Synchronous/Asynchronous Buckling of Double-Layered Microplate Systems北大核心CSCD

采用修正的偶应力理论和双变量高阶剪切变形理论,发展了层间填充弹性介质的双层微板系统在面内压缩荷载作用下的屈曲模型.基于Euler-Lagrange方程推导了系统屈曲的控制微分方程,运用Navier法获得了上下层均为四边简支时系统同步/异步屈曲的解析解.通过数值算例讨论了系统各参数对其屈曲特性的影响.结果表明：系统的异步屈曲特性依赖于材料尺度参数、长宽比和弹性介质模量,而同步屈曲特性仅依赖于前两项,并且异步屈曲荷载高于同步屈曲荷载;弹性介质的Pasternak模量较之于Winkler模量对系统的屈曲特性影响更显著.     

弹性介质中受轴向冲击载荷作用的圆柱壳的屈曲临界载荷计算分析

建立并求解了弹性介质中圆柱壳的径向位移控制方程,考虑边界条件及相容条件,得到了应力波传播及反射过程中圆柱壳的动力屈曲分叉条件.通过计算得到了不同时间段屈曲临界载荷与应力波波阵面到达圆柱壳的位置、弹性介质的刚度、壳体未嵌入弹性介质部分的长度与总长之比的关系.数值计算结果表明,弹性介质中的圆柱壳发生轴对称屈曲和非轴对称屈曲趋势一致;嵌入弹性介质部分越深、弹性介质刚度越大圆柱壳越难屈曲;屈曲临界载荷随着弹性介质刚度的增大经历了增长缓慢、增长迅速以及增长较慢3个阶段;应力波反射前波阵面通过分界面后,屈曲仅发生在应力波传播区域,反射波波阵面通过分界面前,临界载荷较小时屈曲先发生在反射端部,随着轴向阶数增大,屈曲覆盖整个圆柱壳区域,反射波波阵面通过分界面后,壳体发生的屈曲始终覆盖整个圆柱壳区域.     

铁磁体磁弹性相互作用的广义变分原理与理论模型 *

定量分析发现:目前对三维弹性体所建立的各类理论模型均不能同时模拟铁磁介质磁弹性相互作用的两类基本实验现象 .进而导致模拟复杂形状结构 ,如壳体的磁弹性力学行为陷入困境 .这样 ,对三维铁磁弹性介质建立能统一描述已有两类典型实验现象的理论模型就成为模拟复杂形状结构磁弹性相互作用力学特征的关键 .从三维铁磁体磁弹性耦合系统的总能量泛函变分原理出发 ,建立了能同时模拟已有铁磁板磁弹性相互作用两类典型实验现象的三维可磁化弹性体的理论模型 .结果表明 :对于各向同性线性铁磁体 ,除了得到与由经典磁偶极子物理模型导出的宏观电磁力计算式一致的作用在铁磁介质上的磁体力外 ,还得到了作用在可磁化介质表面的边界面分布磁力 ,其值为两侧磁介质Faraday磁应力在表面处的跳变值 .这是各经典模型 ,如磁偶极子模型和公理化模型等所没有的 .     

在受轴向周期扰动作用下双壁碳纳米管动力屈曲的研究

对双壁碳纳米管受轴向周期扰动的动力响应进行了研究．采用连续体模型研究双壁碳纳米管的动力屈曲问题,考虑了壁间van der Waals力和周围弹性介质对轴向动力屈曲的影响．给出了受轴向周期扰动的屈曲模型及临界应变和临界频率．发现双壁碳纳米管由于壁间van der Waals力的作用较单壁碳纳米管具有较低的临界应变．van der Waals力和周围弹性介质将影响双壁碳纳米管不稳定区,van der Waals力使受轴向周期性扰动的双壁碳纳米管的临界频率增大,周围弹性介质对双壁碳纳米管的临界频率影响不大．     

一维纳米准晶层合梁的非局部振动、屈曲与弯曲研究

基于非局部理论,建立了一维纳米准晶层合简支深梁模型,研究了其自由振动、屈曲行为及其弯曲变形问题.采用伪Stroh型公式,导出了纳米梁的控制方程,并通过传递矩阵法获得简支边界条件下纳米准晶层合梁固有频率、临界屈曲载荷及弯曲变形广义位移和广义应力的精确解.通过数值算例,分析了高跨比、层厚比、叠层顺序及非局部效应对一维纳米准晶层合简支梁固有频率、临界屈曲载荷和弯曲变形的影响.结果表明：固有频率和临界屈曲载荷随着非局部参数增大而减小;外层准晶弹性常数更高时,固有频率和临界屈曲载荷更大;叠层顺序对纳米准晶梁的力学行为有较大影响.所得的精确解可为纳米尺度下梁结构的各种数值方法和实验结果提供参考.     

