应用数学和力学

2006 Vol. 27, No. 6

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Numerical Investigation of the Evolution of Disturbances in Supersonic Sharp Cone Boundary Layers

DONG Ming, LUO Ji-sheng, CAO Wei

2006, 27(6): 631-636.

Abstract(2742) PDF(694)

Abstract:
The spatial evolution of 2_D disturbances in supersonic sharp cone boundary layers was investigated by direct numerical simulation(DNS)in high order compact difference scheme.The results obtained suggested that although the normal velocity in the sharp cone boundary layer is not small, the evolution of amplitude and phase for small amplitude disturbances would be well in accord with the results obtained by the linear stability theory(LST)which supposes the flow is parallel.The evolution of some finite amplitude disturbances was also investigated,and the characteristic of the evolution was shown.And shocklets were also found when the amplitude of the disturbances over some value.

Solutions for Second Order Impulsive Integro-Differential Equation on Unbounded Domains in Banach Spaces

CHEN Fang-qi, TIAN Rui-lan, CHEN Yu-shu

2006, 27(6): 637-645.

Abstract(2355) PDF(581)

Abstract:
Under loose conditions,the existence of solutions to initial value problem are studied for second order impulsive integro_differential equation with infinite moments of impulse effect on the positive half real axis in Banach spaces.By the use of recurrence method,Tonelii sequence and the locally convex topology,the new existence theorems are achieved,which improve the related results obtained by GUO Da_jun.

Scattering of Anti-Plane Shear Waves in a Functionally Graded Material Strip With an Off-Center Vertical Crack

LI Lin, ZHOU Zhen-gong, WANG Biao

2006, 27(6): 646-654.

Abstract(2419) PDF(733)

Abstract:
The scattering problem of anti_plane shear waves in a functionally graded material strip with an off_center crack is investigated by use of Schmidt method.The crack is vertically to the edge of the strip.By using the Fourier transform,the problem can be solved with the help of a pair of dual integral equations that the unknown variable is the jump of the displacement across the crack surfaces.To solve the dual integral equations,the jump of the displacement across the crack surfaces was expanded in a series of Jacobi polynomials.Numerical examples were provided to show the effects of the parameter describing the functionally graded materials,the position of the crack and the frequency of the incident waves upon the stress intensity factors of the crack.

Viscoelastic Constitutive Model Related to Deformation of Insect Wing Under Loading in Flapping Motion

BAO Lin, HU Jin-song, YU Yong-liang, CHENG Peng, XU Bo-qing, TONG Bing-gang

2006, 27(6): 655-662.

Abstract(2753) PDF(607)

Abstract:
Flexible insect wings deform passively under periodic loading during flapping flight.The wing flexibility is considered as one of the specific mechanisms on improving insect flight perfor mance.The constitutive relation of the insect wing material plays a key role on the wing deformation, but has not been clearly understood yet.A viscoelastic constitutive relation model was established based on the experimental results:the stress relaxation experiment of a dragonfly wing(in vitro). This model was examined by the finite element analysis of the dynamic deformation response for a model insect wing under the action of the periodical inertial force in flapping.It is revealed that the viscoelastic constitutive relation is rational to characterize the biomaterial property of insect wings in contrast to the elastic one.The amplitude and form of the passive viscoelastic deformation of the wing is evidently dependent on the viscous parameters in the constitutive relation.

Thickness-Shear Vibration of a Circular Crystal Plate in a Cylindrical Shell as a Pressure Sensor

HU Yuan-tai, CUI Zhi-jian, JIANG Shu-nong, YANG Jia-shi

2006, 27(6): 663-668.

Abstract(2953) PDF(522)

Abstract:
Based on the theory for small fields superposed on relatively larger fields in an electroelastic body,a theoretical analysis was performed on a circular plate thickness-shear crystal resonator sealed in a circular cylindrical shell for pressure measurement.A simple expression is obtained for pressure induced frequency shifts in the resonator,which is examined for design optimization.Numerical results show that the frequency shifts depend linearly on the pressure,and that a pressure sensor with a softer outer shell or a smaller thickness ratio of the crystal plate to the outer shell has higher sensitivity.

Unified Computation of Flow With Compressible and Incompressible Fluid Based on Roe’s Scheme

HUANG Dian-gui

2006, 27(6): 669-674.

