Principal Radii Of Curvature Research Articles

An analytical model for nanoscale flexoelectric doubly curved shells

The phenomenon of flexoelectricity, wherein the generation of electrical polarization results from strain gradients, has garnered significant attention in electromechanics. This phenomenon holds immense potential for diverse applications in nanoelectromechanical systems. Doubly curved shells present a particularly compelling structure for implementing flexoelectric devices, representing the most general scenario. To comprehensively understand the mechanical behavior of nanoscale flexoelectric doubly curved shells, we present a novel analytical model in this study. Our model incorporates a reformulated strain gradient elasticity theory, a modified expression of electric enthalpy density considering Maxwell’s self-field gradient, and a moderately thick shell configuration. To further elucidate this model, we establish complete formulations and analytical solutions of static bending and free vibration problems. This study reveals the crucial roles that various parameters such as principal curvature radii, length scale parameters, transverse structural sizes, and the Winkler–Pasternak foundation play in controlling the mechanical behavior of nanoscale flexoelectric doubly curved shells. These theoretical findings offer valuable insights into the design of nanodevices based on the direct flexoelectric effect.

A Dynamic Analysis of Randomly Oriented Functionally Graded Carbon Nanotubes/Fiber-Reinforced Composite Laminated Shells with Different Geometries

The present study demonstrates the free vibration behavior of composite laminated shells reinforced by both randomly oriented single-walled carbon nanotubes (SWCNTs) and functionally graded fibers. The shell structures with different principal radii of curvature are considered, such as cylindrical, spherical, elliptical–paraboloid shell, hyperbolic–paraboloid shell, and plate. The volume fraction of the fibers has a linear variation along the shell thickness from layer to layer, while the volume fraction of CNTs is constant in all shell layers and uniformly distributed. The fiber-reinforced elements are distributed with three functions which are V-distribution, O-distribution, and X-distribution in addition to the uniform distribution. A numerical analysis was carried out systematically to validate the proposed solution. A new analytical solution is presented based on the Galerkin approach for shells and is exploited to illustrate the influence of some factors on the free vibration behavior of CNTs/fibe-reinforced composite (CNTs/F-RC) laminated shells, including the distributions and volume fractions, various boundary conditions, and geometrical properties of the reinforcement materials. The proposed solution is shown to be an effective theoretical tool to analyze the free vibration response of shells.

An Automatic and Forward Method to Establish 3-D Parametric Scattering Center Models of Complex Targets for Target Recognition

In this article, an automatic and forward method is realized to establish attributed scattering center models directly from the computer-aided design (CAD) model of the complex target. The main steps include the preprocessing of the CAD model, the separation of scattering sources, the selection of strong scattering sources, and the automatic determination of model parameters. With the proposed method, the scattering sources, scattering mechanisms, and model parameters of the scattering centers can be identified and derived, such that the complicated manual intervention is completely avoided. Moreover, the method establishes the distributed scattering center models formed by curved surfaces with large curvature radii. Therefore, the formation mechanism of the distributed scattering center is extended from typical scattering structures to a general case. Thus, the model of the attributed scattering center is extended and can be applied to describe the scattering from the real structures of the complex target. The geometric shape of the scattering source is distinguished based on the principal curvature radii, which are calculated by differential geometry theory. Thus, the frequency dependence parameter is obtained according to its corresponding relationship with the geometric shape. In addition, based on the automatic method, a technology is studied to diagnose and correct the scattering center models of a target whose CAD model is unknown or partially known. Finally, parametric models of several targets in the Moving and Stationary Target Acquisition Recognition (MSTAR) program are established, and then compared with the measured data. The results validate the effectiveness of the proposed method.

Rotational surfaces of constant astigmatism in space forms

A surface in a Riemannian space is called surface of constant astigmatism if the difference between the principal radii of curvature at each point is a constant function. In this paper we give a classification of all rotational surfaces of constant astigmatism in space forms. We also prove that generating curves of such surfaces are critical points of a variational problem for a curvature energy. Using the description of these curves, we locally construct all rotational surfaces of constant astigmatism as the associated binormal evolution surfaces from the generating curves.

