应用数学和力学

2001 Vol. 22, No. 1

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Dynamical Behavior of Viscoelastic Cylindrical Shells Under Axial Pressures

CHENG Chang-jun, ZHANG Neng-hui

2001, 22(1): 1-8.

Abstract(2531) PDF(634)

Abstract:
The hypotheses of the Kûrmûn-Donnell theory of thin shells with large deflections and the Boltzmann laws for isotropic linear,viscoelastic materials,the constitutive equations of shallow shells are first derived.Then the governing equations for the deflection and stress function are formulated by using the procedure similar to establishing the Kûrmûn equations of elastic thin plates.Introducing proper assumptions,an approximate theory for viscoelastic cylindrical shells under axial pressures can be obtained.Finally,the dynamical behavior is studied in detail by using several numerical methods. Dynamical properties,such as,hyperchaos,chaos,strange attractor,limit cycle etc.,are discovered.

The Problem of an External Circular Crack Under Asymmetric Loadings

WANG Yin-bang

2001, 22(1): 9-15.

Abstract(2540) PDF(543)

Abstract:
Using the boundary integral equation method,the problem of an external circular crack in a three-dimensional infinite elastic body under asymmetric loadings is investigated.The two-dimen-sional singular boundary integral equations of the problem were reduced to a system of Abel integral equations by means of Fourier series and hyper geometric functions.The exact solutions of stress intensity factors are obtained for the problem of an external circular crack under asymmetric loadings, which are even more universal than the results obtained by the use of Hankel transform method.The results demonstrate that the boundary integral equation method has great potential as a new analytic method.

On the Bending, Vibration and Stability of Laminated Rectangular Plates With Transversely Isotropic Layers

DING Hao-jiang, CHEN Wei-qiu, XU Rong-qiao

2001, 22(1): 16-22.

Abstract(2490) PDF(640)

Abstract:
A method based on newly presented state space formulations is developed for analyzing the bending,vibration and stability of laminated transversely isotropic rectangular plates with simply supported edges.By introducing two displacement functions and two stress functions,two independent state equations were constructed based on the three-dimensional elasticity equations for transverse isotropy.The original differential equations are thus decoupled with the order reduced that will facilitate obtaining solutions of various problems.For the simply supported rectangular plate,two relations between the state variables at the top and bottom surfaces were established.In particular,for the free vibration(stability)problem,it is found that there exist two independent classes:One corresponds to the pure in-plane vibration(stability)and the other to the general bending vibration(stability).Numerical examples are finally presented and the effects of some parameters are discussed.

On the Iterative Approximation Problem of Fixed Points for Asymptotically Nonexpansive Type Mappings in Banach Spaces

ZHANG Shi-sheng

2001, 22(1): 23-31.

Abstract(2311) PDF(661)

Abstract:
Some iterative approximation theorems of fixed points for asymptotically nonexpansive type mappings in Banach spaces are obtained.

Analysis of a Partially Debonded Conducting Rigid Elliptical Inclusion in a Piezoelectric Matrix

WANG Xu, SHEN Ya-peng

2001, 22(1): 32-46.

Abstract(1880) PDF(598)

Abstract:
A closed-form full-field solution for the problem of a partially debonded conducting rigid elliptical inclusion embedded in a piezoelectric matrix is obtained by employing the eight-dimensional Stroh formalism in conjunction with the techniques of conformal mapping,analytical continuation and singularity analysis.Some new identities and sums for anisotropic piezoelectric media are also derived,through which real-form expressions for the stresses and electric displacements along the interface as well as the rotation of the rigid inclusion can be obtained.As is expected,the stresses and electric displacements at the tips of the debonded part of the interface exhibit the same singular behavior as in the case of a straight Griffith interface crack between dissimilar piezoelectric media.Some numerical examples are presented to validate the correctness of the obtained solution and also to illustrate the generality of the exact solution and the effects of various electromechanical loading conditions,geometry parameters and material constants on the distribution of stresses and electric displacements along the interface.

Nonlinear Theory of Dynamic Stability for Laminated Composite Cylindrical Shells

ZHOU Cheng-ti, WANG Lie-dong

2001, 22(1): 47-55.

Abstract(2314) PDF(792)

Abstract:
Hamilton Principle was used to derive the general governing equations of nonlinear dynamic stability for laminated cylindrical shells in which,factors of nonlinear large deflection,transverse shear and longitudinal inertiaforce were concluded.Equations were solved by variational method. Analysis reveals that under the action of dynamic load,laminated cylindrical shells will fall into a state of parametric resonance and enter into the dynamic unstable region that causes dynamic instability of shells.Laminated shells of three typical composites w ere computed:i.e.T300/5208 graphite epoxy E-glass epoxy,and ARALL shells.Results show that all factors will induce important influence for dynamic stability of laminated shells.So,in research of dynamic stability for laminated shells,to consider these factors is important.

