Solutions of half-space and half-plane contact problems based on surface elasticity
Solutions of half-space and half-plane contact problems based on surface elasticity
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DOI：
10.1007/s00033-012-0205-0
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发表时间：
2012-04
期刊：
影响因子：
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通讯作者：
Songsheng Zhou;Xin-Lin Gao
中科院分区：
文献类型：
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作者：
Songsheng Zhou;Xin-Lin Gao
本文利用Gurtin和Murdoch的线性化表面弹性理论的一般形式，统一地导出了受分布法向力作用的弹性半空间和弹性半平面问题的解析解。在公式中使用了Papkovitch Neuber势函数、傅里叶变换和贝塞尔函数。新得到的解是一般的，并且在不考虑表面效应的情况下退化为基于经典线弹性理论的半空间和半平面接触问题的解。此外，现有的解决方案的半空间和半平面接触问题的基础上简化版本的Gurtin和默多克的表面弹性理论恢复为当前解决方案的特殊情况。直接应用这些新的解，解决了Boussinesq的平头冲头问题、Hertz的球形冲头问题和锥形冲头问题，得到了与经典弹性力学不同的深度相关的硬度公式。数值结果表明，更平滑的弹性场和更小的位移预测的当前解决方案比经典的弹性为基础的解决方案。此外，它表明，离面位移和应力分量强烈依赖于表面残余应力。此外，它被发现，新的解决方案的基础上的表面弹性理论预测更大的值的压痕硬度比基于经典弹性的解决方案。
Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.