Soft modes near the buckling transition of icosahedral shells

Icosahedral shells undergo a buckling transition as the ratio of Young's modulus to bending stiffness increases. Strong bending stiffness favors smooth, nearly spherical shapes, while weak bending stiffness leads to a sharply faceted icosahedral shape. Based on the phonon spectrum of a simplified mass-and-spring model of the shell, we interpret the transition from smooth to faceted as a soft-mode transition. In contrast to the case of a disclinated planar network where the transition is sharply defined, the mean curvature of the sphere smooths the transitition. We define elastic susceptibilities as the response to forces applied at vertices, edges and faces of an icosahedron. At the soft-mode transition the vertex susceptibility is the largest, but as the shell becomes more faceted the edge and face susceptibilities greatly exceed the vertex susceptibility. Limiting behaviors of the susceptibilities are analyzed and related to the ridge-scaling behavior of elastic sheets. Our results apply to virus capsids, liposomes with crystalline order and other shell-like structures with icosahedral symmetry.Comment: 28 pages, 6 figure

Self-similar impulsive capillary waves on a ligament

We study the short-time dynamics of a liquid ligament, held between two solid cylinders, when one is impulsively accelerated along its axis. A set of one-dimensional equations in the slender-slope approximation is used to describe the dynamics, including surface tension and viscous effects. An exact self-similar solution to the linearized equations is successfully compared to experiments made with millimetric ligaments. Another non-linear self-similar solution of the full set of equations is found numerically. Both the linear and non-linear solutions show that the axial depth at which the liquid is affected by the motion of the cylinder scales like $\sqrt{t}$. The non-linear solution presents the peculiar feature that there exists a maximum driving velocity $U^\star$ above which the solution disappears, a phenomenon probably related to the de-pinning of the contact line observed in experiments for large pulling velocities

Phonon-phonon interactions and phonon damping in carbon nanotubes

We formulate and study the effective low-energy quantum theory of interacting long-wavelength acoustic phonons in carbon nanotubes within the framework of continuum elasticity theory. A general and analytical derivation of all three- and four-phonon processes is provided, and the relevant coupling constants are determined in terms of few elastic coefficients. Due to the low dimensionality and the parabolic dispersion, the finite-temperature density of noninteracting flexural phonons diverges, and a nonperturbative approach to their interactions is necessary. Within a mean-field description, we find that a dynamical gap opens. In practice, this gap is thermally smeared, but still has important consequences. Using our theory, we compute the decay rates of acoustic phonons due to phonon-phonon and electron-phonon interactions, implying upper bounds for their quality factor.Comment: 15 pages, 2 figures, published versio

Analytical Solution for the Deformation of a Cylinder under Tidal Gravitational Forces

Quite a few future high precision space missions for testing Special and General Relativity will use optical resonators which are used for laser frequency stabilization. These devices are used for carrying out tests of the isotropy of light (Michelson-Morley experiment) and of the universality of the gravitational redshift. As the resonator frequency not only depends on the speed of light but also on the resonator length, the quality of these measurements is very sensitive to elastic deformations of the optical resonator itself. As a consequence, a detailed knowledge about the deformations of the cavity is necessary. Therefore in this article we investigate the modeling of optical resonators in a space environment. Usually for simulation issues the Finite Element Method (FEM) is applied in order to investigate the influence of disturbances on the resonator measurements. However, for a careful control of the numerical quality of FEM simulations a comparison with an analytical solution of a simplified resonator model is beneficial. In this article we present an analytical solution for the problem of an elastic, isotropic, homogeneous free-flying cylinder in space under the influence of a tidal gravitational force. The solution is gained by solving the linear equations of elasticity for special boundary conditions. The applicability of using FEM codes for these simulations shall be verified through the comparison of the analytical solution with the results gained within the FEM code.Comment: 23 pages, 3 figure

The constitutive tensor of linear elasticity: its decompositions, Cauchy relations, null Lagrangians, and wave propagation

In linear anisotropic elasticity, the elastic properties of a medium are described by the fourth rank elasticity tensor C. The decomposition of C into a partially symmetric tensor M and a partially antisymmetric tensors N is often used in the literature. An alternative, less well-known decomposition, into the completely symmetric part S of C plus the reminder A, turns out to be irreducible under the 3-dimensional general linear group. We show that the SA-decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor. The MN-decomposition fails to have these desirable properties and is such inferior from a physical point of view. Various applications of the SA-decomposition are discussed: the Cauchy relations (vanishing of A), the non-existence of elastic null Lagrangians, the decomposition of the elastic energy and of the acoustic wave propagation. The acoustic or Christoffel tensor is split in a Cauchy and a non-Cauchy part. The Cauchy part governs the longitudinal wave propagation. We provide explicit examples of the effectiveness of the SA-decomposition. A complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions, similarly as in an isotropic medium.Comment: 1 figur

Conformal invariance: from Weyl to SO(2,d)

The present work deals with two different but subtilely related kinds of conformal mappings: Weyl rescaling in $d>2$ dimensional spaces and SO(2,d) transformations. We express how the difference between the two can be compensated by diffeomorphic transformations. This is well known in the framework of String Theory but in the particular case of $d=2$ spaces. Indeed, the Polyakov formalism describes world-sheets in terms of two-dimensional conformal field theory. On the other hand, B. Zumino had shown that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS). This allows us to assert that a classical $SO(2,d)$-invariant field does not distinguish, at least locally, between two different $d$-dimensional CFSs.Comment: 5 pages, no figures. There are slight modifications to match with the published versio

High-frequency homogenization for periodic media

This article is available open access through the publisher’s website at the link below. Copyright @ 2010 The Royal Society.An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.NSERC (Canada) and the EPSRC

Twirling and Whirling: Viscous Dynamics of Rotating Elastica

Motivated by diverse phenomena in cellular biophysics, including bacterial flagellar motion and DNA transcription and replication, we study the overdamped nonlinear dynamics of a rotationally forced filament with twist and bend elasticity. Competition between twist injection, twist diffusion, and writhing instabilities is described by a novel pair of coupled PDEs for twist and bend evolution. Analytical and numerical methods elucidate the twist/bend coupling and reveal two dynamical regimes separated by a Hopf bifurcation: (i) diffusion-dominated axial rotation, or twirling, and (ii) steady-state crankshafting motion, or whirling. The consequences of these phenomena for self-propulsion are investigated, and experimental tests proposed.Comment: To be published in Physical Review Letter

Exact Effective Action for (1+1 Dimensional) Fermions in an Abelian Background at Finite Temperature

In an effort to further understand the structure of effective actions for fermions in an external gauge background at finite temperature, we study the example of 1+1 dimensional fermions interacting with an arbitrary Abelian gauge field. We evaluate the effective action exactly at finite temperature. This effective action is non-analytic as is expected at finite temperature. However, contrary to the structure at zero temperature and contrary to naive expectations, the effective action at finite temperature has interactions to all (even) orders (which, however, do not lead to any quantum corrections). The covariant structure thus obtained may prove useful in studying 2+1 dimensional models in arbitrary backgrounds. We also comment briefly on the solubility of various 1+1 dimensional models at finite temperature.Comment: A few clarifying remarks added;21 page