Herberthson Magnus, Özarslan Evren, Westin Carl-Fredrik
Department of Mathematics, Linköping University, Linköping, Sweden.
Department of Biomedical Engineering, Linköping University, Linköping, Sweden.
Math Vis. 2021;2021:3-22. doi: 10.1007/978-3-030-56215-1_1. Epub 2021 Feb 11.
Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors and . In terms of components, such an equivalence means that components of the first tensor will transform into the components of the second tensor for some change of the coordinate system.
计算一族张量的方差，每个张量由一个对称半正定二阶张量/矩阵表示，这涉及到一个四阶张量的形成。为了形成这个张量，需将每个二阶张量与其自身进行张量积运算，然后对这些乘积求和，从而得到一个与连续介质力学中的弹性张量具有相同对称性质的张量。针对这个张量，人们已经研究了许多性质：表示、不变量、分解、等价问题等等。在本文中，我们聚焦于二维情形，给出了一组不变量，这些不变量确保了两个这样的四阶张量(A)和(B)的等价性。就分量而言，这种等价性意味着对于某个坐标系的变换，第一个张量的分量(A_{ijkl})将变换为第二个张量的分量(B_{ijkl})。
