Triangular buckling patterns of twisted inextensible strips

When twisting a strip of paper or acetate under high longitudinal tension, one observes, at some critical load, a buckling of the strip into a regular triangular pattern. Very similar triangular facets have recently been observed in solutions to a new set of geometrically-exact equations describing the equilibrium shape of thin inextensible elastic strips. Here we formulate a modified boundary-value problem for these equations and construct post-buckling solutions in good agreement with the observed pattern in twisted strips. We also study the force-extension and moment-twist behaviour of these strips by varying the mode number n of triangular facets

On the Equilibrium of a Thin Elastic Spherical Bowl

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Sudden collapse of a colloidal gel

Metastable gels formed by weakly attractive colloidal particles display a distinctive two-stage time-dependent settling behavior under their own weight. Initially a space-spanning network is formed that for a characteristic time, which we define as the lag time \taud, resists compaction. This solid-like behavior persists only for a limited time. Gels whose age \tw is greater than \taud yield and suddenly collapse. We use a combination of confocal microscopy, rheology and time-lapse video imaging to investigate both the process of sudden collapse and its microscopic origin in an refractive-index matched emulsion-polymer system. We show that the height $h$ of the gel in the early stages of collapse is well described by the surprisingly simple expression, h(\ts) = \h0 - A \ts^{3/2}, with \h0 the initial height and \ts = \tw-\taud the time counted from the instant where the gel first yields. We propose that this unexpected result arises because the colloidal network progressively builds up internal stress as a consequence of localized rearrangement events which leads ultimately to collapse as thermal equilibrium is re-established.Comment: 14 pages, 11 figures, final versio

The Effective Potential And Additional Large Radius Compactified Space-Time Dimensions

The consequences of large radius extra space-time compactified dimensions on the four dimensional one loop effective potential are investigated for a model which includes scalar self interactions and Yukawa coupling to fermions. The Kaluza-Klein tower of states associated with the extra compact dimensions shifts the location of the effective potential minimum and modifies its curvature. The dependence of these effects on the radius of the extra dimension is illustrated for various choices of coupling constants and masses. For large radii, the consequence of twisting the fermion boundary condition on the compactified dimensions is numerically found to produce but a negligible effect on the effective potential.Comment: 14 pages, LaTeX, 6 Postscript figure

Relativistic theory of tidal Love numbers

In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body deformed by an external tidal field, and provide precise and meaningful definitions for electric-type and magnetic-type Love numbers; and these are computed for polytropic equations of state. The theory applies to black holes as well, and we find that the relativistic Love numbers of a nonrotating black hole are all zero.Comment: 25 pages, 8 figures, many tables; final version to be published in Physical Review

Elastic Instability Triggered Pattern Formation

Recent experiments have exploited elastic instabilities in membranes to create complex patterns. However, the rational design of such structures poses many challenges, as they are products of nonlinear elastic behavior. We pose a simple model for determining the orientational order of such patterns using only linear elasticity theory which correctly predicts the outcomes of several experiments. Each element of the pattern is modeled by a "dislocation dipole" located at a point on a lattice, which then interacts elastically with all other dipoles in the system. We explicitly consider a membrane with a square lattice of circular holes under uniform compression and examine the changes in morphology as it is allowed to relax in a specified direction.Comment: 15 pages, 7 figures, the full catastroph

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

Multistability of free spontaneously-curved anisotropic strips

Multistable structures are objects with more than one stable conformation, exemplified by the simple switch. Continuum versions are often elastic composite plates or shells, such as the common measuring tape or the slap bracelet, both of which exhibit two stable configurations: rolled and unrolled. Here we consider the energy landscape of a general class of multistable anisotropic strips with spontaneous Gaussian curvature. We show that while strips with non-zero Gaussian curvature can be bistable, strips with positive spontaneous curvature are always bistable, independent of the elastic moduli, strips of spontaneous negative curvature are bistable only in the presence of spontaneous twist and when certain conditions on the relative stiffness of the strip in tension and shear are satisfied. Furthermore, anisotropic strips can become tristable when their bending rigidity is small. Our study complements and extends the theory of multistability in anisotropic shells and suggests new design criteria for these structures.Comment: 20 pages, 10 figure

Fluorescence Imaging of Underexpanded Jets and Comparison with CFD

An experimental study of underexpanded and highly underexpanded axisymmetric nitrogen free jets seeded with 0.5% nitric oxide (NO) and issuing from a sonic orifice was conducted at NASA Langley Research Center. Reynolds numbers based on nozzle exit conditions ranged from 770 to 35,700, and nozzle exit-to-ambient jet pressure ratios ranged from 2 to 35. These flows were non-intrusively visualized with a spatial resolution of approximately 0.14 mm x 0.14 mm x 1 mm thick and a temporal resolution of 1 s using planar laser-induced fluorescence (PLIF) of NO, with the laser tuned to the strongly-fluorescing UV absorption bands of the Q1 band head near 226.256 nm. Three laminar cases were selected for comparison with computational fluid dynamics (CFD). The cases were run using GASP (General Aerodynamic Simulation Program) Version 4. Comparisons of the fundamental wavelength of the jet flow showed good agreement between CFD and experiment for all three test cases, while comparisons of Mach disk location and Mach disk diameter showed good agreement at lower jet pressure ratios, with a tendency to slightly underpredict these parameters with increasing jet pressure ratio

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