应用数学和力学

Orientation Distribution Functions for Microstructures of Heterogeneous Materials(Ⅰ)-Directional Distribution Functions and Irreducible Tensors

ZHENG Quan-shui, ZOU Wen-nan

2001, 22(8): 773-789.

Abstract(2516) PDF(1095)

Abstract:
In this two-part paper,a thorough investigation is made on Fourier expansions with irreducible tensorial coefficents for orientation distribution functions(ODFs) and crystal orientation distribution functions(CODFs),which are scalar functions defined on the unit sphere and the rotation group,respectively.Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically-based approaches to mechanical and physical properties of heterogeneous materials.The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors.The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size,shape,phase,position of the material constitutions and defects.In Part(Ⅰ),the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N-dimensional(N-D) unit sphere is carried out.Attention is particularly paid to constructing simple expressions for 2and 3-D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point-group(the synonym of subgroup of the full orthogonal group) symmetries.In the continued work(Part Ⅱ),the explicit expression for the irreducible tensorial expansions of CODFs is established.The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point-group symmetries are derived.

Orientation Distribution Functions for Microstructures of Heterogeneous Materials(Ⅱ)-Crystal Distribution Functions and Irreducible Tensors Restricted by Various Material Symmetries

ZHENG Quan-shui, FU Yi-bin

2001, 22(8): 790-805.

Abstract(2788) PDF(757)

Abstract:
The explicit representations for tensorial Fourier expansion of 3-D crystal orientation distribution functions(CODFs) are established.In comparison with that the coefficients in the m th-term of the Fourier expansion of a 3-D ODF make up just a single irreducible m th order tensor,the coefficients in the m th term of the Fourier expansion of a 3-D CODF constitute generally so many as 2m+1 irreducible m th order tensors.Therefore,the restricted forms of tensorial Fourier expansions of 3-D CODFs imposed by various microand macro-scopic symmetries are further established,and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3-D CODFs contain remarkably reduced numbers of m th order irreducible tensors than the number 2m+1.These results are based on the restricted forms of irreducible tensors imposed by various pointgroup symmetries,which are also thoroughly investigated in the present part in both 2and 3-D spaces.

Effects of Surface Waves and Marine Soil Parameters on Seabed Stability

LIN Mian, LI Jia-chun

2001, 22(8): 806-816.

Abstract(2124) PDF(676)

Abstract:
Based on the Yamamoto's soil model by considering the Coulomb friction effects,the wave-induced seabed instability has been investigated.An analytical solution is derived for soil response of a finite depth seabed under surface water wave.The effects of wave parameters and soil characteristics on the seabed instability are addressed for three types of soil.Finally,the roles of Coulomb friction stability are then analyzed as well.

1:2 Internal Resonance of Coupled Dynamic System With Quadratic and Cubic Nonlinearities

CHEN Yu-shu, YANG Cai-xia, WU Zhi-qiang, CHEN Fang-qi

2001, 22(8): 817-824.

Abstract(2552) PDF(652)

Abstract:
The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied.The normal forms of this system in 1:2 internal resonance were derived by using the direct method of normal form.In the normal forms,quadratic and cubic nonlinearities were remained.Based on a new convenient transformation technique,the 4-dimension bifurcation equations were reduced to 3-dimension.A bifurcation equation with one-dimension was obtained.Then the bi furcation behaviors of a universal unfolding were studied by using the singularity theory.The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.

Study on Exact Analystical Solutions for Two Systems of Nonlinear Evolution Equations

YAN Zhen-ya, ZHANG Hong-qing

2001, 22(8): 825-833.

Abstract(2404) PDF(1165)

Abstract:
The homogeneous balance method was improved and applied to two systems of nonlinear evolution equations.As a result,several families of exact analytic solutions are derived by some new ansatzs.These solutions contain Wang's and Zhang's results and other new types of analytical solutions,such as rational fraction solutions and periodic solutions.The way can also be applied to solve more nonlinear partial differential equations.

An Accurate Solution of the Poisson Equation by the Finite Difference-Chebyshev Tau Method

Hani I. Siyyam

2001, 22(8): 834-838.

Abstract(2531) PDF(985)

Abstract:
A new finite difference-Chebyshev Tau method for the solution of the two-dimensional Poisson equation is presented.Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods.

Stretch and Rotation of a Suspended Cable Subject to Transverse Fluid Excitation

JIN Dong-ping, HU Hai-yan

2001, 22(8): 839-844.

