应用张量分析推导柱坐标系和球坐标系中弹性力学几何方程和平衡微分方程

Applicationof tensor analysis in deriving geometric equations and equilibrium differential equations of elasticity in cylindrical coordinates and spherical coordinates

利用正交曲线坐标系与笛卡儿坐标系单位矢量的关系,以及笛卡儿坐标系单位矢量为常矢量的特性,从单位矢量变换的角度,推导柱坐标系和球坐标系中的梯度算子,以及单位矢量对坐标的偏导数.并根据张量的场论基础,通过微分运算,推导出位移矢量的梯度和应力张量的散度.再根据几何方程和平衡微分方程的张量表达形式,推导出柱坐标系和球坐标系中的应变几何方程和应力平衡微分方程.

First,using the relations of the unit vectors between orthogonal curvilinear coordinates and Cartesian coordinates,and the invariability of the unit vectors in Cartesian coordinates,the gradient operator in both coordinates,and the partial derivative of the unit vectors to the coordinates are derived from the perspective of the transformation of the unit vectors.Then,using the fundamentals of the theorems of tensor field,the gradient of displacement vector and the divergence of stress tensor are derived by derivation.Last,the geometric equations of strain and the differential equations of equilibrium are derived in details in cylindrical coordinates and spherical coordinates from their forms expressed by tensor.