Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor
Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor
复制标题
四阶三维对称无迹张量各向同性不变量的两个不可约泛函基
DOI：
10.1177/1081286519835246
复制
发表时间：
2019
影响因子：
2.6
通讯作者：
Wennan Zou
中科院分区：
文献类型：
--
作者：
Zhongming Chen;Yannan Chen;Liqun Qi;Wennan Zou
弹性张量是力学中最重要的四阶张量之一。四阶三维对称无痕张量在弹性张量的研究中起着至关重要的作用。在本文中，我们提出了四阶三维对称无痕张量的两个各向同性不可约函数基。其中之一就是Smith和Bao在1997年提出的最小完整性基。它有九个齐次多项式不变量，分别为二、三、四、五、六、七、八、九和十次。我们证明它也是一个不可约的泛函基础。第二个不可约泛函基也有九个齐次多项式不变量。它没有四次不变量，但有两个六次不变量。其他七个不变量与 Smith-Bao 基相同。因此，第二个不可约泛函基不包含在任何最小完整性基中。
The elasticity tensor is one of the most important fourth-order tensors in mechanics. Fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith–Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.