Abstract(2236) PDF(589)

Abstract:
On the basis of analysis of the fluid drag acting on a suspended cable subjected to transverse fluid excitation,an expression is established for the fluid forces applied on the cable.By using a coordinate transform,the equations of motion of the cable are simplified into those in terms of the stretch and rotation coordinates.In the case of small ratio of sag-to-span,a study is made on the fluid-induced vibration behavior of the cable,including the critical fluid flow speed,the stability of equilibrium position,as well as the influences of gravitational parameter on the cable dynamics.

Equivalent Linearization Method Based on Energy-to-cth-Power Difference Criterion in Nonlinear Stochastic Vibration Analysis of Multi-Degree-of-Freedom Systems

WANG Guo-yan, DAI Min

2001, 22(8): 845-852.

Abstract(3652) PDF(539)

Abstract:
Basic equations of energy-tocth-power difference criterion were derived for multi-degreeof-freedom(MDOF) systems subjected to stationary Gaussian excitations with non-zero mean.Modal transform technique was used in order to reduce unknowns.Main computational formulae were presented and suggested values of c were given.Numerical results show that the method of this paper prevails over equation difference criterion both in accuracy and in simplicity.

The Discrete Models on a Frictional Single Degree of Freedom System

FENG Qi, ZHANG Xiang-ting

2001, 22(8): 853-861.

Abstract(2212) PDF(577)

Abstract:
Two stochastic models on simple random system with friction were developed.One of them was a discrete model by a two-dimensional mean map applied to describe random stick-slip motion.The numerical examples show that external noise can reduce the complexity of the system behavior.Secondly,a probability model described was established with coexistence of stick-slip and slip motions.The numerical results point out that this model possesses pure stochastic behavior.

Research on Equivalent Processes of Rock Mass Parameters With Anchor Piles

CHEN Hong-kai, TANG Hong-mei, WANG Rong, TANG Fen, WANG Kai, YUAN Jian-yi

2001, 22(8): 862-868.

Abstract(1815) PDF(744)

Abstract:
Using normal and shear rigid coefficients of intact rock and fracture plane,rigidly normal,shear equivalent rigid coefficients of fissure rock mass are conducted.On the basis of hypotheses of small displacement of rock mass,principle of superposition,irrelevance of strength parameters C and U and Coulomb theory,formulas to calculate equivalent strength parameters C and U of equivalent continuous mass from fissured rock mass with anchor piles are given.The achievement is extremely valuable in integral stability analysis of the rock mass slope and important in promoting the research of the rock mass's constitutive relation.

Singular Characteristics of Nonlinear Normal Modes in a Two Degrees of Freedom Asymmetric System With Cubic Nonlinearities

XU Jian, LU Qi-shao, HUANG Ke-lei

2001, 22(8): 869-878.

Abstract(2352) PDF(588)

Abstract:
Nonlinear normal modes in a two degrees of freedom asymmetric system with cubic nonlinearities as singularity occurs in the system are studied,based on the invariant space in nonlinear normal modes and perturbation technique.Emphasis is placed on singular characteristics as the linear coupling between subsystems degenerates.For non-resonances,it is analytically presented that a single-mode motion and localization of vibrations occur in the system,and the degree of localization relates not only to the coupling stiffness between oscillators,but also to the asymmetric parameter.The parametric threshold value of localization is analytically given.For 11 resonance,there exist bifurcations of normal modes with nonlinearly coupling stiffness and asymmetric parameter varying.The bifurcating set on the parameter and bifurcating curves of normal modes are obtained.

The Decay of Swirling Flows in a Type of Cross-Section-Varying Pipes

XIONG Ao-kui, WEI Qing-ding

2001, 22(8): 879-884.

Abstract(2158) PDF(785)

Abstract:
The decay of weakly swirling flows in a type of cross-section-varying pipes was discussed analytically.For laminar swirling flow,the feature of exponential decay was demonstrated.For turbulent swirling flow,in spite of the decay of circulation flux,a necessary condition for local circulation to amplify along downstream was obtained under the Boussinesq's hypothesis.

Study on the Authorship of “Applied Mathematics and Mechanics”

QIN Zhi-qiang

2001, 22(8): 885-890.

Abstract(2314) PDF(728)

Abstract:
Based on the theory of the metrology of literature,a careful survey and analysis of the papers and their authors included in the magazine of "Applied Mathematics and Mechanics"(1996~2000) are made.The author attempts to reflect some important features concerning this magazine as well as its authorship.It is shown that this magazine is of high quality.

2001 Vol. 22, No. 8