各向异性介质中有衬砌的非圆形结构与SH波的相互作用

本文利用复变函数方法求解各向异性介质中有衬砌的非圆形结构与ＳＨ波的相互作用问题．各向异性介质可以用来模拟结构周围的地质条件．利用作者在文献［５，７］中的方法决定介质和衬砌结构中的散射波，结合Ｓａｖｉｎ在文献［６］中求解有衬砌结构所采用的对结构轮廓线进行近似处理的方法，并利用结构与介质的连续性条件与衬砌内边界上的边界条件，对问题进行求解．作为算例．本文讨论了在同一种各向异性介质中，两种不同材料组成的正方形有衬砌的结构与ＳＨ波的相互作用问题．给出了数值分析结果．     

关于分数阶Maxwell淤泥模型及其在水波衰减问题中的应用

分数阶Maxwell模型可用来模拟粘弹性的海床淤泥．与传统的有理分式模型相比,分数阶Maxwell模型可以用更少的自由参数,较好地描述某些真实淤泥的流变特性．将该分数阶Maxwell模型用于研究淤泥与自由表面水波的相互作用,并得到了线形单色波的衰减率．从水波衰减率曲线中可观测到淤泥层的共振现象,共振时衰减率将达到峰值．对于线形单色波,其衰减率还可表示为各模态衰减率之和,从而可研究某一模态的运动对水波衰减的影响．模态分析表明,当某一模态运动引发共振时,总衰减率由该模态的模态衰减率决定．     

一种适用于非均匀地形的高阶Boussinesq水波模型

推导了适用于变地形情况的高阶Boussinesq波浪模型．该模型采用自由表面边界条件作为时间步进方程,利用势函数满足的Laplace方程的解析解形式建立了自由表面边界速度和底面边界速度之间的关系,使得问题封闭．以0.5倍相对水深处的速度为基本未知量,在对Laplace方程解析解进行级数求逆时保留水深梯度的高阶项,改进了速度场的Taylor展开式．对于线性特性,进行了线性浅化和Booij反射的验证性计算．为了检验有背景流动情况下拓展的Boussinesq模型的性态,对波-流相互作用问题进行了数值模拟．数值计算结果与现有理论解或其他完全势流的数值解吻合良好,表明该模型的应用范围可以扩展到含有非均匀变化地形的问题．     

对部分线性模型用惩罚最小二乘估计时最优光滑化的注记

部分线性模型也就是响应变量关于一个或者多个协变量是线性的, 但对于其他的协变量是非线性的关系\bd 对于部分线性模型中的参数和非参数部分的估计方法, 惩罚最小二乘估计是重要的估计方法之一\bd 对于这种估计方法, 广义交叉验证法提供了一种确定光滑参数的方法\bd 但是, 在部分线性模型中, 用广义交叉验证法确定光滑参数的最优性还没有被证明\bd 本文证明了利用惩罚最小二乘估计对于部分线性模型估计时, 用广义交叉验证法选择光滑参数的最优性\bd 通过模拟验证了本文中所提出的用广义交叉验证法选择光滑参数具有很好的效果, 同时, 本文在模拟部分比较了广义交叉验证和最小二乘交叉验证的优劣.     

平面P波在弹性介质和非饱和多孔弹性介质分界面上的传播

使用线性粘滞的多孔弹性介质模型,解决在弹性介质和非饱和多孔弹性介质分界面上平面P波的反射与透射问题,这里的非饱和多孔介质中固体骨架被两种相互耦合的流体（液体和气体）所充满．通过势函数的方法得到了振幅反射系数与振幅透射系数．然后推导得到入射波与反射波、透射波之间能量转换情况．研究发现:用振幅比和能量比所表示的反射系数与透射系数是与入射角度、饱和度、入射频率以及上下层介质的弹性常数有关的方程式．数值计算通过图形的形式表达出来,而且入射角度、频率及饱和度对振幅和能量的反射与透射系数的影响分别进行了讨论．证明了在整个波的传播过程中分界处并没有发生能量的耗散．     

Estimation of Weak Lensing Parameters by Stochastic Integration

Abstract

A procedure for computing distance–redshift statistics in inhomogeneous universes is described. We interpret the generalized Dyer–Roeder equation [Dyer, C. C.; Roeder, R. C. The distance-redshift relation for universes with no intergalactic medium. Astrophys. J. 1972, 174, L115–L117; Dyer, C. C.; Roeder, R. C. Distance-redshift relations for universes with some intergalactic medium. Astrophys. J. 1972, 180, L31–L34] as a stochastic differential equation which treats the shearing forces as Brownian movement and yields a Monte-Carlo procedure. Shear distributions taken from N-body simulations produce a redshift dependent diffusion coefficient.     

