Curvature Of Plate Research Articles

Curvature changes during thermal cycling of a compositionally graded NiAl 2O 3 multi-layered material

The elastoplastic deformation characteristics of a plasma-sprayed, tri-layered composite plate subjected to thermal cycling from 20°C up to 800°C were studied experimentally and numerically. The tri-layered solid comprised polycrystalline Ni and Al 2O 3 outer layers and a 2.2 mm thick compositionally graded NiAl 2O 3 composite interlayer (FGM) wherein the composition varied approximately linearly along the layer thickness. The experiments involved in situ and ex situ measurements, employing a scanning laser technique, of the changes in the overall curvature of the unconstrained plate arising from the thermal mismatch between the constituent phases. The variations of curvature, accumulated plastic strains and thermal stresses at different locations in the layered solid were also assessed numerically with the aid of available continuum formulations, and the numerical predictions were compared with experiments, wherever appropriate. It is shown that when only small plastic strains exist in the Ni layer or in the Ni-rich end of the FGM layer, known formulations are capable of providing approximate predictions of the cyclic variations in curvature, the onset of plasticity and some features of the initiation of cracking (and its location). We also examine possible sources of error in the experimental measurements of curvature and in the interpretation of thermally induced deformation due to some processing conditions.

Experimental study of the effect of surface defects on the finesse and contrast of a Fabry–Perot interferometer

While investigating the instrumental function of a Fabry-Perot interferometer [Appl. Opt. 34, 58 (1995), we noticed some variation in finesse and contrast in the measured spectra when a 1.5-mm-diameter aperture was used at various spots within the standard 8-mm aperture. By comparing experimentally determined finesse versus contrast plots for many such spectra with calculated plots, we found spots on the plates that gave non-Airy-function line shapes over the entire order of interference, unlike the Airy line shape we determined previously by using the entire 8-mm aperture. We have reviewed several models that describe the effects of various types of surface defects, such as Gaussian-height distribution of roughness, curvature and tilt of plates, sinusoidal roughness, and asymmetrical roughness on the finesse and contrast. Our experimental results can be accounted for if we assume that the reflectivity is nonuniform over the Fabry-Perot plates and that there is some reasonable contribution that is due to Gaussian roughness, curvature, or tilt.

Isostasy, equivalent elastic thickness, and inelastic rheology of continents and oceans

The equivalent elastic thickness (EET) is used to estimate lithospheric strength expressed in response to loading by topography and subsurface loads. The data on EET allow comparisons between different plates and detection of thermal events. In oceans, the EET corresponds to the mechanical “core” of the lithosphere, i.e., a geotherm (400–600 °C). In continents, the EET has no relation to any depth. This has led to doubts in applicability of a unique approach to the continents and oceans, and in the utility of estimates of EET for continents. Rheological data suggest that most rocks are inelastic in the long term (>0.1 m.y.). This requires interpretation of the EET in terms of real rheology. We propose an analytical model that gives rheological interpretation of both the oceanic and continental EET. It also allows estimates of the mechanical thickness of the lithosphere. The EET depends upon three parameters: geotherm age, crustal thickness, and flexural plate curvature. Any one of these values can be estimated if the others are known. Comparisons of model predictions with the observed EET suggest that most continental plates have a weak lower crust, allowing mechanical decoupling between the upper crust and the mantle lithosphere. Such decoupling leads to strong reduction in the EET and thus can be easily detected. Flow of rocks in the weak lower crust may have a significant influence on the temporal evolution of relief (mountain building, erosion). Differences in the mechanical behavior of oceans and continents can be explained by domination of different parameters: geotherm age has a major control in the oceans, whereas in the continents crustal thickness is equally important. Additional local variations of the EET result from weakening by flexural stresses.

