Showing papers in "Applied Mathematics and Mechanics-english Edition in 2001"
TL;DR: In this article, a mathematical model of a kind of complicated financial system, and all possible things that the model shows in the operation of our country's macro-financial system are analyzed.
Abstract: Based on the work discussed on the former study, this article first starts from the mathematical model of a kind of complicated financial system, and analyses all possible things that the model shows in the operation of our country's macro-financial system: balance, stable periodic, fractal, Hopf-bifurcation, the relationship between parameters and Hopf-bifurcation, and chaotic motion etc. By the changes of parameters of all economic meanings, the conditions on which the complicated behaviors occur in such a financial system, and the influence of the adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior have been analyzed. This study will deepen people's understanding of the lever function of all kinds of financial policies.
147 citations
TL;DR: In this article, the authors investigated the laws of these factors varying with the slope gradient by using the kinematic wave theory and showed that the critical slope gradient of soil erosion is dependent on grain size, soil bulk density, surface roughness, runoff length, net rain excess, and the friction coefficient of soil.
Abstract: The main factors influencing soil erosion include the net rain excess, the water depth, the velocity, the shear stress of overland flows, and the erosion-resisting capacity of soil. The laws of these factors varying with the slope gradient were investigated by using the kinematic wave theory. Furthermore, the critical slope gradient of erosion was driven. The analysis shows that the critical slope gradient of soil erosion is dependent on grain size, soil bulk density, surface roughness, runoff length, net rain excess, and the friction coefficient of soil, etc. The critical slope gradient has been estimated theoretically with its range between 41.5° ∽ 50°.
48 citations
TL;DR: In this paper, a static linear interval finite element method was presented to solve the non-random uncertain structures, where the two parameters, median and deviation, were used to represent the uncertainties of interval variables.
Abstract: When the uncertainties of structures may be bounded in intervals, through some suitable discretization, interval finite element method can be constructed by combining the interval analysis with the traditional finite element method (FEM). The two parameters, median and deviation, were used to represent the uncertainties of interval variables. Based on the arithmetic rules of intervals, some properties and arithmetic rules of interval variables were demonstrated. Combining the procedure of interval analysis with FEM, a static linear interval finite element method was presented to solve the non-random uncertain structures. The solving of the characteristic parameters of n-freedom uncertain displacement field of the static governing equation was transformed into 2n- order linear equations. It is shown by a numerical example that the proposed method is practical and effective.
39 citations
TL;DR: In this article, the authors studied two types of evolution equations, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term.
Abstract: The “trial function method” (TFM for short) and a routine way in finding traveling wave solutions to some nonlinear partial differential equations (PDE for short) wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.
35 citations
TL;DR: In this paper, the dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated, and a new numerical method is proposed to avoid storing all history data.
Abstract: The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The governing equation of motion was derived as a weakly singular Volterra integro-partial-differential equation, and it was simplified into a weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. In terms of the averaging method, the dynamical stability was analyzed. A new numerical method is proposed to avoid storing all history data. Numerical examples are presented and the numerical results agree with the analytical ones.
34 citations
TL;DR: In this article, the explicit representations for tensorial Fourier expansion of 3-D crystal orientation distribution functions (CODFs) are established, and it is shown that in most cases of symmetry the restricted forms of tensor Fourier expansions of 3D CODFs contain remarkably reduced numbers of mth-order irreducible tensors than the number 2m + 1.
Abstract: The explicit representations for tensorial Fourier expansion of 3-D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3-D ODF make up just a single irreducible mth-order tensor, the coefficients in the mth term of the Fourier expansion of a 3-D CODF constitute generally so many as 2m + 1 irreducible mth-order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3-D CODFs imposed by various micro- and macro-scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3-D CODFs contain remarkably reduced numbers of mth-order irreducible tensors than the number 2m + 1. These results are based on the restricted forms of irreducible tensors imposed by various point-group symmetries, which are also thoroughly investigated in the present part in both 2- and 3-D spaces.
23 citations
Journal Article•
TL;DR: In this paper, a mathematical model of a kind of complicated financial system, and all possible things that the model shows in the operation of our country's macro-financial system are analyzed.
