应用数学和力学

2016 Vol. 37, No. 7

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Mathematics Mechanization and Tao of Mathematics

ZHANG Hong-qing, MEI Jian-qin

2016, 37(7): 665-677. doi: 10.21656/1000-0887.370115

Abstract(1349) PDF(1343)

Abstract:
The Chinese traditional mathematics, with the Chinese traditional culture as the background and characterized by mechanization and Tao (principle and law) of mathematics, once made glorious achievements. These ideas, used in the fields such as the computer science, the soliton theory and the quantum field theory, can not only yield the existent results systematically, but also open up some new research areas and breed major breakthroughs.

Wave Lengths of Periodic Waves for the Vakhnenko Equation

GUO Li-na, CHEN Ai-yong, HUANG Wen-tao

2016, 37(7): 678-690. doi: 10.21656/1000-0887.370020

Abstract(900) PDF(876)

Abstract:
The wave lengths of smooth periodic traveling wave solutions to the Vakhnenko equation were studied. The Vakhnenko equation was reduced to a planar polynomial differential system through the transformation of variables. The polynomial differential system was treated with the critical period bifurcation method based on the dynamical system theory. The main results involve the monotonicity properties of periodic function T(h) (or wave length function λ(a)). In comparison with the wave length for the KdV equation, wave length function λ(a) monotonically decreases to a finite value rather than monotonically increases to infinity. This shows that, for fixed wave speed c, there exist no smooth periodic wave solutions with arbitrarily small wave lengths or arbitrarily large wave lengths, to the Vakhnenko equation.

Asymptotic Solutions to a Class of Singular Perturbation Burning Models

SHI Juan-rong, MO Jia-qi

2016, 37(7): 691-698. doi: 10.21656/1000-0887.360293

Abstract(964) PDF(544)

Abstract:
A class of nonlinear singularly perturbed burning models with two parameters were discussed. Firstly, the outer solution to the burning model was constructed with the perturbation method. Secondly, through the introduction of a stretched variable, the initial layer correction term of the solution to the burning model was constructed. Then the multi-scale method and the composite expansion method were used to build the boundary layer correction term of the model solution and find the asymptotic solution to the original initial boundary value problem. Finally, the uniform validity of the obtained asymptotic solution was proved according to the theory of differential inequalities. The proposed solving method for this class of nonlinear singularly perturbed burning models is convenient and practicable.

Kinds of Periodic Contact Problems of 1D Hexagonal Quasicrystals

MA Xiao-dan, LI Xing

2016, 37(7): 699-709. doi: 10.21656/1000-0887.370029

Abstract(1138) PDF(575)

Abstract:
With the complex variable method, 2 kinds of periodic contact problems (the frictionless and the adhesive periodic contact problems) of 1D hexagonal quasicrystals in the aperiodic plane were discussed. Based on the Hilbert kernel integral formula, the closed-form solutions were obtained to the 2 kinds of periodic contact problems. In the frictionless case, the explicit solutions of contact stresses were given under the actions of 3 common basal punches (the straight horizontal, the straight inclined and the circular basal punches). In the adhesive case, the analytic solutions of contact stresses were given with the wedge-shaped periodic displacement at the contact boundary. If the effect of the phason field is neglected, the obtained results will match well with the corresponding solutions to the periodic contact problems of orthogonal anisotropic materials.

Dual Projective Synchronization of Fractional-Order Chaotic Systems With a Linear Controller

ZHANG Wei-wei, WU Ran-chao

2016, 37(7): 710-717. doi: 10.21656/1000-0887.360356

Abstract(960) PDF(658)

Abstract:
The dual synchronization of fractional-order chaotic systems is a new method of synchronization. There was few study on the dual projective synchronization of fractional-order chaotic systems. With a linear signal, the dual projective synchronization of fractional-order chaotic systems was investigated. Based on the stability theory of the fractional-order systems, a general method was proposed. Furthermore, the work extends the previous research of dual synchronization. Finally, the dual projective synchronizations of the fractional-order Van der Pol system and the fractional-order Willis system were numerically simulated. The corresponding results show the effectiveness of the present method.

Sensitivity Analysis of Production in Fractured Shale Gas Reservoirs

ZHAO Tian-wu, SHEN Yong-kuan, CHEN Jie, LIU Chuang, LIU He, WU Heng-an

2016, 37(7): 718-728. doi: 10.21656/1000-0887.360357

Abstract(1035) PDF(2660)

Abstract:
In view of gas seepage and effect of desorption, shale compressibility and slippage effect, the full coupling mathematical model for production of the fracture system in shale gas reservoirs was established. The matrix and fracture flow field was solved with the finite element method. The simulation results were consistent with the published field data from the Barnett Shale in the Newark East field. Based on this model, the effects of the fracture’s half length, number, spacing and gas desorption on gas production were simulated. The numerical results show that the fracture parameters are interactive. The influence of the half length on production enhancement is dominant. The optimal number of fractures is related to the fracture’s half length. In the middle and late parts of the gas production period, the effect of desorption provides considerable production. The presented model makes an effective theoretical tool for the optimization of shale fracturing design.

