力学的几何化

本文介绍了力学发展过程中,力学理论系统的几何化.把力学的动力系统归结于黎曼几何、辛几何的几何课题,以及对偶概念、对称概念、变换和不变量概念在力学中的普遍应用.     

超细长弹性杆精确模型的运动学问题

作为研究DNA等生物大分子链力学行为的模型,考察弹性细杆精确模型截面随弧坐标和时间的运动学问题.给出了基本假定,用映射的概念阐明离散化思想,得到了杆的运动学方程,导出了存在拉/压变形时弯扭度和角速度的关系;定义了截面的虚位移,表示为弧坐标和时间变分均为零的变分法则,在微分和变分运算次序可以交换的前提下,导出了截面虚角位移的导数与截面弯扭度以及角速度变分的关系,为建立超细长弹性杆精确模型动力学的分析力学方法准备理论基础.     

壳体计算的位势——有限元方法

本文提出了壳体计算中的一种位势——有限元方法。这种方法的特点是首先利用位势理论将壳体理论的边值问题化成一组积分微分方程,然后再将之离散而求解。与一般有限元方法相比,位势——有限元法的形状函数只需保证位移本身的连续性;另外,离散化过程并不需改变壳体原来的几何形状,因而可以方便地计算各种变曲率壳体。     

从刚体力学引出结构稳定性的几个基本概念

在材料力学、结构力学等课程的教学中,由于弹性力学理论本身过于复杂,压杆或结构的稳定性主要概念难以全面引入.本文通过刚体力学模型和能量法,引入了临界力、平衡路径分叉、非线性后屈曲平衡路径稳定性、临界点稳定性、缺陷结构稳定性等重要概念,由于模型简单导致了所有概念都可解析讨论,力学概念清晰,易于理解,供相关人员参考.     

轴对称物体内三维残余应力场的确定

三维残余应力场的测量理论是国内外未能解决的问题.本文建立了新的轴对称空间力学模型,介绍了对45钢140×200圆柱体进行内剥层测量的实验.在分析中引入了罐载和非罐载应力的概念,为正确处理变动的边界条件给出了思路、使得空间力学模型真实地反映了轴对称体内残余应力的实质.在求解中提出了变应力函数及其解法,使得轴对称三维残余应力场的测量理论得到了真正解决.     

三维编织复合材料圆柱壳在外压作用下的屈曲分析

提出一种三维四向编织复合材料圆柱壳的细观力学模型,宏观上基于Reddy高阶剪切变形理论和广义Kármán型方程,采用奇异摄动法求解在固支边界条件下三维四向编织复合材料圆柱壳在外压作用下的屈曲和后屈曲.分析中同时考虑非线性前屈曲、大挠度和初始几何缺陷的影响.讨论了纤维体积含量、壳体几何参数等因素对圆柱壳屈曲行为的影响.     

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简要综述了生物膜力学与几何的新进展. 在生物膜力学中,着重介绍了基于微分算子的平衡 理论和几何约束理论;在生物膜几何中,重点评述了源于生物膜力学的新梯度算子及其积分 性质. 指出：新梯度算子可能在生物膜曲面上诱发新的驱动力;生物膜力学与几何是一个有 机整体,其背后存在着一个对称的几何体系,包括对称的微分算子以及对称的积分定理系统.     

常子午线曲率薄壳轴对称几何非线性问题的复变量方程

1.几何非线性问题的基本方程在本世纪初,Reissner H.和Meissner E.利用在线性薄壳理论中存在的静力-几何比拟关系,将线弹性薄壳轴对称问题,归结为以应力函数和转角为未知量的两个常微分方程。以后,人们利用这两个方程的相似性,引入复未知函数,把一些典型壳体的方程简化为一个二阶变系数常微分方程,为这些问题的求解带来极大的便利。本文将这一方法推广到薄壳大位移问题,导出用复未知函数表示的常子午线曲率壳体轴对称变形的非线性微分方程。从这个一般方程可以直接得到关于柱壳,锥壳,圆球壳,环壳和圆板几何非线性问     

基于SVM回归的圆柱壳体抗脉冲强度预估模型

在小子样结构响应试验数据样本的基础上,利用支持向量机回归的方法模拟了圆柱壳体动态极限应变峰值同壳体几何尺寸和外加脉冲载荷大小的非线性函数关系,同时通过改进的模拟退火单纯形混合算法优化了支持向量机的性能参数,并将支持向量机回归分析的预测性能同BP人工神经网络方法做了比较,验证了具有优化性能参数组合的支持向量机在小样本条件下更好的预测和推广能力. 最后,从支持向量机回归模型导出了大尺寸圆柱壳体抗脉冲载荷的强度极限同自身几何尺寸的多元函数关系,从而为该类型壳体设备抗脉冲载荷的强度分析提供了一个可借鉴的预估模型. 研究结果表明了支持向量机在机械结构的强度预估和可靠性分析等力学领域具有广泛的应用前景.      