Abstract: Based on the work discussed on the former study, this article first starts from the mathematical model of a kind of complicated financial system, and analyses all possible things that the model shows in the operation of our country's macro_financial system: balance, stable periodic, fractal, Hopf_bifurcation, the relationship between parameters and Hopf_bifurcation, and chaotic motion etc. By the changes of parameters of all economic meanings, the conditions on which the complicated behaviors occur in such a financial system, and the influence of the adjustment of the macro_economic policies and adjustment of some parameter on the whole financial system behavior have been analyzed. This study will deepen people's understanding of the lever function of all kinds of financial policies.
19 citations
TL;DR: In this article, the conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed by decomposing the singular term from its solution and combining an asymptotic expansion of the equation.
Abstract: The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.
18 citations
TL;DR: In this article, a new method was proposed to treat the sum of independent unbounded random variables by truncating the original probability space (Ω, T, P) and the probability exponential inequalities for sums of independent bounded random variables were given.
Abstract: The tail probability inequalities for the sum of independent unbounded random variables on a probability space (Ω, T, P) were studied and a new method was proposed to treat the sum of independent unbounded random variables by truncating the original probability space (Ω, T, P). The probability exponential inequalities for sums of independent unbounded random variables were given. As applications of the results, some interesting examples were given. The examples show that the method proposed in the paper and the results of the paper are quite useful in the study of the large sample properties of the sums of independent unbounded random variables.
18 citations
TL;DR: In this article, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (oDFs) and crystal orientation distribution function (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively.
Abstract: In this two-part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (oDFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically-based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (I), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N-dimensional (N-D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2- and 3-D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point-group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work — Part (II), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point-group symmetries are derived.
17 citations
TL;DR: In this paper, a method based on newly presented state space formulations is developed for analyzing the bending, vibration and stability of laminated transversely isotropic rectangular plates with simply supported edges.
Abstract: A method based on newly presented state space formulations is developed for analyzing the bending, vibration and stability of laminated transversely isotropic rectangular plates with simply supported edges. By introducing two displacement functions and two stress functions, two independent state equations were constructed based on the three-dimensional elasticity equations for transverse isotropy. The original differential equations are thus decoupled with the order reduced that will facilitate obtaining solutions of various problems. For the simply supported rectangular plate, two relations between the state variables at the top and bottom surfaces were established. In particular, for the free vibration (stability) problem, it is found that there exist two independent classes: One corresponds to the pure in-plane vibration (stability) and the other to the general bending vibration (stability). Numerical examples are finally presented and the effects of some parameters are discussed.
TL;DR: In this article, an adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed, based on a combination of finite element and finite difference techniques.
Abstract: Adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed. The approach is based on a combination of finite element and finite difference techniques. The existence and uniqueness of the weak solution are proved without any assumptions on choice of the space-time meshes. Basic error estimates in L∞ (L2) norm, that is maximum-norm in time, L2-norm in space are obtained. The numerical results are given in the last part and the analysis between theoretic and experimental results are obtained.
TL;DR: In this article, an auto-Darboux transformation for the Brusselator reaction diffusion model is used to obtain exact solutions which contain some authors' results known, and then using sine-cosine method, more exact solutions are found which contain soliton solutions.
Abstract: Firstly, using the improved homogeneous balance method, an auto-Darboux transformation (ADT) for the Brusselator reaction diffusion model is found. Based on the ADT, several exact solutions are obtained which contain some authors' results known. Secondly, by using a series of transformations, the model is reduced into a nonlinear reaction diffusion equation and then through using sine-cosine method, more exact solutions are found which contain soliton solutions.
TL;DR: Based on the layered visco-elastic soil model, according to the Terzaghi's one dimensional consolidation theory, by the method of Laplace transform and matrix transfer technique, the problems about the consolidation of layered and saturated viscoelastic soils under arbitrary loading were solved as mentioned in this paper.
