Rigidity Percolation in Disordered Fiber Systems: Theory and Applications Public Deposited

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  • March 19, 2019
    • Affiliation: College of Arts and Sciences, Department of Mathematics
  • Nanocomposites, particularly carbon nanocomposites, find many applications spanning an impressive variety of industries on account of their impressive properties and versatility. However, the discrepancy between the performance of individual nanoparticles and that of nanocomposites suggests continued technological development and better theoretical understanding will provide much opportunity for further property enhancement. Study of computational renderings of disordered fiber systems has been successful in various nanocomposite modeling applications, particularly toward the characterization of electrical properties. Motivated by these successes, I develop an explanatory model for `mechanical' or `rheological percolation,' terms used by experimentalists to describe a nonlinear increase in elastic modulus/strength that occurs at particle inclusion volume fractions well above the electrical percolation threshold. Specifically, I formalize a hypothesis given by \\citet*{penu}, which states that these dramatic gains result from the formation of a `rigid CNT network.' Idealizing particle interactions as hinges, this amounts to the network property of \\emph{rigidity percolation}---the emergence of a giant component (within the inclusion contact network) that is not only connected, but furthermore the inherent contacts are patterned to constrain all internal degrees of freedom in the component. Rigidity percolation has been studied in various systems (particularly the characterization of glasses and proteins) but has never been applied to disordered systems of three-dimensional rod-like particles. With mathematically principled arguments from \\emph{rigidity matroid theory}, I develop a scalable algorithm (\\emph{Rigid Graph Compression}, or \\emph{RGC}), which can be used to detect rigidity percolation in such systems by iteratively compressing provably rigid subgraphs within the rod contact networks. Prior to approaching the 3D system, I confirm the usefulness of \\emph{RGC} by using it to accurately approximate the rigidity percolation threshold in disordered systems of 2D fibers---achieving $<1\\%$ error relative to a previous exact method. Then, I develop an implementation of \\emph{RGC} in three dimensions and determine an upper bound for the rigidity percolation threshold in disordered 3D fiber systems. More work is required to show that this approximation is sufficiently accurate---however, this work confirms that rigidity in the inclusion network is a viable explanation for the industrially useful mechanical percolation. Furthermore, I use \\emph{RGC} to quantitatively characterize the effects of interphase growth and spatial CNT clustering in a real polymer nanocomposite system of experimental interest.
Date of publication
Resource type
  • Adalsteinsson, David
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2018

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