Parameter estimation in flexural plate motion; elastic, plastic-elastic and thermo-elastic models applied to the Chukchi Basin, Alaska

Models are presented of flexural plate basement motion (elastic, plastic-elastic and thermo-elastic) due to: (1) compressive or extensional stress; (2) variation of lithospheric basement density along the flexural plate; (3) spatial variations of plate rigidity; and (4) variations of initial plate angle, hinge point position, plate length, initial end load and plate curvature. A non-linear inverse method determines parameters. Synthetic test results and a case history from the Central Chukchi Basin illustrate the accuracy and resolution of the inversion procedure. The Central Chukchi Basin has an unloaded geometry mainly controlled by a Palaeozoic fault system. Plastic or thermal effects produce a narrow unloaded basin geometry not conforming to the Paleozoic fault distribution or to the isopach map of the Upper Ellesmerian sediments. Inverse elastic model results agree with both the known fault system development (from seismic data) and the direction of the Brooks Range motion.

Nonlinear flexure and equivalent mechanical thickness of the lithosphere

The elastic flexural thickness of the lithosphere depends on plate curvature. As curvature increases, the elastic core of the plate is thinned from above (frictional yielding) and from below (plastic creep). Sometimes decoupling between crust and mantle elastic cores can also occur. The mechanical thickness of the lithosphere, on the other hand, is the thickness of the rheologically layered plate having a strength above a critical threshold. In order to estimate the mechanical thickness from a simple uniform-rheology plate model, we adopt a plastic work-hardening constitutive equation, which results in lower stresses in the regions of high strain, relative to the elastic model. It is, therefore, a better approximation to the actual rheology of the flexed lithosphere where there is no lower-crustal decoupling (e.g., in oceanic and in cold continental lithosphere). The equivalent mechanical thickness of the nonlinear plate can be directly obtained, if the curvature is known, from the estimated elastic flexural rigidity or thickness. Comparison with numerical integration of bending moment in rheologically layered lithosphere shows that equivalent mechanical thickness is a good estimation of mechanical thickness. Examination of both oceanic and continental data suggests that mechanical thickness is controlled by the 900 ± 100° C isotherm. This corresponds to a creep strength of the order of 10 MPa in upper mantle material.

Geometrically Nonlinear Stress-Deflection Relations for Thin Film/Substrate Systems With a Finite Element Comparison

A new higher order geometrically nonlinear relation is developed to relate the deflection of a thin film /substrate system to the intrinsic film stress when these deflections are larger than the thickness of the substrate. Using the Rayleigh-Ritz method, these nonlinear relations are developed by approximating the out-of-plane deflections by a second-order polynomial and midplane normal strains by sixthorder polynomials. Several plate deflection configurations arise in an isotropic system: at very low intrinsic film stresses, a single, stable, spherical plate configuration is predicted; as the intrinsic film stress increases, the solution bifurcates into one unstable spherical shape and two stable ellipsoidal shapes; in the limit as the intrinsic film stress approaches infinity, the ellipsoidal configurations develop into cylindrical plate curvatures about either one of the two axes. Curvatures predicted by this new relation are significantly more accurate than previous theories when compared to curvatures calculated from three-dimensional nonlinear finite element deflection results. Furthermore, the finite element results display significant transverse stresses in a small boundary region near the free edge.

Theoretical and experimental analysis of plates asymmetric rolling

A bending of a front end of a plate in a process of rolling in a reversing finishing mill has been analyzed in the work. An asymmetric process of rolling has been investigated by application of finite element method. An effect of a plate entry angle and other process parameters for curvature of a rolled plate has been described. Results of investigations of the asymmetric process of a plate rolling in a quarto stand of a rolling plant of Czestochowa Steel Work are presented. An implementation of the asymmetric proces in industrial conditions has alowed for getting the higher quality of the final product, the output increase and high economical effects.