The Non-Axisymmetrical Dynamic Response of Transversely Isotropic Saturated Poroelastic Media

ZHANG Yin-ke, HUANG Yi

2001, 22(1): 56-70.

Abstract(2584) PDF(699)

Abstract:
The Biot's wave equations of transversely isotropic saturated poroelastic media excited by non-axisymmetrical harmonic source were solved by means of Fourier expansion and Hankel transform.Then the components of total stress in porous media are expressed with the solutions of Biot's wave equations.The method of research on non-axisymmetrical dynamic response of saturated porous media is discussed,and a numerical result is presented.

Method to Calculate Bending Center and Stress Intensity Factors of Cracked Cylinder Under Saint-Venant Bending

TANG Ren-ji, TANG Xin-yan

2001, 22(1): 71-78.

Abstract(1690) PDF(526)

Abstract:
Using the single crack solution and the regular solution of plane harmonic function,the problem of Saint-Venant bending of a cracked cylinder by a transverse force was reduced to solving two sets of integral equations and its general solution was then obtained.Based on the obtained solution,a method to calculate the bending center and the stress intensity factors of the cracked cylinger whose cross section is not thin-walled,but of small torsion rigidity is proposed.Some numerical examples are given.

Some Misunderstandings on Rotation of Crystals and Reasonable Plastic Strain Rate

ZHAO Zu-wu

2001, 22(1): 79-84.

Abstract(2217) PDF(716)

Abstract:
It is pointed out that crystals are discrete but not continuous materials.Hence the rotation R in decomposition F=RU and spin Win R F>F^-1 are not correct.Errors will arise in plastic deformation rate if it is directly expressed with amounts of velocity of slips in glide systems such as γ>ν≠n. The geometrical figure of crystal lattices does not change after slips and based on this idea a simple way in mechanics of continuous media to get the plastic deformations rate induced by slips is proposed.Constitutive equations are recommended.

An Interface Inclusion Between Two Dissimilar Piezoelectric Materials

GAO Cun-fa, FAN Wei-xun

2001, 22(1): 85-92.

Abstract(2228) PDF(450)

Abstract:
The generalized two-dimensional problem of a dielectric rigid line inclusion,at the interface between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity,is studied by means of the Stroh formalism.The problem was reduced to a Hilbert problem,and then closed-form expressions were obtained,respectively,for the complex potentials in piezoelectric media,the electric field inside the inclusion and the tip fields near the inclusion.It is shown that in the media,all field variables near the inclusion-tip show square root singularity and oscillatory singularity, the intensity of which is dependent on the material constants and the strains at infinity.In addition,it is found that the electric field inside the inclusion is singular and oscillatory too,when approaching the inclusion-tips from inside the inclusion.

Non-Linear and Elasto-Plasticity Consolidation Models of Unsaturated Soil and Applications

CHEN Zheng-han, HUANG Hai, LU Zai-hua

2001, 22(1): 93-103.

Abstract(2822) PDF(1182)

Abstract:
The non-linear constitutive model suggested by the authors and the Alonso.s elasto-plasticity model of unsaturated soil modified by the authors are introduced into the consolidation theory of unsaturated soil proposed by CHEN Zheng-han,and the non-linear and the elasto-plasticity consolidation models of unsaturated soil are obtained.Programs related to the two consolidation models are designed,and a 2-D consolidation problem of unsaturated soil is solved using the programs,the consolidation process and the development of plastic zone under multi-grade load are studied.The above research develops the consolidation theory of unsaturated soil to a new level.

Analysis of Financial Derivatives by Mechanical Method (Ⅰ)——Basic Equation of Price of Index Futures

YUN Tian-quan

2001, 22(1): 104-110.

Abstract(1999) PDF(681)

Abstract:
Similar to the method of continuum mechanics,the variation of the price of index futures is viewed to be continuous and regular.According to the characteristic of index futures,a basic equation of price of index futures was established.It is a differential equation,its solution shows that the relation between time and price forms a logarithmic circle.If the time is thought of as the probability of its corresponding price,then such a relation is perfectly coincided with the main assumption of the famous formula of option pricing,based on statistical theory,established by Black and Scholes,winner of 1997 Nobel.prize on economy.In that formula,the probability of price of basic assets(they stand for index futures here)is assummed to be a logarithmic normal distribution.This agreement shows that the same result may be obtained by two analytic methods with different bases.However, the result,given by assumption by Black-Scholes,is derived from the solution of the differential equation.