Nonlinear dynamics of doubly curved shallow microshells

The nonlinear dynamical characteristics of a doubly curved shallow microshell are investigated thoroughly. A consistent nonlinear model for the microshell is developed on the basis of the modified couple stress theory (MCST) in an orthogonal curvilinear coordinate system. In particular, based on Donnell’s nonlinear theory, the expressions for the strain and the symmetric rotation gradient tensors are obtained in the framework of MCST, which are then used to derive the potential energy of the microshell. The analytical geometrically nonlinear equations of motion of the doubly microshell are obtained for in-plane displacements as well as the out-of-plane one. These equations of partial differential type are reduced to a large set of ordinary differential equations making use of a two-dimensional Galerkin scheme. Extensive numerical simulations are conducted to obtain the nonlinear resonant response of the system for various principal radii of curvature and to examine the effect of modal interactions and the length-scale parameter.

Applicability of half-space-based methods to non-conforming elastic normal contact problems

The half-space assumption has been employed in many solution methods for non-conforming contact problems in elasticity such as the Hertz theory and the Kalker's variational theory. It is generally believed that to guarantee acceptable accuracy in these half-space-based methods, the characteristic size (twice length of one semi-axis) of the contact patch should be much smaller than the significant dimensions (i.e. the height, width, length and the principal radii of curvature) of each body in contact. In engineering practice, the 3x rule is often employed, which requires that the significant dimensions be at least three times as large as the characteristic size. However, this requirement has not been justified. In this paper, the applicability of half-space-based methods is investigated by comparing the solutions obtained using the Hertz theory and the Kalker's theory with those of the Finite Element (FE) method which is not limited to the half-space assumption. Different combinations of significant dimensions in terms of height, width and length are studied. Various contact patch eccentricities and contact body shapes are considered. It is found that the half-space-based methods yield high-accuracy calculation for non-conforming contact problems. Even when the significant dimensions are as small as 1.1x the characteristic size, the differences between the solutions of the half-space-based methods and the FE method are within 9%. The findings of this paper indicate that the typically assumed 3x restriction can be greatly relaxed. Since a clear estimation of the deviation of the results of half-space-based methods from those of the FE method is provided, the applicability of half-space-based methods in mechanical engineering can be much better understood.

Improvements, testing and development of the ADM-τ sub-grid surface tension model for two-phase LES

In this paper, a specific subgrid term occurring in Large Eddy Simulation (LES) of two-phase flows is investigated. This and other subgrid terms are presented, we subsequently elaborate on the existing models for those and re-formulate the ADM-τ model for sub-grid surface tension previously published by these authors. This paper presents a substantial, conceptual simplification over the original model version, accompanied by a decrease in its computational cost. At the same time, it addresses the issues the original model version faced, e.g. introduces non-isotropic applicability criteria based on resolved interface's principal curvature radii. Additionally, this paper introduces more throughout testing of the ADM-τ, in both simple and complex flows.

Contact Kinematics in the Roller Screw Mechanism

This paper investigates the nature of the contact between the load transferring surfaces in the roller screw mechanism, i.e., between the screw and roller threads and between the nut and roller threads. The analysis is applied to both planetary roller screws and recirculating roller screws. Prior work has neglected to take a fundamental approach toward understanding the kinematics of the contact between these components and, as a consequence, detailed analyses of aspects such as contact mechanics, friction, lubrication, and wear are not correctly carried out. Accordingly, in this paper, the principle of conjugate surfaces is used to establish contact at the screw/roller and nut/roller interfaces. The in-plane angles to the contact points are derived and it is shown that for the screw/roller interface, the contact point cannot lie on the bodies' line of centers, as has been the assumption in previous papers. Then, based on the curved profile of the roller thread, the radii of contact on the roller, screw, and nut bodies are also derived. Knowledge of the contact point locations is necessary in order to understand the interaction forces between the key components of the roller screw mechanism. The principal radii of curvature at the contact points and the angle between the principal axes are also derived. Lastly, a brief example is developed showing how the developed theory may be used to design a roller screw for improved stiffness and decreased contact stresses.

