Past Differential Equations and Applications Seminar

29 October 2009
16:30
Abstract
Frank-Read sources are among the most important mechanisms of dislocation multiplication, and their operation signals the onset of yield in crystals. We show that the critical stress required to initiate dislocation production falls dramatically at high elastic anisotropy, irrespective of the mean shear modulus. We hence predict the yield stress of crystals to fall dramatically as their anisotropy increases. This behaviour is consistent with the severe plastic softening observed in alpha-iron and ferritic steels as the alpha − gamma martensitic phase transition is approached, a temperature regime of crucial importance for structural steels designed for future nuclear applications.
  • Differential Equations and Applications Seminar
15 October 2009
16:30
Ricardo Carretero
Abstract
Traditional Faraday waves appear in a layer of liquid that is shaken vertically. These patterns can take the form of horizontal stripes, close-packed hexagons, or even squares or quasiperiodic patterns. Faraday waves are commonly observed as fine stripes on the surface of wine in a wineglass that is ringing like a bell when periodically forced. Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement's trap. We offer a fully analytical explanation of the observed parametric resonance yielding the pattern periodicity versus the driving frequency. These results, corroborated by numerical simulations, match extremely well with the experimental observations.
  • Differential Equations and Applications Seminar
18 June 2009
16:30
Mark McGuinness
Abstract
Platelet ice may be an important component of Antarctic land-fast sea ice. Typically, it is found at depth in first-year landfast sea ice cover, near ice shelves. To explain why platelet ice is not commonly observed at shallower depths, we consider a new mechanism. Our hypothesis is that platelet ice eventually appears due to the sudden deposition of frazil ice against the fast ice-ocean interface, providing randomly oriented nucleation sites for crystal growth. Brine rejected in plumes from land-fast ice generates stirring sufficient to prevent frazil ice from attaching to the interface, forcing it to remain in suspension until ice growth rate and brine rejection slow to the point that frazil can stick. We calculate a brine plume velocity, and match this to frazil rise velocity. We consider both laminar and turbulent environments. We find that brine plume velocities are generally powerful enough to prevent most frazil from sticking in the case of laminar flow, and that in the turbulent case there may be a critical ice thickness at which most frazil suddenly settles.
  • Differential Equations and Applications Seminar
11 June 2009
16:30
Rachel Kuske
Abstract
Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior. Coherence resonance is one relatively simple example of this transformation, and we consider others such as noise-driven synchronization in networks, recurrence of diseases, and stochastic stabilization in systems with delay. The challenge is to identify common features in these phenomena, leading to new approaches for other systems of this type. Some recurring themes include the influence of multiple time scales, cooperation of both discrete and continuous aspects in the dynamics, and the remnants of underlying bifurcation structure visible through the noise.
  • Differential Equations and Applications Seminar
4 June 2009
16:30
Karima Khusnutdinova
Abstract
Layered (or laminated) structures are increasingly used in modern industry (e.g., in microelectronics and aerospace engineering). Integrity of such structures is mainly determined by the quality of their interfaces: poor adhesion or delamination can lead to a catastrophic failure of the whole structure. Can nonlinear waves help us to detect such defects? We study the dynamics of a nonlinear longitudinal bulk strain wave in a split, layered elastic bar, made of nonlinearly hyperelastic Murnaghan material. We consider a symmetric two-layered bar and assume that there is perfect interface for x < 0 and splitting for x > 0, where the x-axis is directed along the bar. Using matched asymptotic multiple-scales expansions and the integrability theory of the KdV equation by the Inverse Scattering Transform, we examine scattering of solitary waves and show that the defect causes generation of more than one secondary solitary waves from a single incident soliton and, thus, can be used to detect the defect. The theory is supported by experimental results. Experiments have been performed in the Ioffe Institute in St. Petersburg (Russia), using holographic interferometry and laser induced generation of an incident compression solitary wave in two- and three-layered polymethylmethacrylate (PMMA) bars, bonded using ethyl cyanoacrylate-based (CA) adhesive.
  • Differential Equations and Applications Seminar
28 May 2009
16:30
Xanthippi Markenscoff
Abstract
In the context of the linear theory of elasticity with eigenstrains, the radiated fields, including inertia effects, and the energy-release rates (“driving forces”) of a spherically expanding and a plane inclusion with constant dilatational eigenstrains are calculated. The fields of a plane boundary with dilatational eigenstrain moving from rest in general motion are calculated by a limiting process from the spherical ones, as the radius tends to infinity, which yield the time-dependent tractions that need to be applied on the lateral boundaries for the global problem to be well-posed. The energy-release rate required to move the plane inclusion boundary (and to create a new volume of eigenstrain) in general motion is obtained here for a superposed loading of a homogeneous uniaxial tensile stress. This provides the relation of the applied stress to the boundary velocity through the energy-rate balance equation, yielding the “equation of motion” (or “kinetic relation”) of the plane boundary under external tensile axial loading. This energy-rate balance expression is the counterpart to the Peach-Koehler force on a dislocation plus the “self-force” of the moving dislocation.
  • Differential Equations and Applications Seminar
21 May 2009
16:30
Dominic Vella
Abstract
An elastic sheet will buckle out of the plane when subjected to an in-plane compression. In the simplest systems the typical lengthscale of the buckled structure is that of the system itself but with additional physics (e.g. an elastic substrate) repeated buckles with a well-defined wavelength may be seen. We discuss two examples in which neither of these scenarios is realized: instead a small number of localized structures are observed with a size different to that of the system itself. The first example is a heavy sheet on a rigid floor - a ruck in a rug. We study the static properties of these rucks and also how they propagate when one end of the rug is moved quickly. The second example involves a thin film adhered to a much softer substrate. Here delamination blisters are formed with a well-defined size, which we characterize in terms of the material properties of the system. We then discuss the possible application of these model systems to real world problems ranging from the propagation of slip pulses in earthquakes to the manufacture of flexible electronic devices."
  • Differential Equations and Applications Seminar

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