Physics:Maxwell material
A Maxwell material is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. [1] 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 is represented by a purely viscous damper and a purely elastic spring connected in series,[2] as shown in the diagram. In this configuration, under an applied axial stress, the total stress, [math]\displaystyle{ \sigma_\mathrm{Total} }[/math] and the total strain, [math]\displaystyle{ \varepsilon_\mathrm{Total} }[/math] can be defined as follows:[1]
- [math]\displaystyle{ \sigma_\mathrm{Total}=\sigma_D = \sigma_S }[/math]
- [math]\displaystyle{ \varepsilon_\mathrm{Total}=\varepsilon_D+\varepsilon_S }[/math]
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:
- [math]\displaystyle{ \frac {d\varepsilon_\mathrm{Total}} {dt} = \frac {d\varepsilon_D} {dt} + \frac {d\varepsilon_S} {dt} = \frac {\sigma} {\eta} + \frac {1} {E} \frac {d\sigma} {dt} }[/math]
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, instead, we connect these two elements in parallel,[2] we get the generalized model of a solid 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:[1]
- [math]\displaystyle{ \frac {1} {E} \frac {d\sigma} {dt} + \frac {\sigma} {\eta} = \frac {d\varepsilon} {dt} }[/math]
or, in dot notation:
- [math]\displaystyle{ \frac {\dot {\sigma}} {E} + \frac {\sigma} {\eta}= \dot {\varepsilon} }[/math]
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 [math]\displaystyle{ \varepsilon_0 }[/math], then the stress decays on a characteristic timescale of [math]\displaystyle{ \frac{\eta}{E} }[/math], known as the relaxation time. The phenomenon is known as stress relaxation.
The picture shows dependence of dimensionless stress [math]\displaystyle{ \frac {\sigma(t)} {E\varepsilon_0} }[/math] upon dimensionless time [math]\displaystyle{ \frac{E}{\eta} t }[/math]:
If we free the material at time [math]\displaystyle{ t_1 }[/math], then the elastic element will spring back by the value of
- [math]\displaystyle{ \varepsilon_\mathrm{back} = -\frac {\sigma(t_1)} E = \varepsilon_0 \exp \left(-\frac{E}{\eta} t_1\right). }[/math]
Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below:
- [math]\displaystyle{ \varepsilon_\mathrm{irreversible} = \varepsilon_0 \left(1- \exp \left(-\frac{E}{\eta} t_1\right)\right). }[/math]
Effect of a sudden stress
If a Maxwell material is suddenly subjected to a stress [math]\displaystyle{ \sigma_0 }[/math], then the elastic element would suddenly deform and the viscous element would deform with a constant rate:
- [math]\displaystyle{ \varepsilon(t) = \frac {\sigma_0} E + t \frac{\sigma_0} \eta }[/math]
If at some time [math]\displaystyle{ t_1 }[/math] we released 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:
- [math]\displaystyle{ \varepsilon_\mathrm{reversible} = \frac {\sigma_0} E, }[/math]
- [math]\displaystyle{ \varepsilon_\mathrm{irreversible} = t_1 \frac{\sigma_0} \eta. }[/math]
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.
Effect of a constant strain rate
If a Maxwell material is subject to a constant strain rate [math]\displaystyle{ \dot{\epsilon} }[/math]then the stress increases, reaching a constant value of
[math]\displaystyle{ \sigma=\eta \dot{\varepsilon} }[/math]
In general
[math]\displaystyle{ \sigma (t)=\eta \dot{\varepsilon}(1- e^{-Et/\eta})
}[/math]
Dynamic modulus
The complex dynamic modulus of a Maxwell material would be:
- [math]\displaystyle{ E^*(\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} }[/math]
Thus, the components of the dynamic modulus are :
- [math]\displaystyle{ E_1(\omega) = \frac {E\eta^2 \omega^2 } {\eta^2 \omega^2 + E^2} = \frac {(\eta/E)^2\omega^2} {(\eta/E)^2 \omega^2 + 1} E = \frac {\tau^2\omega^2} {\tau^2 \omega^2 + 1} E }[/math]
and
- [math]\displaystyle{ E_2(\omega) = \frac {\omega E^2\eta} {\eta^2 \omega^2 + E^2} = \frac {(\eta/E)\omega} {(\eta/E)^2 \omega^2 + 1} E = \frac {\tau\omega} {\tau^2 \omega^2 + 1} E }[/math]
The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is [math]\displaystyle{ \tau \equiv \eta / E }[/math].
| Blue curve | dimensionless elastic modulus [math]\displaystyle{ \frac {E_1} {E} }[/math] |
| Pink curve | dimensionless modulus of losses [math]\displaystyle{ \frac {E_2} {E} }[/math] |
| Yellow curve | dimensionless apparent viscosity [math]\displaystyle{ \frac {E_2} {\omega \eta} }[/math] |
| X-axis | dimensionless frequency [math]\displaystyle{ \omega\tau }[/math]. |
See also
- Burgers material
- Generalized Maxwell model
- Kelvin–Voigt material
- Oldroyd-B model
- Standard linear solid model
- Upper-convected Maxwell model
References
- ↑ 1.0 1.1 1.2 Roylance, David (2001). Engineering Viscoelasticity. Cambridge, MA 02139: Massachusetts Institute of Technology. pp. 8–11. http://web.mit.edu/course/3/3.11/www/modules/visco.pdf.
- ↑ 2.0 2.1 Christensen, R. M (1971). Theory of Viscoelasticity. London, W1X6BA: Academic Press. pp. 16–20. ISBN 9780121742508. https://archive.org/details/theoryofviscoela0000chri.
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