应用数学和力学

2011 Vol. 32, No. 5

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Shape Gradient and Classical Gradient of Curvatures: Driving Forces on Micro/Nano Curved Surfaces

YIN Ya-jun, CHEN Chao, LV Cun-jing, ZHENG Quan-shui

2011, 32(5): 509-521. doi: 10.3879/j.issn.1000-0887.2011.05.001

Abstract(2056) PDF(1507)

Abstract:
Recent experiment and molecule dynamics simulation showed that adhesion droplet on conical surface could move spontaneously and directionally. Besides, this spontaneous and directional motion was independent of the hydrophilicity and hydrophobicity of the conical surface. Aimed at this important phenomenon, a general theoretical explanation was provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard-curved-surface were st udied, and the geometric foundation for the interactions between the particle and the hard-curved-surface were analyzed. The following results are derived: (a) The potential of the particle/hard-curved-surface is of the unified curvature-form (i. e. the potential is always a unified function of the mean curvature and Gauss curvature of the curved surface); (b) On the basis of the curvature-based potential, the geometrization of the micro/nano mechanics on hard-curved-surfaces can be realized; (c) Curvatures and the intrinsic gradients of curvatures form the driving forces on curved spaces; (d) The direction of the driving force is independent of the hydrophilicity and hydroph obicity of the curved surface, which explains the experimental phenomenon of spontaneous and directional motion.

Two-Dimensional Polynomial Eigenstrain Formulation of Boundary Integral Equation With Numerical Verification

MA Hang, GUO Zhao, QIN Qing-hua

2011, 32(5): 522-532. doi: 10.3879/j.issn.1000-0887.2011.05.002

Abstract(1649) PDF(867)

Abstract:
he low-order polynomial distributed eigenstrain formulation of boundary integral equation (BIE) and the corresponding definition of Eshelby tensors were proposed for elliptical-shaped inhomogeneities in a two-dimensional elastic medium. Taking the results from traditional sub-domain boundary element method (BEM) as the control, effectiveness of the present algorithm was verified for an elastic medium with a single elliptical inhomogeneity. It is shown that, with the present computational model and algorithm, significant improvements are achieved in terms of efficiency as compared with the traditional BEM and in terms of accuracy as compared with the constant eigenstrain formulation of the BIE.

General Solutions of Plane Problem for Power Function Curved Cracks

GUO Jun-hong, YUAN Ze-shuai, LU Zi-xing

2011, 32(5): 533-540. doi: 10.3879/j.issn.1000-0887.2011.05.003

Abstract(2043) PDF(946)

Abstract:
A new, exact and universal conformal mapping was proposed. Using the Muskhelishvili's complex potential method, the plane elasticity problem of power function curved cracks with an arbitrary power of natural number was investigated and the general solutions of stress intensity factors (SIFs) for mode Ⅰ and mode Ⅱ at the crack tip were obtained under the remotely uniform tensile loads. The present results can be reduced to the well-known solutions when the power of power function is prescribed to different natural numbers. Numerical examples are conducted to reveal the effects of opening orientation, opening size, power and projected length along x-axis of the power function curved crack on the SIFs for mode Ⅰ and mode Ⅱ.

Dispersion Equation of Magnetoelastic Shear Waves in an Irregular Monoclinic Layer

A. Chattopadhyay, S. Gupta, S. A. Sahu, A. K. Singh

2011, 32(5): 541-555. doi: 10.3879/j.issn.1000-0887.2011.05.004

Abstract(1466) PDF(788)

Abstract:
The propagation of horizontally polarised shear waves in an internal magnetoelastic monoclinic stratum with irregularity in lower interface was studied. The stratum was sandwiched between two magnetoelastic monoclinic semi-infinite media. Dispersion equation was obtained in closed form. In absence of magnetic field and irregularity of the medium, the dispersion equation agrees with the equation of classical case in three layered media. The effect of magnetic field and size of irregularity on the phase velocity has been depicted by means of graphs.

