任意厚度具有自由边叠层板的精确解析解

自由边问题一直是三维弹性力学中的难题,通常很难满足自由边上一个正应力和两个剪应力都等于0．基于三维弹性力学基本方程和状态空间方法,引入自由边界位移函数并考虑全部弹性常数,建立了正交异性具有自由边单层和叠层板的状态方程．对状态方程中的变量以级数形式展开,通过边界条件的满足精确求解任意厚度具有自由边叠层板的位移和应力,此解满足层间应力和位移的连续条件．算例计算表明,采用引入的位移函数形式,简化了计算过程并且采用较少的级数项可以获得收敛解．与有限元方法计算结果进行了对比,可以得到较高精度的数值结果．其解可以作为其它数值方法和半解析方法的参考解．     

抽象算子在偏微分方程中的应用（I）

根据解析函数和线性算子的基本性质定义了一类线性算子，建立了关于这种算子的完整理论，然后把一般形式的高阶常系数线性偏微分方程初值问题的解析解用这种算子表示出来；通过把这种算子表示成积分形式，这种算子形式的偏微分方程解就转化为积分形式的解，我们就彻底解决了把任意阶常系数线性偏微分方程初值问题的解析解求出并表示成给定函数的积分这一重要课题，而无需传统的对方程进行分类和讨论。     

抽象算子在偏微分方程中的应用(Ⅰ)

根据解析函数和线性算子的基本性质定义了一类线性算子，建立了关于这种算子的完整理论，然后把一般形式的高阶常系数线性偏微分方程初值问题的解析解用这种算子表示出来；通过把这种算子表示成积分形式，这种算子形式的偏微分方程解就转化为积分形式的解，我们就彻底解决了把任意阶常系数线性偏微分方程初值问题的解析解求出并表示成给定函数的积分这一重要课题，而无需传统的对方程进行分类和讨论     

两种材料组成空间的弹性力学基本解

本文研究了由两种不同材料的半空间所组成弹性体的弹性力学基本解。应用三维弹性理论中的Papkovich-Neuber通解以及Kelvin特解,求解出了在空间内部作用有集中力时空间的弹性力学位移场。该位移场在两个半空间内部分别满足各自的位移平衡方程,在其交接面上满足位移及面力的连续条件。作为本文结果的几种特殊情况,半空间的Lorentz问题与Mindlin问题的解,以及Stokes流中类似问题的解均可从该解答中导出。     

正交各向异性弹性力学正交关系的研究

新的正交关系被推广到正交各向异性三维弹性力学．将弹性力学新正交关系中构造对偶向量的思路推广到正交各向异性问题．将弹性力学求解辛体系的对偶向量重新排序后,提出了一种新的对偶向量．由混合变量求解法直接得到对偶微分方程．所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点．由于对偶微分矩阵的这一特点,对于正交各向异性三维弹性力学发现了2个独立的、对称的正交关系．采用分离变量法求解对偶微分方程．从正交各向异性弹性力学求解体系的积分形式出发,利用一些恒等式证明了新的正交关系．新的正交关系不但包含原有的辛正交关系,而且比原有的关系简洁．新正交关系的物理意义是对偶方程的解关于z坐标的对称性的体现．辛正交关系是一个广义关系,但辛正交关系可以在一定的条件下以狭义的强形式出现．新的研究成果将为研究正交各向异性三维弹性力学的解析解和有限元解提供新的有效工具．     

复合材料力学的Hamilton体系和辛几何方法(Ⅱ)——平面问题

把在本文第(Ⅰ)部分[8]中讲述的基本原理和方法用于求解各向异性平面问题.先建立可进入Hamilton体系的广义变分原理,求出Hamilton微分算子矩阵,再求解横向本征解,可得到矩形域各向异性线性弹性平面问题的级数解和半解析解.     

基于应力形式的二维弹性问题的本征展开法

给出求解基于应力形式的二维弹性问题的本征函数展开方法．通过引入适当的状态函数,将该问题的基本偏微分方程等价地转化为上三角微分系统,导出相应的上三角算子矩阵．证明了该矩阵的两个对角块算子均具有规范的正交本征函数系,并得到它们在相应空间中的完备性．此外,基于本征函数系的完备性,应用本征函数展开法给出了二维弹性问题的一般解．     

上三角算子矩阵的本征函数展开法在应力形式的二维弹性问题中的应用

深入研究了求解基于应力形式的二维弹性问题的本征函数展开法．根据已有的研究结果,将基于应力形式的二维弹性问题的基本偏微分方程组等价地转化为上三角微分系统,并导出了相应的上三角算子矩阵．通过深入研究,分别获得了该算子矩阵的两个对角块算子更为简洁的正交本征函数系,并证明了它们在相应空间中的完备性,进而应用本征函数展开法给出了该二维弹性问题的更为简洁实用的一般解．此外,对该二维弹性问题,还指出了什么样的边界条件可以应用此方法求解．最后应用具体的算例验证了所得结论的合理性．     