9.计算力学

计算方法在固体力学中起着关键作用.它是模拟各种基本力学行为的一种方法,是把这种改进了的模拟能力转变成新的工程工具的一种媒介,是把这些工具应用于工程实践的一种手段.现代计算方法能将力学系统的符合实际的模型加以系统地表达而不管是否容易得到分析解.日益增大的计算能力也是发展更精确的理论的一个动力,因为它使得用这些理论来解决复杂的工程问题成为可能.      

Restrictions on nonlinear constitutive equations for elastic shells

Constitutive equations for the resultant forces and moments applied to a shell-like body necessarily couple the influences of the shell geometry and the constitutive nature of the three-dimensional material from which the shell is constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the complicated influence of the shell geometry on the constitutive response of the shell is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic shells which ensure that exact solutions of the shell equations are consistent with exact nonlinear solutions of the three-dimensional equations for homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of shells. Examples of the linear theories of a plate and a spherical shell are considered.     

ON EXACT SOLUTION OF KÁRMÁN’S EQUATIONS OF RIGID CLAMPED CIRCULAR PLATE AND SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

It is extremely difficult to obtain an exact solution of von Karman’s equations because the equations are nonlinear and coupled. So far many approximate methods have been used to solve the large deflection problems except that only a few exact solutions have been investigated but no strict proof on convergence is presented yet. In this paper, first of all, we reduce the von KÁrmÁn’s equations to equivalent integral equations which are nonlinear, coupled and singular. Secondly the sequences of continuous function with general form are constructed using iterative technique. Based on the sequences to be uniformly convergent, we obtain analytical formula of exact solutions to von Karman’s equations related to large deflection problems of circular plate and shallow spherical shell with clamped boundary subjected to a concentrated load at the centre.     

NONLINEAR DYNAMICAL BIFURCATION AND CHAOTIC MOTION OF SHALLOW CONICAL LATTICE SHELL

The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.     

A unified nonlinear formulation for plate and shell theories

A general geometrically exact nonlinear theory for the dynamics of laminated plates and shells under-going large-rotation and small-strain vibrations in three-dimensional space is presented. The theory fully accounts for geometric nonlinearities by using the new concepts of local displacements and local engineering stress and strain measures, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle. Moreover, the model accounts for shear coupling effects, continuity of interlaminar shear stresses, free shear-stress conditions on the bonding surfaces, and extensionality. Because the only differences among different plates and shells are the initial curvatures of the coordinates used in the modeling and all possible initial curvatures are included in the formulation, the theory is valid for any plate or shell geometry and contains most of the existing nonlinear and shear-deformable plate and shell theories as special cases. Five fully nonlinear partial-differential equations and corresponding boundary and corner conditions are obtained, which describe the extension-extension-bending-shear-shear vibrations of general laminated two-dimensional structures and display linear elastic and nonlinear geometric coupling among all motions. Moreover, the energy and Newtonian formulations are completely correlated in the theory.     

Nonlinear stress and deformation analysis of thin current-carrying strip shells

The paper analyzes the nonlinear deformation of a current-carrying thin shell in coupled electromagnetic and mechanical fields. The nonlinear magnetoelastic kinetic equations, physical equations, geometric equations, electrodynamic equations, expressions for the Lorentz force of a current-carrying thin shell in a coupled field are given. The normal Cauchy form nonlinear differential equations that include ten basic unknown functions are obtained by the variable replacement method. The difference and quasi-linearization methods are used to reduce the nonlinear magnetoelastic equations to a sequence of quasilinear differential equations that can be solved by discrete orthogonalization. Numerical solutions for the stresses and strains in a current-carrying thin strip shell with two edges simply supported are obtained as an example. The dependence of the stresses and strains in the current-carrying thin strip shell on the electromagnetic parameters is discussed. In a special case, it is shown that the deformation of the shell can be controlled by changing the electromagnetic parameters     

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green's function. To solve the integral-differential equations, expansion method is used to obtain Green's function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells     

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study, the first-order shear deformation theory(FSDT) is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC). The vibration analyses of the FG-GPLRC truncated conical shell are presented. Considering the graphene platelets(GPLs) of the FG-GPLRC truncated conical shell with three different distribution patterns, the modified Halpin-Tsai model is used to calculate the effective Young's modulus. Hamilton's principle, the FSDT, and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell. The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell. Then, the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method. The effects of the weight fraction and distribution pattern of the GPLs, the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed. This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.     

On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel’s classes of equations

Second-order ordinary differential equations (ODEs) with strong nonlinear stiffness terms (cubic nonlinearities) governing wave motions, dynamic crack propagations, nonlinear oscillations etc. in physics and nonlinear mechanics are analyzed. Selecting as guide line a second-order nonlinear ODE of the form of the forced Duffing equation and using admissible functional transformations it is possible to reduce it to an equivalent first-order nonlinear integrodifferential equation. The reduced equation is exact. In the limits of small or large values of the parameter characterizing this nonlinear problem, it is shown that further reductions lead to a nonlinear ODE of the Abel classes. Taking into account the known exact analytic solutions of this equivalent equation it is proved that there does not exist an exact analytic solution of this type of equations. However, in cases when convenient functional relations connecting all parameters of the corresponding null equation and the characteristics of the driving force exist, approximate analytic solutions to the problem under consideration are provided.     

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.