Attribute grammar

Template:Ref improve 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 fluid.

Definition

The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. In this configuration, under an applied axial stress, the total stress, ${\sigma }_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ and the total strain, ${ϵ}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ can be defined as follows:

${\sigma }_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\sigma }_D={\sigma }_S$

${ϵ}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={ϵ}_D+{ϵ}_S$

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:

$\frac{d{ϵ}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{dt}=\frac{d{ϵ}_D}{dt}+\frac{d{ϵ}_S}{dt}=\frac{\sigma }{\eta }+\frac{1}{E}\frac{d\sigma }{dt}$

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:

$\frac{1}{E}\frac{d\sigma }{dt}+\frac{\sigma }{\eta }=\frac{dϵ}{dt}$

or, in dot notation:

$\frac{\dot{\sigma }}{E}+\frac{\sigma }{\eta }=\dot{ϵ}$

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 and held to a strain of ${ϵ}_0$, then the stress decays with a characteristic time of $\frac{\eta }{E}$.

The picture shows dependence of dimensionless stress $\frac{\sigma (t)}{E{ϵ}_0}$ upon dimensionless time $\frac{E}{\eta }t$:

File:Maxwell deformation.PNG

Dependence of dimensionless stress upon dimensionless time under constant strain

If we free the material at time $t_1$, then the elastic element will spring back by the value of

${ϵ}_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}}=-\frac{\sigma (t_1)}{E}={ϵ}_0\exp (-\frac{E}{\eta }{t}_1).$

Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:

${ϵ}_{\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}}={ϵ}_0(1-\exp (-\frac{E}{\eta }{t}_1)).$

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:

$ϵ(t)=\frac{{\sigma }_0}{E}+t\frac{{\sigma }_0}{\eta }$

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:

${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}}=\frac{{\sigma }_0}{E},$

${ϵ}_{\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}}=t_1\frac{{\sigma }_0}{\eta }.$

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. Thus, Maxwell material is a type of liquid.

Dynamic modulus

The complex dynamic modulus of a Maxwell material would be:

$E^{\ast }(\omega )=\frac{1}{1/E-i/(\omega \eta )}=\frac{E{\eta }^2{\omega }^2+i\omega E^2\eta }{{\eta }^2{\omega }^2+E^2}$

Thus, the components of the dynamic modulus are :

$E_1(\omega )=\frac{E{\eta }^2{\omega }^2}{{\eta }^2{\omega }^2+E^2}$

and

$E_2(\omega )=\frac{\omega E^2\eta }{{\eta }^2{\omega }^2+E^2}$

File:Maxwell relax spectra.PNG

Relaxational spectrum for Maxwell material

The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is $\lambda \equiv \eta /E$.

Blue curve	dimensionless elastic modulus $\frac{E_1}{E}$
Pink curve	dimensionless modulus of losses $\frac{E_2}{E}$
Yellow curve	dimensionless apparent viscosity $\frac{E_2}{\omega \eta }$
X-axis	dimensionless frequency $\omega \lambda$.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

See also

• Generalized Maxwell material
• Kelvin–Voigt material
• Oldroyd material
• Standard Linear Solid Material
• Upper-convected Maxwell model