球体的弹性动力学解和动应力集中现象

The Elastodynamic Solution for a Solid Sphere and Dynamic Stress-Focusing Phenomenon

本文提出了一种解析方法求解球体的弹性动力学问题.将球体弹性动力学基本解,分解为一个满足给定非齐次混合边界条件的准静态解和一个仅满足齐次混合边界条件的动态解的叠加.利用变量替换将动态解需满足的动态方程变换为贝塞尔方程,并通过定义一个有限汉克尔变换,就可以容易地求得非齐次动态方程的动态解,从而,得到球体弹性动力学的精确解.从计算结果中可以发现,在冲击外压作用下的球体圆心处具有动应力集中现象,并导致很高的动应力峰值,这对球体的动强度研究有一定的实际意义.

This paper presents an analytical method of solving the elastodynamic problem of a solid sphere. The basic solution of the elastodynamic problem is decomposed into a quasi-static solution satisfying the in homogeneous compound boundary conditions and a dynamic solution satisfying the homogeneous compound boundary conditions. By utilizing thi variable transform, the dynamic equation may be transformed into Bassel equation. By defining a finite Hankel transform, we can easily obtain the dynamic solution for the in homogeneous dynamic equation. Thereby, the exact elastodynamic solution for a solid sphere can be obtained. From the results carried out, we have observed that there exists the dynamic stress-focusing phenomenon at the center of a solid sphere Under shock load and it results in very high dynamic stress-peak.