摘要
考虑应变梯度和速度梯度的影响,建立薄板控制微分方程及给出其边值问题的提法,修正了前人给出的薄板角点条件.采用Levy法,给出受分布力作用下简支板的挠度及自由振动频率的解析解.通过与文献中分子动力学数据对比,验证了该文模型的有效性并提出校核材料参数的一种方法.研究结果表明,增大弹性地基和应变梯度参数可以有效提高板的等效刚度,而速度梯度参数则相反.该文提出的板的边值问题为研究薄板在复杂支撑边界及外荷载等条件提供了理论依据.同时,有望为其有限元法、有限差分法和基于能量原理的Galerkin法等数值方法提供理论依据.
A new type of thin plate model and the related nonclassical boundary value problems were established within the framework of strain gradient and velocity gradient elasticity.The closed-form solutions of deflections and free vibrational frequencies of a simply supported plate resting on an elastic foundation were obtained.The results of the present model agree well with those predicted by the molecular dynamics.Numerical results show that,the elastic foundation and the strain gradient parameter have a stiffness-hardening effect,while the velocity gradient parameter has a stiffness-softening effect.The proposed boundary value problems are of great significance to the study of the mechanical behaviors of plates under complex boundary conditions and external loadings.Furthermore,it will be useful for developing effective numerical methods such as the finite element method,the finite difference method and the Garlerkin method.
作者
徐晓建
邓子辰
XU Xiaojian;DENG Ziehen(Key Laboratory for Special Area Highway Engineering ofMinistry of Education,School of Highway,Chang^an University,Xi'an 710064,P.R.China;Department ofEngineering Mechanics,Northwestern Polytechnical University,Xi'an 710072,P.R.China)
出处
《应用数学和力学》
CSCD
北大核心
2022年第4期363-373,共11页
Applied Mathematics and Mechanics
基金
国家自然科学基金(12072266)
中央高校基本科研业务费(300102219315)
陕西省自然科学基础研究计划(2020JQ-337)。
关键词
板
振动
位移
应变梯度理论
变分原理
plate
vibration
deflection
strain gradient theory
variational principle
