应用数学和力学

2003 Vol. 24, No. 12

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Renewal of Basic Laws and Principles for Polar Continuum Theories(Ⅵ)——Conservation Laws of Mass and Inertia

DAI Tian-min

2003, 24(12): 1211-1216.

Abstract(2367) PDF(754)

Abstract:
The purpose is to reestablish the coupled conservation laws, the local conservation equations and the jump conditions of mass and inertia for polar continuum theories. In this connection the new material derivatives of the deformation gradient, the line element, the surface element and the volume element were derived and the generalized Reynolds. transport theorem was presented. Combining these conservation laws of mass and inertia with the balance laws of momentum, angular momentum and energy derived in our previous papers of this series, a rather complete system of coupled basic laws and principles for polar continuum theories is constituted on the whole. From this system the coupled nonlocal balance equations of mass, inertia, momentum, angular momentum and energy may be obtained by the usual localization.

Renewal of Basic Laws and Principles for Polar Continuum Theories(Ⅶ)——Incremental Rate Type

DAI Tian-min

2003, 24(12): 1217-1222.

Abstract(2378) PDF(745)

Abstract:
The purpose is to establish the rather complete equations of motion, boundary conditions and equation of energy rate of incremental rate type for micropolar continua. To this end the rather complete definitions for rates of deformation gradient and its inverse are made. The new relations between various stress and couple stress rate tensors are derived. Finally, the coupled equations of motion, boundary conditions and equation of energy rate of incremental rate type for continuum mechanics are obtained as a special case.

Nonlinear Vibration of Thin Shallow Conic Shells Under Combined Action of Peripheral Moment and Transverse Loads

ZHAO Yong-gang, WANG Xin-zhi, YEH Kai-yuan

2003, 24(12): 1223-1230.

Abstract(2376) PDF(629)

Abstract:
Based on the variation and harmonic equations and by taking the maximum amplitude of the shell center as the perturbation parameter, nonlinear vibration of thin shallow conic shells under combined action of peripheral moment and transverse loads was solved. The linear natural frequency can be got by the first-order approximation and the more accurate nonlinear frequency is got by the second-order approximation under the action of static loads. Meanwhile the third-order approximate analytic expression is given for describing the nonlinear relation between nature frequency and peripheral moment, transverse loads, amplitude, base angle under the small deformation. Within some range, the complex and regularity of the nonlinear relation can be directly observed from the numeric results.

Equivalent Boundary Integral Equations With Indirect Variables for Plane Elasticity Problems

ZHANG Yao-ming, WEN Wei-dong, ZHANG Zuo-quan, SUN Huan-chun, LÜ He-xiang

2003, 24(12): 1231-1237.

Abstract(2711) PDF(704)

Abstract:
The exact form of the exterior problem for plane elasticity problems was produced and fully proved by the variational principle. Based on this, the equivalent boundary integral equations(EBIE) with direct variables, which are equivalent to the original boundary value problem, were deduced rigorously. The conventionally prevailing boundary integral equation with direct variables was discussed thoroughly by some examples and it is shown that the previous results are not EBIE.

Analysis of Transient Thermal Stress in Cylindrically Orthotropic Tubes

LING Dao-sheng

2003, 24(12): 1238-1242.

Abstract(2315) PDF(837)

Abstract:
The incorrect deduction of equations in the research works devoted to the studies of transient stress in cylindrically orthotropic tubes and done by Kardomateas(Journal of Applied Mechanics, 1989, 1990) leads to the wrong results. The errata(1991) correct the deduction error, but do not give the right numerical results. All errors are corrected, and the Mathematica is adopted to solve the large argument problem for Bessel function. A theoretical solution of the transient thermal stresses in tubes with uniform form is presented, and a numerical example is studied.

Objectivity Requirement for Fluid Dynamics

ZOU Wen-nan

2003, 24(12): 1243-1248.

Abstract(2240) PDF(728)

Abstract:
A new flow theory is established through the objectivity requirement on the fluid dynamics. It was known that inhomogeneous fluid motion gave rise to viscous force while the selection of observers on different space-time points would change such an inhomogeneous character. Therefore, when the viscous force was considered as an objective existence foreign to the selection of observers, the form invariances of viscous force and momentum equation under local rotation transformation required a new dynamic field, namely the vortex field to be introduced. Then the dynamical equations of all flow fields were obtained through constructing the Lagrangian density of fluid system and using the variational approach of energy.

Internal Resonant Interactions of Three Free Surface-Waves in a Circular Cylindrical Basin

MA Chen-ming

2003, 24(12): 1249-1257.

Abstract(2753) PDF(769)

Abstract:
The basic equations of free capillary-gravity surface-waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevation, two-order perturbation equations were derived by use of expansion of multiple scale. The nonlinear interactions with two-order internal resonance of three free surface-waves were discussed based on the above. The results include:derivation of the couple equations of resonant interactions among three waves and the conservation laws; analysis of the positions of equilibrium points in phase plane; study of the resonant parameters and the non-resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of second-order interaction equations corresponding to different parameters and analysis of the stability property of the solutions; discussion of the effective solutions only in the limited time range. The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance. The energy may either exchange among three waves periodically or damp or increase in single waves.

Iterative Approximation of Fixed Points for Almost Asymptotically Nonexpansive Type Mappings in Banach Spaces

ZENG Lu-chuan

2003, 24(12): 1258-1266.

