不可压缩弹性薄膜球形压痕问题的一种渐近解析解

An asymptotic analytical solution to the spherical indentation problem of incompressible elastic thin film

针对刚性基底上不可压缩弹性薄膜的轴对称球形压痕问题,采用了一种基于Kerr模型的简单解析求解方法。在该方法中,薄膜上表面的接触压强与位移为线性微分关系。之后利用贝蒂互等定理,求解了该问题的高阶渐近解,推导了接触力、压痕深度和接触半径之间的显式关系。当忽略高阶项时,得出的高阶渐近解与现有研究中的低阶解相同。此外还建立了有限元模型来验证渐近解的精度。结果显示,与已有的低阶渐近解相比,高阶渐近解与现有的数值计算结果和有限元分析结果吻合得更好。

In order to solve the problem of axisymmetric indentation of an incompressible elastic film on a rigid substrate,a simple analytical method based on Kerr-model is derived,in which the differential relation between the contact pressure and the displacement of the film’s upper surface is established.Then,the high-order asymptotic solution to the problem is solved by using Betti’s reciprocal theorem and the explicit relation between contact pressure,indentation depth and contact radius is built.When the high-order term is ignored,the present asymptotic solution is the same as the existing low-order solution.In addition,a finite element model is established to verify the accuracy of the asymptotic solution.The result shows that,compared with the existing low-order asymptotic solutions,the higher-order asymptotic solution agrees better with the existing numerical results and the newly developed numerical simulation results.