Diffraction of elastic waves in the plane multiply-connected region and dynamic stress concentration

This paper deals with the problem of diffraction of elastic waves in the plane multiply-connected regions by the theory of complex functions. The complete function series which approach the solution of the problem and general expressions for boundary conditions are given.’ Then the problem is reduced to the solution to infinite series of algebraic equations and the solution can be directly obtained by using electronic computer. In particular, for the case of weak interaction, an asymptotic method is presented here, by which the problem ofp waves diffracted by a circular cavities is discussed in detail. Based on the solution of the diffracted wave field the general formulas for calculating dynamic stress concentration factor for a cavity of arbitrary shape in multiply-connected region are given.     

Second-order torsion problem of a cylinder consisting of different isotropic homogeneous compressible elastic materials

Some second-order solutions of the torsion problem for simply-connected regions are available based on the theory given by Green and others both for compressible and incompressible materials. Bhargava and Gupta [1] have recently extended the theory for torsion problem of multiply-connected regions. In the present paper these theories are extended further to account for the composite regions. The complex variable formulation is employed. As an illustration, results for the torsion problem of a composite cylinder of concentric circular cross-section are given.     

Photothermoelastic analysis of bonded propellant grains

A two-dimensional model of the transversal cross section of a bonded rocket propellant grain was subjected to uniform and steady thermal loading and, alternatively, to mechanically applied uniform radial displacements on the outer boundary. The optimization of perforation contours (attained in previous research programs by applying uniform pressure on the outer boundary of the grain model) was confirmed for both types of thermal loading. The concentration factor at the fillets of the inner contour was determined. An attempt was made to predict the maximum strain in the actual propellant subjected to the same thermal conditions. The material used for the model was a urethane rubber. The thermoelastic properties of the model material were determined.     

Minimum-weight design of hollow cylinders for given lower bounds on torsional and flexural rigidities

An approximate analytical solution is obtained for the problem of minimizing the cross-sectional area of elastic, doubly-connected cylindrical bars for given lower bounds on torsional and flexural rigidities. The bars are designed to withstand either a twisting or a bending moment, but not both, at any given time. The shape of the inner contour of the doubly-connected (hollow) cross-section is specified while that of the outer contour is determined as a result of the optimization. The economy achieved by optimization is estimated.     

An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section

In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions.     

Dynamic Out of Plane Displacement Fields for an Inclined Crack in Graded Materials

A generalized elastic solution for an arbitrarily propagating crack in Functionally Gradient Materials (FGMs) is obtained through an asymptotic analysis. The shear modulus and mass density of the FGM are assumed to vary exponentially along the gradation direction. The mode mixity due to the inclination of property gradient is accommodated in the analysis through superposition of the opening and shear modes. Using this asymptotic solution, contours of constant out of plane displacement are generated. The effect of inclination of property gradation direction and the crack speed on these contours is discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

变截面多稳态梁结构减振吸能分析与优化设计

基于多稳态梁结构具有吸能且可重复使用的特点,本文研究包含变截面多稳态梁的单胞结构及其周期性排布的减振吸能效应及其优化设计方法。对多稳态结构进行考虑几何非线性的位移加载/卸载有限元仿真,根据其载荷-位移曲线分析多稳态结构的减振吸能原理,并研究串联与并联周期性排布形式对结构整体吸能特性的影响规律。研究基于多参数调控的变截面梁结构形状表征方法,根据多稳态结构储能特点建立变截面多稳态单胞结构的结构优化模型,通过求解优化问题获得总质量不变条件下最优变截面梁结构形状。进一步地通过对优化结果的有限元分析验证优化的有效性,并对结构进行瞬态冲击荷载下动响应分析,证明多稳态结构的冲击保护作用。     

An asymptotic linear thin-walled rod model coupling twist and bending

A linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods is presented. A dimensional analysis of the linear three-dimensional equilibrium equations yields dimensionless numbers that reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions. In the case of low force levels, we obtain a one-dimensional model whose kinematics, traction, and twist equations correspond to the Vlassov ones. However, this model couples twist and bending effects in the bending equations, unlike the Vlassov model where the twist angle and the bending displacement are uncoupled     

