Maxwell material

A Maxwell material is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. ^[1] It is named for James Clerk Maxwell who proposed the model in 1867. It is also known as a Maxwell fluid.

Definition

The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,^[2] as shown in the diagram. In this configuration, under an applied axial stress, the total stress, $\sigma _{\mathrm {Total} }$ and the total strain, $\varepsilon _{\mathrm {Total} }$ can be defined as follows:^[1]

\sigma _{\mathrm {Total} }=\sigma _{D}=\sigma _{S}

\varepsilon _{\mathrm {Total} }=\varepsilon _{D}+\varepsilon _{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:

{\frac {d\varepsilon _{\mathrm {Total} }}{dt}}={\frac {d\varepsilon _{D}}{dt}}+{\frac {d\varepsilon _{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.

If, instead, we connect these two elements in parallel,^[2] we get the generalized model of a solid Kelvin–Voigt material.

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:^[1]

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

or, in dot notation:

{\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.

Effect of a sudden deformation

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

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

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

\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:

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

Effect of a sudden stress

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

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

If at some time $t_{1}$ we would release 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:

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

\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.

Effect of a constant strain rate

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

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

In general

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

Dynamic modulus

The complex dynamic modulus of a Maxwell material would be:

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 :

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

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 $\tau \equiv \eta /E$ .

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

References

^ ^a ^b ^c Roylance, David (2001). Engineering Viscoelasticity (PDF). Cambridge, MA 02139: Massachusetts Institute of Technology. pp. 8–11.{{cite book}}: CS1 maint: location (link)
^ ^a ^b Christensen, R. M (1971). Theory of Viscoelasticity. London, W1X6BA: Academic Press. pp. 16–20.{{cite book}}: CS1 maint: location (link)

[roylance_EV-1] Roylance, David (2001). Engineering Viscoelasticity (PDF). Cambridge, MA 02139: Massachusetts Institute of Technology. pp. 8–11.{{cite book}}: CS1 maint: location (link)

[christensen-2] Christensen, R. M (1971). Theory of Viscoelasticity. London, W1X6BA: Academic Press. pp. 16–20.{{cite book}}: CS1 maint: location (link)

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