一般动力学系统的精确不变量和绝热不变量

在增广相空间中研究一般动力学系统的精确不变量与绝热不变量，建立了该空间中力学系统的对称性与不变量之间的关系，提出了一般动力学系统的高阶绝热不变量的概念及其构造方法，揭示了高阶绝热不变量与无穷小对称变换之间的正反关系，讨论了ＶａｎｄｅｒＰｏｌ方程和Ｄｕｆｆｉｎｇ－ＶａｎｄｅｒＰｏｌ方程的一阶绝热不变量与相应的无穷小对称变换．     

关于力学的作用量原理──分析力学札记之三

由Ｈａｍｉｌｔｏｎ作用量的非等时变分导出Ｈａｍｉｌ－ｔｏｎ原理,Ｌａｇｒａｎｇｅ原理,Ｐｏｉｎｃａｒｅ－Ｃａｒｔａｎ积分不变量以及Ｎｏｅｔｈｅｒ定理．     

关于完整非保守系统的基本积分变量关系

本文证明,完整非保守系统不存在 Poincare-Cartan 积分不变量和 Poincare 通用积分不变量。取而代之,给出非保守系统的 Poincare-Cartan 型和 Poincare 型积分变量关系,并将这种积分变量关系用于求解非线性振动问题。我们还证明了文[1]、[2]所谓的积分不变量只是我们所引入的基本积分变量关系。     

包含伺服约束的非完整系统的对称性摄动与绝热不变量

建立包含伺服约束系统的运动微分方程,基于包含伺服约束系统对称性与不变量理论研究包含伺服约束系统的对称性摄动与绝热不变量 问题,提出了包含伺服约束系统的高阶绝热不变量概念,证明了精确不变量与绝热不变量存在的条件及形式,并举例说明结果的应用。     

非线性横观各向同性弹性材料的本构方程及其势函数

研究了非线性Green弹性材料弹性张量独立分量,归纳推导出横观各向同性Green弹性材料、各向同性非线性弹性材料独立的弹性常数个数.从张量函数出发,用含有高阶弹性张量的张量多项式,推导出四阶非线性横观各向同性,各向同性材料Green弹性材料本构方程及其势函数.并将本构方程及其势甬数用张量不变量,标量不变量表示.证明了方程是完备的,不可约的,满足张量函数表示定理.     

含裂纹三点弯曲梁起始扩展的动态分析

本文利用高速摄影捕捉含裂纹三点弯曲梁在冲击下裂纹扩展过程，借助动态焦散线对裂纹尖端初始曲线范围内的应力第二不变量分布进行了试验研究，并通过Ｄｅｔ．－准则对动载下裂纹起始扩展的条件和方向进行了分析。结果表明：裂纹扩展的实际方向与理论预测相一致；Ｄｅｔ．－准则与焦散线相结合在动态断裂力学研究中具有广泛的使用价值     

非线性正交各向异性弹性材料的本构方程及其势函数

研究了非线性Green弹性材料弹性张量独立分量,归纳推导出各向异性Green弹性材料、具有一个对称面Green弹性材料、 正交各向异性非线性弹性材料独立的弹性常数个数.从张量函数出发,用含有高阶弹性张量的张量多项式,推导出三阶非线性正交各向异性Green弹性材料本构方程及其势函数.并将本构方程及其势函数用张量不变量,标量不变量表示.证明了方程是完备的,不可约的,满足张量函数表示定理.详细研究Green弹性材料势函数存在的充分和必要条件,给出并证明了具有普适性的势函数存在定理.     

张量函数的表示理论──本构方程统一不变性研究（续上期）

张量函数的表示理论──本构方程统一不变性研究（续上期）郑泉水（Ｑ－ＳＺｈｅｎｇ）清华大学工程力学系，北京１０００８４４矢量和二阶张量的二维所有种类各向异性张量函数如前所述，理论和应用力学的研究者对平面问题一直怀有浓厚的兴趣．三维空间各向异性物理行为通...     

