Maxwell material

A Maxwell material is a viscoelastic material having the properties both of elasticity and viscosity.^[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 can be 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 we connect these two elements in parallel,^[2] we get a generalized model of 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 with a characteristic time of $\frac{\eta}{E}$ .

The picture shows dependence of dimensionless stress $\frac {\sigma(t)} {E\varepsilon_0}$ upon dimensionless time $\frac{E}{\eta} t$ :

Dependence of dimensionless stress upon dimensionless time under constant strain

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.

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

Relaxational spectrum for Maxwell material

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

1 2 3 Roylance, David (2001). Engineering Viscoelasticity (PDF). Cambridge, MA 02139: Massachusetts Institute of Technology. pp. 8–11.
1 2 Christensen, R. M (1971). Theory of Viscoelasticity. London, W1X6BA: Academic Press. pp. 16–20.