Abstract(2562) PDF(1243)

Abstract:
A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm.The primitive variables were pressure,velocities and temperature.The time integration scheme was used in conjunction with a finite volume discretization.The preconditioning was coupled with a high order implicit upwind scheme based on the definition of a Roe's type matrix.Computational capabilities are demonstrated through computations of high Mach number,middle Mach number,very low Mach number,and incompressible flow.It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe's scheme.

A Class of Compact Upwind TVD Difference Schemes

TU Guo-hua, YUAN Xiang-jiang, XIA Zhi-qiang, HU Zhen

2006, 27(6): 675-682.

Abstract(3650) PDF(679)

Abstract:
A new method was proposed for constructing total variation diminishing(TVD)upwind schemes in conservation forms.Two limiters were used to prevent non-physical oscillations across discontinuity.Both limiters can ensure the nonlinear compact schemes TVD property.Two compact TVD(CTVD)schemes were tested,one is third-order accuracy,and the other is fifth-order.The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems,as well as a two-dimensional shock-vortex interaction and a shock-boundary flow interaction.Numerical results show their high-order accuracy and high resolution,and low oscillations across discontinuities.

Anti-Plane Fracture Analysis of a Functionally Gradient Material Infinite Strip With Finite Width

LI Yong-dong, JIA Bin, ZHANG Nan, DAI Yao, TANG Li-qiang

2006, 27(6): 683-689.

Abstract(3007) PDF(620)

Abstract:
The special case of a crack under mode conditions was treated,lying parallel to the edges of an infinite strip with finite width and with the shear modulus varying exponentially perpendicular to the edges.By using Fourier transforms the problem was formulate dinterms of a singular integral equation.It was numerically solved by representing the unknown dislocation density by a truncated series of Chebyshev polynomials leading to a linear system of equations.The stress intensity factor (SIF)results were discussed with respect to the influences of different geometric parameter sand the strength of the non-homogeneity.It was indicated that the SIF increases with the increase of the crack length and decreases with the increase of the rigidity of the material in the vicinity of crack.The SIF of narrow strip is very sensitive to the change of the non-homog eneity parameter and its variation is complicated.With the increase of the non-homogeneity parameter,the stress intensity factor may increase,decrease or keep constant,which is mainly determined by the strip width and the relative crack location.If the crack is located at the midline of the strip or if the strip is wide,the stress intensity factor is not sensitive to the material non-homogeneity parameter.

Deformation Due to Time Harmonic Sources in Micropolar Thermoelastic Medium Possessing Cubic Symmetry With Two Relaxation Times

Rajneesh Kumar, Praveen Ailawalia

2006, 27(6): 690-700.

Abstract(2954) PDF(532)

Abstract:
The response of a micropolar thermoelastic medium possessing cubic symmetry with two relaxation times due to time harmonic sources has been investigated.Fourier transform was employed and the transform was inverted by using a numerical inversion technique.The components of displacement,stress,microrotation and temperature distribution in the physical domain were obtained numerically.The results of normal displacement,normal force stress,tangential couple stress and temperature distribution were compared for micropolar cubic crystal and micropolar isotropic solid.The numerical results were illustrated graphically for a particular material.Some special cases were also deduced.

Flexural Wave Propagation in Narrow Mindlin’s Plate

HU Chao, HAN Gang, FANG Xue-qian, HUANG Wen-hu

2006, 27(6): 701-708.

Abstract(3179) PDF(554)

Abstract:
Appling Mindlin's theory of thick plates and Hamilton formulism to propagation of elastic waves under free boundary condition,a solution of the problem was given.Dispersion equations of propagation mode of strip plates were deduced from eigenfunction expansion method.It was compared with the dispersion relation that was gained through solution of thick plate theory proposed by Mindlin.Based on the two kinds of theories,the dispersion curves show great difference in the region of short waves,and the cutoff frequencies are higher in Hamiltonian systems.However,the dispersion curves are almost the same in the region of long waves.

Thermal Post-Buckling of Functionally Graded Material Timoshenko Beams

LI Shi-rong, ZHANG Jing-hua, ZHAO Yong-gang

2006, 27(6): 709-715.