Generation of microstripe cylindrical and toroidal mirrors by localized laser evaporation of fused silica

We report a new technique for the rapid fabrication of microstripe cylindrical and toroidal mirrors with a high ratio (>10) of the two principal radii of curvature (RoC(1)/RoC(2)), and demonstrate their effectiveness as mode-selecting resonator mirrors for high-power planar waveguide lasers. In this process, the larger radius of curvature (RoC(1)) is determined by the planar or cylindrical shape of the fused silica substrate selected for laser processing, whilst the other (RoC(2)) is produced by controlled CO(2) laser-induced vaporization of the glass. The narrow stripe mirror aperture is achieved by applying a set of partially overlapped laser scans, with the incident laser power, the number of laser scans, and their spacing being used to control the curvature produced by laser evaporation. In this work, a 1 mm diameter laser spot is used to produce grooves of cylindrical/toroidal shape with 240 μm width and 16 mm length. After high reflectance coating, these grooves are found to provide excellent mode selectivity as resonator mirrors for a 150 μm core Yb:YAG planar waveguide laser, producing high brightness output at more than 300 W. The results show clearly that the laser-generated microstripe mirrors can improve the optical performance of high-power planar waveguide lasers when applied in a low-loss mode-selective resonator configuration.

Contact Stress Analysis of Multilayered Strands

The calculation method of contact force in contact-zone between adjacent layer wires has been analyzed. The principal radii of curvatures of wires were taken into consideration while obtaining the analytical expressions for contact stesses and sizes of contact surface. Meanwhile, a formular for shear stress of arbitrary point in half-space under contact-zone was derived on basis of the Boussinesq problem and it was simplified by using Gaussian quadrature. According to the results, the stress distribution could be unsderstood more thoroughly and the results is of great importance for studying looseness, fatigue and fretting wear of multilayered strands.

Designing Fresnel microlenses for focusing astigmatic multi-Gaussian beams by using fractional order Fourier transforms

According to a scalar theory of diffraction, light propagation can be expressed by two-dimensional fractional order Fourier transforms. Since the fractional Fourier transform of a chirp function is a Dirac distribution, focusing a light beam is optically achieved by using a diffractive screen whose transmission function is a two-dimensional chirp function. This property is applied to designing Fresnel microlenses, and the orders of the involved Fourier fractional transforms depend on diffraction distances as well as on emitter and receiver radii of curvature. If the emitter is astigmatic (with two principal radii of curvature), the diffraction phenomenon involves two one-dimensional fractional Fourier transforms whose orders are different. This degree of freedom allows us to design microlenses that can focus astigmatic Gaussian beams, as produced by a line-shaped laser diode source.

Forces and deformations of the abdominal wall—A mechanical and geometrical approach to the linea alba

Force-elongation responses of the human abdominal wall in the linea alba region were determined by tensile tests in which the linea alba was seen to exhibit a nonlinear elastic, anisotropic behavior as is frequently observed in soft biological tissues. In addition, the geometry of the abdominal wall was determined, based on MRI data. The geometry can be specified by principal radii of curvature in longitudinal of approximately 470mm and in the transverse direction of about 200mm. The determined radii agree with values found in other studies. Mechanical stresses, deformations and abdominal pressures for load cases above 6% elongation can be related using Laplace's formula and our constitutive and geometrical findings. Results from uni- and biaxial tensile tests can thus be compared using this model. Calculations confirm that abdominal pressures of approximately 20kPa correspond to related biaxial forces of about 3.4N/mm in the transverse and 1.5N/mm in the longitudinal direction. Young's moduli can be calculated with respect to the uniaxial as well as the biaxial loading. At these physiological loadings, a compliance ratio of about 2:1 between the longitudinal and transversal directions is found. Young's moduli of about 50kPa occur in transversal direction and of about 20kPa in longitudinal direction at transverse and longitudinal strains both in the order of 6%. These findings coincide with results from other investigations in which the properties of the abdominal wall have been examined.

