Maxwell material

From Academic Kids

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:

<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>:

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

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>

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

<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>.

See also

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