Abstract(3358) PDF(783)

Abstract:
Analysis of thermal post-buckling of FGM Timoshenko beams subjected to transversely non-uniform temperature rise is presented.By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam,geometrically nonlinear governing equations,including seven basic unknown functions,for functionally graded beams subjected to mechanical and thermal loads were formulated.In the analysis,it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate.By using a shooting method,the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely non-uniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained.Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted.The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details.The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.

Bifurcations of Travelling Wave Solutions in Variant Boussinesq Equations

YUAN Yu-bo, PU Dong-mei, LI Shu-min

2006, 27(6): 716-726.

Abstract(2871) PDF(625)

Abstract:
The bifurcations of solitary waves and kink waves for variant Boussinesq equations were studied by using the bifurcation theory of planar dynamical systems.The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented.Several types explicit formulas of solitary wave solutions and kink wave solutions are obtained.In the end, several formulas of periodic wave solutions are presented.

Dynamic Characteristic Analysis of Fuzzy Stochastic Truss Structures Based on Fuzzy Factor Method and Random Factor Method

MA Juan, CHEN Jian-jun, XU Ya-lan, JIANG Tao

2006, 27(6): 727-734.

Abstract(2623) PDF(549)

Abstract:
A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented.Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously,the structural stiffness and mass matrices were constructed based on the fuzzy factor method and random factor method;from the Rayleigh's quotient of structural vibration,the structural fuzzy random dynamic characteristic was obtained by means of the interval arithmetic;the fuzzy numeric characteristics of dynamic characteristic were then derived by using the random variable's moment function method and algebra synthesis method.Two examples were used to illustrate the validity and rationality of the method given.The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameter on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.

Theoretic Solution of Rectangular Thin Plate on Foundation With Four Edges Free by Symplectic Geometry Method

ZHONG Yang, ZHANG Yong-shan

2006, 27(6): 735-740.

Abstract(2989) PDF(645)

Abstract:
The theoretic solution for rectangular thin plate on foundation with four edges free was derived by symplectic geometry method.In the analysis proceeding,the elastic foundation was presented by the Winkler model.Firstly,the basic equations for elastic thin plate were transferred into Hamilton canonical equations.The symplectic geometry method was used to separate the whole variables and eigenvalues were obtained simultaneously.Finally,according to the method of eigen function expansion,the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees were developed.Since the basic elasticity equations of thin plate is only used and it is not need to select the deformation function arbitrary.Therefore,the solution is theoretical and reasonable.In order to show the correction of formulations derived,a numerical example was given to demonstrate the accuracy and convergence of the current solution.

Mechanism and Catastrophe Theory Analysis of Circular Tunnel Rockburst

PAN Yue, ZHANG Yong, YU Guang-ming

2006, 27(6): 741-749.

Abstract(2516) PDF(636)

Abstract:
Mechanism of circular tunnel rockburst is that,when the carrying capacity of the centralized zone of plastic deformation in limiting state reduces,the comparatively intact part in rock mass unloads by way of elasticity;rockburst occurs immediately when the elastic energy released by the comparatively intact part exceeds the energy dissipated by plastic deformation.The equivalent strain was taken as a state variable to establish a catastrophe model of tunnel rockburst,and the computation expression of the earthquake energy released by tunnel rockburst was given.The analysis shows that,the conditions of rockburst occurrence are relative to rock's ratio of elastic modulus to descendent modulus and crack growth degree of rocks;to rock mass with specific rockburst tendency,there exists a corresponding critical depth of softened zone,and rockburst occurs when the depth of softened zone reaches critical depth of softened zone.

Orthogonal Polynomials and Determinant Formulas of Function-Valued Padé-Type Approximation Using for Solution of Integral Equations

GU Chuan-qing, PAN Bao-zheng, WU Bei-bei

2006, 27(6): 750-756.

Abstract(2798) PDF(850)

Abstract:
To solve Fredholm integral equations of the second kind,a generalized linear functional is introduced and a new function-valued Pad-type approximation was defined.By means of the power series expansion of the solution,this method can construct an approximate solution to solve the given integral equation.On the basis of the orthogonal polynomials,two useful deter-minant expressions of the numerator polynomial and the denominator polynomial for Pad-type approximation were explicitly given.