Series solutions of unsteady free convection flow in the stagnation-point region of a three-dimensional body

The unsteady three-dimensional free convection flow in the stagnation-point region on a general curved isothermal surface placed in an ambient fluid is studied. By introducing new similarity transformations, the momentum and energy balance equations are reduced to a set of three fully coupled nonlinear partial differential equations. These equations are solved analytically for some various values of the ratio of the two principal radii of curvature. The accurate series solutions are obtained which are uniformly valid for all dimensionless time in the whole spatial region 0 ⩽ η < ∞ .

Underwater acoustic imaging: image due to a specular reflector in the geometrical-acoustics limit

In underwater acoustic imaging, which is used to produce high-quality images in turbid waters, a specular reflector can produce a “pseudoimage” of the receiving array at the reflecting surface. Based on the “geometrical approximation” (which is similar to geometrical acoustics), formulae are derived for the size and shape of the pseudoimage for both flat and curved reflectors. For curved reflectors, described by two principal radii of curvature, the formulae also assume the “large-range approximation.” The formulae allow radii of curvature to be determined from an image. Also discussed briefly are some possible extensions and the role of nongeometrical effects.

Automatic Polishing of Super Accuracy Mirror Mold with Free-Form Surface by Curvature Analysis

This paper proposed a polishing path planning method of super accuracy mirror mold with free-form surface by curvature analysis. First, IGES files of free-form surfaces are read and the mold geometry is regenerated as B-spline surface by the Automatic Mold Polishing System (AMPS). By using the derivative properties of B-spline surface, normal vector and principal curvatures at any point of the surface are calculated. In addition, the effective contact width between polishing tool and mold surface based on the grain size and the principal radii of curvature is also determined. The minimum contact width in 3-D is mapped onto the (u, v) parameters of B-spline surface. Then a modified Peano fractal path with weaving function is calculated based on the effective contact width in the (u, v) coordinate. This Peano-weaving path was tested on an optical mold with free-form surface. The polishing result shows the method is very effective and achieves the level of mirror surface with roughness Ra 29nm.

Theoretical aerothermal concepts for configuration design of hypersonic vehicles

Convection coefficients and heat fluxes due to aerodynamic heating on critical surfaces of hypersonic vehicle are obtained analytically. The applicability of recovery temperature for stagnation regions is discussed. Convection coefficient for the bicurvature forward stagnation region is obtained directly from 2-D stagnation region correlation, using the two principal radii of curvatures. Convective heat flux to swept-back leading edge (SBLE) surface is obtained from the 2-D stagnation region and flat plate heat fluxes, using the respective velocity vector components. Results reveal the concepts of temperature-minimised-sweepback, and the thermally-benign sharp SBLE effect at high sweepback angles.

On the coalescence speed of bubbles

Two bubbles brought into contact will coalesce rapidly. We study experimentally the initial coalescence motions in the neck region connecting the two bubbles, using an ultrahigh-speed video camera. The air bubbles are grown in ethyl alcohol, at the ends of two opposing vertical needles, until they come into contact. The growth rate of the neck is measured and modeled with a capillary/inertial balance. The instantaneous profile in the neck forms a circular arc, with the ratio of the two principal radii of curvatures increasing linearly in time.

