Modeling of Flexible Beam Networks and Morphing Structures by Geometrically Exact Discrete Beams

Abstract We demonstrate how a geometrically exact formulation of discrete slender beams can be generalized for the efficient simulation of complex networks of flexible beams by introducing rigid connections through special junction elements. The numerical framework, which is based on discrete differential geometry of framed curves in a time-discrete setting for time- and history-dependent constitutive models, is applicable to elastic and inelastic beams undergoing large rotations with and without natural curvature and actuation. Especially, the latter two aspects make our approach a versatile and efficient alternative to higher-dimensional finite element techniques frequently used, e.g., for the simulation of active, shape-morphing, and reconfigurable structures, as demonstrated by a suite of examples.</jats:p

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Mobius and Other Strips

We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order one-dimensional problem posed on the centreline of the strip. We derive Euler–Lagrange equations for this problem in Euler–Poincaré form and formulate boundary-value problems for closed symmetric one- and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami

Super Space Clothoids

Special Issue: SIGGRAPH 2013 ConferenceInternational audienceThin elastic filaments in real world such as vine tendrils, hair ringlets or curled ribbons often depict a very smooth, curved shape that low-order rod models -- e.g., segment-based rods -- fail to reproduce accurately and compactly. In this paper, we push forward the investigation of high-order models for thin, inextensible elastic rods by building the dynamics of a G2-continuous piecewise 3D clothoid: a smooth space curve with piecewise affine curvature. With the aim of precisely integrating the rod kinematic problem, for which no closed-form solution exists, we introduce a dedicated integration scheme based on power series expansions. It turns out that our algorithm reaches machine precision orders of magnitude faster compared to classical numerical integrators. This property, nicely preserved under simple algebraic and differential operations, allows us to compute all spatial terms of the rod kinematics and dynamics in both an efficient and accurate way. Combined with a semi-implicit time-stepping scheme, our method leads to the efficient and robust simulation of arbitrary curly filaments that exhibit rich, visually pleasing configurations and motion. Our approach was successfully applied to generate various scenarios such as the unwinding of a curled ribbon as well as the aesthetic animation of spiral-like hair or the fascinating growth of twining plants

Equilibrium shapes with stress localisation for inextensible elastic möbius and other strips

© Springer Science+Business Media Dordrecht 2015. We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order onedimensional problem posed on the centreline of the strip. We derive Euler-Lagrange equations for this problem in Euler-Poincaré form and formulate boundary-value problems for closed symmetric one-and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami

Reduced-order modeling of a sliding ring on an elastic rod with incremental potential formulation

Mechanical interactions between rigid rings and flexible cables are widespread in both daily life (hanging clothes) and engineering system (closing a tether net). A reduced-order method for the dynamic analysis of sliding rings on a deformable one-dimensional (1D) rod-like object is proposed. In contrast to discretize the joint rings into multiple nodes and edges for contact detection and numerical simulation, a single point is used to reduce the order of the numerical model. In order to achieve the non-deviation condition between sliding ring and flexible rod, a novel barrier functional is derived based on incremental potential theory, and the tangent frictional interplay is later procured by a lagged dissipative formulation. The proposed barrier functional and the associated frictional functional are $C^{2}$ continuous, hence the nonlinear elastodynamic system can be solved variationally by an implicit time-stepping scheme. The numerical framework is first applied to simple examples where the analytical solutions are available for validation. Then, multiple complex practical engineering examples are considered to showcase the effectiveness of the proposed method. The simplified ring-to-rod interaction model can provide lifelike visual effect for picture animations, and also can support the optimal design for space debris removal system.Comment: 15 pages, 9 figure

Geometry, mechanics and actuation of intrinsically curved folds

We combine theory and experiments to explore the kinematics and actuation of intrinsically curved folds (ICFs) in otherwise developable shells. Unlike origami folds, ICFs are not bending isometries of flat sheets, but arise via non-isometric processes (growth/moulding) or by joining sheets along curved boundaries. Experimentally, we implement both, first making joined ICFs from paper, then fabricating flat liquid crystal elastomer (LCE) sheets that morph into ICFs upon heating/swelling via programmed metric changes. Theoretically, an ICF's intrinsic geometry is defined by the geodesic curvatures on either side, $\kappa_{g_i}$. Given these, and a target 3D fold-line, one can construct the entire surface isometrically, and compute the bending energy. This construction shows ICFs are bending mechanisms, with a continuous family of isometries trading fold angle against fold-line curvature. In ICFs with symmetric $\kappa_{g_i}$, straightening the fold-line culminates in a fully-folded flat state that is deployable but weak, while asymmetric ICFs ultimately lock with a mechanically strong finite-angle. When unloaded, freely-hinged ICFs simply adopt the (thickness $t$ independent) isometry that minimizes the bend energy. In contrast, in LCE ICFs a competition between flank and ridge selects a ridge curvature that, unusually, scales as $t^{-1/7}$. Finally, we demonstrate how multiple ICFs can be combined in one LCE sheet, to create a versatile stretch-strong gripper that lifts $\sim$40x its own weight.Comment: The supplemental movies are available at https://drive.google.com/drive/folders/1CR5TdbZNhveHiDYt0_a20O7_nQYS6xZ

A computational model of twisted elastic ribbons

We develop an irregular lattice mass-spring-model (MSM) to simulate and study the deformation modes of a thin elastic ribbon as a function of applied end-to-end twist and tension. Our simulations reproduce all reported experimentally observed modes, including transitions from helicoids to longitudinal wrinkles, creased helicoids and loops with self-contact, and transverse wrinkles to accordion self-folds. Our simulations also show that the twist angles at which the primary longitudinal and transverse wrinkles appear are well described by various analyses of the F\"oppl-von K\'arm\'an (FvK) equations, but the characteristic wavelength of the longitudinal wrinkles has a more complex relationship to applied tension than previously estimated. The clamped edges are shown to suppress longitudinal wrinkling over a distance set by the applied tension and the ribbon width, but otherwise have no apparent effect on measured wavelength. Further, by analyzing the stress profile, we find that longitudinal wrinkling does not completely alleviate compression, but caps the magnitude of the compression. Nonetheless, the width over which wrinkles form is observed to be wider than the near-threshold analysis predictions -- the width is more consistent with the predictions of far-from-threshold analysis. However, the end-to-end contraction of the ribbon as a function of twist is found to more closely follow the corresponding near-threshold prediction as tension in the ribbon is increased, in contrast to the expectations of far-from-threshold analysis. These results point to the need for further theoretical analysis of this rich thin elastic system, guided by our physically robust and intuitive simulation model.Comment: 19 pages, 15 figure

Modeling DNA Structure, Elasticity and Deformations at the Base-pair Level

We present a generic model for DNA at the base-pair level. We use a variant of the Gay-Berne potential to represent the stacking energy between neighboring base-pairs. The sugar-phosphate backbones are taken into account by semi-rigid harmonic springs with a non-zero spring length. The competition of these two interactions and the introduction of a simple geometrical constraint leads to a stacked right-handed B-DNA-like conformation. The mapping of the presented model to the Marko-Siggia and the Stack-of-Plates model enables us to optimize the free model parameters so as to reproduce the experimentally known observables such as persistence lengths, mean and mean squared base-pair step parameters. For the optimized model parameters we measured the critical force where the transition from B- to S-DNA occurs to be approximately $140{pN}$. We observe an overstretched S-DNA conformation with highly inclined bases that partially preserves the stacking of successive base-pairs.Comment: 15 pages, 25 figures. submitted to PR