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# [SCG49-P08] Rheology of rocks: The equivalence of viscoelastic models

Keywords:rheology, viscoelastic model, equivalence, network theory, system theory

Modelling the viscoelastic behavior of rocks has classically been done using mechanical circuits consisting of springs and dashpots. Such mechanical circuits are called viscoelastic models and the most elementary examples are Maxwell model and Voigt model, a dashpot and a spring connected serially and parallelly respectively. Although these basic models are only capable of predicting rather simple viscoelastic behaviors, by combining them one can model complicated behaviors. Well known examples of such models are generalized Maxwell model (GMM), a model consisting of n-parallelly connected Maxwell models, and generalized Voigt model (GVM), a model of n-serially connected Voigt models.

A notable fact is that there are always multiple types of viscoelastic models that can reproduce any given viscoelastic behavior, i.e. some viscoelastic models are equivalent. For instance, experimentally observed time-dependent relaxation functions of viscoelastic materials exhibit an algebraic decay, and by supposing appropriate scaling laws between constants, GMM, GVM and a certain ladder-type model (e.g. Schiessel and Bulmen, 1995) are known to reproduce the same decaying pattern. The equivalence of certain viscoelastic models, such as GMM and GVM, has already been studied, however, the equivalence among more general cases seemed to remain unclear.

We proved that any mechanical circuit consisting of springs and dashpots are equivalent to GMM, GVM and certain ladder-like mechanical circuits called Cauer-1

^{st}and 2

^{nd}models. This result suggests that it is sufficient to only consider certain types of models in discussing the mechanical properties of rocks using viscoelastic models. Note that we obtained this result by applying electrical network theory to viscoelastic models, based on the mathematical equivalence between mechanical and electrical circuits. In addition, we proposed a method to write down the transformations between equivalent viscoelastic models in the form of nonsingular matrices. This approach enables us to understand the equivalent transformations as change of basis of the state-space of the system.