Maxwell model
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A Maxwell model is the most simple model viscoelastic material showing properties of a typical liquid.<ref>Template:Cite book</ref> It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations.<ref name=roylance_EV>Template:Cite reportTemplate:Self-published inline</ref> It is named for James Clerk Maxwell who proposed the model in 1867.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> It is also known as a Maxwell fluid. A generalization of the scalar relation to a tensor equation lacks motivation from more microscopic models and does not comply with the concept of material objectivity. However, these criteria are fulfilled by the Upper-convected Maxwell model.Template:Fact
Definition
The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series,<ref name="christensen">Template:Cite book</ref> as shown in the diagram. If, instead, we connect these two elements in parallel,<ref name="christensen" /> we get the generalized model of a solid Kelvin–Voigt material.
In Maxwell configuration, under an applied axial stress, the total stress, <math>\sigma_\mathrm{Total}</math> and the total strain, <math>\varepsilon_\mathrm{Total}</math> can be defined as follows:<ref name="roylance_EV" />
- <math>\sigma_\mathrm{Total}=\sigma_{\rm D} = \sigma_{\rm S}</math>
- <math>\varepsilon_\mathrm{Total}=\varepsilon_{\rm D}+\varepsilon_{\rm 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>\frac {d\varepsilon_\mathrm{Total}} {dt} = \frac {d\varepsilon_{\rm D}} {dt} + \frac {d\varepsilon_{\rm 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.
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:<ref name=roylance_EV />
- <math>\frac {1} {E} \frac {d\sigma} {dt} + \frac {\sigma} {\eta} = \frac {d\varepsilon} {dt}</math>
or, in dot notation:
- <math>\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>\varepsilon_0</math>, then the stress decays on a characteristic timescale of <math>\frac{\eta}{E}</math>, known as the relaxation time. The phenomenon is known as stress relaxation.
The picture shows dependence of dimensionless stress <math>\frac {\sigma(t)} {E\varepsilon_0} </math> upon dimensionless time <math>\frac{E}{\eta} t</math>:
If we free the material at time <math>t_1</math>, then the elastic element will spring back by the value of
- <math>\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>\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>\sigma_0</math>, then the elastic element would suddenly deform and the viscous element would deform with a constant rate:
- <math>\varepsilon(t) = \frac {\sigma_0} E + t \frac{\sigma_0} \eta </math>
If at some time <math>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>\varepsilon_\mathrm{reversible} = \frac {\sigma_0} E, </math>
- <math>\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>\dot{\epsilon}</math>then the stress increases, reaching a constant value of
<math>\sigma=\eta \dot{\varepsilon}
</math>
In general
<math>\sigma (t)=\eta \dot{\varepsilon}(1- e^{-Et/\eta})
</math>
Dynamic modulus
The complex dynamic modulus of a Maxwell material would be:
- <math>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>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>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> \tau \equiv \eta / E </math>.
| Blue curve | dimensionless elastic modulus <math>\frac {E_1} {E}</math> |
| Pink curve | dimensionless modulus of losses <math>\frac {E_2} {E}</math> |
| Yellow curve | dimensionless apparent viscosity <math>\frac {E_2} {\omega \eta}</math> |
| X-axis | dimensionless frequency <math> \omega\tau</math>. |
See also
- Burgers material
- Generalized Maxwell model
- Kelvin–Voigt material
- Oldroyd-B model
- Standard linear solid model
- Upper-convected Maxwell model