由大变形引起的有限转动的描述

The Description of Finite Rotation for Large Deformation

极分解定理[1]中的R是一个正交张量。一个正交张量对不共面的三个矢量的作用,可以使这三个矢量发生刚性转动而不改变各矢量的大小和它们之间的夹角[2]。但是,在大变形中,变形体上一点的任意三个不共面的物质线索矢量,变形后,每一个矢量的大小和它们之间的夹角都要发生变化。所以用一个正交张量只能描述变形体主向的有限转动,而不可能描述非主向的有限转动。但是,对于每一个物质线素矢量,却可以用正交张量或有限转动矢量来描述[3],i=1、2、3。本文将这三个彼此有联系的正交张量或三个有限转动矢量进行适当的组合,就可以构成一个二阶张量H,张量H就描述了于由变形引起的三个非主向的有限转动。 此外,文中还证明,当小变形时,这三个转动张量退化为由剪切变形确定的反对称张量。

In the polar decomposition theorem, R is an orthogonal tensor. The orthogonal tensor R can describe the rigid rotations of the deformed body by the three vectors in noncoincident plane. It cannot change the magnftute of vectors and their angles between them. However, when the body is in the case of large deformation, the arbitrary three material element vectors of a point will be changed their magnitute and the angles between these vectors. So that the finite rotation cannot be described in terms of an orthogonal tensor in the nonprincipal directions. Nevertheless, the finite rotation of every matterial element vector can be described by the corresponding orthogonal tensor or the finite rotatary vector.In this paper, a two order tensor H is constituted from the suitable components of these three orthogonal tensors, the rotations for deformation in the non-principal directions can be described by tensor H.In the case of the small deformation, the finite rotation tensors R can be reduced to an antisymmetrical tensor. The components of the antisymmetrical tensor are only concerned with shearing components of the strain tensor.