Abstract(2163) PDF(678)

Abstract:
A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced, which includes a number of known classes of nonlinear Lipschitzian mappings and non-Lipschitzian mappings in Banach spaces as special cases; for example, the known classes of nonexpansive mappings, asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings. The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered. Not only S. S. Chang's inequality but also H. K. Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one. Moreover, ZHANG Shi-sheng's method(Applied Mathema tics an d Mechanics(English Edition), 2001, 22(1):25-34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also, the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria. The above results unify, improve and generalize ZHANG Shi-sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.

Multiple Reciprocity Method With Two Series of Sequences of High-Order Fundamental Solution for Thin Plate Bending

DING Fang-yun, DING Rui, LI Bing-jie

2003, 24(12): 1267-1275.

Abstract(2559) PDF(760)

Abstract:
The boundary value problem of plate bending problem on two-parameter foundation was discussed. Using two series of the high-order fundamental solution sequences, namely the fundamental solution sequences for the multi-harmonic operator and Laplace operator, applying the multiple reciprocity method(MRM), the MRM boundary integral equation for plate bending problem was constructed. It proves that the boundary integral equation derived from MRM is essentially identical to the conventional boundary integral equation. Hence the convergence analysis of MRM for plate bending problem can be obtained by the error estimation for the conventional boundary integral equation. In addition this method can extend to the case of more series of the high-order fundamental solution sequences.

Singularly Perturbed Nonlinear Boundary Value Problem for a Kind of Volterra Type Functional Differential Equation

LU Shi-ping

2003, 24(12): 1276-1284.

Abstract(2819) PDF(623)

Abstract:
By employing the theory of differential inequality and some analysis methods, a nonlinear boundary value problem subject to a general kind of second order Volterra functional differential equation was considered first. Then, by constructing the right-side layer function and the outer solution, a nonlinear boundary value problem subject to a kind of second order Volterra functional differential equation with a small parameter was studied further. By using the differential mean value theorem and the technique of upper and lower solution, a new result on the existence of the solutions to the boundary value problem is obtained, and a uniformly valid asymptotic expansions of the solution is given as well.

Singularly Perturbed Boundary Value Problems for Semi-Linear Retarded Differential Equations With Nonlinear Boundary Conditions

REN Jing-li, GE Wei-gao

2003, 24(12): 1285-1290.

Abstract(2351) PDF(730)

Abstract:
A boundary value problems for functional differenatial equations, with nonlinear boundary condition, is studied by the theorem of differential inequality. Using new method to construct the upper solution and lower solution, sufficient conditions for the existence of the problems. solution are established. A uniformly valid asymptotic expansions of the solution is also given.

Coarse-Mesh-Accuracy Improvement of Bilinear Q₄-Plane Element by the Combined Hybrid Finite Element Method

XIE Xiao-ping, ZHOU Tian-xiao

2003, 24(12): 1291-1300.

Abstract(2451) PDF(619)

Abstract:
The combined hybrid finite element method is of an intrinsic mechanism of enhancing coarse-mesh-accuracy of lower order displacement schemes. It was confirmed that the combined hybrid scheme without energy error leads to enhancement of accuracy at coarse meshes, and that the combination parameter plays an important role in the enhancement. As an improvement of conforming bilinear Q₄-plane element, the combined hybrid method adopted the most convenient quadrilateral displacements-stress mode, i. e. the mode of compatible isoparametric bilinear displacements and pure constant stresses. By adjusting the combined parameter, the optimized version of the combined hybrid element was obtained and numerical tests indicated that this parameter-adjusted version behaves much better than Q₄-element and is of high accuracy at coarse meshes. Due to elimination of stress parameters at the elemental level, this combined hybrid version is of the same computational cost as that of Q₄-element.

Lower Bound Limit Analysis of Three-Dimensional Elastoplastic Structures by Boundary Element Method

LIU Ying-hua, ZHANG Xiao-feng, CEN Zhang-zhi

2003, 24(12): 1301-1308.

Abstract(2745) PDF(653)

Abstract:
Based on the lower bound theorem of limit analysis, a solution procedure for limit analysis of three-dimensional elastoplastic structures was established using conventional boundary element method(BEM). The elastic stress field for lower bound limit analysis was computed directly by three-dimensional boundary element method(3-D BEM). The self-equilibrium stress field was constructed by the linear combination of several self-equilibrium "basis vectors" which can be computed by elastic-plastic incremental iteration of 3-D BEM analysis. The lower bound limit analysis problem was finally reduced to a series of nonlinear programming sub-problems with relatively few optimal variables. The complex method was used to solve the nonlinear programming sub-problems. The numerical results show that the present solution procedure has good accuracy and high efficiency.

Fluid-Solid Coupling Mathematical Model of Contaminant Transport in Unsaturated Zone and Its Asymptotical Solution

XUE Qiang, LIANG Bing, LIU Xiao-li, LI Hong-yan

2003, 24(12): 1309-1318.

Abstract(3125) PDF(739)

Abstract:
The process of contaminant transport is a problem of multicomponent and multiphase flow in unsaturated zone. Under the presupposition that gas existence affects water transport, a coupled mathematical model of contaminant transport in unsaturated zone has been established based on fluid- solid interaction mechanics theory. The asymptotical solutions to the nonlinear coupling mathematical model were accomplished by the perturbation and integral transformation method. The distribution law of pore pressure, pore water velocity and contaminant concentration in unsaturated zone has been presented under the conditions of with coupling and without coupling gas phase. An example problem was used to provide a quantitative verification and validation of the model. The asymptotical solution was compared with Faust model solution. The comparison results show reasonable agreement between asymptotical solution and Faust solution, and the gas effect and media deformation has a large impact on the contaminant transport. The theoretical basis is provided for forecasting contaminant transport and the determination of the relationship among pressure-saturation-permeability in laboratory.