Maxwell model

In fluid mechanics, a Maxwell model is the simplest model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for James Clerk Maxwell who proposed the model in 1867. 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.

In fluid mechanics, a Maxwell model is the simplest model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for James Clerk Maxwell who proposed the model in 1867. 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. \begin{align} \sigma_\mathrm{total} &= \sigma_{\rm D} = \sigma_{\rm S} \\[2pt] \varepsilon_\mathrm{total} &= \varepsilon_{\rm D}+\varepsilon_{\rm S } \end{align} where the subscript indicates the stress–strain in the damper and the subscript indicates the stress–strain in the spring. Taking the derivative of strain with respect to time, we obtain: \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} where is the elastic modulus and 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 , strain and their rates of change with respect to time are governed by equations of the form: \frac{1}{E} \frac{d\sigma}{dt} + \frac{\sigma}{\eta} = \frac{d\varepsilon}{dt}, or, in dot notation: \frac{\dot\sigma}{E} + \frac{\sigma}{\eta}= \dot\varepsilon. 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. \varepsilon_\mathrm{back} = -\frac {\sigma(t_1)} E = \varepsilon_0 \exp \left(-\frac{E}{\eta} t_1\right). Since the viscous element would not return to its original length, the irreversible component of deformation can be simplified to the expression below: \varepsilon_\mathrm{irreversible} = \varepsilon_0 \left[1- \exp \left(-\frac{E}{\eta} t_1\right)\right].

Effect of a sudden stress

If a Maxwell material is suddenly subjected to a stress , then the elastic element would suddenly deform and the viscous element would deform with a constant rate: \varepsilon(t) = \frac {\sigma_0} E + t \frac{\sigma_0} \eta If at some time 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: \begin{align} \varepsilon_\mathrm{reversible} &= \frac {\sigma_0} E, \\[2pt] \varepsilon_\mathrm{irreversible} &= t_1 \frac{\sigma_0} \eta. \end{align} 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. \sigma=\eta \dot{\varepsilon} In general \sigma (t)=\eta \dot{\varepsilon}(1- e^{-Et/\eta}) E^*(\omega) = \frac 1 {\frac{1}{E} - \frac{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: \begin{align} E_1(\omega) &= \frac{ E\eta^2 \omega^2 }{ \eta^2 \omega^2 + E^2 } \\[2pt] &= E \frac{ \left(\frac{\eta}{E}\right)^2\omega^2 }{ \left(\frac{\eta}{E}\right)^2 \omega^2 + 1} = E \frac{ \tau^2\omega^2 }{ \tau^2 \omega^2 + 1} \\[10pt] E_2(\omega) &= \frac{ \omega E^2\eta }{ \eta^2 \omega^2 + E^2} \\[2pt] &= E \frac{ \left(\frac{\eta}{E}\right)\omega }{ \left(\frac{\eta}{E}\right)^2 \omega^2 + 1} = E \frac{ \tau\omega }{ \tau^2 \omega^2 + 1} \end{align} As shown, can be factored out to put the components in terms of the relaxation time constant .

See also

Burgers material Generalized Maxwell model Kelvin–Voigt material Oldroyd-B model Standard linear solid model Upper-convected Maxwell model

References

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