Stigmatic Imaging by Spherical Fresnel Zone Plates

Abstract We present a succinct and convenient formulation for the zone radii of a spherical Fresnel zone plate that produces the stigmatic image of an axial object at a specific finite/infinite distance from the zone plate. By ‘image’ we imply the first order or primary image produced by the zone plate. The variation in zone radii for different conjugate positions and for different values of curvature of the spherical zone plate is demonstrated with the help of numerical results. In order to bring forth the significance of this variation for Fresnel zone plates operating at different regions of the electromagnetic spectrum, the variation is shown for one representative wavelength at each of the four regions of the electromagnetic spectrum, namely, the infrared, the visible, the extreme ultraviolet and the soft X-rays.

超砥粒砥石による薄板の研削変形(第3報)

It is important to know the temperature distribution in a workpiece to understand the curvature of thin plate in surface grinding. In this paper, the relationship between curvature and calculated temperature distribution by FEM was clarified. As a result, it is evident that the steeper temperature gradient the larger concave curvature is induced. This can be understood as an effect of thermal stress. In the case of gentle temperature gradient, the effect of thermal stress is not clear, and consequently burnishing by cutting edge becomes effective. Summarizing these results, the curvature is well explained with temperature gradient and burnishing effect. The transformation of 304 stainless steel is confirmed by means of X-ray diffraction test, and the influence of transformation on curvature is studied. Key words: surface grinding;superabrasive wheels;thin plate;curvature, temperature distribution;force ratio;transformation.

Determination of Optimum Forming Path in Three-Roll Bending by Combination of Fuzzy Reasoning and Finite Element Simulation

Summary A method for determining the optimum forming paths in the roll-bending process of plates by combining the finite element simulation with the fuzzy reasoning is proposed. In this method, the forming conditions in the finite element simulation are automatically changed to attain the desired shape of the plate with the help of the fuzzy reasoning by monitoring the calculated results such as the curvature of the plate, the stress and strain distributions, etc. in each deformation step. To form the front and the rear ends of a plate into a proper curvature by three-roll bending, it is essential to give an appropriate path of the top roll. For this purpose, the optimum motion of the top roll is successively searched for in the finite element simulation. The fuzzy reasoning is used to determine the motion of the top roll by taking account of the effects of the plate thickness, the flow stress and the difference between the desired and the calculated curvatures. It is shown that the decrease in the curvature near both ends of the bent plate is prevented by the use of the present method.

Measurement of curvature by moir� shearing interferometry and mirror surface translation

In this paper, the curvature of the deformed object plate is measured by means of moire shearing interferometry and mirror surface translation. The curvature of a clamped steel plate with a concentrated force in its center is measured.

Flexural ridges, trenches, and outer rises around coronae on Venus

High‐resolution altimetry collected by the Magellan spacecraft reveals trench and outer rise topographic signatures around major coronae (e.g. Eithinoha, Heng‐O, Artemis, and Latona). In addition, Magellan synthetic aperature radar images show circumferential fractures in areas where the plates are curved downward. Both observations suggest that the lithosphere around coronae is flexed downward by the weight of the overriding coronal rim or by the negative buoyancy of subducted lithosphere. We have modelled the trench and outer rise topography as a thin elastic plate subjected to a line load and bending moment beneath the corona rim. The approach was tested at northern Freyja Montes where the best fit elastic thickness is 18 km, in agreement with previously published results. The elastic thicknesses determined by modelling numerous profiles at Eithinoha, Heng‐O, Artemis, and Latona are 15, 40, 37, and 35 km, respectively. At Eithinoha, Artemis, and Latona where the plates appear to be yielding, the maximum bending moments and elastic thicknesses are similar to those found at the Middle America, Mariana, and Aleutian trenches on Earth, respectively. Estimates of effective elastic thickness and plate curvature are used with a yield strength envelope model of the lithosphere to estimate lithospheric temperature gradients. At Heng‐O, Artemis, and Latona, temperature gradients are less than 10 K/km, which correspond to conductive heat losses of less than one half the expected average planetary value. We propose two scenarios for the creation of the ridge, trench, and outer rise topography: differential thermal subsidence and lithospheric subduction. The topography of Heng‐O is well matched by the differential thermal subsidence model. However, at Artemis and Latona the amplitudes of the trench and outer rise signatures are a factor of 5 too large to be explained by thermal subsidence alone. In these cases we favor the lithospheric subduction model wherein the lithosphere outboard of the corona perimeter subducts (rolls back) and the corona diameter increases.

