Towards data-driven methods for complex systems: Unfolding the crumpling dynamics of thin sheets

Abstract

As researchers aim to understand increasingly complex, nonlinear systems, traditional physics-based approaches to modeling can become mathematically or computationally intractable. This has created a demand for new methods to model and predict the behavior of such systems. Meanwhile, the rise of big data has prompted the development of complementary tools and methodologies for identifying patterns and structure in high-dimensional datasets. Thus, there is pronounced interest in applying data-driven methods to complex systems as a means of obtaining new physical insight. In this thesis, we investigate the crumpling dynamics of thin sheets as a test case in using data-driven methods towards scientific discovery. Crumpled sheets are prime examples of complexity and disorder: As a thin sheet is confined, stresses spontaneously localize to produce a complex network of vertices and ridges in the sheet. However, our studies reveal unexpected order in the way thin sheets accumulate damage in the form of creases. First, we introduce an experimental study of thin sheets crumpled repeatedly via uniaxial loading, which reveals a remarkable property: The total length of creases grows logarithmically as a function of crumpling repetitions, and does so with striking statistical reproducibility across different strengths of compaction. Furthermore, the incremental change in total crease length with each new crumple exhibits insensitivity to the precise history of crumpling, dependent only on the current state of the sheet. We develop a theoretical explanation for this experimental finding through a correspondence between crumpling and fragmentation processes. Over time, crumpling partitions the surface of a thin sheet into facets delineated by creases. We find that the statistics of facet area in sequentially crumpled sheets are consistent with an area-conserving fragmentation rate equation used previously to describe fracture phenomena such as the breakup of rocks. We propose a model for how the facet area distribution changes incrementally over one crumpling cycle based on geometric frustration between existing facets and the confining container. Our model captures the gradual compliance of the sheet which slows damage accumulation over time, consistent with the observed logarithmic scaling. Notably, the connection between crumpling and fragmentation suggests the possibility of universal behavior uniting a broader class of disordered systems. We next investigate order in the spatial structure of damage networks. Using simulation-augmented machine learning, we find promising evidence that data-driven models can learn underlying geometric patterns of crease networks. By separating a crease pattern into contributions of positive (valleys) and negative (ridges) curvature, we show that a neural network supplied only with valleys can suitably reconstruct the corresponding ridges. A key to this approach is the use of high quality experimental data, which is limited in quantity, in combination with computer-simulated flat-folding patterns, which are simple to generate in large volumes, during the training process. In contrast to crumpling, rigid flat-folding is governed by mathematical rules that impose strict relationships between creases locally at each vertex, and we find that these rules are necessary to more faithfully recover the correlations in crumpled patterns as well. Finally, to further aid data-driven studies, we conclude with the development of a computational model for thin elastoplastic sheets. Motivated by the physical behavior of thin sheets during crumpling, we adopt two different formulations of the governing equations of motion: a quasistatic formulation that effectively describes smooth deformations, and a fully dynamic formulation that captures large changes in the sheet’s velocity. The former is a differential-algebraic system solved implicitly, while the latter is a purely differential system solved explicitly, using a hybrid integration scheme that adaptively alternates between the two representations. We discuss several numerical considerations of this approach, including preconditioned iterative methods employed at each implicit integration step. Finally, we demonstrate the capacity of this method to effectively simulate a variety of crumpling phenomena, as well as recover the logarithmic scaling and fragmentation statistics of crease patterns found experimentally.