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

Force dipoles and stable local defects on fluid vesicles

An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal invariance of the two-dimensional bending energy is used to identify the distribution of energy as well as the stress established in the vesicle. While these states are local minima of the energy, this energy is degenerate; there is a zero mode in the energy fluctuation spectrum, associated with area and volume preserving conformal transformations, which breaks the symmetry between the two points. The volume constraint fixes the distance $S$, measured along the surface, between the two points; if it is relaxed, a second zero mode appears, reflecting the independence of the energy on $S$; in the absence of this constraint a pathway opens for the membrane to slip out of the defect. Logarithmic curvature singularities in the surface geometry at the points of contact signal the presence of external forces. The magnitude of these forces varies inversely with $S$ and so diverges as the points merge; the corresponding torques vanish in these defects. The geometry behaves near each of the singularities as a biharmonic monopole, in the region between them as a surface of constant mean curvature, and in distant regions as a biharmonic quadrupole. Comparison of the distribution of stress with the quadratic approximation in the height functions points to shortcomings of the latter representation. Radial tension is accompanied by lateral compression, both near the singularities and far away, with a crossover from tension to compression occurring in the region between them.Comment: 26 pages, 10 figure

Hábitat popular, renovación, infiltración y desarrollo

La propuesta entonces consiste en replantear la ubicación de los habitantes de la calle que viven en el Bronx, desarrollar un gran proyecto que sea enfático en materia de reciclaje y que con las asesorías necesarias, sean ellos mismos las personas que se encarguen de darle un optimo funcionamiento a dicho tema proyectual. De igual forma, una vez se tenga la disponibilidad terrenal para hacer nuestro proyecto de renovación, se procederá a evaluar el estado de las edificaciones que actualmente configuran el lugar para así determinar cuáles vale la pena mantener y comenzar el desarrollo de nuestro complejo de vivienda social y en altura que la ciudad necesita de manera inmediata. Finalmente, nuestra propuesta puntual, va integrada y direccionada a un planteamiento urbano y ambiental que integrará de manera muy concreta las localidades de los Mártires y de Puente Aranda, desarrollando un eje estructurador que va ligado con el trazado vial que denota la conocida avenida de los comuneros la cual es el eje vital de nuestro proyecto.The proposal is to rethink the location of street people living in the Bronx, developing a large project that is emphatic and recycling with the necessary advice, people are themselves which is responsible for giving optimal performance to that topic of design. Similarly, once you have the availability to make our earthly renovation project, it shall assess the condition of the buildings that currently constitute the place in order to determine what is worth keeping and begin the development of our social housing complex and height that the city needs immediately. Finally, our proposal on time, is directed to integrated urban and environmental approach that will integrate very specifically the towns of Martires and Puente Aranda, developing a structuring axis that is linked with the roadway known denoting Avenue community members which is the linchpin of our project.Arquitecto (a)Pregrad