有限变形弹性体J积分守恒及其对偶形式

基于势能原理与余能原理的对偶关系,给出了有限变形弹性体的J积分形式及其对偶形式,并在弹塑性有限变形的前提下给出其路径守恒的证明.     

J积分在多层介质中的守恒性和其应用

首先证明和讨论了当裂纹平行于多层介质界面时,其裂纹尖端J积分的守恒性;然后建立了具有界面裂纹的由4种介质构成的材料分析模型,根据J积分在多层介质中的守恒性原理,提出了该模型的I型断裂能量释放率守恒假设,并经过了数值实验的验证,通过能量分析和量纲分析,用渐近分析方法得到了计算该模型结构界面裂纹尖端能量释放率和复应力强度因子的分析公式,并且进行了讨论。     

广义Heisenberg方程的无穷守恒律

本文给出了Heisenberg方程的无穷守恒律,并具体给出了其前三个守恒律,特别得到了铁磁链方程无穷多个微分形式与积分形式的守恒律。     

不受本构关系限制的守恒积分

Rice定义的J积分受介质本构关系的限制.本文对J积分的Rice定义加以推广,提出不受介质本构关系限制的守恒积分,并证明其具有通常J积分的性质.在弹性介质情况下,这种推广的守恒积分就是Rice定义的J积分.     

AKNS族与Geng族的规范等价的约束流

先由ＡＫＮＳ族出发用规范变换建立了Ｇｅｎｇ族的约束流及它的第一组守恒积分和Ｌａｘ表示,再用其Ｌａｘ表示建立了Ｇｅｎｇ族约束流的ｒ－矩阵。并且还给出了Ｇｅｎｇ族约束流的第二组守恒积分及其对合性的证明。     

非线性对流扩散方程的守恒律

利用直接方法研究了非线性对流扩散方程的守恒律,得到了关于非线性对流扩散方程的守恒律乘子性质的一个定理.利用这个定理,可以简化守恒律乘子的确定方程.随后通过对确定方程中的变量函数进行分析,发现在四种情况下乘子的确定方程是可解的.最后解出这些守恒律乘子,利用积分公式法分别得到了四种情况下对应于各个守恒律乘子的守恒律.     

动力系统的Poisson结构和可积变形

利用动力系统的守恒积分构造Poisson结构,将动力系统表示为广义Hamilton系统的形式,并以一个三维动力系统为例,通过添加任意可微函数推广守恒积分,构造系统的可积变形,并给出变形后系统的Poisson结构,由此得到了新的刘维尔可积系统.     

规范变换与有限维可积系统间的变换

本文表明，利用两个特征值问题的规范变换，不仅可以建立和它们相联系的势的约束之间以及相应的有限维Ｈａｍｉｌｔｏｎ系统间的变换关系式，而且可以由一个可积系统的对合守恒积分导出另一个系统的守恒积分     

复合材料平面断裂中的J积分

本文采用复变函数方法,首先将裂纹尖端应力和位移代入J积分的一般公式得到了线弹性正交异性复合材料单向板复合型裂纹尖端的J积分的复形式,其次证明了该J积分的路径无关性,最后推出了该J积分的计算公式.作为特例,给出了线弹性正交异性复合材料单向板Ⅰ,Ⅱ型裂纹尖端的J积分的复形式,路径无关性和计算公式.     

层状弹性材料界面J积分的产生和特征

层状弹性材料的裂纹方向垂直于界面时,沿围绕裂尖的任意一条封闭路径Γ的J积分（J_Г）由两部分组成,J_Г=J_tip+J_int,这里J_tip表示裂尖产生的J积分,J_int表示Γ所包围的界面产生的J积分.裂尖产生的J积分不随Γ变化,物理含义是裂纹扩展能量释放率;界面产生的J积分随Γ变化,物理含义与裂纹扩展能量释放率无关.界面J积分的产生使J_Г失去了路径无关特性,也失去了实际物理意义.为了有助于理解非均匀材料J积分的含义和局限性,分析了层状弹性材料界面J积分的产生原因和特点.由不同均匀弹性材料组成的层状材料中,应变能密度的跳跃是界面J积分产生的原因,而弹性模量和残余应力在界面处的跳跃可使应变能密度在界面处产生跳跃.层状弹性材料的界面J积分之间具有相互抵消的作用.     

描述低周疲劳裂纹扩展速率的循环J积分新参量

探讨了低周疲劳加载条件下的应力增量．应变增量关系，提出了模拟裂纹疲劳扩展的二维模型以建立新的循环．积分参量，详细阐述了该积分参量的定义、主要特点、物理意义以及数值计算方法，并通过紧凑拉伸试样的疲劳试验检验该积分参量的有效性．结果表明：该积分参量能够较好描述恒幅低周疲劳裂纹的扩展速率．此外，基于积分参量体系，从能量的角度解释了疲劳迟滞现象．     

The role of transverse stretching in the delamination fracture of softcore sandwich plates

This paper presents the semi-layerwise analysis of structural sandwich plates with through-width delamination. The mechanical model of rectangular plates is based on the method of four equivalent single layers and the system of exact kinematic conditions. An important improvement compared to a previous formulation is the consideration of linear and quadratic stretching term in the transverse displacement component. Three different delamination scenarios are investigated: core-core failure, face-core delamination and the face-face failure. By applying the first- and second-order laminated plate theories and the principle of virtual work the governing equations are derived. The equilibrium equations are solved under Lévy type boundary conditions using the state-space approach. Solutions for the mechanical fields are provided and compared to 3D finite element results. The energy release rate distributions along the delamination front are also determined using the J-integral. Although the stress resultants by transverse stretching do not influence directly the J-integral, the results indicate that this effect improves the accuracy of the model in general, and substantially influences the results of the first-order plate theory in the case of the face-face delamination.     

