平面应变 Biot 固结的解析层元

提出用解析层元法有效地解决任意深度单层土的平面应变 Biot 固结问题. 从 Biot 固结问题的控制方程出发, 采用特征值法在 Laplace-Fourier 变换域内推导出一个精确对称的解析层元刚度矩阵. 通过表示单层土广义力和广义位移之间关系的解析层元, 并结合土层的边界条件, 推导出土层任意点的解答; 物理域内的真实解可以通过 Laplace-Fourier 数值逆变换进一步获得. 通过数值计算验证理论的正确性, 研究了土层性质及时间因素对固结的影响.}      

非均质流固耦合介质轴对称动力问题时域解

将地基视为流固两相介质并考虑其非均质成层特性，推导了多层地基动力问题时域解．文中首先建立了一组解耦的两相介质动力控制方程；而后利用Ｌａｐｌａｃｅ－Ｈａｎｋｅｌ变换推导了单层地基象空间初参数解答；再利用初参数法及传递矩阵技术导出了任意多层地基瞬态解的一般解析算式．本文获得的解答可方便地退化为现有理想弹性介质的解答     

Mode i and mixed mode fields with incomplete crack tip plasticity

Plane strain slip line fields, in which plasticity does not fully surround the crack tip have been developed for mode I and mixed mode I\II cracks under contained yielding. Analytical solutions have been assembled using slip line theory for the plastic sectors and semi-infinite wedge solutions for the elastic sectors. These solutions are compared with finite element solutions based on modified boundary layer formulations. The analytical solutions agree well with numerical solutions, and form a family of fields with incomplete plasticity around the crack tip.     

Viscous boundary layers in flows through a domain with permeable boundary

The effect of small viscosity on nearly inviscid flows of an incompressible fluid through a given domain with permeable boundary is studied. The Vishik–Lyusternik method is applied to construct a boundary layer asymptotic at the outlet in the limit of vanishing viscosity. Mathematical problems with both consistent and inconsistent initial and boundary conditions at the outlet are considered. It is shown that in the former case, the viscosity leads to a boundary layer only at the outlet. In the latter case, in the leading term of the expansion there is a boundary layer at the outlet and there is no boundary layer at the inlet, but in higher order terms another boundary layer appears at the inlet. To verify the validity of the expansion, a number of simple examples are presented. The examples demonstrate that asymptotic solutions are in quite good agreement with exact or numerical solutions.     

Axisymmetrical analytical solution for vertical vibration of end-bearing pile in saturated viscoelastic soil layer

Based on elasticity and the theory of saturated porous media, and regarding the pile and the soil as a single phase elastic and a saturated viscoelastic media, respectively, the dynamical behavior of vertical vibration of an end-bearing pile in a saturated viscoelastic soil layer is investigated in the frequency domain using the Helmholtz decomposition and variable separation method. The axisymmetrical analytical solutions for vertical vibrations of the pile in a saturated viscoelastic soil layer are obtained, and the analytical expression of the dynamical complex stiffness of the pile top is presented. Responses of dynamic stiffness factor and equivalent damping of pile top with respect to the frequency are shown in figures using a numerical method. Effects of the saturated soil parameters, modulus ratio of the pile to soil, slenderness ratio of pile and pile＇s Poisson ratio, etc. on the stiffness factor and damping are examined. It is shown that, due to the effect of the transversal deformation of the pile and the radial force of the saturated viscoelastic soil acting on the pile, the dynamic stiffness factor and the damping derived from the axisymmetrical solution are greatly different from those derived from the classical Euler-Bernoulli rod model, especially at some specific excitation frequencies. Therefore, there are limitations on applicability of the Euler-Bernoulli rod model in analyzing verticai vibration of the pile. More accurate analysis should be based on a three-dimensional model.     

Natural convection in an inclined enclosure with a fluid layer and a heat-generating porous bed

Multiple steady-state solutions of natural convection in an inclined enclosure with a fluid layer and a heat-generating porous bed is investigated numerically by the finite volume method. The conservation equations for the porous layer are based on a general flow model which includes both the effects of flow inertia and friction. The flow in fluid layer is modeled by Navier–Stokes equations. The method of pseudo arc-length continuation is adapted in studying the effects of tilt angle on flow pattern and heat transfer. It is found that, in the whole domain of tilt angle, there exist two groups of solutions with quite different flow pattern and heat transfer behavior. The effects of aspect ratio on flow pattern and heat transfer have also been studied. Received on 04 March 1997     

