应用数学和力学

2003 Vol. 24, No. 1

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Investigation of the Dynamic Behavior of Two Collinear Anti-Plane Shear Cracks in a Piezoelectric Layer Bonded to Two Half Spaces by a New Method

ZHOU Zhen-gong, WANG Biao

2003, 24(1): 1-11.

Abstract(1949) PDF(632)

Abstract:
The dynamic behavior of two collinear anti-plane shear cracks in a piezoelcetric layer bonded to two half spaces subjected to the harmonic waves is investigated by a new method. The cracks are parallel to the interfaces in the mid-plane of the piezoelectric layer. By using the Fourier transform, the problem can be solved with two pairs of triple integral equations. These equations are solved by using Schmidt's method. This process is quite different from that adopted previously. Numerical examples are provided to show the effect of the geometry of cracks, the frequency of the incident wave, the thichness of the piezoelectric layer and the constants of the material upon the dynamic stress intensity factor of cracks.

On the Calculation of Energy Release Rate for Viscoelastic Cracked Laminates

LIU Yu-lan, WANG Biao, WANG Dian-fu

2003, 24(1): 12-18.

Abstract(1970) PDF(625)

Abstract:
The energy release rate(ERR) of crack growth as the energy change at the same time t between the two states of the structure is redefined, one is with crack length a under the loading σ(t), the other is the state with crack lengtha +Δa under the same loading condition. Thus the defined energy release rate corresponds to the released energy when a crack grows from a to a+Δa in an infinitesimal time. It is found that under a given loading histroy, the ERR is a function of time, and its maximum value should correspond with the critical state for delamination to propagate. Following William's work, the explicit expressions of ERR for DCB experimental configurations to measure the interfacial fracture toughness have been obtained through the classical beam assumption.

The Hamiltonian Equations in Some Mathematics and Physics Problems

CHEN Yong, ZHENG Yu, ZHANG Hong-qing

2003, 24(1): 19-24.

Abstract(2164) PDF(935)

Abstract:
Some new Hamiltonian canonical system are discussed for a series of partial differential equations in Mathematics and Physics. It includes the Hamiltonian formalism for the symmetry 2-order equation with the variable coefficients, the new nonhomogeneous Hamiltonian representation for 4-order symmetry equation with constant coefficients, the one of MKdV equation and KP equation.

Large Deformation of Circular Membrane Under the Concentrated Force

CHEN Shan-lin, ZHENG Zhou-lian

2003, 24(1): 25-28.

Abstract(4632) PDF(862)

Abstract:
Making use of basic equation of large deformation of circular membrane under the concetrated force and its boundary conditions and Hencky transformation, the problems of nonlinear boundary condition were solved. The Hencky transformation was extended and a precise solution of large deformation of circular membrane under the concentrated force has been obtained.

The Alternating Segment Crank-Nicolson Method for Solving Convection-Diffusion Equation With Variable Coeffcient

WANG Wen-qia

2003, 24(1): 29-38.

Abstract(2888) PDF(868)

Abstract:
A new discrete approximation to the convection term of the covection diffusion equation was constructed in Saul. yev type difference scheme,then the alternating segment Crank-Nicolson (ASC-N) method for solving the convection-diffusion equation with variable coefficient was developed. The ASC-N method is unconditionally stable. Numerical experiment shows that this method has the obvious property of parallelism and accuracy. The method can be used directly on parallel computers.

Dynamic Response Optimization Design for Engineering Structures Based on Reliability

DAI Jun, CHEN Jian-jun, LI Yong-gong, ZHAO Zhu-qing, MA Hong-bo

2003, 24(1): 39-46.

Abstract(2078) PDF(633)

Abstract:
In many practical structures, physical parameters of material and applied loads have random property. To optimize this kind of structures, an optimum mathematical model was built. This model has reliability constranints on dymanic stress and displacement and upper & lower limits of the design variables. The numerical characteristic of dynamic response and sensitivity of dynamic response based on probability of structure were deduced repectively. By equivalent disposing, the reliability constraints were changed into conventional forms. The SUMT method was used in the optimization process. Two examples illustrate the correctness and practicability of the optimum model and solving approach.

On a Generalized Taylor Theorem:a Rational Proof of the Validity of the So-Called Homotopy Analysis Method

LIAO Shi-jun

2003, 24(1): 47-54.

