两种坐标系中旋转体的体积计算

分别在直角坐标系和极坐标系情形下,计算笛卡尔叶形线、心形线、四叶玫瑰线和伯努利双纽线所围区域绕轴旋转所得旋转体体积。     

Conserved quantities and Hamiltonization of nonholonomic systems

This paper studies hamiltonization of nonholonomic systems using geometric tools, building on [1], [5]. The main novelty in this paper is the use of symmetries and suitable first integrals of the system to explicitly define a new bracket on the reduced space that codifies the nonholonomic dynamics and carries, additionally, an almost symplectic foliation (determined by the common level sets of the first integrals); in particular cases of interest, this new bracket is a Poisson structure that hamiltonizes the system. Our construction of the new bracket is based on a gauge transformation of the nonholonomic bracket by a global 2-form that we explicitly describe. We study various geometric features of the reduced brackets and apply our formulas to obtain a geometric proof of the hamiltonization of a homogeneous ball rolling without sliding in the interior side of a convex surface of revolution.     

Hamiltonization of the generalized Veselova LR system

We revise the solution to the problem of Hamiltonization of the n-dimensional Veselova nonholonomic system studied previously in [1]. Namely, we give a short and direct proof of the hamiltonization theorem and also show the trajectorial equivalence of the problem with the geodesic flow on the ellipsoid.     

用混合法求解带有离散支承的旋转体

本文采用超静定结构力学中力法处理带有离散支承的旋转体.将离散固定支承反力看成超静定未知力.用旋转体半解析有限元法去计算力法正则方程的系数阵(柔度阵)和右端项,求解力法正则方程.然后,将支承反力与外荷载叠加起来作为旋转体外荷载,而得到离散支承的旋转体有限元解.     

Finsler Geodesics of Lagrangian Systems Through Routh Reduction

Mediterranean Journal of Mathematics - We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler–Lagrange equations of a strongly convex...     

Hamiltonization of non-holonomic systems in the neighborhood of invariant manifolds

The problem of Hamiltonization of non-holonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in non-holonomic mechanics.     

Reduction of the Two-Dimensional Thermoelasticity Problems for Solids with Corner Points to Key Integrodifferential Equations

Ukrainian Mathematical Journal - We propose a generalization of the method of direct integration of the original equations of two-dimensional problems of thermoelasticity for solids with corner...     

Hamiltonization of Higher-Order Nonlinear Ordinary Differential Equations and the Jacobi Last Multiplier

It is known that Jacobi’s last multiplier is directly connected to the deduction of a Lagrangian via Rao’s formula (Madhava Rao in Proc. Benaras Math. Soc. (N.S.) 2:53–59, 1940). In this paper we explicitly demonstrate that it also plays an important role in Hamiltonian theory. In particular, we apply the recent results obtained by Torres del Castillo (J. Phys. A Math. Theor. 43:265202, 2009) and deduce the Hamiltonian of a second-order ODE of the Lienard type, namely, [(x)\ddot]+f(x)[(x)\dot]²+g(x)=0\ddot{x}+f(x){\dot{x}}^{2}+g(x)=0. In addition, we consider cases where the coefficient functions may also depend on the independent variable t. We illustrate our construction with various examples taken from astrophysics, cosmology and the Painlevé-Gambier class of differential equations. Finally we discuss the Hamiltonization of third-order equations using Nambu-Hamiltonian mechanics.     

Controlled Deformation of Solids

A model of the control of elastic deformation of solids with a gradual variation in the position of the load is described. The search for a constrained extremum of the objective function is reduced to an unconstrained extremum problem. Possible engineering applications are suggested.     

非线性固体的各向导性程度

这篇文章目的在于将在Rychlewski和张进敏(1988)中得到详细研究的线弹性体各向异性度的概念推广到非线性非弹性体.所定义的各向异性度的良好特性说明了其合理性.     

Atomic Vibrations in Solids

This paper is an introductory survey of the basic theory ofatomic vibrations in solids. Crystalline solids, solids withdefects, and disordered materials are considered both generallyand by means of illustrative models. The emphasis throughoutis on a unified approach, using the formalism and theorems ofmatrix algebra.     

非局部非对称弹性固体理论

本文基于非局部连续统场论和非线性连续体力学理论,建立了非局部非对称弹性固体的非线性理论.它完善和发展了Eringen等人所建立的非局部弹性场论.将文献[1]中所建立的非局部非对称弹性力学的线性理论推广到有限变形.证明了在非局部弹性固体中存在着非局部体力矩.非局部体力矩引起应力的非对称性,而非局部体力矩则由原子晶格相互作用形成的共价键所产生的.并应用本文建立的理论合理地解释了平面横波和纵波色散系关的不相似性.     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Elastic Solids with Micro-Structure.

Abstract. Based on the general framework of evolutionary equations in the sense of [6,8] and referring to well-posedness results in the context of evolutionary problems from mathematical physics we consider two problems from elasticity theory as particular applications. Our structural perspective is illustrated by showing that the theory of elastic solids with micro-structure can be transparently embedded into the abstract solution theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

饱和多孔介质耦合系统的变分原理

本文采用变积方法,建立了等温准静态下饱和多孔介质的六类变量的广义变分原理.在此基础上,通过引入约束条件得到各级变分原理,其中包括五类变量,四类变量,三类变量和二类变量的变分原理.除得到文献中已有的变分原理外,本文给出了许多新的变分原理,为建立饱和多孔介质的有限元模型提供了基础.     

固体的统一弹、粘、塑性理论

提出了一个弹、粘、塑性统一理论,可用以计算在任意受力过程下物体各点弹、粘、塑性的变化情况.理论的基础是热力学定律及虚弹性假设.文中导出本构关系以及有关的变分原理,由此容易推导空间-时间的有限元构式.值得指出,适当选取文中的物质常数,可以得出类似于当前习用的塑性本构关系.