Theoretical And Computational Challenges With Rods
A rod is a long and slender object whose lateral dimension is very small compared to its length. In solid mechanics, the theory of rods can be thought of as a generalized and geometrically exact version of the classical beam theory. There are two major variants of rod theory which are used commonly: Kirchoff rods and Cosserat rods. In Kirchoff rod theory, a rod is assumed to be unstretchable as well as unshearable characterized by linear elasticities, whereas in Cosserat rod theory, these restrictions are done away with. Due to its one-dimensional character, a rod serves as an excellent and efficient tool for theoretical as well as computational modeling of several biomolecules, arteries, cables, carbon nanotubes as well as several bacteria and viruses. The present dissertation deals with addressing the theoretical and computational challenges associated with rods so that its area of applicability can be further broadened. Broadly speaking, this dissertation addresses three important issues: (1) development of a general and efficient computational framework to determine stability of equilibria of constrained elastic rods, (2) extension of the Cosserat rod theory in a mathematically consistent way to allow deformation of a rod's cross-section and (3) explanation of some peculiar atomistic simulation data of carbon nanotubes using an extended version of the special Cosserat rod theory. It is found that the determination of stability of constrained elastic systems leads to a generalized and singular eigenvalue problem. A new numerical algorithm is developed to remove the singularity present and at the same time maintain efficiency of the algorithm. The present state-of-the-art for determination of stability of rods was limited to Dirichlet problems and in the presence of integral constraints, while the algorithm developed here has the capacity to address any general boundary conditions, general loadings and equality constraints of all types. A new variational principle for extensible and unshearable rods is also proposed to facilitate application of the developed numerical algorithm for extensible rods. This is followed by development of a novel formulation of a rod model that allows in-plane deformation of its cross-section. The resulting theory has the potential to bridge the gap between 1-d rod theory and 2-d shell theory, efficiently. It also opens the door for modeling and analysis of hollow tubes such as arteries and nanotubes using a one-dimensional theory. The proposed model also explains a new coupling effect: extension, twist and cross-sectional shrinkage coupling of chiral carbon nanotubes. The peculiarity of a (9,6) carbon nanotube such as rotation of its neighboring cross-sections in alternate directions and fluctuation in twist and axial stretch along its axis at exactly two levels, when the ends of a nanotube are axially moved apart, are also explained using the proposed rod model.
dissertation or thesis