Maxwell model

A Maxwell model is the most simple model viscoelastic material showing properties of a typical liquid.^[1] It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations.^[2] It is named for James Clerk Maxwell who proposed the model in 1867.^[3][4] It is also known as a Maxwell fluid. A generalization of the scalar relation to a tensor equation lacks motivation from more microscopic models and does not comply with the concept of material objectivity. However, these criteria are fulfilled by the Upper-convected Maxwell model.^{[citation needed]}

The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,^[5] as shown in the diagram. If, instead, we connect these two elements in parallel,^[5] we get the generalized model of a solid Kelvin–Voigt material.

In Maxwell configuration, under an applied axial stress, the total stress, ${\displaystyle \sigma _{\mathrm {Total} }}$ and the total strain, ${\displaystyle \varepsilon _{\mathrm {Total} }}$ can be defined as follows:^[2]

${\displaystyle \sigma _{\mathrm {Total} }=\sigma _{\rm {D}}=\sigma _{\rm {S}}}$

${\displaystyle \varepsilon _{\mathrm {Total} }=\varepsilon _{\rm {D}}+\varepsilon _{\rm {S}}}$

where the subscript D indicates the stress–strain in the damper and the subscript S indicates the stress–strain in the spring. Taking the derivative of strain with respect to time, we obtain:

${\displaystyle {\frac {d\varepsilon _{\mathrm {Total} }}{dt}}={\frac {d\varepsilon _{\rm {D}}}{dt}}+{\frac {d\varepsilon _{\rm {S}}}{dt}}={\frac {\sigma }{\eta }}+{\frac {1}{E}}{\frac {d\sigma }{dt}}}$

where E is the elastic modulus and η is the material coefficient of viscosity. This model describes the damper as a Newtonian fluid and models the spring with Hooke's law.

In a Maxwell material, stress σ, strain ε and their rates of change with respect to time t are governed by equations of the form:^[2]

${\displaystyle {\frac {1}{E}}{\frac {d\sigma }{dt}}+{\frac {\sigma }{\eta }}={\frac {d\varepsilon }{dt}}}$

or, in dot notation:

${\displaystyle {\frac {\dot {\sigma }}{E}}+{\frac {\sigma }{\eta }}={\dot {\varepsilon }}}$

The equation can be applied either to the shear stress or to the uniform tension in a material. In the former case, the viscosity corresponds to that for a Newtonian fluid. In the latter case, it has a slightly different meaning relating stress and rate of strain.

The model is usually applied to the case of small deformations. For the large deformations we should include some geometrical non-linearity. For the simplest way of generalizing the Maxwell model, refer to the upper-convected Maxwell model.

If a Maxwell material is suddenly deformed and held to a strain of ${\displaystyle \varepsilon _{0}}$, then the stress decays on a characteristic timescale of ${\displaystyle {\frac {\eta }{E}}}$, known as the relaxation time. The phenomenon is known as stress relaxation.

The picture shows dependence of dimensionless stress ${\displaystyle {\frac {\sigma (t)}{E\varepsilon _{0}}}}$ upon dimensionless time ${\displaystyle {\frac {E}{\eta }}t}$:

If we free the material at time ${\displaystyle t_{1}}$, then the elastic element will spring back by the value of

${\displaystyle \varepsilon _{\mathrm {back} }=-{\frac {\sigma (t_{1})}{E}}=\varepsilon _{0}\exp \left(-{\frac {E}{\eta }}t_{1}\right).}$

Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:

${\displaystyle \varepsilon _{\mathrm {irreversible} }=\varepsilon _{0}\left[1-\exp \left(-{\frac {E}{\eta }}t_{1}\right)\right].}$

If a Maxwell material is suddenly subjected to a stress ${\displaystyle \sigma _{0}}$, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:

${\displaystyle \varepsilon (t)={\frac {\sigma _{0}}{E}}+t{\frac {\sigma _{0}}{\eta }}}$

If at some time ${\displaystyle t_{1}}$ we released the material, then the deformation of the elastic element would be the spring-back deformation and the deformation of the viscous element would not change:

${\displaystyle \varepsilon _{\mathrm {reversible} }={\frac {\sigma _{0}}{E}},}$

${\displaystyle \varepsilon _{\mathrm {irreversible} }=t_{1}{\frac {\sigma _{0}}{\eta }}.}$

The Maxwell model does not exhibit creep since it models strain as linear function of time.

If a small stress is applied for a sufficiently long time, then the irreversible strains become large. Thus, Maxwell material is a type of liquid.

If a Maxwell material is subject to a constant strain rate ${\displaystyle {\dot {\epsilon }}}$then the stress increases, reaching a constant value of

${\displaystyle \sigma =\eta {\dot {\varepsilon }}}$

In general

${\displaystyle \sigma (t)=\eta {\dot {\varepsilon }}(1-e^{-Et/\eta })}$

The complex dynamic modulus of a Maxwell material would be:

${\displaystyle E^{*}(\omega )={\frac {1}{1/E-i/(\omega \eta )}}={\frac {E\eta ^{2}\omega ^{2}+i\omega E^{2}\eta }{\eta ^{2}\omega ^{2}+E^{2}}}}$

Thus, the components of the dynamic modulus are :

${\displaystyle E_{1}(\omega )={\frac {E\eta ^{2}\omega ^{2}}{\eta ^{2}\omega ^{2}+E^{2}}}={\frac {(\eta /E)^{2}\omega ^{2}}{(\eta /E)^{2}\omega ^{2}+1}}E={\frac {\tau ^{2}\omega ^{2}}{\tau ^{2}\omega ^{2}+1}}E}$

and

${\displaystyle E_{2}(\omega )={\frac {\omega E^{2}\eta }{\eta ^{2}\omega ^{2}+E^{2}}}={\frac {(\eta /E)\omega }{(\eta /E)^{2}\omega ^{2}+1}}E={\frac {\tau \omega }{\tau ^{2}\omega ^{2}+1}}E}$

The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is ${\displaystyle \tau \equiv \eta /E}$.

Blue curve	dimensionless elastic modulus ${\displaystyle {\frac {E_{1}}{E}}}$
Pink curve	dimensionless modulus of losses ${\displaystyle {\frac {E_{2}}{E}}}$
Yellow curve	dimensionless apparent viscosity ${\displaystyle {\frac {E_{2}}{\omega \eta }}}$
X-axis	dimensionless frequency ${\displaystyle \omega \tau }$.

• Burgers material
• Generalized Maxwell model
• Kelvin–Voigt material
• Oldroyd-B model
• Standard linear solid model
• Upper-convected Maxwell model