Similarity solutions for non-Newtonian power-law fluid flow

The problem of the boundary layer flow of power law non-Newtonian fluids with a novel boundary condition is studied.The existence and uniqueness of the solutions are examined,which are found to depend on the curvature of the solutions for different values of the power law index n.It is established with the aid of the Picard-Lindel¨of theorem that the nonlinear boundary value problem has a unique solution in the global domain for all values of the power law index n but with certain conditions on the curvature of the solutions.This is done after a suitable transformation of the dependent and independent variables.For 0 n 1,the solution has a positive curvature,while for n 1,the solution has a negative or zero curvature on some part of the global domain.Some solutions are presented graphically to illustrate the results and the behaviors of the solutions.     

二维水下声辐射问题的改进径向基点插值法研究

径向基点插值法是一种典型的无网格数值计算方法,在分析声学问题时,相比于传统有限元法能更好地抑制频散误差,且在相同的节点分布下通常可以得到更精确的数值解。本文提出一种改进的节点选取方案用于构造插值形函数,即改进径向基点插值法。该方案采取一个简单而直接的格式,可确保在进行数值积分时同一背景积分单元中的被积函数是连续可微的,从而减小数值积分误差,得到比原始径向基点插值法更精确的数值解。同时,为了处理外声场问题,本文采用DtN映射技术将无限域截断为有界计算域,满足索默菲尔德辐射条件。数值试验表明,相比于传统有限元法和原始径向基点插值法,本文改进方法具有更高的计算精度和计算效率,在研究水下声辐射问题时具有良好的应用前景。     

A new method for solving Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system

A new method is developed to solve Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system. Based on the governing equations of Biot＇s consolidation and the technique of Laplace transform, Fourier expansions and Hankel transform with respect to time t, coordinate θ and coordinate r, respectively, a relationship of displacements, stresses, excess pore water pressure and flux is established between the ground surface （z = 0） and an arbitrary depth z in the Laplace and Hankel transform domain. By referring to proper boundary conditions of the finite soil layer, the solutions for displacements, stresses, excess pore water pressure and flux of any point in the transform domain can be obtained. The actual solutions in the physical domain can be acquired by inverting the Laplace and the Hankel transforms.     

Singularities of the boundary layer equations and the structure of the flow in the vicinity of the convergence plane on conical bodies

The singularities of the boundary layer equations and the laminar viscous gas flow structure in the vicinity of the convergence plane on sharp conical bodies at incidence are analyzed. In the outer part of the boundary layer the singularities are obtained in explicit form. It is shown that in the vicinity of a singularity a boundary domain, in which the flow is governed by the shortened Navier-Stokes equations, is formed; their regular solutions are obtained. The viscous-inviscid interaction effect predominates in a region whose extent is of the order of the square root of the boundary layer thickness, in which the flow is described by a two-layer model, namely, the Euler equations in the slender-body approximation for the outer region and the three-dimensional boundary layer equations; the pressure is determined from the interaction conditions. On the basis of an analysis of the solutions for the outer part of the boundary layer it is shown that interaction leads to attenuation of the singularities and the dependence of the nature of the flow on the longitudinal coordinate, but does not make it possible to eliminate the singularities completely.     

Steady thermocapillary-buoyant convection in a shallow annular pool. Part 1: Single layer fluid

This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios,which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt,the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio,the applicability of the present asymptotic solutions decreases gradually.     

粘性阻尼土中变截面桩的纵向振动特性与应用研究

考虑土体轴对称波动效应,对变截面桩在任意激振力作用下的纵向振动特性进行了研究。假定桩为竖直、弹性、变截面体,土为线性粘弹性体,其材料阻尼为粘性阻尼。利用拉普拉斯变换,将定解问题转化到拉普拉斯域内求解,通过引入势函数并结合阻抗函数的传递性,得到了拉普拉斯域内的桩顶阻抗函数解析解,进而可得到频域内的桩顶阻抗函数和速度导纳的解析解,利用卷积定理和傅里叶逆变换,求得了半正弦脉冲激振力作用下桩顶速度时域响应半解析解。基于所得解对桩的纵向振动特性进行了分析,重点讨论了桩身截面变化情况对速度导纳曲线和反射波曲线的影响,得到了许多重要结论。     

Finite element analysis of smart functionally graded plates

This paper deals with the derivation of a finite element model for the static analysis of functionally graded (FG) plates integrated with a layer of piezoelectric fiber reinforced composite (PFRC) material. The layer of PFRC material acts as the distributed actuator of the FG plates. The Young’s modulus of the FG plate is assumed to vary exponentially along the thickness of the plate while the Poisson’s ratio is assumed to be constant over the domain of the plate. The finite element model has been verified with the exact solutions for both thick and thin plates. Emphasis has been placed on investigating the effect of variation of piezoelectric fiber angle in the PFRC layer on its actuating capability of the FG plates. The finite element solutions also revealed that the activated PFRC layer is more effective in controlling the deformations of the FG plates when the layer is attached to the surface of the FG plate with minimum stiffness than when it is attached to the surface of the same with maximum stiffness.     

