地形变应变张量矩阵的不变量分析

ANALYSIS OF INVARIANTS IN STRAIN TENSOR MATRIXES OF CRUSTAL DEFORMATION

在给出正交曲线坐标系的有关位移向量及其全微分、位移梯度矩阵、应变张量矩阵的普适表达式的基础上,又给出了任意两种正交曲线坐标系下的应变张量矩阵的普适转换表达式,并指出:由于该变换矩阵为正交矩阵,故应变张量矩阵为相似矩阵。并对应变张量矩阵的几何物理性质进行了分析,指出任何一种正交曲线坐标系的应变张量矩阵都具有唯一不变的主应变特征多项式,由该矩阵的主应变特征值方程皆可求得地壳质点处的主应变及其主方向,由主方向单位向量又可把该矩阵化为以主应变为对角元素的对角矩阵,该矩阵及其对角矩阵的迹皆为该质点处的体应变,该矩阵的行列式等于该质点处3个主应变的乘积,这些几何物理量皆为该质点处的地应变不变量。

On the basis of deducing the universal expressions in an orthogonal curvilinear coordinate system of the displacement gradient matrix and the strain tensor matrix of displacement vectors, we further derive the univer- sal expression of the conversional matrix between two partial coodinates in random different orthogonal curvilinear coordinate systems,meanwhile, indicats that this conversional matrix also belongs to the orthogonal matrixes so that the strain tensor matrixes are similar matrixes. With this understanding, the after deply analysis of the geometric and physical natures of the strain tensor matrixes, discovers the invariant code mystery of crustal deformation hidden in these matrixes, which is, whatever the orthogonal curvilinear coordinate system is, the strain tensor matrix has a u- nique and invariant principle strain characteristic polynomial and by its corresponding strain eigenvalue equation we can derive the principle strains with their directions at any crustal particle, then the matrix can be turned into a di- agonal matrix by putting the principle strains values as its diagonal elements, and the matrix trace equals the bodystrain, the matrix determinant equals a product of the principe strains. All these geometric and physical quantities are the crustal deformation invariants at the spot.