有限元法的力学基础中空间一点应力状态矩阵化思维教学研究

Teaching research on matrix representation of spatial stress state in the mechanics foundation of finite element method

弹性力学是有限元法的力学基础,空间一点应力状态是实际中我们常常遇到的问题。然而,目前这部分的教学内容都是基于微小单元通过力学平衡的角度来进行严密推导的,计算上显得繁琐,教学效果并不理想。对此,该文在严谨力学推导的基础上,总结出空间一点应力在不同斜面上的应力状态,实质上就是应力矩阵和斜面外法线的数学运算,而主应力和主平面即为求解应力矩阵特征值和特征向量的问题。通过空间一点应力状态的矩阵化思维,概念清晰、容易记忆、且运算简单,可以很方便地结合现代计算机技术进行计算求解。

Elasticity is the mechanical basis of the finite element method. Space stress state is a common problemin practice. However, the current teaching content of this part is based on the micro-units to perform a rigorousderivation from the perspective of mechanical equilibrium. The calculation is cumbersome so that the teaching effect isnot ideal. Then, based on the rigorous mechanics derivation, the spatial stress state on different planes is analyzed andsummarized in this paper and essentially the mathematical operation of the stress matrix and the outer normal vector ofthe plane. In fact, calculating the principal stress and the principal plane is converted to solve matrix eigenvalues andeigenvectors. Through the matrix representation of a point stress state in space, the concept is clear, the expression iseasy to remember, and the operation is simple, which can be conveniently combined with modern computer technologyto solve.