Abstract: Based on the layered visco-elastic soil model, according to the Terzaghi's one dimensional consolidation theory, by the method of Laplace transform and matrix transfer technique, the problems about the consolidation of layered and saturated visco-elastic soils under arbitrary loading were solved. Through deductions, the general solution, in the terms of layer thickness, the modulus and the coefficients of permeability and Laplacian transform's parameters was obtained. The strain and deformation of the layered and saturated visco-elastic soils under arbitrary loading can be calculated by Laplace inversion. According to the results of several numerical examples, the consolidation of visco-elastic soils lags behind that of elastic soils. The development of effective stress and the displacement is vibrant process under cyclic loading. Finally, an engineering case is studied and the results prove that the methods are very effective.
TL;DR: In this paper, a three-dimensional hydraulic fracture propagating and dynamical production predicting models for coalbed methane well is put forward, where the fracture propagation model takes the variation of rock mechanical properties and in-situ stress distribution into consideration.
Abstract: In accordance with the fracturing and producing mechanism in coalbed methane well, and combining the knowledge of fluid mechanics, linear elastic fracture mechanics, thermal transfer, computing mathematics and software engineering, the three-dimensional hydraulic fracture propagating and dynamical production predicting models for coalbed methane well is put forward. The fracture propagation model takes the variation of rock mechanical properties and in-situ stress distribution into consideration. The dynamic performance prediction model takes the gas production mechanism into consideration. With these models, a three-dimensional hydraulic fracturing optimum design software for coalbed methane well is developed, and its practicality and reliability have been proved by example computation.
TL;DR: In this paper, the scattering of harmonic waves by two collinear symmetric cracks was studied using the non-local theory, where a one-dimensional non-linear kernel was used to replace a twodimensional one for the dynamic problem to obtain the stress occurring at the crack tips.
Abstract: The scattering of harmonic waves by two collinear symmetric cracks is studied using the non-local theory. A one-dimensional non-local kernel was used to replace a twodimensional one for the dynamic problem to obtain the stress occurring at the crack tips. The Fourier transform was applied and a mixed boundary value problem was formulated. Then a set of triple integral equations was solved by using Schmidt's method. This method is more exact and more reasonable than Eringe's for solving this problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the lattice parameter and the circular frequency of incident wave.
TL;DR: By using the method of stress functions, the problem of mode-II Griffith crack in decagonal quasicrystals was solved as discussed by the authors, where the crack problem of two-dimensional quasi-crystals was decomposed into a plane strain state problem superposed on anti-plane state problem.
Abstract: By using the method of stress functions, the problem of mode- II Griffith crack in decagonal quasicrystals was solved First, the crack problem of two-dimensional quasi-crystals was decomposed into a plane strain state problem superposed on anti-plane state problem and secondly, by introducing stress functions, the 18 basic elasticity equations on coupling phonon-phason field of decagonal quasicrystals were reduced to a single higher-order partial differential equations The solution of this equation under mixed boundary conditions of mode- II Griffith crack was obtained in terms of Fourier transform and dual integral equations methods All components of stresses and displacements can be expressed by elemental functions and the stress intensity factor and the strain energy release rate were determined
TL;DR: In this paper, the authors derived general expressions of constitutive equations for isotropic elastic damaged materials from the basic law of irreversible thermodynamics and showed that the classical damage constitutive equation based on the strain equivalence hypothesis is only a first-order approximation of the general expression.
Abstract: The general expressions of constitutive equations for isotropic elastic damaged materials were derived directly from the basic law of irreversible thermodynamics. The limitations of the classical damage constitutive equation based on the well-known strain equivalence hypothesis were overcome. The relationships between the two elastic isotropic damage models (i. e. single and double scalar damage models) were revealed. When a single scalar damage variable defined according to the microscopic geometry of a damaged material is used to describle the isotropic damage state, the constitutive equations contain two “damage effect functions”, which describe the different influences of damage on the two independent elastic constants. The classical damage constitutive equation based on the strain equivalence hypothesis is only the first-order approximation of the general expression. It may be unduly simplified and may fail to describe satisfactorily the damage phenomena of practical materials.
TL;DR: In this paper, the general steps of elastic-plastic analysis near crack line for mode I crack in elastic-perfectly plastic solids under plane stress condition, and in turn given out the solving process and result for a specific problem.