Abstract(2199) PDF(1012)

Abstract:
A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can be applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.

Visco-Elastic Systems Under Both Deterministic Harmonic and Random Excitation

XU Wei, RONG Hai-wu, FANG Tong

2003, 24(1): 55-61.

Abstract(2112) PDF(919)

Abstract:
The response of visco-elastic system to combined deterministic harmonic and random excitation was investigated. The method of harmonic balance and the method of stochastic averaging were used to determine the response of the system. The theoretical analysis was verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increase, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions and jumps may exist.

Poincaré-Cartan Integral Invariants of Birkhoffian Systems

GUO Yong-xin, SHANG Mei, LUO Shao-kai

2003, 24(1): 62-66.

Abstract(2400) PDF(719)

Abstract:
Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincar -Cartan's type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincar's type is found.

Switched Processes Generalized Mandelbrot Sets for Complex Index Number

WANG Xing-yuan

2003, 24(1): 67-73.

Abstract(2353) PDF(548)

Abstract:
According to the switched complex mapping proposed by the author, the method constructing the switched processes generalized M (Mandelbrot ) sets was elaborated, and a series of the switched processes generalized M sets for complex index number were constructed. The construction characteristics of the generalized M sets were expounded according to the analysis of the algorithm constructing the generalized M sets. On the basis of what has already been achieved, the trajectories of a starting point in the complex C-plane under the switched mapping were researched into. The results show that the switched processesgeneralized M sets hare te fractal feature the construction characteristics of the switched processes generalized M sets are dependent on the complex index number w and the switched variable r₀, and the reason which results in the discontinuity of the switched processes generalized M sets is the discontinuity of choice of the principal range of the phase angle.

The Asymptotic Theory of Initial Value Problems for Semilinear Perturbed Wave Equations in Two Space Dimensions

LAI Shao-yong, FU Qing-long

2003, 24(1): 74-82.

Abstract(2376) PDF(687)

Abstract:
The asymptotic theory of initial value problems for semilinear wave equations in two space dimensions was dealt with. The well-posedness and vaildity of formal approximations on a long time scale were discussed in the twice continuous classical space. These results describe the behavior of long time existence for the validity of formal approximations. And an application of the asymptotic theory is given to analyze a spacial wave equation in two space dimensions.

Sensitivity Analysis Based on Lanczos Algorithm in Structural Dynamics

LI Shu, WANG Bo, HU Ji-zhong

2003, 24(1): 83-88.

Abstract(2433) PDF(707)

Abstract:
The sensitivity calculating formulas in structural dynamics was developed by utilizing the mathematical theorem and new definitions of sensitivities. So the singularity problem of sensitivity with repeated eigenvalues is solved completely. To improve the computational efficient, the reduction system is obtained based on Lanczos vectors. After incorporating the mathematical theory with the Lanczos algorithm, the approximate sensitivity solution can be obtained. A numerical example is presented to illustrate the performance of the method.

Existence of Solutions for Nonlinear Elliptic Boundary Value Problems

SHAO Rong, NIU xin, SHEN Zu-he

2003, 24(1): 89-97.

Abstract(1965) PDF(701)

Abstract:
Hilbert space method is applied to a class of semilinear second order elliptic boundary value problems and the existence of solutions is obtained with some restrictions.

Fluid Boundary Element Method and Orthogonal Transform of Double Complex Variables

LUO Yi-ying

2003, 24(1): 98-104.

Abstract(2288) PDF(577)

Abstract:
A concept of orthogonal double complex variables space was put forward, Its corresponding operation rules, the concept of analytic function and conformal transform and its foreground for application of fluid boundary element method are explained. The results indicate that the concept and special mark may develop the plane complex into space, and extensive application in physics is got.

On the Existence of Periodic Solutions for Nonlinear System With Multiple Delays

CAO Xian-bing

2003, 24(1): 105-110.

Abstract(2340) PDF(707)

Abstract:
The existence of T periodic solutions of the nonlinear system with multiple delays is studied. By using the topological degree method, sufficient conditions are obtained for the existence of T-periodic solutions. As an application, the existence of positive periodic solution for a logarithmic population model is established under some conditions.