6] This leads to a Prony series representation of the relaxation modulus:

${\displaystyle G(t)=\sum _{i=1}^{N}G_{i}\exp \left(-{\frac {t}{\tau _{i}}}\right)}$

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.^[9]

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:^[10]

Three Parameter Maxwell Fluid Model (6)

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

Comparison of Linear Viscoelastic Models

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[9]	Requires fitting many parameters for accurate behavior