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

作　　者：焦志安 吴剑[1] 万玲[1] JIAO Zhian;WU Jian;WAN Ling(College of Aerospace Engineering,Chongqing University,Chongqing 400044,P.R.China)

机构地区：[1]重庆大学航空航天学院

出　　处：《重庆大学学报（自然科学版）》2019年第12期74-80,共7页Journal of Chongqing University

基　　金：国家自然科学基金项目(11802043);重庆市留学人员创新资助项目(51204067);重庆市基础科学与前沿技术研究专项项目(cstc2016jcyjA0058)~~

摘　　要：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.

关 键 词：球形压痕 不可压缩 Kerr模型 贝蒂互等定理 弹性薄膜 接触 

分 类 号：O343.3[理学—固体力学]