11:00 AM - 1:00 PM
[SCG49-P08] Rheology of rocks: The equivalence of viscoelastic models
Keywords:rheology, viscoelastic model, equivalence, network theory, system theory
The mechanical properties of rocks forming crust and mantle differ depending on the time scale of the deformation. In time scales short as few hours to few days rocks behave elastically as seen in seismic wave propagation and tidal deformation, while they behave viscously in time scales long as tens of millions of years as seen in mantle convection. Transient deformations that occur in ten to thousands of years, such as post-seismic deformation and post-glacial rebound, tend to exhibit both elastic and viscous properties and thus understood as viscoelastic deformations. The problem is that since such deformations take too long, it is difficult to accurately observe the viscous and viscoelastic behaviors of rocks in laboratory experiments. Therefore, the study of such deformation is compelled to largely rely on the comparison between field observation and model prediction.
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-1st and 2nd 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.
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-1st and 2nd 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.