Effect of alkalis, phosphorus, and water on the surface tension of haplogranite melt

The sessile drop method has been used for measurements of the surface tension of haplogranite (HPG) melts containing an excess of alkalis and phosphorous (HPG8, HPG8 + 5 wt% Li 2 O, 5 wt% Na 2 O, 20 wt% Na 2 O, 5 wt% K 2 O, 5 wt% Rb 2 O, 5 wt% Cs 2 O, 10 wt% P 2 O 5 ) and of Armenian rhyolite in the temperature interval, 650–1665 °C, and at 1 bar pressure. Sessile drops were placed on graphite substrates in a Pyrox tube furnace purged with Ar. Drop shape was monitored with a videocamera and stored in a videorecorder. The surface tension was calculated by measuring the two principal radii of curvature of the drop shape in vertical cross section. The precision of the method was checked against the surface tension of water. The surface tension of HPG and rhyolite melt is ~280–300 ± 5 mN/m in the temperature interval 1200–1400 °C. Temperature dependence of the surface tension of haplogranite melts and rhyolite is weak and positive (dσ/d T = 0.06 to 0.09 mN/m/°C). Addition of 5 wt% of alkali oxides (except Li 2 O) results in a decrease of the surface tension of haplogranite melts. The HPG melts with 10 wt% P 2 O 5 have 30% higher surface tension than haplogranite melts with excess alkalis, and a negative temperature derivative (dσ/d T = −0.1 mN/m/°C). The HPG melts with 20 wt% Na 2 O and 5 wt% Li 2 O exhibit a decrease in surface tension with temperature (dσ/d T = −0.02 and −0.10 mN/m/°C, respectively). The surface tension of HPG8 melt saturated with water at 1–4 kbar was measured on sessile drops quenched at high pressure in an internally heated gas vessel at temperatures of 800–1200 °C. Water pressure significantly decreases the surface tension of melt from 270 mN/m at 1 bar (1000 °C) to 65 mN/m at 4 kbar. At 1 bar in “dry” conditions, dσ/d T = +0.056 mN/m/°C and at 3 kbar of water pressure, dσ/d T = +0.075 mN/m/°C. The decrease in the surface tension of HPG melt at a water pressure of several kbars is from −10 to −30 mN/m/wt% H 2 O. The increase of water content to more than 10 wt% in granite melts may not result in any significant decrease in the surface tension, which may be explained by formation of a surface sublayer having physical properties very distinct from those of the bulk.

Computation of 3D curvatures on a wet snow sample

The map of 3D curvatures of a porous medium characterizes most of its capillary properties. A model for directly computing curvatures from a three-dimensional image of the solid matrix of a porous medium is presented. A precise distance map of the object is built using the “chamfer” distance of discrete geometry. The set of local maxima of the distance map is used for quick location of the normal to each point P of the object's surface. The normal being known, principal radii of curvature are computed in 2D and lead to 3D curvature. This model was validated on geometric shapes of known curvature, then applied on a natural snow sample. The snow image was obtained from a serial cut (performed in cold laboratory) observed under specularly reflected light. Views of both fresh and sublimated sections were taken for each of the 64 section planes: this allowed easier distinction between snow and filling medium and made possible automatic contouring of section plane images. Curvature maps computed from pore and grain phases respectively were found to be in excellent agreement for each tested object shape, including the snow sample.

The Effect of Motion Curves on the Curvatures of VPLS Transmission Mechanisms with Ruled- and Involute-Revolution Surface Meshing Elements

AbstractThe purpose of this research is to study the effect of motion curves on the curvature of variable pitch lead screw (hereinafter VPLS) transmission mechanisms with different meshing elements. The surfaces of VPLS are generated based on various meshing elements with different specified motion curves. There are three kinds of ruled- and one involute-revolution surface meshing elements, and four types of motion curves are studied, which including modified trapezoidal (MT) curve, modified sine (MS) curve, modified constant velocity (MCV) curve, and polynomial (PL) curve. The curvature analyses include the determination of direction of contact line, induced principal radii of curvature, and lubrication angles between the surfaces of VPLS and various meshing elements. The results of this work provide a comparison between the effect of motion curve on the curvature of VPLS transmission mechanisms with different meshing elements.