Exact solutions of MHD second Stokes flow of generalized Burgers fluid

This work concerns with the exact solutions of magnetohydrodynamic (MHD) flow of generalized Burgers fluid describing the second Stokes problem. The modified Darcy law is taken into account. The related velocity distribution and shear stress are expressed as a combination of steady-state and transient solutions computed by means of integral transformations. The effects of various parameters on the flow field are investigated. The MHD flow results in reduction of velocity distribution and associated thickness of the boundary layer.     

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

Nonsimilar Boundary Layer Analysis of Free Convection Heat Transfer Over a Vertical Cylinder in Bidisperse Porous Media

This work presents a boundary layer analysis for the free convection heat transfer from a vertical cylinder in bidisperse porous media with constant wall temperature. A boundary layer analysis and the two-velocity two-temperature formulation are used to derive the nonsimilar governing equations. The transformed governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The effects of inter-phase heat transfer parameter, modified thermal conductivity ratio, and permeability ratio on the heat transfer and flow characteristics are studied. Results show that an increase in the modified thermal conductivity ratio and the permeability ratio can effectively enhance the free convection heat transfer of the vertical cylinder in a bidisperse porous medium. Moreover, the thermal nonequilibrium effects are strong for low values of the inter-phase heat transfer parameter.     

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K_I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.     

Explosive loading of deformable gas-permeable axisymmetric structural elements

A mathematical model is proposed which describes the interrelated processes of unsteady elastoplastic deformation of stacks of woven metal wire mesh and wave processes in pore gas in a two-dimensional axisymmetric approximation. The nonlinear equations of the dynamics of two interpenetrating continua are solved numerically using a modified Godunov’s scheme. The problem of explosive loading of a multilayer shell with an internal permeable deformable layer is solved. The results of numerical solutions are compared with experimental data. The influence of the gas-permeable layer on shell deformation is determined.     

Wave propagation in anisotropic media with non-local elasticity

In an effort to understand and quantify the effect of non-local elasticity on the wave propagation response of laminated composite layered media, a frequency-wavenumber domain based finite element method is employed. The developed elements are based on the exact solution in the transformed domain and thus exactly represent the dynamics of a layer. This feature enables to model a layer of any thickness by a single element and drastically reduces the cost of computation. The effect of non-locality on the dispersion relation and in turn on the wave response is compared with local (classical) elasticity solutions. A procedure and sample example is outlined to estimate the magnitude of the non-locality parameter by comparing the dispersion relation with lattice dynamics. The effect of non-locality, in terms of the mode-shift and appearance of dispersion on the modes of Lamb waves is further demonstrated.     

A weak formulation for interior acoustic analysis of enclosures with inclined walls and impedance boundary

A weak variational principle based approach is presented in this paper to study the sound field inside the acoustic enclosures with walls in arbitrary inclination and impedance conditions. The whole acoustic domain is firstly divided into several sub-cavities with trapezoidal and rectangular faces, and each sub-cavity is coupled with adjacent ones by matching the required continuity constraints on the interfaces on the basis of a modified variational principle and least-squares weighted residual method. By using this domain partitioning strategy, high-order acoustic modes and responses can be easily achieved. Chebyshev orthogonal polynomials of the first kind are employed as the wholly admissible unknown sound pressure functions for each sub-cavity without meshing process like FEM/BEM does, and then each physical domain is mapped into a square spectral domain. To demonstrate the convergence, accuracy and stability of the approach, the modal and sound response analyses of several configurations of cavities are examined and compared with available analytical solutions, or those obtained by using FEM. Effects of the weighted parameters together with the number of truncated polynomial terms and the divided cavity segments on the accuracy of present solutions are investigated. Key parametric studies concerning the influences of the geometrical properties as well as the impedance boundary of enclosing walls are also performed. It is demonstrated that the present method is a computationally efficient way to achieve interior sound predictions in mid-frequency range with a satisfactory accuracy of solutions.