The Mechanics of Ribbons and Möbius Bands

Flashcards covering key terminology and concepts from the book 'The Mechanics of Ribbons and Möbius Bands'.

Roger Fosdick, Eliot Fried

Editors of 'The Mechanics of Ribbons and Möbius Bands' published by Springer.

The Mechanics of Ribbons and Möbius Bands

A compilation of works, including translations and current research, focusing on the mechanics of Möbius bands and related topics.

Journal of Elasticity, Volume 119, Issues 1-2, 2015

A previous publication where materials for 'The Mechanics of Ribbons and Möbius Bands' were initially featured.

Möbius Band

A surface with only one side and one edge.

Developable Surface

A type of ruled surface that can be flattened into a planar form without stretching or tearing.

Isometric Mapping

A transformation that preserves distances between points.

Michael Sadowsky

Author of landmark papers from the 1930s concerning the mechanics of the Möbius band, originally written in German.

W. Wunderlich

Author of a paper published in 1962 on a developable Möbius band.

Energy Functional

A mathematical expression that represents the total energy of a system, often used in variational problems to find equilibrium states.

Bending Energy

The energy stored in a material due to its bending deformation.

Kinematic Properties

describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move.

Isotropy with respect to bending

describes a material's uniform resistance to bending forces, regardless of the direction of applied force relative to the material.

Anisotropy under bending

describes a material's varying resistance to bending forces dependent on the direction of applied force relative to the material.

Frenet-Serret Formulas

A set of equations describing the kinematic properties of a particle moving along a curve in three-dimensional Euclidean space

Virtual Torsion

infinitesimal rotation applied to the tangent vector of a curve

Mean Curvature

A measure of the average curvature of a surface at a given point.

Torsion

The degree to which a curve twists out of its osculating plane.

Rectifying Surface

The rectifying surface is the envelope of all rectifying planes; cf. the work by the author cited in the footnote on the present page

Symmetry axis of a MÖBIUS band

defined such that it is congruent with itself after a rotation of 180◦ about that axis.

Gamma-Limit

a concept used to define a notion of convergence for functionals, especially in the calculus of variations.

Low-dimensional media

Materials or structures where one or more dimensions are significantly smaller than the others, leading to unique physical properties. Examples include thin films, nanowires, and in this case, ribbons.

Dimensional reduction

A general method where lower-dimensional geometries are used to describe higher-dimensional objects to reduce the complexity of the physical models, and computational effort to simulate them.

Euclidean Ribbons

Thin, long strips of elastic material that do not contain any form of internal residual stresses.

Non-Euclidean Ribbons

Thin, long strips of elastic material that contain some form of internal residual stresses

Darboux frame

A moving frame on a surface adapted to a curve on the surface.

“Wunderlich, Meet Kirchhoff”

A phrase that captures a method unifing the description of thin ribbons and thin rods.

Geometric Instability

Morphological instabilities arising from the relationship between geometry and forces.

“An Elementary Proof for the Existence of a Developable MÖBIUS Band and the Attribution of the Geometric Problem to a Variational Problem”

Michael Sadowsky’s original paper appeared in Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 17. Juli 1930