关于张量函数表示理论的标量不变量的讨论

发现文献[1,2]提出的张量函数表示理论中的完备而不可约的不变量不是互相完全独立的．分别对一个任意二阶张量和两个对称二阶张量的标量不变量进行了计算,证明前者只有六个不变量是独立的,后者只有九个是独立的．     

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”     

New mixed plate-bending elements with shear strain interpolations

This paper is concerned with two mixed plate-bending elements with shear strain interpolations, a quadrilateral element and an 8-node serendipity-type element based on discussions on the element proposed in Ref.[1]. The shear strains and inner-forces in the natural coordinates are interpolated in an element and then transformed into Cartesian coordinates in accordance with covariant and contravariant tensor transformations, respectively. The quadrilateral element coincides with the element in Ref.[1] when it is rectangular. Numerical examples show that the two new elements are free from shear locking and spurious kinematic modes under regular and irregular meshes and have the advantages of being insensitive to element distortion and able to give fairly accurate results.The Project supported by National Natural Science Foundation of China.     

极分解的一类近似计算与比较

基于级数展开给出了极分解中右伸长张量U的级数表示,通过对级数项的选取得到右伸长张量的不同近似表达式.针对不同级数展开表示,得到表达式最小误差的级数展开形式.进而结合一些简单实例,验证了近似公式的有效性.最后与文献[1]关于计算右伸长张量U和转动张量R的近似表达式进行了比较,本文的级数展开方式得到的右伸长张量U和转动张量R的近似表达式不但简洁,而且计算精度更高、适用范围更广.     

Problem on circular-arc cracks between bonded dissimilar materials under concentrated force and moment

The method is very efficient by applying extended Schwarz principle integrated with the analysis of the singularity of complex stress functions to solve some plane-elastic problems under concentrated loads, in Ref.[1], this method is used to deal with the elastic problems of homogeneous plane. In this paper, it is extended to the case of dissimilar materials with co-circular cracks under concentrated force and moment. For several typical cases the solutions of complex stress function in closed form are built up and the stress intensity factors are given. From these solutions, we provide a series of particular results, in which two of them coincide with those in Refs. [1] and [6].     

Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

Constitutive equations relating the components of the stress tensor in a Eulerian coordinate system and the linear components of the finite-strain tensor are derived. These stress and strain measures are energy-consistent. It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator and that the first invariants of the stress and strain tensors are in a nonlinear relationship. In the case of combined elastoplastic deformation of elements of the body, this relationship, as well as the relationship between the second invariants of the stress and strain deviators, is determined from fundamental tests on a tubular specimen subjected to proportional loading at several values of stress mode angle (the third invariant of the stress deviator). Methods to individualize these relationships are proposed. The initial assumptions are experimentally validated. The constitutive equations derived underlie an algorithm for solving boundary-value problems __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 43–55, June 2007.     

A failure of a class of K-BKZ equations based on principal stretches

The invariants in the K-BKZ constitutive equation for an incompressible viscoelastic fluid are usually taken to be the trace of the Finger strain tensor and its inverse. The basis for this choice of invariants is not derived from the K-BKZ theory, but rather is due to the perception that this is the most natural choice. Research into using other sets of invariants in the K-BKZ equation, such as the principal stretches or the eigenvalues of the Finger strain tensor (i.e., the squares of the principal stretches) is relatively new. We attempt here to derive a K-BKZ equation based on the squares of the principal stretches that models the behavior of a low-density polyethylene melt in simple shear and uniaxial elongational deformation. In doing so, two assumptions are made as to the form of the strain-dependent energy function: first, that there is a function f(q) such that the energy function can be written as the sum of f(q _i ),i = 1, 2, 3, where the q _i 'sare the squares of the principal stretches, and second that f is a power law. We find that the K-BKZ equation resulting from these two assumptions is inadequate to describe both the shear and elongational behavior of our material and we conclude that the second of the above assumptions is not valid. Further investigation, including predictions of the second normal stress difference and some finite element calculations reveals that the first assumption is also invalid for our material.     

On Isotropic Invariants of the Elasticity Tensor

Isotropic invariants of the elasticity tensor always yield the same values no matter what coordinate system is concerned and therefore they characterize the linear elasticity of a solid material intrinsically. There exists a finite set of invariants of the elasticity tensor such that each invariant of the elasticity tensor can be expressed as a single-valued function of this set. Such a set, called a basis of invariants of the elasticity tensor, can be used to realize a parametrization of the manifold of orbits of elastic moduli, i.e. to distinguish different kinds of linear elastic materials. Seeking such a basis is an old problem in theory of invariants and seems to have been unsuccessful until now. In this paper, by means of the unique spectral decomposition of the elasticity tensor every invariant of the elasticity tensor is shown to be a joint invariant of the eigenprojections of the elasticity tensor, and then by utilizing some properties of the eigenprojections a basis for each case concerning the multiplicity of the eigenvalues of the elasticity tensor is presented in terms of joint invariants of the eigenprojections. In addition to the foregoing properties, the presented invariants may also be used to form invariant criteria for identification of elastic symmetry axes.     

Extension of covariant derivative(Ⅲ): From classical gradient to shape gradient

This paper further extends the generalized covariant derivative from the first covariant derivative to the second one on curved surfaces. Through the linear transformation between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the second class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric structure of the tensor analysis on curved surfaces are revealed.     

THE ASYMPTOTIC SOLUTION OF THE UNSTEADY PLANAR PARALLEL GAS FLOW FOR THE CASE THAT THE ADIABATIC INDEX NUMBER γ IS BIG

In ref I, under the condition that the components of velocity are only the functions of time and polar angle θ, Drornikov solved eqss. (1.1) (1.3) of the ideal gas unsteady planar parallel potential flow. It was pointed out in ref. [1] that in general cases, the evident solutions could not he obtained. Only for two especial cases, the evident solutions were obtained.In this paper, the author studies the same prohlein as that in ref. [1]. In the first section we obtain the evident solution of equations (1.1)-(1.3) under the condition that the sonic velocity is restricted by some complemental conditions. In the second section, we obtain the first-order approximate solutions of the fundamental equation for the case that γ>>1