Abstract: Crack line field analysis method has become an independent method for crack elastic-plastic analysis, which greatly simplifies the complexity of crack elastic-plastic problems and overcomes the corresponding mathematical difficulty. With this method, the precise elastic-plastic solutions near crack lines for variety of crack problems can be obtained. But up to now all solutions obtained by this method were for different concrete problems, no general steps and no general form of matching equations near crack line are given out. With crack line analysis method, this paper proposes the general steps of elastic-plastic analysis near crack line for mode I crack in elastic-perfectly plastic solids under plane stress condition, and in turn given out the solving process and result for a specific problem.
TL;DR: In this article, sufficient conditions are given to assert that two differentiable mappings between Banach spaces have common values, and the proof is essentially based upon continuation methods, which is essentially the same as in this paper.
Abstract: Sufficient conditions are given to assert that two differentiable mappings between Banach spaces have common values. The proof is essentially based upon continuation methods.
TL;DR: In this article, the authors presented new principles of work and energy as well as power and energy rate with cross terms for polar and nonlocal polar continuum field theories, and from them all corresponding equations of motion and boundary conditions and complete equations of energy and energy rates with the help of generalized Piola's theorems were naturally derived in all and without any additional requirement.
Abstract: New principles of work and energy as well as power and energy rate with cross terms for polar and nonlocal polar continuum field theories were presented and from them all corresponding equations of motion and boundary conditions as well as complete equations of energy and energy rate with the help of generalized Piola’s theorems were naturally derived in all and without any additional requirement. Finally, some new balance laws of energy and energy rate for generalized continuum mechanics were established. The new principles of work and energy as well as power and energy rate with cross terms presented in this paper are believed to be new and they have corrected the incompleteness of all existing corresponding principles and laws without cross terms in literatures of generalized continuum field theories.
TL;DR: In this paper, the Hamilton Principle was used to derive the general governing equations of nonlinear dynamic stability for laminated cylindrical shells in which, factors such as large deflection, transverse shear and longitudinal inertia force were concluded.
Abstract: Hamilton Principle was used to derive the general governing equations of nonlinear dynamic stability for laminated cylindrical shells in which, factors of nonlinear large deflection, transverse shear and longitudinal inertia force were concluded. Equations were solved by variational method. Analysis reveals that under the action of dynamic load, laminated cylindrical shells will fall into a state of parametric resonance and enter into the dynamic unstable region that causes dynamic instability of shells. Laminated shells of three typical composites were computed: i. e. T300/5 208 graphite epoxy E-glass epoxy, and ARALL shells. Results show that all factors will induce important influence for dynamic stability of laminated shells. So, in research of dynamic stability for laminated shells, to consider these factors is important.
TL;DR: In this paper, the prior estimate and decay property of positive solutions were derived for a system of quasi-linear elliptic differential equations first, and the result of non-existence for differential equation system of radially nonincreasing positive solutions was implied.
Abstract: The prior estimate and decay property of positive solutions are derived for a system of quasi-linear elliptic differential equations first. Hence, the result of non-existence for differential equation system of radially nonincreasing positive solutions is implied. By using this non-existence result, blow-up estimates for a class quasi-linear reaction-diffusion systems (non-Newtonian filtration systems) are established, which extends the result of semi-linear reaction-diffusion (Fujita type) systems.
TL;DR: In this article, the constitutive equations of shallow shells with large deflection and stress functions were derived by using the procedure similar to establishing the Karman equations of elastic thin plates, and an approximate theory for viscoelastic cylindrical shells under axial pressures was obtained.
Abstract: The hypotheses of the Karman-Donnell theory of thin shells with large deflections and the Boltzmann laws for isotropic linear, viscoelastic materials, the constitutive equations of shallow shells ae first derived. Then the governing equations for the deflection and stress function are formulated by using the procedure similar to establishing the Karman equations of elastic thin plates. Introducing proper assumptions, an approximate theory for viscoelastic cylindrical shells under axial pressures can be obtained. Finally, the dynamical behavior is studied in detail by using several numerical methods. Dynamical properties, such as, hyperchaos, chaos, strange attractor, limit cycle etc., are discovered.
TL;DR: In this article, an exact analytical solution for free vibration of composite shell structure-hermetic capsule was presented for axisymmetric vibration were based on the Love classical thin shell theory and derived for shells of revolution with arbitrary meridian shape.
