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1.5  Conclusion

For a given complex modulus, it is difficult to validate the fact that general thermodynamic principles are verified. Ref. [] gives an analysis based on the fact that each component of a generalized maxwell model must verify the second principle of thermodynamics and thus be associated with positive stiffness and damping coefficients.

In practice, the criteria that are used to judge the quality of a constitutive model are the accurate reproduction of experimental complex modulus measurements within the tested frequency/temperature range, and the likelihood of extrapolations. Figure ?? illustrates the fact that with rational fractions one can fairly easily satisfy the first objective (reproduce the modulus in a narrow band in bold), but extrapolations can be difficult (thin lines are here of poor quality).

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Figure 1.12: Complex modulus of the TA viscoelastic (—) and estimation with a 3 pole model (- - -) whose frequencies are indicated as vertical lines

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On the other hand, one should note that the determination of the complex modulus is often difficult and that the frequency / temperature superposition principle is an hypothesis. It just happens to be reasonably valid in many cases.

In conclusion, for the material stand-point, accurate representations are either tabulated (our point of view is that this is actually the most practical representation), rational fractions with a sufficient degree or fractional derivative models. The following chapters will address difficulties in using these representations for modeling damped structures.