Numerical Simulation of High-Speed Crack Propagating and Branching Phenomena

GU Xin-bao, ZHOU Xiao-ping, XU Xiao

2016, 37(7): 729-739. doi: 10.21656/1000-0887.360310

Abstract(1343) PDF(786)

Abstract:
The peridynamic theory was first introduced, then 2 examples of highspeed crack propagating and branching phenomena were given and investigated. The effects of peridynamic parameters including the neighbourhood radius and the grid spacing, and such external parameters as the material elastic modulus, the material density and the temperature difference, on the crack propagating velocity and the crack branching angle were analyzed. It is found from the numerical results that the crack propagating velocity decreases and the crack branching angle increases with the neighbourhood radius; both the crack propagating velocity and the crack branching angle decrease with the grid spacing; the crack branching length in the material of a smaller elastic modulus and a larger density is longer; the crack propagating velocity increases with the elastic modulus difference; the crack propagating velocity increases as the materials’density difference decreases, and decreases with the temperature difference. Moreover, the crack propagating and branching process can be simulated with the peridynamic method spontaneously, without any outer criterion and preset crack propagating paths. Therefore, peridynamics has natural advantages in the simulation of highspeed crack propagating and branching phenomena.

Solutions for a Circular Hole With Edge Cracks Under Shear Load

DUAN Shi-jie, LIU Shu-hong

2016, 37(7): 740-747. doi: 10.21656/1000-0887.360352

Abstract(1035) PDF(705)

Abstract:
A 2D mechanical analysis was performed on an infinite plate containing a circular hole with two collinear edge cracks of unequal lengths, under uniformly distributed shear load at infinity. Based on the complex variable function method, the analytic solutions of stress functions and stress intensity factors were obtained. Through an numerical example, the stress distributions along the coordinate axes and the hole edge were given in the graphical form, and the stress intensity factors were also calculated. The results show that, there is obvious stress concentration near the hole and the cracks, and the stress values far from the defects tend towards the applied load, which conforms to the Saint-Venant’s principle. In addition, the results from the finite element simulation agree well with the analytic solutions, to prove the correctness of the theoretical derivation.

A Method for Evaluating Material Forces and Crack Forces in Ceramic Laminates

CHEN Chang-rong

2016, 37(7): 748-755. doi: 10.21656/1000-0887.370088

Abstract(958) PDF(987)

Abstract:
Characteristics of J_far(0), J_far(a), J_far(a)-J_far(0)and J_tip were analyzed for ceramic laminates under bending loads based on the J-integral theory. Here J_far(0) and J_far(a) were the far-field J-integrals corresponding to crack lengths 0 and a respectively. The crack was perpendicular to the interfaces. A basic assumption was that the crack length was small compared with the laminate thickness, and the stress and strain fields in the region far from the crack were little influenced by the crack. Both J_far(0) and J_far(a) were path-dependent, because the lengths of the interfaces enclosed by the path of integration varied with the path. However, J_far(a)-J_far(0) became path-independent when the path was far from the crack. J_far(a)-J_far(0) was seen as a parameter to represent the global driving force for fracture. The purpose is to make the present method available to evaluate the inhibiting or boosting effects of material inhomogeneities on the crack tip driving force by J_tip-(J_far(a)-J_far(0)).

Refined Equations for Functionally Graded Material Plates Under Bending-Tension Coupling

HU Chao, ZHENG Ri-heng, SUN Xu-feng, ZHOU Chuan-ping

2016, 37(7): 756-765. doi: 10.21656/1000-0887.370097

Abstract(834) PDF(658)

Abstract:
Based on the theory of elasticity for inhomogeneous media, the spectral compositions of operators and the Vieta’s theorem of algebra were applied, and the bending-tension coupling problem of plates of functionally graded material (FGM) was investigated. The refined equations for FGM plates under bending-tension coupling were given. It is shown that, unlike those for the isotropic plate under bending and tension, both the generalized displacement function and the shear function describing the bending stress state and the tension stress state for FGM plates are coupled. Since the derivation of the governing equations was conducted without prior assumptions, the proposed equations for FGM plates can be regarded as exact ones. The work also found out the coupling mechanism and the response structure. The proposed governing equations can be used to analyze the stress of the plate-like FGM structures for thermal protection, and to advance the lightweight design.

The Collision Model for Indoor Fine Particles and the Collision Results

WANG Xiu-juan, LI Can

2016, 37(7): 766-774. doi: 10.21656/1000-0887.360349

Abstract(1120) PDF(568)

Abstract:
In order to study the coagulation mechanism for indoor fine particles, the number of particles colliding with walls and the collision probability between particles were calculated based on the idea of statistical mean and random motion of particles. The collision results were also studied with the qualitative and quantitative analysis methods. The relationship between the final compressive deformation and the initial velocity, and that between the collision efficiency and the particle diameter, were obtained for 0.3 μm DOP particles. The results show that the coagulation possibility of particles decreases with the particle velocity, the finer particles are easier to coagulate after collision, and the collision efficiency increases with the particle diameter.