Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical centerlines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance.biomolecules ͉ differential geometry ͉ elasticity ͉ filaments ͉ rods S cientists have long held a fascination, sometimes bordering on mystical obsession, for helical structures in nature (1, 2). Helices arise in nanosprings, carbon nanotubes, ␣-helices, DNA double and collagen triple helices, lipid bilayers, bacterial flagella in Salmonella and Escherichia coli, aerial hyphae in actynomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, and helical staircases (3-13). Helical structures can be understood from a discrete point of view as a regular, periodic stacking of rigid blocks, such as the bases in DNA strands (14), the tail sheaths of bacteriophages (15), the packing of flagellin subunits (16), or simply the stairs in spiral staircases (1). However, it also can be beneficial to adopt a continuous description in which a filamentary structure is represented by a central space curve along with a frame that captures the orientation of the material cross sections at each point along the curve. Although this description neglects some fine features of deformations in the cross section, it is nevertheless suitable to describe the large-scale geometrical and physical properties of long, thin structures. If the deformations are small enough with respect to the appropriate characteristic length scales of the problem, the physical attributes of the filament such as the stresses acting across cross sections can be averaged and represented as a resultant force and moment acting on the centerline. We will refer to such a description of a filamentary structure as a rod model (see, for instance, refs 5, 7, 8, and 10-13 for examples of such rod theories).In continuum mechanics, a semi-inverse problem is generally taken to mean the study of a special class of solutions w...
It is demonstrated that a uniform and hyperelastic, but otherwise arbitrary, nonlinear Cosserat rod subject to appropriate end loadings has equilibria whose center lines form two-parameter families of helices. The absolute energy minimizer that arises in the absence of any end loading is a helical equilibrium by the assumption of uniformity, but more generally the helical equilibria arise for non-vanishing end loads. For inextensible, unshearable rods the two parameters correspond to arbitrary values of the curvature and torsion of the helix. For non-isotropic rods, each member of the two-dimensional family of helical center lines has at least two possible equilibrium orientations of the director frame. The possible orientations are characterized by a pair of finite-dimensional, dual variational principles involving pointwise values of the strain-energy density and its conjugate function. For isotropic rods, the characterization of possible equilibrium configurations degenerates, and in place of a discrete number of two-parameter families of helical equilibria, typically a single four-parameter family arises. The four continuous parameters correspond to the two of the helical center lines, a one-parameter family of possible angular phases, and a one-parameter family of imposed excess twists. (2000): 74K10. Mathematics Subject Classifications
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