Characterizing chaotic dynamics from simulations of large strain behavior of a granular material under biaxial compression

Abstract

For a given observed time series, it is still a rather difficult problem to provide a useful and compelling description of the underlying dynamics. The approach we take here, and the general philosophy adopted elsewhere, is to reconstruct the (assumed) attractor from the observed time series. From this attractor, we then use a black-box modelling algorithm to estimate the underlying evolution operator.We assume that what cannot be modeled by this algorithm is best treated as a combination of dynamic and observational noise. As a final step, we apply an ensemble of techniques to quantify the dynamics described in each model and show that certain types of dynamics provide a better match to the original data. Using this approach, we not only build a model but also verify the performance of that model. The methodology is applied to simulations of a granular assembly under compression. In particular, we choose a single time series recording of bulk measurements of the stress ratio in a biaxial compression test of a densely packed granular assembly-observed during the large strain or so-called critical state regime in the presence of a fully developed shear band. We show that the observed behavior may best be modeled by structures capable of exhibiting (hyper-) chaotic dynamics. Inferring dynamical structure of a deterministic nonlinear system from noisy time series data is a long standing problem. Here, we demonstrate the application of a range of methods to a problem of particular interest in the study of granular materials. An initially dense assembly of grains under biaxial compression with constant confining pressure reaches a near steady-state behavior (modulo fluctuations) in the large strain regime. The dynamics of this system at this limit are the focus of this paper-and of significant interest within the soil mechanics literature. Understanding the dynamical response of the granular structure in this regime is crucial for prediction and control in many industrial and geophysical systems. Our analysis indicates that the dynamics of this system in the presence of a fully developed shear band are consistent with a form of transient chaos-characterized by a bistable system which switches between two chaotic regimes, one with large slow oscillations and one with smaller and faster dynamics.

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