This shows that an untwisted rod is a minimizer of bending energy – an elastica. Now assume the rod is closed of length L, and (for convenience) that the material frame M satisfies M(0) = M(L). That is, we take a rod, ‘twist’ it n times and weld the ends together. The Frenet frame automatically closes up, but the natural frame need not.
An elastic rod is an equilibrium configuration for the energy with appropriate bound-ary conditions (usually, having each end fixed in position and clamped.) Note that we can change the rod without changing its centerline by changing q. We can fix the ends of the rod and its centerline and reduce the energy by minimizing the second term.
The concept of the energy stored elastically U has been introduced earlier. For a 3-D body (4.87). The total potential energy energy and potential energy Consider for a while that the material is rigid, for which U 0. Imagine a rigid ball being displaced by an nitesimal amount a at ( = 0) and inclined ( 6= 0) surface, Fig. (8.3).
Then in the Bernoulli-Euler model the bending energy is proportional to the total squared curvature. If we now release the rod and it moves in such a way as to reduce its energy as efficiently as possible, it will want to follow the “negative gradient” of the energy.
We consider a thin elastic rod with circular cross-section and uniform density – the uniform symmetric (linear) Kirchhoff rod.2 For a thorough treatment of the elastic theory of rods, see . The configurations of the rod are described abstractly using adapted framed curves:
0 t(s)ds given by the quantity 2pn + mL. Using this, it is possible to formulate a variational problem whose solutions are exactly the elastic rod centerlines. with k and t the curvature and torsion and l3 6= 0. Then an extremal of F is an elastic rod centerline. Clearly, when l2 = 0, this is an elastic curve.
DISCRETE ELASTIC RODS IN JULIA 1. Introduction.
twisting strains to calculate a rod''s elastic energy. The discrete bending strain is related to the curvature of the centerline, and is de ned by the curvature binormal vector at each vertex: (2.1) ( b) i= 2t iiti 1 1 + titi 1 = 2e e 1 keikkei 1k+ eiei 1: This vector is perpendicular to the plane that locally contains the rod (the osculating. DISCRETE ELASTIC RODS IN JULIA 3 plane) and …
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Lectures on Elastic Curves and Rods
These five lectures constitute a tutorial on the Euler elastica and the Kirchhoff elastic rod. We consider the classical variational problem in Euclidean space and its generalization to Riemannian manifolds. We describe both the Lagrangian and the Hamiltonian formulation of the rod, with the goal of examining the (Liouville-Arnol ...
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On the Bending and Twisting of Rods with Misfit
We derive a one-dimensional variational problem representing the elastic energy of a rod with misfit, starting from a nonlinear, three-dimensional elastic energy with nontrivial …
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Coiling of elastic rods on rigid substrates
Here, we conduct a hybrid experimental and numerical in-vestigation of the coiling of a thin elastic rod onto a moving substrate and characterize the resulting patterns. We perform precision …
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Mechanical analysis of flexible integrated energy storage devices …
Here, we systematically and thoroughly investigated the mechanical behaviors of flexible all-in-one ESDs under bending deformation by the finite element method. The influences of …
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Energy required to bend an elastic rod
The energy required to bend an elastic rod is calculated using the formula E = 1/2 * k * x^2, where E is the energy, k is the stiffness of the rod, and x is the amount of deflection. …
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Energy required to bend an elastic rod
The energy required to bend an elastic rod is calculated using the formula E = 1/2 * k * x^2, where E is the energy, k is the stiffness of the rod, and x is the amount of deflection. What factors affect the energy required to bend an elastic rod?
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On the Bending and Twisting of Rods with Misfit
We derive a one-dimensional variational problem representing the elastic energy of a rod with misfit, starting from a nonlinear, three-dimensional elastic energy with nontrivial preferred strain. Our approach to dimension reduction is to find a Gamma-limit as the thickness of the rod tends to 0.
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Chapter 4 Elastic rods
(Figure 4.1). A linear elastic body occupying the domain B in its stress-free s Figure 4.1: A rod. undeformed state is called an elastic rod, the curve c(x) its central line, and S its cross section. Let z = r(x) be the equation of the central line, with x being the arc length. Then the unit tangent vector to the central line can be obtained as
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Discrete Elastic Rods
Bend & Twist Interaction How do bending and twisting interact? Both terms affect centerline • bending force moves centerline toward straighter curve • twisting force aligns twist-free & material frames
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Lectures on Elastic Curves and Rods
According to (2.5), (5.1) and (6.1), the linear energy density of the rod in which just terms on the order of GAV are retained is given by (1.6). The first term in (1.6) …
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for Discrete elastic rods and ribbons
The Discrete elastic rod method (Bergou et al., 2008) is a numerical method for simulating slender elastic bodies. It works by representing the center-line as a polygonal chain, attaching two perpendicular directors to each segment, and de ning discrete stretching, bending and twisting deformation measures and a discrete strain energy. Here, we ...
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Swelling-driven soft elastic catapults
Adhesive forces act to maintain the rod''s attachment to the substrate, while the elastic energy stored within the bent rod counteracts this adhesion, ultimately driving detachment. This interplay between forces leads to a progressive bending distortion that culminates in the rapid and complete release of the rod.
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Elastic energy storage technology using spiral spring devices and …
Elastic energy storage technology has the advantages of wide-sources, simple structural principle, renewability, high effectiveness and environmental-friendliness. This paper elaborates the operational principles and technical properties and summarizes the applicability of elastic energy storage technology with spiral springs. Elastic energy storage using spiral …
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Energy-minimizing configurations for an elastic rod with self …
We study self-contact configurations of elastic rods by adding a repulsive energy to the bend, twist, shear, and stretch energies of a classical elastic rod. We use a discretized …
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Discrete Elastic Rods
Bend & Twist Interaction How do bending and twisting interact? Both terms affect centerline • bending force moves centerline toward straighter curve • twisting force aligns twist-free & …
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7.5: Rod Bending
Figure 7.9. A global picture of rod bending: (a) the forces acting on a small fragment of a rod, and (b) two bending problem examples, each with two typical but different boundary conditions. First of all, we may write a differential static relation for the average vertical force (mathbf{F}=mathbf{n}_{x} F_{x}(z)) exerted on the part of the rod located to the left of its …
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Twist-Induced Snapping in a Bent Elastic Rod and Ribbon
Although several mechanisms for elastic energy storage and rapid release have been studied in detail, a general understanding of the approach to design such a kinetic system is a key challenge in ...
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Lecture 8: Energy Methods in Elasticity
Lecture 8: Energy Methods in Elasticity The energy methods provide a powerful tool for deriving exact and approximate solutions to many structural problems. 8.1 The Concept of Potential Energy From high school physics you must recall two equations E= 1 2 Mv2 kinematic energy (8.1a) W= mgH potential energy (8.1b)
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8.2 Elastic Strain Energy
The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for stored energy will then be used to solve some elasticity problems using the energy …
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Elastic Energy Storage in Soft Robots
1 Elastic Energy Storage Enables Rapid and Programmable Actuation in Soft Machines Aniket Pal, Debkalpa Goswami, and Ramses V. Martinez* A. Pal, D. Goswami, Prof. R. V. Martinez
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On the energy of an elastic rod
According to (2.5), (5.1) and (6.1), the linear energy density of the rod in which just terms on the order of GAV are retained is given by (1.6). The first term in (1.6) characterizes the tensile and bending energy, the second the tor- sional energy and the additional contribution to the tensile and bending energy from the trans ...
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