等厚双层涂层材料受法向集中力作用的三维理论解

Three Dimensional Theoretical Solution of a Normal Force on the Surface of Two Coating Materials with the Same Thickness

基于三维轴对称弹性理论,通过利用镜像点方法和半无限体表面受法向集中力作用的基本解,推导了等厚双层涂层材料受法向集中力作用的显式理论解.该理论解既可用作格林函数进一步求解复杂问题的理论解,也可用作边界元法的基本解以提高数值计算的精度和效率.算例表明,对其无穷多个镜像点只需考虑前3个即可获得足够精度的解.该理论解以固定在各镜像点上局部坐标系下位移函数的形式给出.高阶镜像点的位移函数,可通过递推的方法由对应于低阶镜像点的位移函数求得.

Based on the theory of spatial axisymmetrical elasticity, the theoretical solution of a normal force acting at the free surface of two bonded dissimilar coating materials with the same thickness was deduced by introducing mirror point method and the fundamental solution of a semi-infinite body suffering a concentrated force. It can be used as Green function to deal with the problem of distributed force and even more complicated ones, and the fundamental solution for boundary element method as well. There are infinite mirror points, but the last numerical analysis indicates that we can get accurate enough solution by only taking the first three mirror points into account. It not only proves the correctness of the theoretical deduction, but also shows only the displacement functions corresponding to the first several mirror points have effects on the accuracy of the solution. This fundamental solution is given by the displacement functions defined under the local coordinate systems with their origins placed at each mirror point. The displacement functions corresponding to the mirror point of higher order can be determined from that to the lower order ones.