Asymptotic long-wave model for an elastic isotropic nanoplate with surface effects derived from the 3D theory of elasticity and its comparison with hypotheses-based models

IF 1.9 4区工程技术 Q3 MECHANICS

Continuum Mechanics and Thermodynamics Pub Date : 2025-03-05 DOI:10.1007/s00161-025-01366-z

Gennadi I. Mikhasev

{"title":"Asymptotic long-wave model for an elastic isotropic nanoplate with surface effects derived from the 3D theory of elasticity and its comparison with hypotheses-based models","authors":"Gennadi I. Mikhasev","doi":"10.1007/s00161-025-01366-z","DOIUrl":null,"url":null,"abstract":"<div><p>The paper deals with the derivation of asymptotically correct equations governing the long-wave bending response of a rectangular ultrathin elastic isotropic plate taking into account surface effects within the framework of the Gurtin-Murdoch theory of surface elasticity. The upper and lower faces are assumed to be pre-stressed by residual surface stresses which can be either tensile or compressive. The original 3D equations of elasticity are split into equations corresponding to the in-plane boundary layer and equations predicting out-of-plane bending deformation. By performing asymptotic integration through the thickness of the 3D equations associated with bending deformations and satisfying the balance equations on both faces, we derive asymptotically correct relations for displacements and stresses, as well as a new Timoshenko-type equation capturing surface stresses and inertia. A comparative analysis of the derived governing equation with similar available equations based on kinematic hypotheses revealed significant differences in the effective bending stiffness and factors of the inertia term. As examples, we studied free low-frequency vibrations and self-buckling of a square nanoplates made of different materials and compared effects of residual stresses on the natural frequencies and the critical value of the plate side using the novel model and the models relying on hypotheses for the normal component of the stress tensor.</p></div>","PeriodicalId":525,"journal":{"name":"Continuum Mechanics and Thermodynamics","volume":"37 2","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Continuum Mechanics and Thermodynamics","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s00161-025-01366-z","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MECHANICS","Score":null,"Total":0}

引用次数: 0

Abstract

The paper deals with the derivation of asymptotically correct equations governing the long-wave bending response of a rectangular ultrathin elastic isotropic plate taking into account surface effects within the framework of the Gurtin-Murdoch theory of surface elasticity. The upper and lower faces are assumed to be pre-stressed by residual surface stresses which can be either tensile or compressive. The original 3D equations of elasticity are split into equations corresponding to the in-plane boundary layer and equations predicting out-of-plane bending deformation. By performing asymptotic integration through the thickness of the 3D equations associated with bending deformations and satisfying the balance equations on both faces, we derive asymptotically correct relations for displacements and stresses, as well as a new Timoshenko-type equation capturing surface stresses and inertia. A comparative analysis of the derived governing equation with similar available equations based on kinematic hypotheses revealed significant differences in the effective bending stiffness and factors of the inertia term. As examples, we studied free low-frequency vibrations and self-buckling of a square nanoplates made of different materials and compared effects of residual stresses on the natural frequencies and the critical value of the plate side using the novel model and the models relying on hypotheses for the normal component of the stress tensor.

查看原文本刊更多论文

基于三维弹性理论的具有表面效应的弹性各向同性纳米板渐近长波模型及其与假设模型的比较

本文在Gurtin-Murdoch表面弹性理论的框架下，推导了考虑表面效应的矩形超薄弹性各向同性板长波弯曲响应的渐近正确方程。假定上、下两个面是由残余表面应力预压的，残余表面应力可以是拉伸的，也可以是压缩的。将原三维弹性方程分解为面内边界层对应方程和面外弯曲变形预测方程。通过对与弯曲变形相关的三维方程的厚度进行渐近积分，并在两个面上满足平衡方程，我们推导出了位移和应力的渐近正确关系，以及一个新的timoshenko型方程，该方程捕获了表面应力和惯性。将导出的控制方程与基于运动学假设的类似现有方程进行对比分析，发现有效弯曲刚度和惯性项因素存在显著差异。以不同材料制成的方形纳米板为例，研究了其自由低频振动和自屈曲，并比较了残余应力对板侧固有频率和临界值的影响，采用新模型和基于应力张量法向分量假设的模型。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Continuum Mechanics and Thermodynamics 物理-力学

CiteScore

5.30

自引率

15.40%

发文量

审稿时长

>12 weeks

期刊介绍： This interdisciplinary journal provides a forum for presenting new ideas in continuum and quasi-continuum modeling of systems with a large number of degrees of freedom and sufficient complexity to require thermodynamic closure. Major emphasis is placed on papers attempting to bridge the gap between discrete and continuum approaches as well as micro- and macro-scales, by means of homogenization, statistical averaging and other mathematical tools aimed at the judicial elimination of small time and length scales. The journal is particularly interested in contributions focusing on a simultaneous description of complex systems at several disparate scales. Papers presenting and explaining new experimental findings are highly encouraged. The journal welcomes numerical studies aimed at understanding the physical nature of the phenomena. Potential subjects range from boiling and turbulence to plasticity and earthquakes. Studies of fluids and solids with nonlinear and non-local interactions, multiple fields and multi-scale responses, nontrivial dissipative properties and complex dynamics are expected to have a strong presence in the pages of the journal. An incomplete list of featured topics includes: active solids and liquids, nano-scale effects and molecular structure of materials, singularities in fluid and solid mechanics, polymers, elastomers and liquid crystals, rheology, cavitation and fracture, hysteresis and friction, mechanics of solid and liquid phase transformations, composite, porous and granular media, scaling in statics and dynamics, large scale processes and geomechanics, stochastic aspects of mechanics. The journal would also like to attract papers addressing the very foundations of thermodynamics and kinetics of continuum processes. Of special interest are contributions to the emerging areas of biophysics and biomechanics of cells, bones and tissues leading to new continuum and thermodynamical models.