Applied Mathematics and Mechanics (English Edition)

Existence and breaking property of real loop-solutions of two nonlinear wave equations

李继彬

2009, 30(5): 537-547. doi:10.1007/s10483-009-0501-4

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Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-soliton solutions are actually visual artifacts. The so-called loopsoliton solution consists of three solutions, and is not a real solution. This paper answers the question as to whether or not nonlinear wave equations exist for which a “real” loopsolution exists, and if so, what are the precise parametric representations of these loop traveling wave solutions.

General forms of elastic-plastic matching equations for mode-III cracks near crack line

易志坚赵朝华杨庆国彭凯黄宗明

2009, 30(5): 549-559. doi:10.1007/s10483-009-0502-2

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Crack line analysis is an effective way to solve elastic-plastic crack problems. Application of the method does not need the traditional small-scale yielding conditions and can obtain sufficiently accurate solutions near the crack line. To address mode- III crack problems under the perfect elastic-plastic condition, matching procedures of the crack line analysis method are summarized and refined to give general forms and formulation steps of plastic field, elastic-plastic boundary, and elastic-plastic matching equations near the crack line. The research unifies mode-III crack problems under different conditions into a problem of determining four integral constants with four matching equations. An example is given to verify correctness, conciseness, and generality of the procedure.

Mixed spectral method for exterior problems of Navier-Stokes equations by using generalized Laguerre functions

焦裕建郭本瑜

2009, 30(5): 561-574. doi:10.1007/s10483-009-0503-z

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In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.

New model of propagation rates of long crack due to structure fatigue

刘建涛杜平安黄明镜周晴

2009, 30(5): 575-584. doi:10.1007/s10483-009-0504-y

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By comparison of the characteristics of existing models for long fatigue crack propagation rates, a new model, called the generalized passivation-lancet model for long fatigue crack propagation rates (GPLFCPR), and a general formula for characterizing the process of crack growth rates are proposed based on the passivation-lancet theory. The GPLFCPR model overcomes disadvantages of the existing models and can describe the rules of the entire fatigue crack growth process from the cracking threshold to the critical fracturing point effectively with explicit physical meaning. It also reflects the influence of material characteristics, such as strength parameters, fracture parameters and heat treatment. Experimental results obtained by testing LZ50 steel, AlZnMgCu0.5, 0.5Cr0.5Mo0.25V steel, etc., show good consistency with the new model. The GPLFCPR model is valuable in theoretical research and practical applications.

Interface crack problems for mode II of double dissimilar orthotropic composite materials

杨维阳张少琴李俊林马玉兰

2009, 30(5): 585-594. doi:10.1007/s10483-009-0505-z

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

Differential characteristic set algorithm for the complete symmetry classification of partial differential equations

特木尔朝鲁白玉山

2009, 30(5): 595-606. doi:10.1007/s10483-009-0506-6

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In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations (PDEs) with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu’s method, in differential equations

Vibrations of FGM thin cylindrical shells with exponential volume fraction law

Abdul Ghafar Shah Tahir Mahmood Muhammad Nawaz Naeem

2009, 30(5): 607-615. doi:10.1007/s10483-009-0507-x

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In this paper, the influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells is studied. Material properties in the shell thickness direction are graded in accordance with the exponential law. Expressions for the strain-displacement and curvature-displacement relationships are taken from Love’s thin shell theory. The Rayleigh-Ritz approach is used to derive the shell eigenfrequency 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.

A micropolar mixture theory of multi-component porous media

黄璐赵成刚

2009, 30(5): 617-630. doi:10.1007/s10483-009-0508-x

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

Theoretical study of void closure in nonlinear plastic materials

张效迅崔振山

2009, 30(5): 631-642. doi:10.1007/s10483-009-0509-7

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Void closing from a spherical shape to a crack is investigated quantitatively in the present study. The constitutive relation of the void-free matrix is assumed to obey the Norton power law. A representative volume element (RVE) which includes matrix and void is employed and a Rayleigh-Ritz procedure is developed to study the deformation-rates of a spherical void and a penny-shaped crack. Based on an approximate interpolation scheme, an analytical model for void closure in nonlinear plastic materials is 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, showing good agreement. The model for void closure provides a novel way for process design and ptimization in terms of elimination of voids in billets because the model for void closure can easily be applied in the CAE analysis.

An eigen theory of static electromagnetic field for anisotropic media

郭少华

2009, 30(5): 643-652. doi:10.1007/s10483-009-0510-y

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Static electromagnetic fields are studied based on standard spaces of the physical presentation, and the modal equations of static electromagnetic fields for anisotropic media are derived. By introducing a new set of first-order potential functions, several novel theoretical results are obtained. It is found that, for isotropic media, electric or magnetic potentials are scalar; while for anisotropic media, they are vectors. Magnitude and direction of the vector potentials are related to the anisotropic subspaces. Based on these results, we discuss the laws of static electromagnetic fields for anisotropic media.

Singularly perturbed solution to semilinear reaction diffusion equations with two parameters

莫嘉琪刘树德

2009, 30(5): 653-658. doi:10.1007/s10483-009-0511-x

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

An SQP algorithm for mathematical programs with nonlinear complementarity constraints

朱志斌简金宝张聪

2009, 30(5): 659-668. doi:10.1007/s10483-009-0512-x

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In this paper, we describe a successive approximation and smooth sequential quadratic programming (SQP) method for mathematical programs with nonlinear complementarity constraints (MPCC). We introduce a class of smooth programs to approximate the MPCC. Using an l1 penalty function, the line search assures global convergence, while the superlinear convergence rate is shown under the strictly complementary and second-order sufficient conditions. Moreover, we prove that the current iterated point is an exact stationary point of the mathematical programs with equilibrium constraints (MPEC) when the algorithm terminates finitely.

Comments on Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

张红娜宇波王艺魏进家李凤臣

2009, 30(5): 669-676. doi:10.1007/s10483-009-0513-7

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The explicit compact difference scheme, proposed in Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD by Lin et al., published in Applied Mathematics and Mechanics (English Edition), 2007, 28(7), 943–953, has the same performance as the conventional finite difference schemes. It is just another expression of the conventional finite difference schemes. The proposed expression does not have the advantages of a compact difference scheme. Nonetheless, we can more easily obtain and implement compared with the conventional expression in which the coefficients can only be obtained by solving equations, especially for higher accurate schemes.

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