正态变量相关情况下可靠性灵敏度分析的新方法

基于独立正态变量情况下可靠性灵敏度分析的线抽样法,提出了一种求解正态相关变量情况下可靠性灵敏度的新方法。在所提方法中,首先将正态相关变量等效变换为正态独立变量,然后利用线抽样方法独立完成等效独立变量情况下失效概率对独立变量的所有分布参数的灵敏度分析,最后依据等效变换前后变量分布参数之间的解析关系和复合函数求导公式,求得...     

关于返回映射算法中应力的四阶张量值函数

针对各向同性材料,基于一组相互正交的基张量,建立了一套有效的相关运算方法.基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴.根据张量函数表示定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组基所表示.推演结果表明:这些张量之间的运算,表现为对应系数矩阵之间的简单关系.其中,四阶张量求逆归结为对应的3×3系数矩阵求逆,它对二阶张量的变换则表现为该矩阵对3×1列阵的变换.最后,对这些变换关系应用于返回映射算法的迭代格式进行了相关讨论.     

A discussion about scale invariants for tensor functions

It is found that in some cases the complete and irreducible scale invariants given by Ref.[1] are not independent. There are some implicit functional relations among them. The scale invariants for two different cases are calculated. The first case is an arbitrary second order tensor. The second case includes a symmetric tensor, an antisymmetric tensor and a vector. By using the eigentensor notation it is proved that in the first case there are only six independent scale invariants rather than seven as reported in Ref.[1] and in the second case there are only nine independent scale invariants which are less than that obtained in Ref.[1].     

Dyadic method for tensor functions

In this paper, we discuss tensor functions by dyadic representation of tensor. Two different cases of scalar invariants and two different cases of tensor invariants are calculated. It is concluded that there are six independent scale invariants for a symmetrical tensor and an antisymmetrical tensor, and there are twelve invariants for two symmetrical tensors and an antisymmetrical tensor. And we present a new list of tensor invariants for the tensor-valued isotropic function. The project supported by the Special Funds for Major State Basic Research Project “Nonlinear Science” and the National Basic Research Project “The Several Key Problems of Fluid and Aerodynamics”     

Nonlinear Orthotropic Elasticity: Only Six Invariants are Independent

It is often declared in the literature that the seven classical invariants used to characterize the strain energy of a compressible orthotropic elastic solid are independent. In this paper, we show that only six of the seven classical invariants are independent, and a syzygy exists between the classical invariants. Consequently, all other sets of seven invariants, proposed in the literature, that are uniquely related to the set of classical invariants, have only six independent invariants.     

Determnation of elasticity constants for microheterogeneous media

The foundations of the theory of stochastically heterogeneous solids were laid a long time ago by Voigt [1J, who developed a method for determining the macroscopic parameters of polycrystalline materials by averaging the appropriate crystallite parameters with respect to orientations. Lifshits and Rozentzveig [2] showed that it was necessary to consider the correlation properties of the field in computations of macroscopic parameters. They calculated the first corrections for the averaged elastic constants of polycrystallites for the case of cubic and hexagonal crystallites. Assuming a low degree of heterogeneity, these authors used an approximation which corresponds to the Born approximation in the theory of scattering [3]. This method and its modifications were subsequently used by several authors for the computation of macroscopic parameters of polycrystallites [4– 6] and of other microheterogeneous materials [8].Moreover, the assumption of a low degree of heterogeneity of the properties is very restrictive. It precludes use of the method in the case of macroscopically isotropic polycrystallites formed from essentially anisotropic crystallite stochastically glass reinforced plastics, and similar microheterogeneous materials. This rises the problem of developing procedures that could be applied in cases of a high degree of heterogeneity. This problem presents serious analytical difficulties, however. It is sufficient to point out that even computation of the second approximation (i.e., the one following the Born approximation) has not yet been completed. Analogous problems in the classical and quantum theories of scattering are also, as a rule, considered only in the Born approximation. More complicated methods (e.g., Feyman's method) make possible only partial summation of infinite sequences in which the result is obtained. A method analogous to that of a selfconsistent field in quantum mechanics [9,10] is promising; however, this method is approximate and the magnitude of its error has not yet been estimated.The possibility of accurate determination of mascroscopic parameters for certain classes of microheterogeneous media was demostrated in [11], in which a detailed analysis was presented of parameters forming a second order tensor and characterizing the distribution in the medium of a certain scalar value obeying an equation similar to the steady-state heat-conduction equation. Accurate formulas for macroscopic coefficients of thermal conductivity (diffusion) were derived for the case of a strongly anisotropic medium and for that of a medium with a high degree of transverse isotropy. We made a comparison with various approximate methods and evaluated their degree of error. This article describes an accurate method of computing macroscopic elastic constants for polycrystalline media with a high degree of anisotropy; for the case of polycrystals with a cubic structure [12] the error margin and range of application of approximate methods are estimated.     

