A Theory of Elastic/Plastic Plane Strain Pure Bending of FGM Sheets at Large Strain

An efficient analytical/numerical method has been developed and programmed to predict the distribution of residual stresses and springback in plane strain pure bending of functionally graded sheets at large strain, followed by unloading. The solution is facilitated by using a Lagrangian coordinate system. The study is concentrated on a power law through thickness distribution of material properties. However, the general method can be used in conjunction with any other through thickness distributions assuming that plastic yielding initiates at one of the surfaces of the sheet. Effects of material properties on the distribution of residual stresses are investigated.

A quasi-Lagrangian coordinate system based on high resolution tracer observations: implementation for the Antarctic polar vortex

In order to quantitatively analyse the chemical and dynamical evolution of the polar vortex it has proven extremely useful to work with coordinate systems that follow the vortex flow. We propose here a two-dimensional quasi-Lagrangian coordinate system {χi, Δχi}, based on the mixing ratio of a long-lived stratospheric trace gas i, and its systematic use with i = N2O, in order to describe the structure of a well-developed Antarctic polar vortex. In the coordinate system {χi, Δχi} the mixing ratio χi is the vertical coordinate and Δχi = χi(Θ)−χivort(Θ) is the meridional coordinate (χivort(Θ) being a vertical reference profile in the vortex core). The quasi-Lagrangian coordinates {χi, Δχi} persist for much longer time than standard isentropic coordinates, potential temperature Θ and equivalent latitude φe, do not require explicit reference to geographic space, and can be derived directly from high-resolution in situ measurements. They are therefore well-suited for studying the evolution of the Antarctic polar vortex throughout the polar winter with respect to the relevant chemical and microphysical processes. By using the introduced coordinate system {χN2O, ΔχN2O} we analyze the well-developed Antarctic vortex investigated during the APE-GAIA (Airborne Polar Experiment – Geophysica Aircraft in Antarctica – 1999) campaign (Carli et al., 2000). A criterion, which uses the local in-situ measurements of χi=χi(Θ) and attributes the inner vortex edge to a rapid change (δ-step) in the meridional profile of the mixing ratio χi, is developed to determine the (Antarctic) inner vortex edge. In turn, we suggest that the outer vortex edge of a well-developed Antarctic vortex can be attributed to the position of a local minimum of the χH2O gradient in the polar vortex area. For a well-developed Antarctic vortex, the ΔχN2O-parametrization of tracer-tracer relationships allows to distinguish the tracer inter-relationships in the vortex core, vortex boundary region and surf zone and to examine their meridional variation throughout these regions. This is illustrated by analyzing the tracer-tracer relationships χi : χN2O obtained from the in-situ data of the APE-GAIA campaign for i = CFC-11, CFC-12, H-1211 and SF6. A number of solitary anomalous points in the CFC-11 : N2O correlation, observed in the Antarctic vortex core, are interpreted in terms of small-scale cross-isentropic dispersion.

A Self-Contained Identification Scheme for Eddies in Drifter and Float Trajectories

It is becoming increasingly recognized that the eddy field plays an important—possibly dominating—role for oceanic motions in many aspects (e.g., transport of properties and risk assessment in the case of extreme events). This motivates the study of individual eddy events. In the Lagrangian coordinate system, vorticity possibly associated with eddies appears in two forms: as shear vorticity between neighboring particles, and as curvature of the trajectory of a single particle. Typical field experiments in physical oceanography using surface drifters or subsurface floats do not reach data densities high enough to produce enough encounters of drifters to calculate shear vorticity between them. However, curvature in individual tracks is easily observed. This study presents a methodology that extracts segments from within a trajectory that are “looping,” which will be interpreted as a drifter being caught in an eddy. The method makes use of autoregressive processes, a simple type of stochastic processes, which easily enables a fit to the nonperfectly shaped trajectory data usually expected from field experiments. These processes also deliver frequency and persistence of the detected eddies by a very simple calculation, which makes the methodology highly suited for automatized scanning of larger datasets.

Slow Dripping of Yield-Stress Fluids

The dripping problem of a viscoplastic (yield-stress) liquid running slowly out of a narrow vertical tube is considered. The volume of the drops which break away is determined. A Lagrangian coordinate system is used to analyze the extension of the thread as it sags under its own weight, neglecting inertia and capillarity. The biviscosity model has been used to characterize viscoplastic fluids; the Newtonian and Bingham models can be recovered as limiting cases. The Bingham limit is of special interest.

The Stretching of a Viscoplastic Thread of Liquid

The problem of stretching a viscoplastic (yield-stress) thread of a liquid hanging vertically is considered. The length of the thread at later times and the time at which it ruptures is determined. A Lagrangian coordinate system is used to analyze the extension of the thread as it sags under its own weight, with negligible inertial effects. The biviscosity model has been used to characterize viscoplastic fluids; the Newtonian and Bingham models can be recovered as limiting cases. The Bingham limit is of special interest.

Drift velocity of spatially decaying waves in a two-layer viscous system

This paper analyses the mass transport velocity in a two-layer system induced by the action of progressive waves. First the movement inside the two layers is obtained. Next the mass transport of spatially decaying waves is calculated by solving the momentum and mass conservation equations in the Lagrangian coordinate system. Two different physical situations are analysed: the first is waves in a closed channel and the second is waves in an unbounded domain, where the steady-state mass flux may be non-zero. The influence of the viscous properties of the lower layer on the mass transport in both layers is studied. Comparison with the experiments of Sakakiyama & Bijker (1989) in a water-mud system shows good agreement. The results show that the mass transport velocity can be quite different from the velocity given by the rigid bed theory, depending on the physical properties of the lower layer.

Local Film Thickness During Transient Voiding of a Liquid-Filled Channel

Using a recently developed method, measurements were obtained of local film thicknesses during transient voiding of a liquid-filled channel. The liquid film remaining on the channel walls was found to vary in thickness over a range of 0.015–0.15 times channel diameter. In a Lagrangian coordinate system, the film thickness at a fixed distance from the head of the void was found to increase with increasing void acceleration. In an Eulerian system, the film thickness at a fixed location on the channel wall was found to decrease with increasing acceleration, when measured at the same time after passage of the head of the void. In all cases, film thickness monotonically decreased with increasing distance from the head of the void. Complete film breakage (dryout) was not observed in these experiments. These experimental measurements of local film thicknesses during transient voiding conditions are pertinent to thermal analyses for reactor safety studies.