Maxwell material

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.

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.

The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,[4] as shown in the diagram.

If, instead, we connect these two elements in parallel,[4] we get the generalized model of a solid Kelvin–Voigt material.

Taking the derivative of strain with respect to time, we obtain: where E is the elastic modulus and η is the material coefficient of viscosity.

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] or, in dot notation: The equation can be applied either to the shear stress or to the uniform tension in a material.

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.

If a Maxwell material is suddenly deformed and held to a strain of

, then the elastic element will spring back by the value of Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below: If a Maxwell material is suddenly subjected to a stress

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: 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.

If a Maxwell material is subject to a constant strain rate

The complex dynamic modulus of a Maxwell material would be: Thus, the components of the dynamic modulus are : and The picture shows relaxational spectrum for Maxwell material.