球面坐标系非连续变形分析的数学模型

The Mathematic Model of Discontinuous Deformation Analysis in Spherical Coordinate System

从球面坐标系的弹性力学基本方程出发 ,推导出球面上块体的位移与 6个位移不变量之间的数学关系 ,进一步建立了联立方程式的球面坐标形式 ,为大范围现代地壳运动的非连续变形分析打下了数学基础。

The discontinuous deformation analysis (DDA) method has been widely used in rock mechanic engineering.It has been proved that DDA is a powerful numerical method in solving discontinuous rock problems.In this paper,the mathematics model of DDA has been discussed.To efficiently use DDA method in the simulation of large scale crustal movement,the ideal of developing DDA model in 2D plane coordinate system to a new DDA model in 2D spherical coordinate system is put forward for the first time.The formulas of DDA method are made in 2D spherical coordinate in detail. Because the surface of the earth is ellipsoid ( e =1/297),it is also soughly taken as round globe.The equation of spherical coordinate system shown in this paper will be properly used to solve the problem of crustal movement. Based on the basic relation formula of displacement and strain on the spherical coordinate system,the 3D spherical coordinate equations are simplified to 2D spherical coordinate equations.In the situation of assuming linear expression of deformation of each block, the function equation of displacement ( u,v ) and its six invariant ( u 0,v 0,r 0,ε φ,ε θ,γ φθ ) of each block on the spherical surface can be set up.That is basic equation to spherical DDA model.The approximation of linear deformation in a block is correct in the situation of relatively small blocks. Depending on the theory of Least Potential Energy,the relation equation of the blocks on spherical surface was established.The coefficient child matrix can be gotten by differencing the equation and then added to the relation equation. We has not shown the problem of no invasion among the blocks on the spherical surface in detail.But a suggestion was given that local plane projection coordinate system be chosen to calculate the distance of invasion,and the problem of spherical surface becomes as easy as the situation of plane coordinate.