Riccati equations of opposite torsions from the Lie-Darboux method for spatial curves and possible applications

A novel formulation of the Lie-Darboux method of obtaining the Riccati equations for the spatial curves in Euclidean three-dimensional space is presented. It leads to two Riccati equations that differ by the sign of torsion. The case of cylindrical helices is used as an illustrative example. Possible applications in Physics are suggested.Comment: 7 pages, 10 references, no figure

A FEM-experimental approach for the development of a conceptual linear actuator based on tendril's free coiling

Within the vastness of the plant species, certain living systems show tendril structures whose motion is of particular interest for biomimetic engineers. Tendrils sense and coil around suitable grips, and by shortening in length, they erect the remaining plant body. To achieve contraction, tendrils rotate along their main axis and shift from a linear to a double-spring geometry. This phenomenon is denoted as the free-coiling phase. In this work, with the aim of understanding the fundamentals of the mechanics behind the free coiling, a reverse-engineering approach based on the finite element method was firstly applied. The model consisted of an elongated cylinder with suitable material properties, boundary, and loading conditions, in order to reproduce the kinematics of the tendril. The simulation succeeded in mimicking coiling faithfully and was therefore used to validate a tentative linear actuator model based on the plant’s working principle. More in detail, exploiting shape memory alloy materials to obtain large reversible deformations, the main tendril features were implemented into a nickel-titanium spring-based testing model. The results of the experimental tests confirmed the feasibility of the idea in terms of both functioning principles and actual performance. It can be concluded that the final set-up can be used as a base for a prototype design of a new kind of a linear actuator

Perversions with a twist

PESS acknowledges grant FCT SFRH/BD/76369/201. MHG acknowledges PTDC/CTM-BIO/6178/2014.Perversions connecting two helices with symmetric handedness are a common occurrence in nature, for example in tendrils. These defects can be found in our day life decorating ribbon gifts or when plants use tendrils to attach to a support. Perversions arise when clamped elastic filaments coil into a helical shape but have to conserve zero overall twist. We investigate whether other types of perversions exist and if they display different properties. Here we show mathematically and experimentally that a continuous range of different perversions can exist and present different geometries. Experimentally, different perversions were generated using micro electrospun fibres. Our experimental results also confirm that these perversions behave differently upon release and adopt different final configurations. These results also demonstrate that it is possible to control on demand the formation and shape of microfilaments, in particular, of electrospun fibres by using ultraviolet light.publishersversionpublishe

Hemihelical local minimizers in prestrained elastic bi-strips

We consider a double layered prestrained elastic rod in the limit of vanishing cross section. For the resulting limit Kirchoff-rod model with intrinsic curvature we prove a supercritical bifurcation result, rigorously showing the emergence of a branch of hemihelical local minimizers from the straight configuration, at a critical force and under clamping at both ends. As a consequence we obtain the existence of nontrivial local minimizers of the $3$-d system.Comment: 16 pages, 2 figure

Rotation, inversion, and perversion in anisotropic elastic cylindrical tubes and membranes

Cylindrical tubes and membranes are universal structural elements found in biology and engineering over a wide range of scales. Working in the framework of nonlinear elasticity we consider the possible deformations of elastic cylindrical shells reinforced by one or two families of anisotropic fibers. We consider both small and large deformations and the reduction from thick cylindrical shells (tubes) to thin shells (cylindrical membranes). In particular, a number of universal regimes can be identified including the possibility of inversion and perversion of rotation

Coiling of elastic rods on rigid substrates

We investigate the deployment of a thin elastic rod onto a rigid substrate and study the resulting coiling patterns. In our approach, we combine precision model experiments, scaling analyses, and computer simulations toward developing predictive understanding of the coiling process. Both cases of deposition onto static and moving substrates are considered. We construct phase diagrams for the possible coiling patterns and characterize them as a function of the geometric and material properties of the rod, as well as the height and relative speeds of deployment. The modes selected and their characteristic length scales are found to arise from a complex interplay between gravitational, bending, and twisting energies of the rod, coupled to the geometric nonlinearities intrinsic to the large deformations. We give particular emphasis to the first sinusoidal mode of instability, which we find to be consistent with a Hopf bifurcation, and analyze the meandering wavelength and amplitude. Throughout, we systematically vary natural curvature of the rod as a control parameter, which has a qualitative and quantitative effect on the pattern formation, above a critical value that we determine. The universality conferred by the prominent role of geometry in the deformation modes of the rod suggests using the gained understanding as design guidelines, in the original applications that motivated the study.National Science Foundation (U.S.) (CMMI-1129894

Rods coiling about a rigid constraint: helices and perversions

Mechanical instabilities can be exploited to design innovative structures, able to change their shape in the presence of external stimuli. In this work, we derive a mathematical model of an elastic beam subjected to an axial force and constrained to smoothly slide along a rigid support, where the distance between the rod midline and the constraint is fixed and finite. Using both theoretical and computational techniques, we characterize the bifurcations of such a mechanical system, in which the axial force and the natural curvature of the beam are used as control parameters. We show that, in the presence of a straight support, the rod can deform into shapes exhibiting helices and perversions, namely transition zones connecting together two helices with opposite chirality. The mathematical predictions of the proposed model are also compared with some experiments, showing a good quantitative agreement. In particular, we find that the buckled configurations may exhibit multiple perversions and we propose a possible explanation for this phenomenon based on the energy landscape of the mechanical system