应用数学和力学

2009 Vol. 30, No. 5

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Existence and Breaking Property of Real Loop-Solutions of Two Nonlinear Wave Equations

LI Ji-bin

2009, 30(5): 505-514. doi: 10.3879/j.issn.1000-0887.2009.05.001

Abstract(1271) PDF(810)

Abstract:
Dynamical analysis revealed that for some nonlinear wave equations, loop-and inverted loop-soliton solutions are merely visual artifacts. So called loop-soliton solution consists of three solutions which is not one real solution. Whether or not there exist some nonlinear wave equations for which there exists a/real0 loop-solution? If yes, what are their precise parametric representations of these loop traveling wave solutions? These problems are answered.

General Forms of Elastic-Plastic Matching Equations for Mode-Ⅲ Cracks Near the Crack Line

YI Zhi-jian, ZHAO Chao-hua, YANG Qing-guo, PENG Kai, HUANG Zong-ming

2009, 30(5): 515-524. doi: 10.3879/j.issn.1000-0887.2009.05.002

Abstract(1305) PDF(900)

Abstract:
To address mode-Ⅲ crack problems under elastic-perfectly plastic condition, the matching procedures of the crack line analysis method was summarized and refined to give the general forms and formulation steps of plastic field, elastic-plastic boundary and elastic-plastic matching equations near the crack line. The work unified the different-condition mode-Ⅲ crack problems as determining 4 integral constants through 4 matching equations. An example was demonstrated to verify the correctness, conciseness and generality of the procedure.

Mixed Spectral Method for Exterior Problem of Navier-Stokes Equations by Using Generalized Laguerre Functions

JIAO Yu-jian, GUO Ben-yu

2009, 30(5): 525-537. doi: 10.3879/j.issn.1000-0887.2009.05.003

Abstract(1640) PDF(897)

Abstract:
The mixed spectral method using generalized Laguerre functions for exterior problems of partial differential equations of fourth order was investigated. A mixed spectral scheme was provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.

Research on New Model of Long Fatigue Crack Propagation Rates for Structures

LIU Jian-tao, DU Ping-an, HUANG Ming-jing, ZHOU Qing

2009, 30(5): 538-546. doi: 10.3879/j.issn.1000-0887.2009.05.004

Abstract(1301) PDF(962)

Abstract:
By comparison of the characteristics of existing models for long fatigue crack propagation rates, a new model called generalized passivation-lancet model for long fatigue crack propagation rates (GPLFCPR) and the general formula for characterizing the process of crack growth rates were put forward based on the deduction of passivation-lancet theory. The GPLFCPR model overcomes the disadvantages of the existing models and could describe the rules of whole fatigue crack growth process from the cracking threshold to the critical fracturing point effectively with explicit physical meaning and also reflects the influence of material characteristics, such as strength parameters, fracture parameters and heat treatment, etc. Experimental results, testing with LZ50 stell, AlZnMgCu 0.5, 0.5Cr 0.5Mo 0.25V steel and so on, are used to verify the new model, which show great consistency. The GPLFCPR model owns much value for theoretical research and practical applications.

Researches on Interface Crack Problems for Mode Ⅱ of Double Dissimilar Orthotropic Composite Materials

YANG Wei-yang, ZHANG Shao-qin, LI Jun-lin, MA Yu-lan

2009, 30(5): 547-555. doi: 10.3879/j.issn.1000-0887.2009.05.005

Abstract(1498) PDF(1230)

Abstract:
The fracture problems near interface crack tip for mode Ⅱ of double dissimilar orthotropic composite materials are studied. The mechanical models of interface crack for mode Ⅱ were given. By translating the governing equations into generalized bi-harmonic equations, the stress functions containing two stress singularity exponents were derived with the help of a complex function method. Based on the boundary conditions, a system of non-homogeneous linear equations was found. Two real stress singularity exponents were determined under appropriate conditions of bi-material engineering parameters through solving this system. According to the theorem of limit uniqueness, both the formulae of stress intensity factors and theoretical solutions of stress field near interface crack tip were deduced. When the two orthotropic materials are the same, the stress singularity exponents, stress intensity factors and stresses for mode Ⅱ crack of orthotropic single material were obtained.

Differential Characteristic Set Algorithm for the Complete Symmetry Classification of (Partial) Differential Equations

Temuer Chaolu, BAI Yu-shan

2009, 30(5): 556-566. doi: 10.3879/j.issn.1000-0887.2009.05.006

Abstract(1718) PDF(1084)

Abstract:
A differential polynomial characteristic set algorithm for the complete symmetry classification of (partial) differe ntial equations with some parameters was given, which made the solution of the complete symmetry classification problem for (partial) differential equations become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter were presented. This is a new application of differential form characteristic set algrithmc (differential form Wu.s method) in field of differential equations.

Vibrations of FGM Thin Cylindrical Shells With Exponential Volume Fraction Law

Abdul Ghafar Shah, Tahir Mahmood, Muhammad Nawaz Naeem

2009, 30(5): 567-574. doi: 10.3879/j.issn.1000-0887.2009.05.007

Abstract(1645) PDF(931)

Abstract:
The influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells was studied. Material properties in the shell thickness direction were graded in a ccordance with the exponential law. Expressions for the strain-displacement and curvature-displacement relationships were taken from Love.s thin shell theory. The Rayleigh-Ritz approach was used to derive the shell eigenfr equency equation. Axial modal dependence is assumed in the characteristic beam functions. Natural frequencies of the shells are observed to be dependent on the constituent volume fractions. The results are compared with those available in the literature for the validity of the present methodology.

Micropolar Mixture Theory of Multicomponent Porous Media

HUANG Lu, ZHAO Cheng-gang

2009, 30(5): 575-586. doi: 10.3879/j.issn.1000-0887.2009.05.008

Abstract(1464) PDF(987)

Abstract:
A mixture theory is developed for multicomponent micropolar porous media by combination of the hybrid mixture theory and micropolar continuum theory. This system was modeled as multicomponent micropolar elastic solids saturated with multicomponent micropolar viscous fluids. Balance equations were given through the mixture theory. Constitutive equations were developed based on the second law of thermodynamic and constitutive assumptions. For taking account of compressibility of solid phase, volume fraction of fluid as an independent state variable was introduced in free energy function, and the dynamic compatibility condition was obtained to restrict the change of pressure difference on solid and fluid interface. The constructed constitutive equations were used to close the field equations. The linear field equations were obtained with the linearization procedure, and the micropolar thermo-hydro-mechanical component transport model was established finally. This model can be applied to some practical problems, such as contaminant, drug and pesticide transport. When the proposed model is supposed to be the porous media, including both fluid and solid are single-component, it will almost agree with Eringen's model.

Theoretical Study of Void Closure in Nonlinear Plastic Materials

ZHANG Xiao-xun, CUI Zhen-shan

2009, 30(5): 587-597. doi: 10.3879/j.issn.1000-0887.2009.05.009

Abstract(1271) PDF(970)

Abstract:
Void closing from a spherical shape to a crack is investigated quantitatively in the present study. The constitutive relation of the void-free matrix was assumed to obey the Norton power law. A representative volume element (RVE) which includes matrix and void was employed and a Rayleigh-Ritz procedure was developed to study the deformation-rates of a spherical void and a pennyshaped crack. Based on an approximate interpolation scheme, an analytical model for void closure in nonlinear plastic materials was established. It is found that the local plastic flows of the matrix material are the main mechanism of void deformation. It is also shown that the relative void volume during the deformation depends on the Norton exponent, on the far-field stress triaxiality, as well as on the far-field effective strain. The predictions of void closure using the present model are compared with the corresponding results in the literature, arriving at good agreement. The model for void closure provides a novel way for process design and optimization in terms of elimination of voids in billets because the model for void closure can be easily applied in the CAE(computer aided engineering) analysis.

Eigen Theory of Static Electromagnetic Field for Anisotropic Media

GUO Shao-hua

2009, 30(5): 598-606. doi: 10.3879/j.issn.1000-0887.2009.05.010

Abstract(1265) PDF(1084)

Abstract:
The static electromagnetic fields were studied here based on the standard spaces of the physical presentation, and the modal equations of static electromagnetic fields for anisotropic media were deduced. By introducing a set of new potential functions of order 1, several novel theoretical results are obtained: the electric or magnetic potentials are scalar for isotropic media, and vector for anisotropic media. The amplitude and direction of the vector potentials are related to the anisotropic subspaces. Based on these results, the laws of static electromagnetic fields for anisotropic media are discussed.

Singularly Perturbed Solution for Semilinear Reaction Diffusion Equations With Two Parameters

MO Jia-qi, LIU Shu-de

2009, 30(5): 607-612. doi: 10.3879/j.issn.1000-0887.2009.05.011

Abstract(1518) PDF(993)

Abstract:
A class of singularly perturbed initial boundary value problem for semilinear reaction diffusion equations with two parameters was considered. Under suitable conditions, using theory of differential inequalities, the existence and asymptotic behavior of solution for initial boundary value problem were studied.

An SQP Algorithm for Mathematical Programs With Nonlinear Complementarity Constraints

ZHU Zhi-bin, JIAN Jin-bao, ZHANG Cong

2009, 30(5): 613-622. doi: 10.3879/j.issn.1000-0887.2009.05.012

Abstract(1307) PDF(1053)

Abstract:
A successive approximation and smooth SQP method for mathematical programs with nonlinear complementarity constraints (MPCC) is described. A class of smooth programs to approximate the MPCC was introduced. Using an l₁ penalty function, the line search assures the global convergence, while superlinear convergence rate is shown under strictly complementary conditions and the second order sufficient condition. Moreover, it was proved that the current iterated point is an exact stationary point of the MPEC when the algorithm terminates finitely.

Comments on the Paper “Three-Point Explicit Compact Difference Scheme With Arbitrary Order of Accuracy and Its Application in CFD”

ZHANG Hong-na, YU Bo, WANG Yi, WEI Jin-jia, LI Feng-chen

2009, 30(5): 623-630. doi: 10.3879/j.issn.1000-0887.2009.05.013

Abstract(1230) PDF(801)

Abstract:
It was proved the explicit compact difference scheme, which was proposed by Lin et al. (Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD. Applied Mathem atics and M echan ics, 2007, 28(7):843-852) has the same performance as the conventional finite difference schemes and it is actually another expression form of the conventional finite difference schemes. Though the proposed expression doesn't have advantages of a compact difference scheme, it is easier to be obtained and implemented in a code compared to the conventional expression in which the coefficients should be obtained by solving equations especially for higher accurate schemes.