You can have a multimodal Maxwell material i.e. you add up a series of Maxwell models , each of which has a different relaxation time.  In the limit of an infinite number of modes you arrive at a continuous distribution of relaxation times, which macroscopically is equivalent to what you appear to be looking for (at least for small strain oscillation) 

The model is what it is, a linear Hookean spring and a Newtonian dashpot in series. Small strain extension and compression will essentially be the same. However in real life the spring can be non-linear and the viscosity can be shear-thinning (for example). Basically anything goes. What makes sense depends on the material you might like to model, eg. beer foam or jello or granite all have their own peculiarities.

Indeed a (liquid) foam can be very complex e.g. when the cells start rearranging (depends on strain), whether or not the gas is able to equilibrate the pressure etc.

The Maxwell-model dashpot parameter will be strain rate dependant because it is the viscous component of the viscoelastic model. The response of the spring element will not be strain rate dependant because the spring is perfectly elastic by definition. Thus the strain rate dependance of the Maxwell element will be as a result of the dashpot element. This can be be mathematically modelled. I have found that a Weibull exponential function is suitable; this is a stretched exponential function (see this topic in Wikipedia), of the same form as the Kohlrausch–Williams–Watts (KWW) model. The response is more rapid over short times and extended over long times compared with a typical exponential function. The KWW function was introduced to model dielectric properties, however it is applied to mechanical properties too. The Weibull function is mostly applied to failure analysis. An extension of the Maxwell model to include extended or stretched response adds a latch element to the model (see publications by Kevin S Fancey, such as: J.Mat.Sci., 40, 4827-4831, 2005). 

The storage and viscous moduli of a polymer depends of the temperature and of the srain rate. Thus, the apparent stiffness and the damping of your material changes with the experimental condition, temperature and strain rate.

Furthermore, a simple Maxwell model (one spring +one dampling in series) is not a adapted for viscoelastic material (It is a viscoplastic model as the strain cannot return to zero if the stress return to zero). You should either use a Kelvin-Voigt model or a generalized Maxwell model (one spring parralel to some Maxwell elements)

Dear all;

Thank you so much for your explanations and clarifications above.

Indeed this subject is very interesting subject.

I will take your recommendations above and continue finishing writing this interesting paper.

Again, thank you so much and looking forward to continue this discussion!

Mohamed,

To continue the discussion, relaxation model of deforming of complex structure such as fibrillar-like structure electrorheological(ER) fluid might be varied upon system. In stress-controlled shear experiment under electric field, we previously observed strain-hardening or repairing of the structure after a critical yield point, Journal of Colloid and Interface Science 325 (2008) 122-129.

Dear Mohamed,

if I may add something to these relevant answers :

- your material will of be time dependant even with a linear Maxwell model, this is the basic idea of adding a damper

- if the answer predicted is different from what you expected from a linear model, first check that you are actually simulating the real testing signal. This could seem obvious but especially for large displacement cases, the experiment is usually controlled by the global force or displacement speed, whereas the real stress or strain rate is evolving along time. To be clearer : for example, if you are driving a tension test at a constant displacement speed, the strain rate is decreasing with time as the sample lentgh is increasing.

- Once this checked, you can try to evaluate to model the dependancy of your material response on strain or stress states. Two main ways are classical. The first one is to add several linear model with a discrete or continuous distribution of characteristic times theta (theta=viscosity/stiffness). The second one is of course to define non linear laws for stifness and/or viscosity .

- to identify these models parameters, you can use DMA test (small amplitudes sollicitations around a given stran or stress) and sweep frequencies to identify your characteristic times. Depending on your testing device you can also use relaxation or creep tests for several strain or stress states. 

Dear Mohamed,

of course you have already some kind of rate-dependency in your dashpot element's viscosity. In addition, in our workgroup, there also exists some work in the field of material modelling of a porous elastomer, where the behaviour of the Maxwell-element, in this case the parameter relaxation time, is formulated with a dependency on the strain rate.

Since you wanted to have some reference, you can search for the articles of Koprowski et al.. Herein, the porous elastomer is investigated. You find some examples of these articles attached.

If you are interested in the formulation of a rate-dependent Maxwell-element, you can also have a look in my article, which deals with the incompressible part of the elastomeric matrix.

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