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Advantages off the Bergstroem-Boyce-Model compared to a Prony-Series  

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I was looking for some reasons, why someone would choose the BB-model instead of a Prony series. In the "Mechanics of Solid Polymere" Bergström states that the BB-model" can overcome some of the main issues with linear viscoelasticity related to both large strain deformations and strain amplitude dependence of the dynamic properties (E’ and E’’)." Are these problems purely of a practical nature or can one explain them theoretically?

Obviously the BB-model uses a nonlinear dampener. Therefore one needs a large number of Prony terms to describe similar material behavior, which can be harder to fit.

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I think you have answered your own question. The Prony series is quite limited because it assumes that, for instance, stiffness monotonically decreases the same over time, regardless of the magnitude of strain. I'm not familiar with the details of the BB-model, but Bergstrom states that his model can address some of these strain amplitude effects.

To determine if this is important in your material, you can conduct stress relaxation tests with different initial (quasi-instantaneous) loads. If the relaxation model that comes out of the data is the same for all strain amplitudes, a Prony-series may suffice. Even if you have a strain-dependent relaxation or creep model, you can sometimes get by with a Prony-series. See, for example, Windslow & Busfield, 2019, "Viscoelastic modeling of extrusion damage in elastomer seals".

I don't understand your question, "...can one explain them theoretically?" Given that the Prony series and BB are both mathematical models, they are by defintion grounded in a theory to explain practical observations.