轴向功能梯度Timoshenko梁弹性接触模型的解析解  

Analytical Solutions for Elastic Contact Models of Axially Gradient Timoshenko Beams

作　　者：向琳 袁江宏 XIANG Lin;YUAN Jiang-hong(Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province,Chengdu 611756,Sichuan,China;School of Mechanics and Aerospace Engineering,Southwest Jiaotong University,Chengdu 611756,Sichuan,China)

机构地区：[1]应用力学与结构安全四川省重点实验室,四川成都611756 [2]西南交通大学力学与航空航天学院,四川成都611756

出　　处：《机械研究与应用》2025年第2期1-6,13,共7页Mechanical Research & Application

基　　金：国家自然科学基金重大项目子课题:基于梯度样品全场分析的高通量表征力学理论(编号:12192211);西南交通大学新型交叉学科培育基金项目:软物质科研培育团队建设(编号:2682022JX001)。

摘　　要：针对高斯函数型轴向功能梯度Timoshenko梁与刚性圆形压头的对称弹性接触问题,该文通过将梁的挠度、转角、弯矩和剪力表示为合流超几何级数形式,解析推导出了以接触半径为参数的压入力-压入深度曲线参数方程。然后利用泰勒展开进一步得到了曲线起始段的线性近似解析表达式,进而发现线性近似解即为跨中集中力作用下梁的跨中挠度解。通过遍历式的参数分析给出了曲线起始线性段长度的近似经验公式,并研究了轴向梯度因子和梁长对曲线起始线性段斜率(即系统的初始接触刚度)的影响规律。结果表明:在给定梁长的条件下,轴向梯度因子与压入力-压入深度曲线起始线性段斜率具有正相关的一一对应关系。该结论对轴向功能梯度样品的高通量压痕表征而言具有一定的理论参考价值。The problem of symmetric elastic contact between an axially gradient Timoshenko beam of Gaussian-function type and a rigid circular indenter was studied in this paper.By expressing the deflection,rotation angle,bending moment,and shearing force of the beam in term of confluent hypergeometric series,the parametric equation of the indentation force-depth curve with the contact radius as the parameter was derived analytically.The analytical linear approximation for the starting part of the curve was further obtained by using Taylor expansion,and it was found that the linear approximation is exactly the solution for the mid-span deflection of the beam under a concentrated force acting also at the mid-span.An empirical formula for the approximate length of the linear starting part of the curve was given through the ergodic parametric analysis,and the influence of the axial gradient factor and the beam length on the slope of the linear starting part(i.e.the initial contact stiffness of the system)was investigated.The results show that,for a given beam length,the axial gradient factor has a positively correlated one-to-one correspondence with the slope of the linear starting part of the indentation force-depth curve.This conclusion is of certain theoretical value for the high-throughput characterization of axially gradient samples via indentation.

关 键 词：轴向功能梯度梁 铁木辛柯梁 弹性接触 压痕表征 

分 类 号：O343.3[理学—固体力学]