Maxwell model

The Maxwell model is represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. If, instead, we connect these two elements in parallel, we get the generalized model of a solid Kelvin–Voigt material. In Maxwell configuration, under an applied axial stress, the total stress, \sigma_\mathrm{Total} and the total strain, \varepsilon_\mathrm{Total} can be defined as follows: :\sigma_\mathrm{Total}=\sigma_{\rm D} = \sigma_{\rm S} :\varepsilon_\mathrm{Total}=\varepsilon_{\rm D}+\varepsilon_{\rm 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\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 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: :\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. == Effect of a sudden deformation ==

If a Maxwell material is suddenly deformed and held to a strain of \varepsilon_0, then the stress decays on a characteristic timescale of \frac{\eta}{E}, known as the relaxation time. The phenomenon is known as stress relaxation. The picture shows dependence of dimensionless stress \frac {\sigma(t)} {E\varepsilon_0} upon dimensionless time \frac{E}{\eta} t: If we free the material at time t_1, then the elastic element will spring back by the value of :\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 \sigma_0, 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 t_1 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: :\varepsilon_\mathrm{reversible} = \frac {\sigma_0} E, :\varepsilon_\mathrm{irreversible} = 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. == Effect of a constant strain rate ==

If a Maxwell material is subject to a constant strain rate \dot{\epsilon}then the stress increases, reaching a constant value of \sigma=\eta \dot{\varepsilon} In general \sigma (t)=\eta \dot{\varepsilon}(1- e^{-Et/\eta}) == Dynamic modulus ==

The complex dynamic modulus of a Maxwell material would be: :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} Thus, the components of the dynamic modulus are : :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 and :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 The picture shows relaxational spectrum for Maxwell material. The relaxation time constant is \tau \equiv \eta / E . ==See also==