张量分析中简化记法在公式推导中的应用及张量分量的计算

APPLICATION OF LACONIC NOTATIONS IN FORMULA DERIVATION AND CALCULATION OF COMPONENTS IN TENSOR ANALYSIS

张量分析在计算力学中应用广泛,但其理论比较复杂,较难掌握和熟练运用。为此,给出了张量简化记法,这种记法与Fortran程序中或商业软件Matlab中的数组形似。将简化后张量的表达式应用于公式推导,可快速准确地得出结论。同时,通过引入矩阵的矢量化概念,将四阶张量的分量用一个方阵表示出来,这有利于符号运算和数值计算。以上两者结合可使得张量分析变得易于掌握,也有利于张量运算的推广。

Tensor analysis plays an important role in computational mechanics. Due to its complexity, the theory is hard to be accepted and applied conveniently. To solve this problem, laconic notations for tensors are presented here to simplify formula derivation in tensor analysis. These notations are different from traditional ones of tensors. They are similar to arrays used in Fortran routine or commercial software Matlab. Laconic notations for the operations of tensors such as dyadic product, inner product and transposition etc. are given, respectively. By this method, one can derive formula easily and accurately. Another point in this work is that a simple method of obtaining the components of a fourth-order tensor is presented by introducing the concept of vectorization of matrix. It makes the components of high-order tensors to be visual and straightforward for symbolic operation and numerical calculation. The theory of tensor analysis will be easy to master if one grasps these two points of this article.