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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. It is also known as a Maxwell solid.

Contents

Definition

Effect of a sudden deformation

Effect of a sudden stress

Dynamic modulus

References

See also

Definition

The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected consecutively, as shown in the diagram. In this configuration, under an applied axial stress, the total stress, $\{sigma\_\{Total\}\}$ and the total strain, $\{epsilon\_\{Total\}\}$ can be defined as follows:
:$\{sigma\_\{Total\}\}=\{sigma\_\{D\}\}=\{sigma\_\{S\}\}$
:$\{epsilon\_\{Total\}\}=\{epsilon\_\{D\}\}+\{epsilon\_\{S\}\}$
where the subscript D indicates the stress/strain in the damper and the subscript S indicates the stress/strain in the spring. 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. This model describes the damper as a Newtonian fluid and models the spring with Hooke's law.

If we connect these two elements in parallel, we get a model of Kelvin-Voigt 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:
:
or, in dot notation:
:
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 generalizing the Maxwell model, refer to the Upper Convected Maxwell Model.

Effect of a sudden deformation

If a Maxwell material is suddenly deformed to a strain of $epsilon\_0$ and is kept under this deformation, then the stresses would decay.
The picture shows dependence of dimensionless stress
upon dimensionless time $lambda\; t$:

If we free the material at time $t\_1$, then the elastic element will spring back by the value of
:
Since the viscous element would stay where it is, the irreversible component of deformation can be simplified to the expression below:
:$epsilon\_mathrm\{irreversible\}\; =\; epsilon\_0\; left(1-\; exp\; (-lambda\; t\_1)$
ight).

Effect of a sudden stress

If a Maxwell material is suddenly subjected to a stress $sigma\_0$, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:
:
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:
:
:
The Maxwell Model is not ideal for predicting the creep behavior of a material since it describes the strain relationship with time as linear.
If 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:
:$E^$
★ (omega) = rac 1 {1/E - i/(omega eta) } = rac {Eeta^2 omega^2 +i omega E^2eta} {omega^2 eta^2 + E^2}
Thus, the components of the dynamic modulus are :
:
and
:

The picture shows relaxational spectrum for Maxwell material.
{| border="1" cellspacing="0"
| Black curve || dimensionless elastic modulus
|-
| Red curve || dimensionless modulus of losses
|-
| Yellow curve || dimensionless apparent viscosity
|-
| X-axis || dimensionless frequency .
|}

References

★ http://stellar.mit.edu/S/course/3/fa06/3.032/index.html

See also

★ Generalized Maxwell material

★ Kelvin-Voigt material

★ Oldroid material

★ Standard Linear Solid Material