Applied Mathematics and Mechanics (English Edition)

INVESTIGATION OF THE DYNAMIC BEHAVIOR OF TWO COLLINEAR ANTI-PLANE SHEAR CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF SPACES BY A NEW METOHD

周振功;王彪

2003, 24(1): 1-13.

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The dynamic behavior of two collinear anti-plane shear cracks in a piezoelectric 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 thickness of the piezoelectric layer and the constants of the materials upon the dynamic stress intensity factor of cracks.

ON THE CALCULATION OF ENERGY RELEASE RATE FOR VISCOELASTIC CRACKED LAMINATES

刘玉岚;王彪;王殿富

2003, 24(1): 14-21.

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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 length a + 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 history, 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

陈勇;郑宇;张鸿庆

2003, 24(1): 22-27.

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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 second-order equation with the variable coefficients, the new nonhomogeneous Hamiltonian representation for fourth-order symmetry equation with constant coefficients, the one of MKdV equation and KP equation.

LARGE DEFORMATION OF CIRCULAR MEMBRANE UNDER THE CONCENTRATED FORCE

陈山林;郑周练

2003, 24(1): 28-31.

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Making use of basic equation of large deformation of circular membrane under the concentrated force and its boundary conditions and Hencky transformation, the problems of nonlinear boundary condition were solved. The Hencky transformation was extended and a exact 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 COEFFICIENT

王文洽

2003, 24(1): 32-42.

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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

戴君;陈建军;李永公;赵竹青;马洪波

2003, 24(1): 43-52.

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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 constraints on dynamic 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 respectively. 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 HOMOTOPY ANALYSIS METHOD

廖世俊

2003, 24(1): 53-60.

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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

徐伟;戎海武;方同

2003, 24(1): 61-67.

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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

郭永新;尚玫;罗绍凯

2003, 24(1): 68-72.

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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 Poincaré-Cartan’s type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré’s type is found.

SWITCHED PROCESSES GENERALIZED MANDELBROT SETS FOR COMPLEX INDEX NUMBER

王兴元

2003, 24(1): 73-81.

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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 processes generalized M sets have the 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

赖绍永;胡青龙

2003, 24(1): 82-91.

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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 special wave equation in two space dimensions.

SENSITIVITY ANALYSIS BASED ON LANCZOS ALGORITHM IN STRUCTURAL DYNAMICS

李书;王波;胡继忠

2003, 24(1): 92-98.

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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 efficiency, 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

邵荣;牛欣;沈祖和

2003, 24(1): 99-108.

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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

罗义银

2003, 24(1): 109-116.

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A concept of orthogonal double function and its complex variables space was put forward. Its corresponding operation rules, the concept of analytic function and conformal transform are established. And using this concept discussed its foreground for application of fluid boundary element method. In results, this concept and special marks may be to enlarge the plane complex into three-dimensional space, and then extensive application may be obtained in physics and mathematics.

ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR NONLINEAR SYSTEM WITH MULTIPLE DELAYS

曹显兵

2003, 24(1): 117-122.

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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.

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