A derivation of Griffith functionals from discrete finite-difference models

We analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $$\delta $$ δ is smaller than the ellipticity parameter $$\varepsilon $$ ε , we show the $$\varGamma $$ Γ -convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $$L^p$$ L p fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.

Effects of Turbulence on the Vortex Modes Carried by Quasi-Diffracting Free Finite Energy Beam in Ocean

By developing new wave structure function of a beam waves, we derive the transmitting probability of signal vortex modes in oceanic turbulence based on Rytov approximation theory. Applying this transmitting probability of signal vortex modes, we study the influence of oceanic turbulence on the transmittance of the vortex modes carried by Mathieu-Gaussian beam. This model shows the transmitting probability of Mathieu-Gaussian beam with narrow initial beam width, long wavelength, and small ellipticity parameter is higher than the transmitting probability of the signal vortex modes in case of the beam with wide initial beam width, short wavelength, and great ellipticity parameter. Furthermore, when Mathieu-Gaussian beam has a suitable semi-cone angle, the effect of weak-turbulence channel on the transmitting probability of signal vortex modes with different topological charge can be ignored. Mathieu-Gaussian beam is a more suitable carrier for high information channel of underwater wireless optical communication than Laguerre-Gaussian beam.

The SAMI Galaxy Survey: rules of behaviour for spin-ellipticity radial tracks in galaxies

ABSTRACT We study the behaviour of the spin-ellipticity radial tracks for 507 galaxies from the Sydney AAO Multiobject Integral Field (SAMI) Galaxy Survey with stellar kinematics out to ≥1.5Re. We advocate for a morpho-dynamical classification of galaxies, relying on spatially resolved photometric and kinematic data. We find the use of spin-ellipticity radial tracks is valuable in identifying substructures within a galaxy, including embedded and counter-rotating discs, that are easily missed in unilateral studies of the photometry alone. Conversely, bars are rarely apparent in the stellar kinematics but are readily identified on images. Consequently, we distinguish the spin-ellipticity radial tracks of seven morpho-dynamical types: elliptical, lenticular, early spiral, late spiral, barred spiral, embedded disc, and 2σ galaxies. The importance of probing beyond the inner radii of galaxies is highlighted by the characteristics of galactic features in the spin-ellipticity radial tracks present at larger radii. The density of information presented through spin-ellipticity radial tracks emphasizes a clear advantage to representing galaxies as a track, rather than a single point, in spin-ellipticity parameter space.

The Influence of the Operating Conditions on the Lubrication Regimes of Ball Screws

In point contact related applications (ball bearings, ball-screws) the minimum and the central film thickness of lubricant have a particular importance. The literature presents various numerical solutions to determine these parameters in the case of elliptical contacts [1-4]. Most of them refer to the fully developed elastohydrodynamic lubrication regime (EHD). Hamrock [5] proposes four different lubrication regimes depending on the size and importance of two physical phenomena that occur in contact: elastic deformation of the bodies in contact under a given load and lubricant viscosity variation of contact pressure. These four lubrication regimes have specific relations for calculating the two parameters. Also, Hamrock [5] develops a methodology for identifying the lubrication regime in a point contact, depending on three dimensionless parameters: the ellipticity parameter; the viscosity parameter; the elasticity parameter. According to the viscosity and elasticity parameters, for each value of the ellipticity parameter a map of lubrication regimes can be built. These parameters are influenced by the materials and geometry of the bodies in contact and operating conditions. By using the methodology [5] a comprehensive analysis regarding the lubrication regimes in contacts between balls and races of a ball screw drive has been done. Thus, a complex program to draw lubrication regimes maps, starting from an imposed geometry and from given operating conditions, has been developed. The developed maps have revealed the different lubrication regimes that can occur in contacts between the balls and races on the screw and nut. It was also revealed, for a given type of screw, which are the load and speed limits that allow transition from an isoviscous-rigid regime of lubrication (IVR) / hydrodynamic (HD) to an elastohydrodynamic lubrication regime (EHD). For each lubrication regime, relations were used for calculating the appropriate minimum thickness of lubricant film, hence the major importance of accurate knowledge of lubrication regime.

Elastohydrodynamic Lubrication of Elliptical Contacts for Materials of Low Elastic Modulus I—Fully Flooded Conjunction

Our earlier studies of elastohydrodynamic lubrication of conjunctions of elliptical form are applied to the particular and interesting situation exhibited by materials of low elastic modulus. By modifying the procedures we outlined in an earlier publication, the influence of the ellipticity parameter k and the dimensionless speed U, load W, and material G parameters on minimum film thickness for these materials has been investigated. The ellipticity parameter was varied from 1 (a ball-on-plate configuration) to 12 (a configuration approaching a line contact). The dimensionless speed and load parameters were varied by 1 order of magnitude. Seventeen different cases were used to generate the following minimum- and central-film-thickness relations: H˜min=7.43(1−0.85e−0.31k)U0.65W−0.21H˜c=7.32(1−0.72e−0.28k)U0.64W−0.22 Contour plots are presented that illustrate in detail the pressure distribution and film thickness in the conjunction.

Isothermal Elastohydrodynamic Lubrication of Point Contacts: Part III—Fully Flooded Results

Utilizing the theory developed by the authors in an earlier publication, the influence of the ellipticity parameter, the dimensionless speed, load, and material parameters on minimum film thickness was investigated. The ellipticity parameter was varied from one (a ball on a plate configuration) to eight (a configuration approaching a line contact). The dimensionless speed parameter was varied over a range of nearly two orders of magnitude. The dimensionless load parameter was varied over a range of one order of magnitude. Conditions corresponding to the use of solid materials of bronze, steel, and silicon nitride and lubricants of paraffinic and naphthenic mineral oils were considered in obtaining the exponent in the dimensionless material parameter. Thirty-four different cases were used in obtaining the minimum film thickness formula given below as H¯min=3.63U0.68G0.49W−0.073(1−e−0.68k) A simplified expression for the ellipticity parameter was found where k=1.03RyRx0.64 Contour plots were also shown which indicate in detail the pressure spike and two side lobes in which the minimum film thickness occurs. These theoretical solutions of film thickness have all the essential features of the previously reported experimental observations based upon optical interferometry.

Isothermal Elastohydrodynamic Lubrication of Point Contacts: Part II—Ellipticity Parameter Results

A numerical solution of the isothermal elastohydrodynamic problem for point contacts has been presented which reproduces all the essential features of the previously reported experimental observations based upon optical interferometry. In particular, the two “side lobes” in which minimum film thickness regions occur are shown to emerge in the theoretical solutions. The influence of the ellipticity parameter upon solutions to the point contact problem has been explored in the present paper. The ellipticity parameter (k) was varied from one (a ball on a plate) to eight (a configuration approaching line contact), and it has been shown that the minimum film thicknesses can be related to the well known line contact solutions by remarkably simple expressions involving either (k) or the effective radius of curvature ratio (Ry/Rx).