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

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

Simple models of the hydrofracture process

Hydrofracturing to recover natural gas and oil relies on the creation of a fracture network with pressurized water. We analyze the creation of the network in two ways. First, we assemble a collection of analytical estimates for pressure-driven crack motion in simple geometries, including crack speed as a function of length, energy dissipated by fluid viscosity and used to break rock, and the conditions under which a second crack will initiate while a first is running. We develop a pseudo-three-dimensional numerical model that couples fluid motion with solid mechanics and can generate branching crack structures not specified in advance. One of our main conclusions is that the typical spacing between fractures must be on the order of a meter, and this conclusion arises in two separate ways. First, it arises from analysis of gas production rates, given the diffusion constants for gas in the rock. Second, it arises from the number of fractures that should be generated given the scale of the affected region and the amounts of water pumped into the rock.Shell UTNational Science FoundationPhysic

Development of materials and process technology for dual alloy disks

Techniques for the preparation of dual alloy disks were developed and evaluated. Four material combinations were evaluated in the form of HIP consolidated and heat treated cylindrical and plate shapes in terms of elevated temperature tensile, stress rupture and low cycle fatigue properties. The process evaluation indicated that the pe-HIP AF-115 rim/loose powder Rene 95 hub combination offered the best overall range of mechanical properties for dual disk applications. The feasibility of this dual alloy concept for the production of more complex components was demonstrated by the scale up fabrication of a prototype CFM-56 disk made from this AF-115/Rene 95 combination. The hub alloy ultimate tensile strength was approximately 92 percent of the program goal of 1520 MPa (220 ksi) at 480 C (900 F) and the rim alloy stress rupture goal of 300 hours at 675 C (1250 F)/925 MPa (134 ksi) was exceeded by 200 hours. The low cycle fatigue properties were equivalent to those exhibited by HIP and heat treated alloys. There was an absence of rupture notch sensitivity in both alloys. The joint tensile properties were approximately 85 percent of the weaker of the two materials (Rene 95) and the stress rupture properties were equivalent to those of the weaker of the two materials (Rene 95)