对莫尔圆极点法的数学理论探源和工程应用探讨

The Exploration of Mathematical Theory of the Pole Method of Mohr Circle and Discussion on Its Application in Engineering

从射影几何学的视角,对莫尔圆极点法的数学理论进行了追根溯源,在引入极点概念的同时,亦引入了与之对应的极线概念;对莫尔圆上极点的唯一性和极点法的合理性进行了简洁的证明;对莫尔圆极点法的内在特性进行了深入的探究,对其工程应用的优越性进行了探讨和例证;从SO(2)变换群的角度看,莫尔圆极点法对于二维情况下的二阶对称张量在坐标轴转动时的分量计算具有普适性,故本文所探讨的原理可类比推广到应变张量、惯性张量和联合概论密度张量中去。

From the perspective of projective geometry,the mathematical theory of the pole method of Mohr circle was explored.The concept of pole was introduced,so was the concept of polar that corresponds to pole.The uniqueness of pole on Mohr circle and the rationality of such a method were proved concisely.The inherent properties of the pole method were investigated in depth,and its advantages in engineering practice were discussed and exemplified.Viewed from transform group SO(2),the pole method of Mohr circle is valid universally for the calculation of components of any second rank symmetric tensor in two dimensions as coordinate axes rotate.And so the principles studied herein also apply to strain tensor,inertia tensors and joint probability density tensors.