A Maxwell material is a viscoelastic material having the properties both of elasticity and viscosity. It is named for James Clerk Maxwell who proposed the model in 1867.

Definition

The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected consecutively, as shown in the diagram.

If we connect these two elements in parallel, we get a model of Kelvin 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:

$\backslash frac\; \{1\}\; \{E\}\; \backslash frac\; \{d\backslash sigma\}\; \{dt\}\; +\; \backslash frac\; \{\backslash sigma\}\; \{c\}\; =\; \backslash frac\; \{d\backslash epsilon\}\; \{dt\}$

$\backslash frac\; \{\backslash dot\; \{\backslash sigma\}\}\; \{E\}\; +\; \backslash frac\; \{\backslash sigma\}\; \{c\}=\; \backslash dot\; \{\backslash epsilon\}$

where E is a modulus of elasticity and c a "viscosity". 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 generalization the Maxwell model see Upper Convected Maxwell Model.

Effect of a sudden deformation

If Maxwell material is suddenly deformed to strain of $\backslash epsilon\_0$ and is kept under this deformation, then the stresses would decay.

The picture shows dependence of dimensionless stress $\backslash frac\; \{\backslash sigma(t)\}\; \{E\backslash epsilon\_0\}$ upon dimensionless time $\backslash lambda\; t$: Maxwell deformation.PNG|right|frame|Dependence of dimesionless stress upon dimensionless time under constant strain|Dependence of dimesionless stress upon dimensionless time under constant strain]]

If we would free the material at time $t\_1$, then the elastic element would spring back by the value of

$\backslash epsilon\_\backslash mathrm\{back\}\; =\; -\backslash frac\; \{\backslash sigma(t\_1)\}\; E\; =\; \backslash epsilon\_0\; \backslash exp\; (-\backslash lambda\; t\_1).$

The viscous element would stay there it was, thus, the irreversible component of deformation is:

$\backslash epsilon\_\backslash mathrm\{irresversible\}\; =\; \backslash epsilon\_0\; \backslash left(1-\; \backslash exp\; (-\backslash lambda\; t\_1)\backslash right).$

Effect of a sudden stress

If a Maxwell material is suddenly subjected to a stress $\backslash sigma\_0$, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:

$\backslash epsilon(t)\; =\; \backslash frac\; \{\backslash sigma\_0\}\; E\; +\; t\; \backslash frac\{\backslash sigma\_0\}\; c$

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:

$\backslash epsilon\_\backslash mathrm\{reversible\}\; =\; \backslash frac\; \{\backslash sigma\_0\}\; E,$

$\backslash epsilon\_\backslash mathrm\{irreversible\}\; =\; t\_1\; \backslash frac\{\backslash sigma\_0\}\; c.$

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

Dynamic modulus

The complex dynamic modulus of Maxwell material would be:

$E^*(\backslash omega)\; =\; \backslash frac\; 1\; \{1/E\; -\; i/(\backslash omega\; c)\; \}\; =\; \backslash frac\; \{Ec^2\; \backslash omega^2\; +i\; \backslash omega\; E^2c\}\; \{\backslash omega^2\; c^2\; +\; E^2\}$

Thus, the components of the dynamic modulus are :

$E\_1(\backslash omega)\; =\; \backslash frac\; \{Ec^2\; \backslash omega^2\; \}\; \{c^2\; \backslash omega^2\; +\; E^2\}$

and

$E\_2(\backslash omega)\; =\; \backslash frac\; \{\backslash omega\; E^2c\}\; \{\backslash omega^2\; c^2\; +\; E^2\}$

The picture shows relaxational spectrum for Maxwell material.

See also

• Generalized Maxwell material
• Kelvin material
• Kelvin-Voigt material
• Oldroid material

Non-Newtonian fluids | Materials science