An obstacle problem for conical deformations of thin elastic sheets

A developable cone ("d-cone") is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance $\epsilon$. Starting from a nonlinear model depending on the thickness $h > 0$ of the sheet, we prove a $\Gamma$-convergence result as $h \rightarrow 0$ to a fourth-order obstacle problem for curves in $\mathbb{S}^2$. We then describe the exact shape of minimizers of the limit problem when $\epsilon$ is small. In particular, we rigorously justify previous results in the physics literature.Comment: 25 page

ME Design and Freeform Fabrication of Compliant Cellular Materials with Graded Stiffness

Typically, cellular materials are designed for structural applications to provide stiffness or absorb impact via permanent plastic deformation. Alternatively, it is possible to design compliant cellular materials that absorb energy via recoverable elastic deformation, allowing the material to spring back to its original configuration after the load is released. Potential applications include automotive panels or prosthetic applications that require repeated, low-speed impact absorption without permanent deformation. The key is to arrange solid base material in cellular topologies that permit high levels of elastic deformation. To prevent plastic deformation, the topologies are designed for contact between cell walls at predetermined load levels, resulting in customized, graded stiffness profiles. Design techniques are established for synthesizing cellular topologies with customized compliance for static or quasi-static applications. The design techniques account for cell wall contact, large scale deformations, and material nonlinearities. Resulting cellular material designs are fabricated with selective laser sintering, and their properties are experimentally evaluated.Mechanical Engineerin

Existence, regularity and structure of confined elasticae

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $\Omega$. We prove existence, regularity and some structural properties of minimizers. In particular, when $\Omega$ is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections

Dynamics of Pulsed Flow in an Elastic Tube

Internal haemorrhage, often leading to cardio-vascular arrest happens to be one of the prime sources of high fatality rates in mammals. We propose a simplistic model of fluid flow to specify the location of the haemorrhagic spots, which, if located accurately, could be operated upon leading to a possible cure. The model we employ for the purpose is inspired by fluid mechanics and consists of a viscous fluid, pumped by a periodic force and flowing through an elastic tube. The analogy is with that of blood, pumped from the heart and flowing through an arte ry or vein. Our results, aided by graphical illustrations, match reasonably well with experimental observations.Comment: 6 pages and 4 figure