1,636 research outputs found
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
Subdivision Shell Elements with Anisotropic Growth
A thin shell finite element approach based on Loop's subdivision surfaces is
proposed, capable of dealing with large deformations and anisotropic growth. To
this end, the Kirchhoff-Love theory of thin shells is derived and extended to
allow for arbitrary in-plane growth. The simplicity and computational
efficiency of the subdivision thin shell elements is outstanding, which is
demonstrated on a few standard loading benchmarks. With this powerful tool at
hand, we demonstrate the broad range of possible applications by numerical
solution of several growth scenarios, ranging from the uniform growth of a
sphere, to boundary instabilities induced by large anisotropic growth. Finally,
it is shown that the problem of a slowly and uniformly growing sheet confined
in a fixed hollow sphere is equivalent to the inverse process where a sheet of
fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless,
quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl
Isometric bending requires local constraints on free edges
While the shape equations describing the equilibrium of an unstretchable thin
sheet that is free to bend are known, the boundary conditions that supplement
these equations on free edges have remained elusive. Intuitively,
unstretchability is captured by a constraint on the metric within the bulk.
Naively one would then guess that this constraint is enough to ensure that the
deformations determining the boundary conditions on these edges respect the
isometry constraint. If matters were this simple, unfortunately, it would imply
unbalanced torques (as well as forces) along the edge unless manifestly
unphysical constraints are met by the boundary geometry. In this paper we
identify the source of the problem: not only the local arc-length but also the
geodesic curvature need to be constrained explicitly on all free edges. We
derive the boundary conditions which follow. Contrary to conventional wisdom,
there is no need to introduce boundary layers. This framework is applied to
isolated conical defects, both with deficit as well, but more briefly, as
surplus angles. Using these boundary conditions, we show that the lateral
tension within a circular cone of fixed radius is equal but opposite to the
radial compression, and independent of the deficit angle itself. We proceed to
examine the effect of an oblique outer edge on this cone perturbatively
demonstrating that both the correction to the geometry as well as the stress
distribution in the cone kicks in at second order in the eccentricity of the
edge.Comment: 25 pages, 3 figure
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
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