Analysis for Cavitation Problem of Elastic Composite Ball Heated

SHANG Xin-chun, ZHANG Rui, REN Hui-lan

2011, 32(5): 556-562. doi: 10.3879/j.issn.1000-0887.2011.05.005

Abstract(1505) PDF(818)

Abstract:
The cavitation problem of composite ball, composed by two elastic solid materials and in uniformtemperature, was investigated. The nonlinear mathematical model of the problem was established by using finite logarithmic strain measure for geometric large deformation and by employing Hooke law for elastic solid. Analytic solutions in the form of parameter were derived for thermal dilatation of the composite ball with large elastic deformation. Solution curves were given to describe variations of the critical temperature in cavitation with the geometric and material parameters. Bifurcation curve was also given to reveal cavity growth after void nucleation. The numeric results for a computational example indicated that radius of cavity would rapidly enlarge over critical temperature, and the loop stress would become infinite with void nucleation. This means the materials near the cavity would produce plastic deformation which leads to local failure and fracture if the material of internal ball is elastoplastic. In addition, the cavitation for the composite ball could appear in a low temperature if elastic property for the material of internal ball is close to be uncompressible.

Influence of Magnetic Field on Wave Propagation at Liquid-Microstretch Solid Interface

Baljeet Singh

2011, 32(5): 563-569. doi: 10.3879/j.issn.1000-0887.2011.05.006

Abstract(1487) PDF(850)

Abstract:
The reflection and refraction of longit udinal wave at an interface between perfectly conducting non-viscous liquid half-space and a perfectly conducting microstretch elastic solid half-space was studied. The appropriate solutions of the governing equations were obtained in both the half-spaces which satisfy the required boundary conditions at the interface to obtain a system of five non-homogeneous equations in the amplitude ratios of various reflected and transmitted waves. The system of equations was solved by Fortran program of Gauss elimination method for a particular example of an interface between water and aluminum-epoxy composite. The numerical values of amplitude ratios were computed for a certain range of the angle of incidence both in presence and absence of impressed transverse magnetic field. The effects of the presence of transverse magnetic field on the amplitude ratios of various reflected and transmitted waves were shown graphically.

Stability Analysis of a Helical Rod Based on Exact Cosserat’s Model

LIU Yan-zhu, XUE Yun

2011, 32(5): 570-578. doi: 10.3879/j.issn.1000-0887.2011.05.007

Abstract(1596) PDF(898)

Abstract:
The helical equilibrium of a thin elastic rod has a practical background such as DNA, fiber, sub-ocean cable and oil-well dill string. The Kirchhoff's kinetic analogy is an effective approach in stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff's theory including no extension of the centerline and no shear deformation of the cross section are not adaptable to the real soft materials of biological fibers. The dynamic equations of a rod with circular cross section were established on the basis of exact Cosserat's model considering tension and shear deformations. The Euler's angles were applied as the attitude representation of the cross section. The deviation of the normal axis of cross section from the tangent of the centerline was considered as the result of shear deformation. The Liapunov's stability of helical equilibrium was discussed in static category and the Euler's critical values of axial force and torque were obtained. The Liapunov's and Euler's stability conditions in space domain are the necessary conditions of Liapunov's stability of the helical rod in time domain.

Low-Diffusion Preconditioning Scheme for Numerical Simulation of Low-Speed Flows Past Airfoil

XIANG Qian, WU Song-ping, LEE Chun-xuan, CAO Ning

2011, 32(5): 579-586. doi: 10.3879/j.issn.1000-0887.2011.05.008

Abstract(1834) PDF(840)

Abstract:
The reconditioning technique addresses the stiffness of low Mach number flow, while the stability was poor. Based on the conventional preconditioning method of Roe's scheme, a new low-diffusion scheme was proposed. An adjustable parameter was introduced to control the numerical dis-sipation in the new scheme, especially the over dissipation in the boundary layer and extremely low speed region. Numerical simulations of low Mach and low Reynolds number flows over a cylinder, low Mach and high Reynolds number flows over NACA0012 and NH02-18 airfoils were performed to validate the new scheme. All results of the three test cases are found to agree well with experiment data, demonstrating the applicability of the suggested scheme to low Mach number flow simulations.