A direct construction of first integrals for certain non-linear dynamical systems

A direct, constructive approach to the problem of finding first integrals of certain non-linear, second order ordinary differential equations is presented. The idea is motivated by the construction of the energy integral for the equations of motion of the corresponding conservative systems. Although the method developed for the class of equations studied herein is elementary, it yields the same results as the more advanced group-theoretical methods, such as the use of symmetries] in the context of Noether's theorem. The approach reveals some interesting features when it is specialized to the case of linear equations. Finally, a two-dimensional example is considered by extending the methodology developed for scalar equations to their vector counterparts. It is shown that, as a consequence, a first integral which is independent of the energy integral exists for a particular Hamiltonian of the Contopoulos type.     

Rate type constitutive equations for fiber reinforced nonlinearly vicoelastic solids using spectral invariants

In this paper we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic solids. It has been shown that constitutive equations for such bodies can be expressed in terms of a complete minimal set of 18 classical invariants associated with deformation and fiber orientation. In this paper, we give an alternative formulation using a set of spectral invariants. It is shown via the use of spectral invariants that only 11 of the 18 classical invariants are independent. We analyze the spectral invariants for two illustrative deformation gradients: (i) simple tension, and (ii) simple shear.     

Constitutive equations of thermoplasticity including the third invariant

Constitutive equations describing the nonisothermal deformation of elements of a body along paths of small curvature are formulated taking into account the stress mode. The equations include two scalar functions, one relating the first invariants of the tensors and the other relating the second invariants of the stress and strain deviators. Both scalar functions are nonlinear, dependent on temperature and stress mode, and determined in tests on tubular specimens. The plastic incompressibility condition is validated in uniaxial-tension tests on tubular and solid specimens. The proposed equations are used to design a loading process that differs from the base ones and proceeds at high temperature. The calculated results are compared with experimental data     

Invariants of Hyperbolic Equations: Solution of the Laplace Problem

This paper gives a solution of the Laplace problem, which consists of finding all invariants of the hyperbolic equations and constructing a basis of the invariants. Three new invariants of the first and second orders are found, and invariantdifferentiation operators are constructed. It is shown that the new invariants, together with the two invariants detected by Ovsyannikov, form a basis such that any invariant of any order is a function of the basis invariants and their invariant derivatives.     

Topological Invariants and Solutions with a High Complexity for Scalar Semilinear PDE

In this paper topological invariants of dynamics of scalar parabolic equations are defined. Conservation of these invariants causes persistence of complex spatially chaotic patterns in the dynamics. It implies, in particular, the existence of extremely many stable essentially different spatially complex solutions of scalar semilinear parabolic, elliptic and hyperbolic equations in the situation when coefficients of equations depend on the spatial variable strongly enough and the domain is large or unbounded, the number of different solutions exponentially depends on the volume of domain. Another implication of the conserved quantities is persistence of complexity of solutions in the dynamics.     

On the derivation for the gradients of the principal invariants

In this notes, it is shown that the method carried out by Truesdell and Noll [1] for the gradients of the principal invariants is available to be used for arbitrary second-order A. Finally, some trace identities for the derivatives of invariants are given.