Abstract: An exact analytical solution was presented for free vibration of composite shell structure-hermetic capsule. The basic equations on axisymmetric vibration were based on the Love classical thin shell theory and derived for shells of revolution with arbitrary meridian shape. The conditions of the junction between the spherical and the cylindrical shell segments are given by the continuity of deformation and the equilibrium relations near the junction point. The mathematical model of problem is reduced to as an eigenvalue problem for a system of ordinary differential equations in two separate domains corresponding to the spherical and the cylindrical shell segments. By using Legendre and trigonometric functions, exact and explicitly analytical solutions of the mode functions were constructed and the exact frequency equation were obtained. The implementation of Maple programme indicates that all calculations are simple and efficient in both the exact symbolic calculation and the numerical results of natural frequencies compare with the results using finite element methods and other numerical methdos. As a benchmark, the exactly analytical solutions presented in this paper is valuable to examine the accuracy of various approximate methods.
TL;DR: In this article, some related fixed point theorems on two metric spaces were established by using functions, and these results generalize some theorem of Fisher, and they are shown to be equivalent to some of the results of the present paper.
Abstract: By using functions, some related fixed point theorems on two metric spaces are established. These results generalize some theorems of Fisher.
TL;DR: In this paper, the Biot's wave equations of transversely isotropic saturated poroelastic media excited by non-axisymmetrical harmonic source were solved by means of Fourier expansion and Hankel transform.
Abstract: The Biot's wave equations of transversely isotropic saturated poroelastic media excited by non-axisymmetrical harmonic source were solved by means of Fourier expansion and Hankel transform. Then the components of total stress in porous media are expressed with the solutions of Biot's wave equations. The method of research on non-axisymmetrical dynamic response of saturated porous media is discussed, and a numerical result is presented.
Journal Article•
TL;DR: In this paper, a three-dimensional hydraulic fracture propagating and dynamical production predicting model for coalbed methane well is proposed, which combines the knowledge of fluid mechanics, linear elastic fracture mechanics, thermal transfer, computing mathematics and software engineering.
Abstract: In accordance with the fracturing and producing mechanism in coalbed methane well, and combining the knowledge of fluid mechanics, linear elastic fracture mechanics, thermal transfer, computing mathematics and software engineering, the three_dimensional hydraulic fracture propagating and dynamical production predicting models for coalbed methane well is put forward. The fracture propagation model takes the variation of rock mechanical properties and in_situ stress distribution into consideration. The dynamic performance prediction model takes the gas production mechanism into consideration. With these models, a three_dimensional hydraulic fracturing optimum design software for coalbed methane well is developed, and its practicality and reliability have been proved by example computation.
TL;DR: In this article, the transition sets and persistent perturbed bifurcation diagrams of 10 elementary Bifurcations of codimension no more than three are given. But the present singularity theory does not contain any analytical methods and results about it.
Abstract: Bifurcations with constraints are open problems appeared in research on periodic bifurcations of nonlinear dynamical systems, but the present singularity theory doesn’t contain any analytical methods and results about it. As the complement to singularity theory and the first step to study on constrained bifurcations, here are given the transition sets and persistent perturbed bifurcation diagrams of 10 elementary bifurcation of codimension no more than three.
TL;DR: In this article, a penny-shaped crack on axially symmetric propagating problems for composite materials was studied and the general representations of the analytical solutions with arbitrary index of self-similarity were presented for fracture elastodynamics problems on axial symmetry by the ways of selfsimilarity under the laddershaped loads.
Abstract: By the theory of complex functions, a penny-shaped crack on axially symmetric propagating problems for composite materials was studied. The general representations of the analytical solutions with arbitrary index of self-similarity were presented for fracture elastodynamics problems on axially symmetry by the ways of self-similarity under the laddershaped loads. The problems dealt with can be transformed into Riemann-Hilbert problems and their closed analytical solutions are obtained rather simple by this method. After those analytical solutions are utilized by using the method of rotational superposition theorem in conjunction with that of Smirnov-Sobolev, the solutions of arbitrary complicated problems can be obtained.