On the Somigliana dislocations associated with applications of the Neuber–Papkovich solution in multiply-connected regions

Summary The study aims to explain the problem of single-valuedness of the mean stress and dilatation in multiply-connected regions. It is shown that in multiply-connected regions, displacement representation potentials, in particular the Neuber–Papkovich potentials, have to be submitted to the integral conditions in order to avoid the many-valuedness. Received 8 September 1998; accepted for publication 8 April 1999     

Effects of the side walls on starting flows in ducts

This article considers the effects of the side walls on the unsteady flow of an incompressible viscous fluid in a duct of uniform cross-section. In order to show the effects of the side walls, three illustrative examples are given. They are: the starting flow in a duct of semicircular cross-section, the starting flow in a duct of rectangular cross-section and the starting flow in a duct of circular cross-section. The velocity distributions and the volume fluxes obtained for these flows are compared and it is shown that the flow in a duct of semicircular cross-section reaches steady state earlier than those for the flow in a duct of circular cross-section and for the flow in a duct of square cross-section. It is found that there are remarkable effects of the side walls of a duct on the required time to attain the asymptotic values of flow properties.     

Study of the effects of inner pressure and surface tension on the fibre drawing process with the aid of an analytical asymptotic fibre drawing model and the numerical solution of the full N.–St. equations

Micro-structured optical fibres (i.e. fibres that contain holes) have assumed a high profile in recent years and given rise to many novel optical devices. The problem of manufacturing such fibres by heating and then drawing a preform is considered for the case of annular capillaries. A fluid mechanics model suggested in the literature that uses asymptotic analysis based on the small aspect ratio of capillaries has been compared with the full 3D set of the N.–St. equations, for modelling the fabrication of capillaries. The final asymptotic equations, analysed in some asymptotic limits, are solved numerically and then compared with the N.–St. solutions, obtained with a commercial finite elements solver. These asymptotic limits give valuable practical information about the control parameters that influence the drawing process, taking into account the effects of surface tension and inner pressure, since those are of essential importance during drawing. It is shown that the asymptotic method delivers reliable results as long as the inner pressure does not exceed too high values.     

An asymptotic non-linear model for thin-walled rodsUn modèle asymptotique non-linéaire de poutres à parois minces

In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions. To cite this article: L. Grillet et al., C. R. Mecanique 332 (2004).     

PROPAGATION OF A LONG WAVE IN A CURVED DUCT (Ⅱ)APPLICATIONS OF MATCHED EXPANSION TO LONG WAVE PROPAGATION THROUGH A HOLE WITH VARIABLE CROSS-SECTION

Only the case in which the parameterε=ka《1 is considered in this paper,where k is the wave number and a is the characteris-tic radius of the cross-section of the hole.The general asymp-totic expansion of the complex velocity potential of a long wave propagating in the hole with variable cross-section is obtained by regular perturbation:The methods of matched asymptotic ex-pansion are employed to calculate the reflection coefficients,scattering coefficients and radiation coefficients at the open ends of the hole when a long wave propagates through it,which may be open at both ends or only at one end.Three examples of different kinds of holes are given to show the way to solve such two-dimensional or three-dimensional problems.     

复杂工程约束下基于NSGA-Ⅲ的车身梁截面优化设计

在汽车概念设计阶段,车身常由薄壁梁简化而成,其截面对轿车白车身刚度有直接影响。多室的薄壁梁截面极其复杂,在设计过程中不仅需考虑其力学性能,还需符合制造加工过程的各种工艺约束。本文基于薄壁梁截面设计的装配性与制造性等实际工程约束,提出一种梁截面形状控制方法以兼容多种工程约束。定义了梁截面多目标优化模型,以基于参考点的遗传算法NSGA-Ⅲ作为高维多目标优化问题算法,实现了相应的软件模块对截面惯性矩和扭转常量等参数进行优化。最终通过算例测试证明此方法的有效性。     