Basin formation behind an active subduction zone: three‐dimensional flexural modelling of Wanganui Basin, New Zealand

AbstractSouth Wanganui Basin is a Plio‐Pleistocene basin formed behind the Hikurangi subduction zone of the North Island, New Zealand. The active Benioffzone lies about 30–40 km beneath the centre of Wanganui Basin. We attribute the development of this basin primarily to frictional shear between the subducted Pacific plate and overriding Australian plate. This frictional interaction harnesses part of the ‘slab pull’ force of plate tectonics to deform the overriding plate and create the basin. The overall shape of the basin is modified to a lesser extent by the effects of uplift around the basin margin and by in‐plane compressional stress. A quantitative three‐dimensional model, based on flexure of a thin elastic sheet with spatially varying rigidity, is used to test the implications of our hypothesis. A maximum vertical loading stress of about 85 MPa is required to produce the vertical deformation in Wanganui Basin, and that in turn implies about 170 MPa of shear stress at the inclined subduction thrust. A vertical stress of 85 MPa applied over a 40‐km‐wide subduction thrust represents about 50% of the magnitude theoretically present in the ‘slab pull’ force. The effective elastic thickness (Te) of the flexed plate that provides the best fit to the observed deformation is 10 ± 5 km; such a low value of 7“e is attributed to the high plate curvature and high strain rates associated with Wanganui Basin. Constraints on our analysis are provided by seismic reflection, seismicity, gravity, geodetic and geological data. A similar situation to Wanganui Basin, in which a high‐friction subduction thrust induces shortening and flexure of the overlying crust, may be found in the Puget Sound area of Washington State, USA.

Curvature patterns of bending plates by speckle-shearing interferometry

This paper describes a method to obtain the curvature and twist patterns of a bending plate by combining optical carrier technique and digital image processing. A carrier consisting of parallel fringes is obtained in double-exposure by changing the illumination form so that the deflection of the plate is recorded as the modulation to the carrier frequency. After optical filtering the modulated carrier is input into a digital image processing system to generate a grey image of ‘deformed grating’, whose digital differentiation can be used to obtain the curvature patterns or the twist patterns.

A Study on Juncture Flows

A computational study of the flow around the strut-plate juncture formed by the inclined strut and curved plate is presented. The characteristics of the flow are discussed by making use of the solution of the Navier-Stokes equations. To investigate the effects of the inclination angle of the strut and the curvature of the plate on the juncture flows, the computation has been made for three different inclination angles of strut and two different curvature of plate. The flow structure is changed greatly due to the inclination of strut and the forces acting on strut are affected greatly by the inclination angle. However the effects of the curvature of plate on the flow and the forces acting on strut are less than thoes of inclined cases. Although the plate is curved, the local angle is still 90° and it can be concluded that the local angle between strut and plate will be an important parameter for the flow structure around a juncture.

The relationship between plate curvature and elastic plate thickness: A study of the Peru‐Chile Trench