Buckling delamination of a sandwich plate-strip with piezoelectric face and elastic core layers

In this study, the buckling delamination problem of a sandwich plate-strip with a piezoelectric face and elastic core layers is studied. It is assumed that the plate-strip is simply supported and grounded along its two parallel ends and is subjected to uniformly-distributed compressive forces on these ends. Moreover, we suppose that the plate-strip has two interface inner cracks between the face and the core layers and it is also supposed that before the plate-strip is loaded (i.e. in the natural state), the surfaces of these cracks have insignificant initial imperfections. Due to compressive forces acting along the cracks we investigate the evolution of the initial imperfections of the cracks’ surfaces. Hence, the values of the critical buckling delamination force of the considered plate-strip are determined from the criteria, according to which, the considered initial imperfections of the cracks’ surfaces grow indefinitely by the compressive forces. Mathematical modeling of the considered problem is formulated within the scope of the exact nonlinear equations of electro-elasticity in the framework of the piecewise homogeneous body model, the solution of which is found numerically by employing the finite elements method. Numerical results showing the influences of the geometrical and material parameters as well as the coupling of the electrical and mechanical fields on the values of the critical force are presented and analyzed.     

Fixed point theorems for generalized contractions in ordered metric spaces

The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505-2517], A. Petru?el, I.A. Rus [A. Petru?el, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411-418] and R.P. Agarwal, M.A. El-Gebeily, D. O'Regan [R.P. Agarwal, M.A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given.     

非完整系统有耗散和循环力时的平衡不稳定性

讨论定常非完整系统在耗散、保守、循环力作用下的不稳定平衡问题．应用方法是基于运动微分方程解的存在性,当t→-∞时,系统渐近地趋于平衡状态．假定在平衡位置附近,动能、Reyleigh耗散函数、位置力都是无限可微函数．结果将通过一个实例说明．部分结果参见Kozlov V V. On the asymptotic motions of systems with dissipation. Prikl Math Mekh, 1994, 58(4): 31-36. (in Russian); Merkin D R. Introduction to the Theory of the Stability of Motion. Moscow: Nauka, 1987. (in Russian); Thomson W, Tait P. Treatise on Natural Philosophy, Part Ⅰ. Cambridge: Cambridge University Press, 1879．     

Local buckling of compressed elements of layered viscoelastic material

The problem of local surface buckling of compressed elements of layered material is solved using the equations of the theory of multilayer plates [1, 2, 5]. The local buckling deformations are described by means of the solutions of differential-difference equations damping into the interior of the medium. The effect of compressive forces, buckling wavelength, and the elastic and viscoelastic constants of the material on the behavior of the deformations and on the thickness of the layer to which significant buckling extends is investigated. Numerical estimates are presented for the buckling parameters of compressed laminated-plastic elements.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 816–821, September–October, 1968.     

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate,a viscoelastic (elastic) bond layer,and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 771–788, November–December, 2007.     

A model combining acid-mediated tumour invasion and nutrient dynamics

We illustrate a mathematical model for the evolution of multicellular tumour spheroids in a host tissue, including the effect of the excess H⁺ ions and the nutrient dynamics. Both the avascular and the vascular case are considered. The model is a nontrivial generalization of the simple scheme proposed in [K. Smallbone, D.J. Gavaghan, R.A. Gatenby, P.K. Maini, The role of acidity in solid tumour growth and invasion, J. Theoret. Biol. 235 (2005) 476–484]. Many different situations may occur, depending on the values of the physical parameters involved. Existence of solutions and qualitative properties are investigated.     

Book reviews

L.N. TREFETHEN and D. BAU, III,Numerical Linear Algebra,SIAM, Philadelphia, 1997G.-C. ROTA,Indiscrete Thoughts,Birkhäuser, Boston, 1997D.E. KEYES, A. SAMEH and V. VENKATAKRISHNAN, eds.Parallel Numerical Algorithms,Kluwer, Dordrecht, 1997A. KIRSCH,An Introduction to the Mathematical Theory of Inverse Problems,Springer, New York, 1996L.F. SHAMPINE, R.C. ALLEN, Jr. and S. PRUESS,Fundamentals of Numerical Computing,Wiley, New York, 1997C.W. UEBERHUBERNumerical Computation, 2 vols.Springer, Berlin, 1997W.G. McCALLUM et al.Multivariate Calculus,Wiley, New York, 1997ZHI-QUAN LUO, JONG-SHI PANG and D. RALPH,Mathematical Programs with Equilibrium Constraints,Cambridge University Press, Cambridge, 1996P.R. POPIVANOV and D.K. PALAGACHEV,The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations,Akademie Verlag, Berlin, 1997