Cosserat rod

This study presents a computational approach to obtain nonlinearly elastic constitutive relations of strip/ribbon-like structures modeled as a special Cosserat rod. Starting with the description of strips as a general Cosserat plate, the strip is first subjected to a strain field which is uniform along its length. The Helical Cauchy-Born rule is used to impose this uniform strain field which deforms the strip into a six-parameter family of helical configurations-the six parameters here correspond to the six strain measures of rod theory.

This work addresses the determination of yield surfaces for geometrically exact elastoplastic rods. Use is made of a formulation where the rod is subjected to an uniform strain field along its arc length, thereby reducing the elastoplastic problem of the full rod to just its cross-section. By integrating the plastic work and the stresses over the rod's cross-section, one then obtains discrete points of the yield surface in terms of stress resultants. Eventually, Lamé curves in their most general form are fitted to the discrete points by an appropriate optimisation method.

A finite element formulation is presented for a direct approach to model elastoplastic deformation in slender bodies using the special Cosserat rod theory. The direct theory has additional plastic strain and hardening variables, which are functions of just the rod's arc-length, to account for plastic deformation of the rod. Furthermore, the theory assumes the existence of an effective yield function in terms of stress resultants, i.e., force and moment in the cross-section and cross-section averaged hardening parameters.

A Cosserat rod based continuum approach is presented to obtain phonon dispersion curves of flexural, torsional, longitudinal, shearing and radial breathing modes in chiral nanorods and nanotubes. Upon substituting the continuum wave form in the linearized dynamic equations of stretched and twisted Cosserat rods, we obtain analytical expression of a coefficient matrix (in terms of the rod's stiffnesses, induced axial force and twisting moment) whose eigenvalues and eigenvectors give us frequencies and mode shapes, respectively, for each of the above phonon modes.

We present an efficient numerical scheme based on asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods. Using quaternions to represent rotation, the equations of static equilibria of special Cosserat rods are posed as a system of thirteen first order ordinary differential equations having cubic nonlinearity. The derivatives in these equations are further discretized to yield a system of cubic polynomial equations.

A general framework is presented to model coupled thermo-elasto-plastic deformations in the theory of special Cosserat rods. The use of the one-dimensional form of the energy balance in conjunction with the one-dimensional entropy balance allows us to obtain an additional equation for the evolution of a temperature-like one-dimensional field variable together with constitutive relations for this theory. Reduction to the case of thermoelasticity leads us to the well known nonlinear theory of thermoelasticity for special Cosserat rods.

We recently published a paper in International Journal of Solids and Structures titled "A rod model for three dimensional deformations of single walled carbon nanotubes".(paper attached)

http://www.sciencedirect.com/science/article/pii/S0020768311002149