Study on Application of Random Walk Method to Water Exchange in Large-Scale Bay

LI Xiao-bao, YUAN De-kui, TAO Jian-hua

2011, 32(5): 587-598. doi: 10.3879/j.issn.1000-0887.2011.05.009

Abstract(1413) PDF(788)

Abstract:
Water exchange matrix is an efficient tool to study the water exchange among sub-areas in large-scale bays. The application of random walk method to calculate water exchange matrix was studied. Compared against the advection-diffusion model, the random walk model is more flexible to calculate the water exchange matrix. The forecast matrix suggested by Thompson was applied to evaluate the water exchange characteristics among the sub-areas fast. According to theoretic analysis, it is found that the precision of the predicted results is mainly affected by three factors, namely the particle number, the generated time of the forecast matrix and the number of the sub-areas. The impact of the above factors was analyzed based on the results of a series of numerical tests. The results show that the precision of the forecast matrix increases with the increase of generated time of the forecast matrix and the number of the particles. If there are enough particles in each sub-area, the precision of the forecast matrix will increase with the number of the sub-areas. On the other hand, if the particles in each sub-area are not enough, excessive number of sub-areas may result in the decrease of the precision of the forecast matrix.

Scattering of Water Waves by a Thin Vertical Plate Submerged Below Ice-Cover Surface

Paramita Maiti, Puspendu Rakshit, Sudeshna Banerjea

2011, 32(5): 599-607. doi: 10.3879/j.issn.1000-0887.2011.05.010

Abstract(1560) PDF(830)

Abstract:
Scattering of water waves by a vertical plate submerged in deep water covered with a thin uniform sheet of ice, modelled as an elastic plate was concerned. The problem was formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem, in terms of difference of potential functions across the barrier. This integral equation was solved by collocation method using a finite series involving Chebyshev polynomials. The reflection and transmission coefficients were obtained numerically and presented graphically for various values of the wave number and ice cover parameter.

Turbulent Flow in Converging Nozzles Part Ⅰ——Boundary Layer Solution

Reza Maddahian, Bijan Farhanieh, Bahar Firoozabadi

2011, 32(5): 608-622. doi: 10.3879/j.issn.1000-0887.2011.05.011

Abstract(1922) PDF(813)

Abstract:
In this research the boundary layer integral method was used to investigate the development of turbulent swirling flow at the entrance region of a conical nozzle. The governing equations in the spherical coordinate system were simplified with the boundary layer assumptions and integrated through the boundary layer. The resulting sets of differential equations were then solved by the forth-order Adams predicto-rcorrector method. The free vortex and uniform velocity profiles were applied for tangential and axial velocities at the inlet region respectively. Due to the lack of experimental data for swirling flow in converging nozzles, the developed model was validated against the numerical simulations. The results of numerical simulations demonstrate the capability of the analytical model in predicting boundary layer parameters, such as boundary layer growth, shear rate and boundary layer thickness, as well as the swirl intensity decay rate for different cone angles. The proposed method introduces a simple and robust procedure in order to investigate the boundary layer parameters inside converging geometries.

Solution and Stability of a Mixed Type Functional Equation in Non-Archimedean Random Spaces

ZHANG Shi-sheng, R. Saadati, G. Sadeghi

2011, 32(5): 623-634. doi: 10.3879/j.issn.1000-0887.2011.05.012

Abstract(1625) PDF(807)

Abstract:
The generalized stability of the Euler-Lagrange quadratic mappings in the framework of non-Archimedean random normed spaces was proved. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces and the theory of functional equations were also presented.