弹性微梁与椭圆柱压头的压痕问题解析解  

Analytical solution of indentation response of an elliptical cylindrical indenter on elastic microbeam with surface tension

作　　者：时朋朋 李艳 丁生虎 李星 SHI PengPeng;LI Yan;DING ShengHu;LI Xing(School of Mathematics and Statistics,Ningxia University,Yinchuan 750021,China;School of Civil Engineering,Xi'an University of Architecture and Technology,Xi'an 710055,China)

机构地区：[1]宁夏大学数学统计学院,银川750021 [2]西安建筑科技大学土木工程学院,西安710055

出　　处：《中国科学：物理学、力学、天文学》2023年第1期101-114,共14页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金(编号:11802225,12272195)资助项目。

摘　　要：压痕技术除被用于块体材料的性能测量,也被广泛用于软材料薄膜和微尺度材料的性能测量.随着材料尺度减小,表面效应对压痕接触问题中的力与变形发挥着重要作用.本文从理论上研究了椭圆柱压头作用下弹性微梁的平面压痕问题.针对具有表面张力影响的微尺度梁,通过求解表面压力分布和法向挠度满足的微分控制方程,获得了接触区域内外的表面压力和表面位移的解析解,解中的待定系数由接触区尺寸、压痕深度所决定.结合微梁在均布载荷下的变形解和Betti功互等定理,求解了不同测量力下的接触区尺寸和压痕深度,实现了弹性微梁压痕问题的解析求解.理论分析了表面张力、椭圆压头参数、微梁尺寸参数等对压痕问题中力与变形的作用规律.该研究有助于了解微尺度材料压痕行为中的表面效应,为相关压痕测试技术提供参考.In addition to the measurement of mechanical properties of bulk materials,the indentation technology is widely used to measure the mechanical properties of soft films and microscale materials.As the material scale decreases,surface effects play an important role in the force and deformation in the indentation contact problem.In this paper,the two-dimensional indentation problem of an elastic microbeam under an elliptical cylinder indenter is studied theoretically.For the microbeam with the surface tension,the differential governing equation satisfied by the surface pressure distribution and the normal deflection is solved.The analytical solutions of the surface pressure and surface displacement inside and outside the contact area are obtained,where the undetermined coefficients are determined by the contact zone half-width and the indentation depth.The deformation solution of the microbeam under uniform load and Betty’s reciprocal theorem of the indentation problem are given.Then,the contact zone half-width and indentation depth under different measurement forces are solved,and the analytical solution of the indentation problem of an elliptical cylindrical indenter on elastic microbeam is obtained.The effect of surface tension,elliptical indenter parameters and microbeam size parameters on surface pressure and deformation in the indentation problem is analyzed theoretically.This study helps to understand the surface effect on the indentation behavior of microscale materials,and provides some reference for related indentation testing techniques.

关 键 词：压痕 弹性微梁 表面张力 椭圆柱压头 解析解 

分 类 号：TB302[一般工业技术—材料科学与工程]