应用数学和力学

2003 Vol. 24, No. 10

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Renewal of Basic Laws and Principles for Polar Continuum Theories(Ⅰ)-Micropolar Continua

DAI Tian-min

2003, 24(10): 991-997.

Abstract(2343) PDF(749)

Abstract:
Based on the restudies of existing polar continuum theories rather complete systems of basic balance laws and equations for micropolar continuum theory are presented.In these new systems not only the additional angular momentum,surface moment and body moment produced by the linear momentum,surface force and body force,respectively,but also the additional velocity produced by the angular velocity are considered.The new coupled balance laws of linear momentum,angular momentum and energy are reestablished.From them the new coupled local and nonlocal balance equations are naturally derived.Via contrast it can be clearly seen that the new results are believed to be rather general and complete.

Renewal of Basic Laws and Principles for Polar Continuum Theories (Ⅱ)-Micromorphic Continuum Theory and Couple Stress Theory

DAI Tian-min

2003, 24(10): 998-1004.

Abstract(2292) PDF(959)

Abstract:
The purpose is to reestablish the balance laws of momentum,angular momentum and energy and to derive the corresponding local and nonlocal balance equations for micromorphic continuum mechanics and couple stress theory.The desired results for micromorphic continuum mechanics and couple stress theory are naturally obtained via direct transitions and reductions from the coupled conservation law of energy for microplar continuum theory,respectively.The basic balance laws and equations for micromorphic continuum mechanics and couple stress theory are constituted by combining these results derived here and the traditional conservation laws and equations of mass and microinertia and the entropy inequality.The incomplete degrees of the former related continuum theories are clarified.Finally,some special cases are conveniently derived.

Renewal of Basic Laws and Principles for Polar Continuum Theories(Ⅲ)-Noether’s Theorem

DAI Tian-min

2003, 24(10): 1005-1011.

Abstract(2353) PDF(663)

Abstract:
The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether.s theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics.The new concrete forms of various conservation laws of couple stress elasticity are derived.The precise nature of these conservation laws which result from the given invariance requirements are established.Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.

Shape Bifurcation of an Elastic Wafer Due to Surface Stress

YAN Kun, HE Ling-hui, LIU Ren-huai

2003, 24(10): 1012-1016.

Abstract(2797) PDF(627)

Abstract:
A geometrically nonlinear analysis was proposed for the deformation of a free standing elastically isotropic wafer caused by the surface stress change on one surface.The link between the curvature and the change in surface stress was obtained analytically from energetic consideration.In contrast to the existing linear analysis,a remarkable consequence is that,when the wafer is very thin or the surface stress difference between the two major surfaces is large enough,the shape of the wafer will bifurcate.

1/3 Subharmonic Solution of Elliptical Sandwich Plates

LI Yin-shan, ZHANG Nian-mei, YANG Gui-tong

2003, 24(10): 1017-1026.

Abstract(2035) PDF(538)

Abstract:
The problem of nonlinear forced oscillations for elliptical sandwich plates is dealt with. Based on the governing equations expressed in terms of five displacement components,the nonlinear dynamic equation of an elliptical sandwich plate under a harmonic force is derived.A superpositive-iterative harmonic balance(SIHB)method is presented for the steady-state analysis of strongly nonlin-ear oscillators.In a periodic oscillation,the periodic solutions can be expressed in the form of basic harmonics and bifurcate harmonics.Thus,an oscillation system which is described as a second order ordinary differential equation,can be expressed as fundamental differential equation with fundamental harmonics and incremental differential equation with derived harmonics.The 1/3 subharmonic solu-tion of an elliptical sandwich plate is investigated by using the methods of SIHB.The SIHB method is compared with the numerical integration method.Finally,asymptotical stability of the 1/3 subhar-monic oscillations is inspected.

Random-Fuzzy Model for the Ductile/Brittle Transition

LÜ Zhen-zhou, YUE Zhu-feng, FENG Yun-wen

2003, 24(10): 1027-1034.

Abstract(2413) PDF(605)

Abstract:
A large sample of experiments was carried out to study influence of stress triaxiality and temperature on the growth of micro voids and the ductile/brittle transition(DBT)behavior of 40Cr steel.A random-fuzzy model was put forward for the transition.It is assumed that the ductile fracture is controlled by the micro void mechanism,and the critical void growth ratio can be used as the criterion of ductile fracture.The brittle fracture is modeled by an embedded penny crack.A micro stress intensity and the fracture characteristic length of the brittle fracture was then presented and calculated. The DBT is completed by the two mechanisms,which exists in the fracture of all specimens simultaneously.The distributions of model parameters were measured experimentally.A random model and a random-fuzzy model for DBT were presented.The comparison between the calculated and experimental results shows that the random-fuzzy model can model the DBT satisfactorily.

Analysis of Chebyshev Pseudospectral Method for Multi-Dimensional Generalized SRLW Equations

SHANG Ya-dong, GUO Bo-ling

2003, 24(10): 1035-1048.

Abstract(2326) PDF(630)

Abstract:
The Chebyshev pseudo-spectral approximation of the homogenous initial boundary value problem for a class of multi-dimensional generalized symmetric of regularized long wave(SRLW)equations is considered.The fully discrete Chebyshev pseudospectral scheme is constructed.The convergence of the approximation solution and the optimum error of approximation solution are obtained.

