Maxwell material
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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.
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Definition
The Maxwell model can be represented by a purely viscous damper and purely elastic spring connected consecutively like shown on the picture:
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:
- <math>\frac {1} {E} \frac {d\sigma} {dt} + \frac {\sigma} {c} = \frac {d\epsilon} {dt}<math>
or, more elegantly:
- <math>\frac {\dot {\sigma}} {E} + \frac {\sigma} {c}= \dot {\epsilon}<math>
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 <math>\epsilon_0<math> and is kept under this deformation, then the stresses would exponentially decay:
<math>\sigma(t)=E\epsilon_0 \exp (-\lambda t) <math>,
where t is time and the rate of relaxation <math> \lambda=\frac {E}{c} <math>
The picture shows dependence of dimensionless stress <math>\frac {\sigma(t)} {E\epsilon_0} <math> upon dimensionless time <math>\lambda t<math>:
Maxwell_deformation.PNG
If we would free the material at time <math>t_1<math>, then the elastic element would spring back by the value of
<math>\epsilon_{back} = -\frac {\sigma(t_1)} E = \epsilon_0 \exp (-\lambda t_1) <math>.
The viscous element would stay there it was, thus, the irreversible component of deformation is:
<math>\epsilon_{irresversible} = \epsilon_0 \left(1- \exp (-\lambda t_1)\right) <math>
Effect of a sudden stress
If Maxwell materiel is suddenly subjected to a stress <math>\sigma_0<math>, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:
<math>\epsilon(t) = \frac {\sigma_0} E + t \frac{\sigma_0} c <math>
If at some time <math>t_1<math> 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:
<math>\epsilon_{back} = \frac {\sigma_0} E <math>
<math>\epsilon_{irreversible} = t_1 \frac{\sigma_0} c <math>
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:
<math>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} <math>
Thus, the components of the dynamic modulus are :
<math>E_1(\omega) = \frac {Ec^2 \omega^2 } {c^2 \omega^2 + E^2} <math>
and
<math>E_2(\omega) = \frac {\omega E^2c} {\omega^2 c^2 + E^2} <math>
Maxwell_relax_spectra.PNG
<math>E_1/E<math>; Red curve - dimensionless modulus of losses <math>E_2/E<math> ; Yellow curve - dimensionless apparent viscosity <math>\frac {E_2} {\omega c}<math>. On the X-axe dimensionless frequency <math>\omega / \lambda<math>.