# Maxwell material

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.

 Contents

## Definition

The Maxwell model can be represented by a purely viscous damper and purely elastic spring connected consecutively like shown on the picture:

Missing image
Maxwell_diagram.PNG

If we will connect these two elements in parallel we will 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:

[itex]\frac {1} {E} \frac {d\sigma} {dt} + \frac {\sigma} {c} = \frac {d\epsilon} {dt}[itex]

or, more elegantly:

[itex]\frac {\dot {\sigma}} {E} + \frac {\sigma} {c}= \dot {\epsilon}[itex]

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.

## Effect of a sudden deformation

If Maxwell material is suddenly deformed to strain of [itex]\epsilon_0[itex] and is kept under this deformation, then the stresses would exponentially decay:

[itex]\sigma(t)=E\epsilon_0 \exp (-\lambda t) [itex],

where t is time and the rate of relaxation [itex] \lambda=\frac {E}{c} [itex]

The picture shows dependence of dimensionless stress [itex]\frac {\sigma(t)} {E\epsilon_0} [itex] upon dimensionless time [itex]\lambda t[itex]:

Missing image
Maxwell_deformation.PNG
Dependence of dimesionless stress upon dimensionless time under constant strain

If we would free the material at time [itex]t_1[itex], then the elastic element would spring back by the value of

[itex]\epsilon_{back} = -\frac {\sigma(t_1)} E = \epsilon_0 \exp (-\lambda t_1) [itex].

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

[itex]\epsilon_{irresversible} = \epsilon_0 \left(1- \exp (-\lambda t_1)\right) [itex]

## Effect of a sudden stress

If Maxwell materiel is suddenly subjected to a stress [itex]\sigma_0[itex], then the elastic element would suddenly deform and the viscous element would deform with a constant rate:

[itex]\epsilon(t) = \frac {\sigma_0} E + t \frac{\sigma_0} c [itex]

If at some time [itex]t_1[itex] 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 spring back:

[itex]\epsilon_{back} = \frac {\sigma_0} E [itex]

[itex]\epsilon_{irreversible} = t_1 \frac{\sigma_0} c [itex]

If even small stress are applied for 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:

[itex]E^*(\omega) = \frac 1 {1/E + i/(\omega c) } = \frac {Ec^2 \omega^2 -i \omega^2 E^2c} {\omega^2 c^2 + E^2} [itex]

Thus, the components of the dynamic modulus are :

[itex]E_1(\omega) = \frac {Ec^2 \omega^2 } {c^2 \omega^2 + E^2} [itex]

and

[itex]E_2(\omega) = \frac {\omega E^2c} {\omega^2 c^2 + E^2} [itex]

Missing image
Maxwell_relax_spectra.PNG
Relaxational spectrum for Maxwell material. Black curve dimensionless E1, Red curve - dimensionless E2, Yellow curve - dimensionless viscosity
The picture shows relaxational spectrum for Maxwell material. Black curve dimensionless elastic modulus

[itex]E_1/E[itex]; Red curve - dimensionless modulus of losses [itex]E_2/E[itex] ; Yellow curve - dimensionless apparent viscosity [itex]\frac {E_2} {\omega c}[itex]. On the X-axe dimensionless frequency [itex]\omega / \lambda[itex].

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