直角坐标系下层状黏弹性地基Biot固结解析刚度矩阵解

基于直角坐标系下Biot固结的基本控制方程,并考虑软土土骨架的黏弹性特性,通过Fourier-Laplace积分变换、解耦变换、微分方程组理论和矩阵理论,推导了黏弹性地基Biot固结三维空间问题和平面应变问题在积分变换域的解析解,进而得到对应问题的单元刚度矩阵.然后根据对号入座原则组装得到层状黏弹性地基Biot固结对应问题的总体刚度矩阵.通过求解总体刚度矩阵形成的线性代数方程,得到层状黏弹性地基Biot固结对应问题在积分变换域内的解答.最后应用Fourier-Laplace逆变换得到其物理域内的解.对比求解黏弹性Biot固结问题退化的弹性Biot固结问题与已有解答,验证了刚度矩阵计算方法的正确性,为层状黏弹性地基Biot固结问题提供了理论基础.     

弹性地基上四边自由矩形薄板的解析解

将弹性地基用Winkler模型来代替,并首先把弹性地基上薄板弯曲问题的控制方程表示成为Hamilton正则方程,然后利用辛几何方法对全状态相变量进行分离变量,求出其本征值后,再按本征函数展开的方法求出弹性地基上四边自由矩形薄板的解析解．由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性地基上薄板弯曲的基本方程出发,直接利用数学的方法求出可以满足四边自由边界条件的解析解,使得问题的求解更加理论化．还给出了计算实例来验证所采用的方法以及所推导出的公式的正确性．     

A general solution for three dimensional steady-state transversely isotropic hygro-thermo-magneto-piezoelectric media

In this article, we extract the general solution of three dimensional (3D) equations using potential theory method (PTM) for steady-state, transversely isotropic, hygro-thermo-magneto-piezoelectric media (HTMPM). The governing equations are simplified by introducing the displacement functions. A general solution is completely determined by advantage of the superposition principle and operator theory, which is connected in terms of two functions, fulfilling a second-order and twelfth-order homogeneous partial differential equation (PDE), separately. With the help of Almansi’s theorem, the general solution can be further shortened, which is stated by seven harmonic functions only. The acquired general solutions are straightforward structure and helpful in boundary value problems of HTMPM. Further, we apply the 3D fundamental solutions inside an infinite and on the surface of semi-infinite of a steady point heat source united with a steady point moisture source transversely isotropic HTMPM. Comprehensive and exact solutions are given in the form of elementary functions, which appear as a standard for various types of approximate solutions and numerical codes. Some numerical simulation is conducted based on the obtained general solutions.     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. I: General solutions and fundamental solutions

Four steady-state general solutions are derived in this paper for the two-dimensional equation of isotropic thermoelastic materials. Using the differential operator theory, three general solutions can be derived and expressed in terms of one function, which satisfies a six-order partial differential equation. By virtue of the Almansi’s theorem, the three general solutions can be transferred to three general solutions which are expressed in terms of two harmonic functions, respectively. At last, a integrate general solution expressed in three harmonic functions is obtained by superposing the obtained two general solutions. The proposed general solution is very simple in form and can be used easily in certain boundary problems. As two examples, the fundamental solutions for both a line heat source in the interior of infinite plane and a line heat source on the surface of semi-infinite plane are presented by virtue of the obtained general solutions.     

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

二维各向同性多孔介质的弹性动力学通解

在二维直角坐标系下,从固体位移和流体流速满足的基本方程出发,研究了二维各向同性多孔介质的弹性动力学通解.首先引入4个物理量,对固体骨架的运动方程、流体流速运动方程、连续性方程进行整理,将方程组分解成膨胀波和扭转波两部分,并利用Lur’e算子矩阵理论,获得由3个类调和函数表示的动力学通解,该通解满足全部基本方程.最后将时间项退化获得稳态通解,并证明了稳态通解的完备性.     

集中载荷作用下层合厚圆板的轴对称弯曲

从三维弹性力学基本方程出发,建立了横观各向同性层合圆板轴对称弯曲问题的状态方程,并将板面的集中载荷展成付里叶贝塞尔级数,从而给出问题的解析解,此解满足弹性力学全部方程,计及了所有独立的弹性常数,并满足层间连续性条件。     

常系数非齐次线性微分方程组的初等解法

利用初等变换将常系数非齐次线性微分方程组化为由若干个相互独立的高阶常系数非齐次线性微分方程组成的方程组,再利用高阶常系数齐次线性微分方程的特征根法和非齐次方程的待定系数法求该方程组的基本解组及特解,最后通过初等变换求原方程组的基本解组及特解,从而可求出其通解.     

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations北大核心CSCD

Burgers方程是一类应用广泛的非线性偏微分方程,方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步:第一步,对三维Burgers方程中的线性导数项(包括时间导数项),求出相应的多项式特解。第二步,将求出的多项式特解作为基函数,对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比,该算法简单易行,得到的近似解精度非常高,算法极其稳定,对于教学过程中提高学生的编程能力,加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。