Generalized Maxwell Model

The Generalized Maxwell model also known as the Maxwell–Wiechert model (after James Clerk Maxwell and E WiechertWiechert, E (1889); "Ueber elastische Nachwirkung", Dissertation, Königsberg University, GermanyWiechert, E (1893); "Gesetze der elastischen Nachwirkung für constante Temperatur", Annalen der Physik, Vol. 286
issue 10, p. 335–348
an
issue 11, p. 546–570
/ref>) is the most general form of the linear model for viscoelasticity. In this model several Maxwell elements are assembled in parallel. It takes into account that the relaxation does not occur at a single time, but in a set of times. Due to the presence of molecular segments of different lengths, with shorter ones contributing less than longer ones, there is a varying time distribution. The Wiechert model shows this by having as many spring–dashpot Maxwell elements as are necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.Roylance, David (2001); "Engineering Viscoelasticity", 14-15Tschoegl, Nicholas W. (1989); "The Phenomenological Theory of Linear Viscoelastic Behavior", 119-126

General model form

Solids

Given $N+1$ elements with moduli $E_i$, viscosities $\eta_i$, and relaxation times $\tau_i=\frac$

The general form for the model for solids is given by :

Example: standard linear solid model

Following the above model with $N+1=2$ elements yields the standard linear solid model:

Fluids

Given $N+1$ elements with moduli $E_i$, viscosities $\eta_i$, and relaxation times $\tau_i=\frac$

The general form for the model for fluids is given by:

Example: three parameter fluid

The analogous model to the standard linear solid model is the three parameter fluid, also known as the Jeffreys model:
{{Equation box 1 , title = Three Parameter Maxwell Fluid Model ({{EquationRef, 6) , equation =
$\sigma+\tau_1\frac{\partial{\sigma{\partial{t=\left({\eta_0+\tau_1 E_1}\right)\frac{\partial{\epsilon{\partial{t$
, cellpadding = 6 , border = 1 , border colour = black , background colour = white

References

Materials science
Non-Newtonian fluids
James Clerk Maxwell