Numerical simulation of magnetohydrodynamic flow in a toroidal duct of square cross-section

We present numerical simulation results of the quasi-static magnetohydrodynamic (MHD) flow in a toroidal duct of square cross-section with insulating Hartmann walls and conducting side walls. Both laminar and turbulent flows are considered. In the case of steady flows, we present a comprehensive analysis of the secondary flow. It consists of two counter-rotating vortex cells, with additional side wall vortices emerging at sufficiently high Hartmann number. Our results agree well with existing asymptotic analysis. In the turbulent regime, we make a comparison between hydrodynamic and MHD flows. We find that the curvature induces an asymmetry between the inner and outer side of the duct, with higher turbulence intensities occurring at the outer side wall. The magnetic field is seen to stabilize the flow so that only the outer side layer remains unstable. These features are illustrated both by a study of statistically averaged quantities and by a visualization of (instantaneous) coherent vortices.     

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将双优化设计变量转化为单优化设计变量,从而建立起预应力方钢管混凝土柱的优化设计数 学模型. 利用罚函数外点法导出了优化设计变量的迭代计算公式,并给出了获得最优解的二 重迭代法计算步骤. 算例结果分析表明,本法对解决预应力方钢管混凝土柱截面优化问题, 具有概念简单,计算方便的特点,并为此类非线性规划问题提供了较好的求解途径.     

A Generalized Piezoelectric Bernoulli–Navier Anisotropic Rod Model

We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli–Navier equilibrium equations and reduced Maxwell–Gauss equations.     

Mathematical model of spiral waves propagating in the aorta

The spiral waves in the viscous incompressible fluid flow within an arterial vessel modeled by a thin elastic isotropic shell are studied. Asymptotic expansions are constructed for two types of spiral waves. The first type is spiral long wall waves generated (owing to the viscous fluid no-slip at the inner shell wall) by the longitudinal and twist harmonic waves that propagate along the wall. For these waves the amplitude distribution over the vessel cross-section has the form of a boundary layer localized near the inner shell surface. The second is short small-amplitude waves that practically fill the entire vessel cross-section. It is shown that for the short waves the transfer mechanismis the steady-state flow, the role of the longitudinal wall waves and the elastic characteristics of the shell being in this case insignificant.     

基于蚁群算法的桁架结构布局离散变量优化方法

提出的布局优化方法是将桁架结构的截面变量、拓扑变量及形状变量统一为离散变量.将离散变量转化为适应于蚁群算法求解TSP问题的离散变量,应用MATLAB语言编写求解桁架结构布局优化程序,最终实现对问题的分析与求解.通过对几个经典的平面、空间桁架结构布局优化算例的验算表明:本文设计的基于蚁群算法的桁架结构布局离散变量优化方法较单独处理截面优化、拓扑优化及形状优化问题具有更大的效益,相对于其他布局优化方法也展现出更好的优化效果.“基于蚁群算法的桁架结构布局离散变量优化方法”在程序设计、求解速度、求解空间及其方法通用性等方面都表现出良好的性能,并且简单、实用,适应于实际工程应用.     

On many-valuedness of the Neuber-Papkovich solution in two-dimensional elasticity

Applying the mean stress approach to the integration of elastostatic equations, we arrive at the following results:

1. 1. known derivations of the Neuber-Papkovich representation are incomplete in multiply-connected regions (cf.[2] and [6]),

2. 2. displacement field is defined by the sum of two component fields (4) and (5a-b) and not by (4) only,

3. 3. mean normal stress and displacement might be many-valued in multiply-connected regions; the conditions of single-valuedness of the solution have the form (2b).

These results show that the usual procedure of applying the Neuber-Papkovich representation to boundary-value problems in a multiply-connected region may lead - contrary to expectation - to an incorrect (many-valued) solution if, in addition, the conditions (2b) are not satisfied identically. So far, this fact was neglected both in theoretical and engineering investigations.