The proposal is divided into three main lines of interconnected research: the nonlinear analysis of filament instabilities, the study of wave propagation and singularity formation in elastic rods, and the analysis and modeling of elastic growth. The analysis of filament instabilities will be carried out through the use and extension of nonlinear analysis techniques developed by the principal investigators. These techniques, which yield systems of nonlinear amplitude equations of considerable mathematical interest, will be applied to a wide variety of cases of practical importance including extensible and shearable rods, tubes conveying fluids, and growing filamentary organisms. The studies of wave propagation in filaments under various conditions will be investigated by a variety of analytical stability techniques, and the investigations of singularity formation will draw on adaptations of techniques used to study wave propagation in inhomogeneous media. The modeling of elastic growth pertinent to biological processes will be developed according to continuum mechanical principles and applied to both filamentary and three dimensional structures.
Filamentary structures are ubiquitous in both the natural and physical worlds at scales ranging from the microscopic to the macroscopic. In nature we see the sinuous motion of vines and plant tendrils; under the optical microscope we can see filamentary bacteria; and through electron microscopy we can see the structure of DNA strands. If the original motivations for developing a rational continuum mechanics of filamentary structures was provided by the mechanical world of cables, springs and struts, contemporary research has often been motivated by the biological context. These problems, including the fundamental natural phenomenon of growth, raise new challenges to the classical formulations. While the static theory of elastic structures has a long and distinguished history, the theory of dynamical effects still presents many important challenges, both theoretically and computationally. Work of the principal investigators over the past few years has shown that the governing equations of elastic filaments, can be reduced to more tractable nonlinear evolution equations capable of describing many dynamical bifurcations including the fundamental twist-to-writhe conversion. If this earlier work concentrated on simplest cases, the extension of this theory to a whole host of important cases of practical importance has yet to be developed. In addition to the effects of structural asymmetry, and inhomogeneity, the effects of internal fluid flow that arise in pipes and hoses, and the incorporation of growth effects required for biological modeling, will be addressed in work supported by this award. Once achieved, these formulations will provide the necessary mathematical tools to study conformation changes, instability, and wave propagation in filamentary structures as varied as whips, hoses and bacterial filaments. The principal investigators will apply these studies to the analysis of filamentary and other elastic structures in a variety of mechanical and biological settings such as the spontaneous change in handedness exhibited by climbing plants and other filamentary organisms, twist wave propagation in bacterial filaments, and morphological changes in aerially growing bacterial microorganisms and fungi. In all these examples, natural growth is a fundamental component in the morphological dynamics, and a variety of investigations will be carried out to improve our understanding of how this fundamental natural effect can be incorporated into self-consistent mathematical models. These will range from the inclusion of growth terms in the constitutive equations for rods, to the development of continuum mechanical models of growing three dimensional elastic structures such as those arising in the description of bulk growth in biological tissues.