The age of the Nazca plate where it enters the Peru and northern Chile trenches varies from 30 Ma in the north to 45 Ma in the south as its dip beneath the South American continent steepens from 13° to 30°. If the elastic thickness Te of oceanic lithosphere depends only on its age, and therefore thermal state, we would expect that Te determined from fitting the flexure of the lithosphere over the outer rise as revealed in the depth, geoid, and gravity anomalies would increase from the Peru Trench in the north to the northern Chile Trench further south. We find that the opposite is true: the lithosphere appears suffer outboard of the Peru Trench than it does offshore Chile. To explain deflections of the lithosphere seaward of Peru, the isotherm controlling the elastic/ductile transition must be near 800°C, providing the thermal structure of the plate is that predicted by the standard thermal plate model. Because the decrease in plate stiffness to the south is correlated with an increase of plate curvature over the outer rise and outer trench wall, we interpret our result in terms of inelastic yielding of the oceanic lithosphere when bent to high strains. The magnitude of the reduction in strength, however, suggests that the standard oceanic yield strength envelope underestimates the amount of failure for lithosphere bent to high strains unless the stress field off northern Chile is characterized by 100‐MPa‐level tension. The more highly bent segment of subducting lithosphere at this trench also dips at a steeper angle at greater depth beneath the continent, but detailed analysis of plate geometry does not support a relationship between slab dip as determined by earthquake hypocenters and elastic behavior over the outer rise.

放電加工における応力分布に関する研究 （第６報） 焼戻し処理による薄板加工面曲率変化と表面組織

When a thin plate is machined by EDM on one side, a certain curvature results owing to tensile residual stress generated on the surface of the workpiece. If the machined plate is tempered, some change in curvature of the thin plate appears to occur due to reduction in residual stress and changes in the degenerated layers.This study was conducted to examine changes in curvature of a machined thin plate under various tempering conditions, in relation to those in the properties (phase changes) of the machined surface, using an X-ray diffractometerReduction in residual stress was concluded to occur due to stress relaxation, phase changes (transformation of γ Fe to α Fe) and the precipitation of carbides (Fe3C, M7C3). Change in curvature of thin plate was essentially the same as that noted by the X-ray stress measuring method. As a result, there occurs considerable plastic deformations (slip, twinning and lattice defects) of the EDM surface, owing to thermal effects.

Practical Application of a Geometrically Nonlinear Stress-Curvature Relation

ABSTRACTA recently developed geometrically nonlinear stress-curvature relation based on a minimization of the total strain energy, which predicts a bifurcation in shape as the magnitude of intrinsic film stress increases, is discussed in this paper. It is compared with the linear theories of Stoney and Brenner & Senderoff for a thin molybdenum film on silicon substrates with various thicknesses. Although the ratio of film to substrate elastic modulus is only 2, Stoney's equation generates significant error for this film/substrate system and the Brenner & Senderoff relation should be used for calculating initial film stress when plate deflections are small. When deflections exceed approximately half the substrate thickness the Brenner & Senderoff equation produces over 10% error and consequently, the nonlinear stress-deflection relation should be used to relate plate curvatures to initial film stress.

A New Method for Determination of Bending Rigidities of Thin Anisotropic Plates

The paper presents a new method to determine the six elastic bending rigidities of thin anisotropic plates. The experimental procedure only requires one plate specimen submitted to several bending tests performed on the same testing device. The data processing is based on the principle of virtual work and it is shown that simple linear relationships between the bending rigidities, applied loads, and plate curvature can be derived using particular virtual displacement fields. The unknown plate rigidities are the solution of a set of linear equations. The numerical simulations of various relevant bending configurations of anisotropic laminated plates point out the efficiency of this new approach.

Stability of a Laminar Boundary Layer Flowing Along a Concave Surface

Go¨rtler instability for incompressible laminar boundary-layer flows over constant curvature concave surfaces is considered. The full linearized disturbance equations are solved by the Galerkin method using Chebyshev polynomials to represent the disturbance functions. Stability curves relating Go¨rtler number, wave number, and vortex amplification for a Blasius mean flow are presented. The effect of streamwise pressure variation is investigated using the Falkner–Skan boundary-layer solutions for the mean flow. The importance of including the normal velocity terms for these flows is shown by their effect on the stability curves. The streamwise velocity distribution in the boundary layer on a 3-m radius of curvature plate was investigated experimentally. The results are compared with the stability curves and predicted disturbance functions.