考虑Steigmann-Ogden表面的纳米圆柱解析模型

Analytical Model of Micro/Nano-cylinders Considering Steigmann-Ogden Surface Effect

基于Steigmann-Ogden(S-O)表面理论,研究了圆柱形纳米材料在径向对压荷载作用下的力学性能.利用级数展开求解材料内部的弹性控制方程,获得了考虑表面效应时的域内解析表达式.当所得结果忽略表面弯曲刚度参数时可退化为Gurtin-Murdoch(G-M)表面模型.用文献中有限元数值结果和实验结果对本理论进行退化验证,结果得到良好一致性.在此基础上,讨论了表面弯曲刚度参数和圆柱尺寸大小对材料特性的影响.结果显示:考虑了表面弯曲刚度参数的S-O模型和G-M模型在应力分布中有很大的不同.另外,随着圆柱尺寸的减小,其表面效应对材料的力学特性的影响逐渐增大.

Due to a sharp increase of surface-to-volume ratio of nano-sized cylindrical subjects,the corresponding surface energy becomes significant.Therefore,it is essential to study the surface effect at the nano-scale.Herein we study the elastic response of nano-cylinders under diametral loading by taking into account the Steigmann-Ogden(S-O)surface theory.The S-O surface is assumed as a zero-thickness film attached to a bulk material with bending stiffness and residual surface tension.Relative to the more popular Gurtin-Murdoch(G-M)interface model,the S-O surface can resist against not only tension but also bending.Based on continuum mechanics theory,the present work develops internal analytical expressions by solving the elastic governing equations through series expansion in the cylindrical coordinate system.The internal unknown coefficients are finally obtained by applying the S-O surface model and extended diametral boundary conditions through mathematical orthogonality.When the surface bending stiffness parameters are ignored,the present results can be degenerated to the G-M model.Finite element simulations and experimental measurements in the literature are employed to verify the present theory with good agreement.Based on the credence of present solutions,the effects of surface bending stiffness parameters,surface residual stress and cylinder dimension on material properties of nano-cylinders are investigated and discussed.The results demonstrate that the S-O model generates different stress distributions relative to the G-M model,indicating that the surface bending stiffness parameters cannot be ignored in the nano-scale.In addition,the surface residual stress plays a certain role in influencing the stress distributions of both solid and hollow nano-cylinders,especially on the surface of hollow ones.What’s more,it is found that the surface effect gradually increases with a decrease of the cylinder’s dimension.Finally,the analytical nature of the present solution offers an attractive alternative to the numerical methods in studying the elastic behavior and surface effects of nano-subjects.