有限元广义伽略金方程,边界变分方程,边界积分方程

在[1]的基础上,我们进一步应用可动边界的变分原理于固体体系的离散分析,得到有限元广义伽略金方程,边界变分方程,边界积分方程.这些方程描述了待解函数在元素内部与元素的边界上应满足的方程.当对固体体系进行离散分析时,可以应用这些方程去建立不同情况下的求解待解函数的离散方程.亦可作为相应情况下的简化计算的依据.由本文得到的边界积分方程可知,在[2]中提出的J积分形式,应用于内部元素边界的围道积分计算是不适宜的.     

Classification of Local Conservation Laws of Maxwell's Equations

A complete and explicit classification of all independent local conservation laws of Maxwell's equations in four dimensional Minkowski space is given. Besides the elementary linear conservation laws, and the well-known quadratic conservation laws associated to the conserved stress-energy and zilch tensors, there are also chiral quadratic conservation laws which are associated to a new conserved tensor. The chiral conservation laws possess odd parity under the electric–magnetic duality transformation of Maxwell's equations, in contrast to the even parity of the stress-energy and zilch conservation laws. The main result of the classification establishes that every local conservation law of Maxwell's equations is equivalent to a linear combination of the elementary conservation laws, the stress-energy and zilch conservation laws, the chiral conservation laws, and their higher order extensions obtained by replacing the electromagnetic field tensor by its repeated Lie derivatives with respect to the conformal Killing vectors on Minkowski space. The classification is based on spinorial methods and provides a direct, unified characterization of the conservation laws in terms of Killing spinors.     

Application of configurational forces in the context of a kinked crack

The application of the configurational force approach in crack problems is often used in order to establish fracture criteria that are adapted to a specific material behaviour. The tangential component of the calculated vectorial quantity that acts at the crack tip is a generalisation of the conventional J-integral and can be interpreted as the energy release rate when the crack extends in this direction. However, the interpretation of nontangential components in the same way, and hence the interpretation of this vectorial quantity as the crack driving force, is not consistent with established kink criteria in the special case of linear elastic fracture mechanics. As a classical example, an in-plane loaded crack in a homogeneous isotropic linear elastic material is considered under the small strain assumption. Using the expansion of stress intensity factors at the extended crack tip, nontangential components of the configurational force can be interpreted as sensitivities to crack deflection. This perspective has the potential of generalisation which can be applied to more complex situations in order to study the interplay between mechanical fields in the vicinity of the crack tip and the microstructural influence within the process zone. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-local conservation laws for the equations of the irrotational isentropic plane-parallel gas motion

By introducing non-local variables, namely, the velocity potential and the stream function, and changing to the hodograph plane, the problem of finding the conservation laws for a non-linear system, describing the plane-parallel steady irrotational isentropic gas motion is reduced to the problem of finding the conservation laws for a linear Chaplygin system. The conservation laws of the zeroth and first orders for the Chaplygin system are obtained. It is established that the set of conservation laws of zeroth order that a Chaplygin system possesses consists of conservation laws that are linear in the velocity potential and the stream function, and a new non-linear conservation law. The linear conservation laws have functional arbitrariness. They produce linearity of this system and are defined by Green's operator formula. It turns out that all the conservation laws in the physical plane, obtained by Rylov, are generated by a linear combination of these linear conservation laws and trivial conservation laws. All the linear conservation laws of the first order for the Chaplygin system, generated by Green's operator formula, that are independent of the stream function, are obtained. It is shown that the Chaplygin system has no more than three first-order conservation laws, independent of the stream function, which are not a linear combination of these linear conservation laws and trivial conservation laws, and their components are quadratic in the velocity potential and its derivatives. All the Chaplygin functions for which the Chaplygin system has three such conservation laws are listed. These conservation laws are obtained.     

Modeling of fracture of fibrous composites with brittle components

Conclusions A numerical analysis of the stress field at the tip of a crack showed high localization, increasing with increase of anisotropy and normal and shear stresses at the crack tip, which can cause various types of fracture. The use of the J-integral for estimates of the flow of elastic energy to the crack tip, simulated by a greatly elongated elliptical notch, is apparently possible in approximate calculations, taking into account localization of the zone of nonlinear behavior of the material. A scheme of estimating the crack resistance of a fibrous material with brittle components, taking into account the contribution of various forms of microfracture to energy dissipation, is proposed. The calculated value of G^* differs from the experimental data by about 20%, which, taking into account the simplicity of the calculation method, can be considered satisfactory.Report presented at the Fourth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, October, 1980).Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 615–619, July–August, 1981.The author thanks S. T. Mileiko for formulating the problem, advice, and discussion of the work.     

New conservation laws obtained directly from symmetry action on a known conservation law

Two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries. The first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation. When the contact transformation is a symmetry of the given system, then the corresponding conservation law is a conservation law of the given system. The second formula checks a priori whether or not the action of a symmetry (continuous or discrete) on a conservation law can yield one or more new conservation laws of the given system. Several examples are considered, including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws.