Crescent Singularities in Crumpled Sheets

We examine the crescent singularity of a developable cone in a setting similar to that studied by Cerda et al [Nature 401, 46 (1999)]. Stretching is localized in a core region near the pushing tip and bending dominates the outer region. Two types of stresses in the outer region are identified and shown to scale differently with the distance to the tip. Energies of the d-cone are estimated and the conditions for the scaling of core region size R_c are discussed. Tests of the pushing force equation and direct geometrical measurements provide numerical evidence that core size scales as R_c ~ h^{1/3} R^{2/3}, where h is the thickness of sheet and R is the supporting container radius, in agreement with the proposition of Cerda et al. We give arguments that this observed scaling law should not represent the asymptotic behavior. Other properties are also studied and tested numerically, consistent with our analysis.Comment: 13 pages with 8 figures, revtex. To appear in PR

Spontaneous curvature cancellation in forced thin sheets

In this paper we report numerically observed spontaneous vanishing of mean curvature on a developable cone made by pushing a thin elastic sheet into a circular container. We show that this feature is independent of thickness of the sheet, the supporting radius and the amount of deflection. Several variants of developable cone are studied to examine the necessary conditions that lead to the vanishing of mean curvature. It is found that the presence of appropriate amount of radial stress is necessary. The developable cone geometry somehow produces the right amount of radial stress to induce just enough radial curvature to cancel the conical azimuthal curvature. In addition, the circular symmetry of supporting container edge plays an important role. With an elliptical supporting edge, the radial curvature overcompensates the azimuthal curvature near the minor axis and undercompensates near the major axis. Our numerical finding is verified by a crude experiment using a reflective plastic sheet. We expect this finding to have broad importance in describing the general geometrical properties of forced crumpling of thin sheets.Comment: 13 pages, 12 figures, revtex

A simplicial gauge theory

We provide an action for gauge theories discretized on simplicial meshes, inspired by finite element methods. The action is discretely gauge invariant and we give a proof of consistency. A discrete Noether's theorem that can be applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a discrete Noether's theorem. v3: Section 4 on Noether's theorem has been expanded with Proposition 8, section 2 has been expanded with a paragraph on standard LGT. v4: Thorough revision with new introduction and more background materia

Rim curvature anomaly in thin conical sheets revisited

This paper revisits one of the puzzling behaviors in a developable cone (d-cone), the shape obtained by pushing a thin sheet into a circular container of radius $R$ by a distance $\eta$ [E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999)]. The mean curvature was reported to vanish at the rim where the d-cone is supported [T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006)]. We investigate the ratio of the two principal curvatures versus sheet thickness $h$ over a wider dynamic range than was used previously, holding $R$ and $\eta$ fixed. Instead of tending towards 1 as suggested by previous work, the ratio scales as $(h/R)^{1/3}$. Thus the mean curvature does not vanish for very thin sheets as previously claimed. Moreover, we find that the normalized rim profile of radial curvature in a d-cone is identical to that in a "c-cone" which is made by pushing a regular cone into a circular container. In both c-cones and d-cones, the ratio of the principal curvatures at the rim scales as $(R/h)^{5/2}F/(YR^{2})$, where $F$ is the pushing force and $Y$ is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.Comment: 25 pages, 12 figures. Added references. Corrected typos. Results unchange

Risk of fractures in half a million survivors of 20 cancers: a population-based matched cohort study using linked English electronic health records.

BACKGROUND: A history of multiple myeloma, prostate cancer, and breast cancer has been associated with adverse bone health, but associations across a broader range of cancers are unclear. We aimed to compare the risk of any bone fracture and major osteoporotic fractures in survivors of a wide range of cancers versus cancer-free individuals. METHODS: In this population-based matched cohort study, we used electronic health records from the UK Clinical Practice Research Datalink linked to hospital data. We included adults (aged ≥18 years) eligible for linkage, and we restricted the study start to Jan 2, 1998, onwards and applied administrative censoring on Jan 31, 2020. The cancer survivor group included survivors of the 20 most common cancers. Each individual with cancer was matched (age, sex, and general practice) to up to five controls (1:5) who were cancer-free. The primary outcomes were any bone fracture and any major osteoporotic fracture (pelvic, hip, wrist, spine, or proximal humeral fractures) occurring more than 1 year after index date (ie, the diagnosis date of the matched individual with cancer). We used Cox regression models, adjusted for shared risk factors, to estimate associations between cancer survivorship and bone fractures. FINDINGS: 578 160 adults with cancer diagnosed in 1998-2020 were matched to 3 226 404 cancer-free individuals. Crude incidence rates of fractures in cancer survivors ranged between 8·39 cases (95% CI 7·45-9·46) per 1000 person-years for thyroid cancer and 21·62 cases (20·18-23·18) per 1000 person-years for multiple myeloma. Compared with cancer-free individuals, the risk of any bone fracture was increased in 15 of 20 cancers, and of major osteoporotic fractures in 17 of 20 cancers. Effect sizes varied: adjusted hazard ratios (HRs) were largest for multiple myeloma (1·94, 95% CI 1·77-2·13) and prostate cancer (1·43, 1·39-1·47); HRs in the range 1·20-1·50 were seen for stomach, liver, pancreas, lung, breast, kidney, and CNS cancers; smaller associations (HR <1·20) were observed for malignant melanoma, non-Hodgkin lymphoma, leukaemia, and oesophageal, colorectal, and cervical cancers. Increased risks of major osteoporotic fracture were noted most substantially in multiple myeloma (2·25, 1·96-2·58) and CNS (2·12, 1·56-2·87), liver (1·62, 1·01-2·61), prostate (1·60, 1·53-1·67), and lung cancers (1·60, 1·44-1·77). Effect sizes tended to reduce over time since diagnosis but remained elevated for more than 5 years in several cancers, such as multiple myeloma and stomach, lung, breast, prostate, and CNS cancers. INTERPRETATION: Survivors of most types of cancer were at increased risk of bone fracture for several years after cancer, with variation by cancer type. These findings can help to inform mitigation and prevention strategies. FUNDING: Wellcome Trust

Grasping the Links in the Chain: Understanding the Unintended Consequences of International Counter-Narcotics Measures for the EU

No abstract available

Synthesis, characterisation, and diffusive properties of functionalised nanomaterials

The aim of this thesis was to assess the diffusive properties of functionalised and unfunctionalised nanomaterials in a variety of different media. The main goal was to gain an insight into the fundamental mechanisms underpinning nanoparticle diffusion and how the surface properties of nanoparticles alter their net movement through different environments. Initially a library of polymer-functionalised silica nanoparticles were synthesised and characterised. The polymers chosen were; poly(ethylene glycol) (PEG), poly(2-oxazolines) (POZ) and poly(n-isopropyl acrylamide) (PNIPAM). Firstly, the diffusion of different sized gold nanoparticles was assessed in concentrations of Pluronic F-127, in order to determine how the solution properties affected diffusion. It was found that as the solution undergoes a transition in response to environmental stimuli, there is an increase in diffusion coefficient; however the area they move in becomes more confined (assessed using a bespoke python script written for use with NTA). PNIPAM- and PNPOZ-silica nanoparticles were assessed for their aggregation and diffusion using DLS, NTA, and SANS. It was found that the position of a nitrogen atom in the amide group, present in both polymers, plays a key role in governing how the particles aggregate in solution, which in turn affects how they diffuse through solvents of varying polarities. POZ-silica nanoparticles were assessed for mucus penetration against a positive control of PEGylated nanoparticles. It was found that POZ-silica was effective at enhancing nanoparticle mucus penetration, and the hydrophilicity of these polymers plays a key role in determining the degree of permeation (with methylated POZ significantly more diffusive than propylated POZ). These finding provide valuable insight into some of the molecular mechanisms governing nanoparticle diffusion and how surface chemistry governs these effects

Curvature condensation and bifurcation in an elastic shell

We study the formation and evolution of localized geometrical defects in an indented cylindrical elastic shell using a combination of experiment and numerical simulation. We find that as a symmetric localized indentation on a semi-cylindrical shell increases, there is a transition from a global mode of deformation to a localized one which leads to the condensation of curvature along a symmetric parabolic crease. This process introduces a soft mode in the system, converting a load-bearing structure into a hinged, kinematic mechanism. Further indentation leads to twinning wherein the parabolic crease bifurcates into two creases that move apart on either side of the line of symmetry. A qualitative theory captures the main features of the phenomena and leads to sharper questions about the nucleation of these defects.Comment: 4 pages, 5 figures, submitted to Physical Review Letter

Light-Cone Quantization of the Liouville Model

We present the quantization of the Liouville model defined in light-cone coordinates in (1,1) signature space. We take advantage of the representation of the Liouville field by the free field of the Backl\"{u}nd transformation and adapt the approch by Braaten, Curtright and Thorn. Quantum operators of the Liouville field $\partial_{+}\phi$, $\partial_{-}\phi$, $e^{g\phi}$, $e^{2g\phi}$ are constructed consistently in terms of the free field. The Liouville model field theory space is found to be restricted to the sector with field momentum $P_{+}=-P_{-}$, $P_{+}> 0$ , which is a closed subspace for the Liouville theory operator algebra.Comment: 16 p, EFI-92-6

Properties of Ridges in Elastic Membranes

When a thin elastic sheet is confined to a region much smaller than its size the morphology of the resulting crumpled membrane is a network of straight ridges or folds that meet at sharp vertices. A virial theorem predicts the ratio of the total bending and stretching energies of a ridge. Small strains and curvatures persist far away from the ridge. We discuss several kinds of perturbations that distinguish a ridge in a crumpled sheet from an isolated ridge studied earlier (A. E. Lobkovsky, Phys. Rev. E. 53 3750 (1996)). Linear response as well as buckling properties are investigated. We find that quite generally, the energy of a ridge can change by no more than a finite fraction before it buckles.Comment: 13 pages, RevTeX, acknowledgement adde