立方晶粒任意集合下的格林函数和有效弹性刚度张量

为了推导多晶体材料的有效弹性刚度张量,给出立方晶粒任意集合的格林函数封闭但近似的表达式,该格林函数表达式包含三个单晶弹性常数和多晶体材料五个织构系数,它考虑取向分布函数的影响直至织构系数的线性项,它适用于弱织构多晶体材料或具有弱各向异性晶粒的多晶体材料(如金属铝),它与Nishioka格林函数近似式的比较通过三个算例给出;Synge的格林函数积分式则直接通过数值计算完成,它可作为问题的精确解供参考.该文还简单介绍了多晶体材料有效弹性刚度张量的推导过程,并把所得结果和有限元计算结果进行比较.     

Invariant characteristic representations for classical and micropolar anisotropic elasticity tensors

By extending and developing the characteristic notion of the classical linear elasticity initiated by Lord Kelvin, a new type of representation for classical and micropolar anisotropic elasticity tensors is introduced. The new representation provides general expressions for characteristic forms of the two kinds of elasticity tensors under the material symmetry restriction and has many properties of physical and mathematical significance. For all types of material symmetries of interest, such new representations are constructed explicitly in terms of certain invariant constants and unit vectors in directions of material symmetry axes and hence they furnish invariants which can completely characterize the classical and micropolar linear elasticities. The results given are shown to be useful. In the case of classical elasticity, the spectral properties disclosed by our results are consistent with those given by similar established results.     

Canonical representations and degree of freedom formulae of orthogonal tensors in n-dimensional Euclidean space

In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part of n/2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.     

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”     

On the derivatives of a subclass of isotropic tensor functions of a nonsymmetric tensor

This paper studies a subclass of isotropic tensor-valued functions of a nonsymmetric tensor, which satisfy the commutative condition, and their derivatives. This subclass of tensor functions includes tensor power series, exponential tensor function, etc., and is more general than those investigated before. In the case of three distinct eigenvalues, the derivatives of these tensor functions are constructed by solving a tensor equation, which is acquired by differentiating the commutative condition. By taking limits, the results are extended to the cases of repeated eigenvalues.     

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].