Mathematical Model of Two-Phase Fluid Nonlinear Flow in Low-Permeability Porous Media With Applications

DENG Ying-er, LIU Ci-qun

2003, 24(10): 1049-1056.

Abstract(2724) PDF(757)

Abstract:
A mathematical model of two-phase fluid nonlinear flow in the direction of normal of ellipse through low-permeability porous media was established according to a nonlinear flow law expressed in a continuous function with three parameters,a mass conservation law and a concept of tur-bulent ellipses.A solution to the model was obtained by using a finite difference method and an extrapolation method.Formulas of calculating development index not only before but also after water breaks through an oil well in the condition of two-phase fluid nonlinear flow in the media were derived.An example was discussed.Water saturation distribution was presented.The moving law of drainage front was found.Laws of change of pressure difference with time were recognized.Results show that there is much difference of water saturation distribution between nonlinear flow and linear flow;that drainage front by water moves faster,water breaks through sooner and the index gets worse because of the nonlinear flow;and that dimensionless pressure difference gets larger at the same dimensionless time and difficulty of oil development becomes bigger by the nonlinear flow.Thus,it is necessary that influence of nonlinear flow on development indexes of the oil fields be taken into account.The results provide water-flooding development of the oil fields with scientific basis.

Nonlinear Faraday Waves in a Parametrically Excited Circular Cylindrical Container

JIAN Yong-jun, E Xue-quan, BAI Wei

2003, 24(10): 1057-1068.

Abstract(2703) PDF(604)

Abstract:
In the cylindrical coordinate system,a singular perturbation theory of multiple-scale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rigid circular cylinder,which is subject to a vertical oscillation.It is assumed that the fluid in the circular cylindrical vessel is inviscid,incompressible and the motion is irrotational, a nonlinear amplitude equation with cubic and vertically excited terms of the vessel was derived by expansion of two-time scales without considering the effect of surface tension.It is shown by numerical computation that different free surface standing wave patterns will be formed in different excited frequencies and amplitudes.The contours of free surface waves are agreed well with the experimental results which were carried out several years ago.

Regularization of Nearly Singular Integrals in the Boundary Element Method of Potential Problems

ZHOU Huan-lin, NIU Zhong-rong, WANG Xiu-xi

2003, 24(10): 1069-1074.

Abstract(2744) PDF(655)

Abstract:
A general algorithm is applied to the regularization of nearly singular integrals in the boundary element method of planar potential problems.For linear elements,the strongly singular and hypersingular integrals of the interior points very close to boundary were categorized into two forms. The factor leading to the singularity was transformed out of the integral representations with integration by parts,so non-singular regularized formulas were presented for the two forms of integrals.Furthermore,quadratic elements are used in addition to linear ones.The quadratic element very close to the internal point can be divided into two linear ones,so that the algorithm is still valid.Numerical examples demonstrate the effectiveness and accuracy of this algorithm.Especially for problems with curved boundaries,the combination of quadratic elements and linear elements can give more accurate results.

Analytical Solution of a Simply Supported Piezoelectric Beam Subjected to a Uniformly-Distributed Loading

ZHANG Lin-nan, SHI Zhi-fei

2003, 24(10): 1075-1082.

Abstract(2612) PDF(616)

Abstract:
Using the inverse method,the analytical solution of a simply supported piezoelcetric beam subjected to a uniformly distributed loading has been studied.First,the polynomials of stress function and induction function are given.Then,considering the gradient properties of the elastic parameter and the potential funciton as well as the piezoelectric parameter,the analytical solution of a simply supported beam subjected to a uniformly distributed loading is obtained and discussed.

Spectrum Distribution of the Second Order Generalized Distributed Parameter Systems

GE Zhao-qiang, ZHU Guang-tian, MA Yong-hao

2003, 24(10): 1083-1089.

Abstract(2402) PDF(681)

Abstract:
Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space.The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator.This is theoretically important for studying the stabilization and asymptotic stability of the second order generalized distributed parameter system.

Numerical Method for the Shape Reconstruction of a Hard Target

YOU Yun-xiang, MIAO Guo-ping

2003, 24(10): 1090-1100.

Abstract(2540) PDF(581)

Abstract:
A nonlinear optimization method was developed to solve the inverse problem of determining the shape of a hard target from the knowlegde of the far-field pattern of the acoustic scattering wave, it was achieved by solving independently an ill-posed linear system and a well-posed minimization problem.Such a separate numerical treatment for the ill-posedness and nonlinearity of the inverse problem makes the numerical implementation of the proposed method very easy and fast since there only involves the solution of a small scale minimization problem with one unknown function in the nonlinear optimization step for determining the shape of the sound-hard obstacle.Another particular feature of the method is that it can reproduce the shape of an unknown hard target efficiently from the knowledge of only one Fourier coefficient of the far-field pattern.Moreover,a two-step adaptive iteration algorithm was presented to implement numerically the nonlinear optimization scheme.Numerical experiments for several two dimensional sound-hard scatterers having a variety of shapes provide an independent verification of the effectiveness and practicality of the inversion scheme.