The shape of the edge of a leaf

Leaves and flowers frequently have a characteristic rippling pattern at their edges. Recent experiments found similar patterns in torn plastic. These patterns can be reproduced by imposing metrics upon thin sheets. The goal of this paper is to discuss a collection of analytical and numerical results for the shape of a sheet with a non--flat metric. First, a simple condition is found to determine when a stretched sheet folded into a cylinder loses axial symmetry, and buckles like a flower. General expressions are next found for the energy of stretched sheet, both in forms suitable for numerical investigation, and for analytical studies in the continuum. The bulk of the paper focuses upon long thin strips of material with a linear gradient in metric. In some special cases, the energy--minimizing shapes of such strips can be determined analytically. Euler--Lagrange equations are found which determine the shapes in general. The paper closes with numerical investigations of these equations.Comment: 15 pages and 6 figure

A Problem with STEM

Striking differences between physics and biology have important implications for interdisciplinary science, technology, engineering, and mathematics (STEM) education. I am a physicist with interdisciplinary connections. The research group in which I work, the Center for Nonlinear Dynamics at the University of Texas at Austin, is converting into the physics department home for biological physics. Many ofmycollaborations have been with faculty in engineering. For the past 15 years, I have been codirector of the program at the University of Texas at Austin that prepares secondary science and mathematics teachers (UTeach, 2012). The future teachers take a course on scientific research I developed and deliver together with colleagues from biology, astronomy, chemistry, and biochemistry (Marder, 2011). This background naturally makes me an enthusiastic advocate of interdisciplinary education at the secondary and undergraduate levels. Yet at the same time, I am worried by some features of what may be coming. These worries have to do with what can happen as we are all lumped together under the heading of STEM.National Science FoundationPhysic

Optically transparent ceramics by spark plasma sintering of oxide nanoparticles

Optical transparency in polycrystalline ceramic oxides can be achieved if the material is fully densified. Spark plasma sintering (SPS) of oxide nanoparticles leads to immediate densification with final-stage sintering. Further densification by annihilation of the isolated pores is associated with diffusional processes, regardless of the densification mechanism during the intermediate stage. Densification equations in conjunction with the concept of grain boundary free volume were used to derive the pore size–grain size–temperature map for designing the nanopowder and SPS process parameters to obtain transparent oxides

Tearing of free-standing graphene

We examine the fracture mechanics of tearing graphene. We present a molecular dynamics simulation of the propagation of cracks in clamped, free-standing graphene as a function of the out-of-plane force. The geometry is motivated by experimental configurations that expose graphene sheets to out-of-plane forces, such as back-gate voltage. We establish the geometry and basic energetics of failure and obtain approximate analytical expressions for critical crack lengths and forces. We also propose a method to obtain graphene's toughness. We observe that the cracks' path and the edge structure produced are dependent on the initial crack length. This work may help avoid the tearing of graphene sheets and aid the production of samples with specific edge structures.CAPESNational Science Foundation DMR 1002428Physic

Field induced phase transitions in the helimagnet Ba2CuGe2O7

We present a theoretical study of the two-dimensional spiral antiferromagnet Ba2CuGe2O7 in the presence of an external magnetic field. We employ a suitable nonlinear sigma model to calculate the T=0 phase diagram and the associated low-energy spin dynamics for arbitrary canted fields, in general agreement with experiment. In particular, when the field is applied parallel to the c axis, a previously anticipated Dzyaloshinskii-type incommensurate-to-commensurate phase transition is actually mediated by an intermediate phase, in agreement with our earlier theoretical prediction confirmed by the recent observation of the so-called double-k structure. The sudden pi/2 rotations of the magnetic structures observed in experiment are accounted for by a weakly broken U(1) symmetry of our model. Finally, our analysis suggests a nonzero weak-ferromagnetic component in the underlying Dzyaloshinskii-Moriya anisotropy, which is important for quantitative agreement with experiment.Comment: 17 pages, 14 figures. Corrected typos in the abstrac

Numerical Method for Shock Front Hugoniot States

We describe a Continuous Hugoniot Method for the efficient simulation of shock wave fronts. This approach achieves significantly improved efficiency when the generation of a tightly spaced collection of individual steady-state shock front states is desired, and allows for the study of shocks as a function of a continuous shock strength parameter, $v_p$. This is, to our knowledge, the first attempt to map the Hugoniot continuously. We apply the method to shock waves in Lennard-Jonesium along the $$ direction. We obtain very good agreement with prior simulations, as well as our own benchmark comparison runs.Comment: 4 pages, 3 figures, from Shock Compression of Condensed Matter 200