Two problems in the study of elastic filaments are considered.First, a reliable numerical algorithm is developed that candetermine the shape of a static elastic rod under a variety ofconditions. In this algorithm the governing equations are writtenentirely in terms of local coordinates and are discretized usingfinite differences. The algorithm has two significant advantages:firstly, it can be implemented for a wide variety of the boundaryconditions and, secondly, it enables the user to work with generalconstitutive relationships with only minor changes to thealgorithm. In the second problem a model is presented describingthe dynamics of an elastic tube conveying a fluid. First weanalyze instabilities that are present in a straight rod or tubeunder tension subject to increasing twist in the absence of afluid. As the twist is increased beyond a critical value, thefilament undergoes a twist-to-writhe bifurcation. A multiplescales expansion is used to derive nonlinear amplitude equationsto examine the dynamics of the elastic rod beyond the bifurcationthreshold. This problem is then reinvestigated for an elastic tubeconveying a fluid to study the effect of fluid flow on thetwist-to-writhe instability. A linear stability analysisdemonstrates that for an infinite rod the twist-to-writhethreshold is lowered by the presence of a fluid flow. Amplitudeequations are then derived from which the delay of bifurcation dueto finite tube length is determined. It is shown that the delayedbifurcation threshold depends delicately on the length of the tubeand that it can be either raised or lowered relative to thefluid-free case. The amplitude equations derived for the case of aconstant average fluid flux are compared to the case where theflux depends on the curvature. In this latter case it is shownthat inclusion of curvature results in small changes in some ofthe coefficients in the amplitude equations and has only a smalleffect on the post-bifurcation dynamics.

Autonomous robots are employed in several important tasks, for example, from health care to military and defense applications involving operations in hazardous and inaccessible environments. Legged autonomous robots can be advantageous due to high adaptability and stability over any terrain, superior obstacle avoidance capability, and advantages through redundancy by utilizing multiple legs. Compared to rigid-legged robots, flexible-legged robots are highly compliant, suitable for non-destructive inspection applications, and possess enhanced gait control with improved energy efficiency. An approach to designing flexible-legged robots is to mimic desirable features evolved via natural selection in biological organisms. Conceptualizing new biologically inspired flexible-legged robots can expand the usability and improve the efficiency of robots in different applications. In this project, the inspiration for locomotion design is the mobility principle utilized by small-scale organisms in the form of beating protrusions referred to as cilia or flagella. Notably, the collective beating dynamics of ciliary arrays reveal essential characteristics such as synchronization, phase locking, and metachronal coordination suitable for terrestrial and aquatic robot locomotion. This thesis presents the formulation, simulation, and analysis of a planar bio-inspired flexible-legged robot for terrestrial locomotion. Each leg of the robot is modeled as a bundle of flexible filaments using constrained Euler elastica that is suitable to describe some of the characteristics of cilia or flagella. The legs/protrusions are mechanically coupled through the base, representing the robot's payload, via linear springs or elastic lumped elements, to produce certain desired collective beating patterns upon individual moment actuations. The locomotion mechanism is illustrated in simulation, wherein the results pave the ground for future work with refined modeling to account for hardware implementation constraints.