{"title":"Elastic sheets on Winkler foundations: Indentation stiffness and nonlinearities","authors":"Erteng Chen, Zhaohe Dai","doi":"10.1016/j.ijsolstr.2025.113346","DOIUrl":null,"url":null,"abstract":"<div><div>The indentation of thin sheets on Winkler’s mattress or elastic foundations offers valuable opportunities to gain quantitative insights into the mechanical properties of both the material and its interface. However, interpreting indentation data is complicated by the interplay of plate bending, sheet pre-tension, and foundation deformation. The challenges are further amplified in recently developed nanoindentation techniques for small-scale systems, such as 2D materials and cell membranes, where indenter size, shape, and foundation nonlinearity have been found to influence the results significantly. Here, we address these challenges by investigating a generalized indentation problem involving a pre-tensioned elastic sheet on a mattress foundation, considering both punch and spherical indenters. By linearizing the Föppl–von Kármán equations and the elastic foundation under small indentation depth, we obtain a set of asymptotic solutions that quantify the effects of pre-tension and indenter geometry on indentation stiffness. These solutions show excellent agreement with numerical solutions in various parameter regimes that we classify. We also discuss sources of nonlinearities arising from the kinematics in sheet stretching and the evolving contact radius in spherical indentation. The results should be of direct use for the nanometrology of layered materials where indentation remains one of the most accessible techniques for characterizing mechanical properties at small scales.</div></div>","PeriodicalId":14311,"journal":{"name":"International Journal of Solids and Structures","volume":"315 ","pages":"Article 113346"},"PeriodicalIF":3.4000,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Solids and Structures","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020768325001325","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
Abstract
The indentation of thin sheets on Winkler’s mattress or elastic foundations offers valuable opportunities to gain quantitative insights into the mechanical properties of both the material and its interface. However, interpreting indentation data is complicated by the interplay of plate bending, sheet pre-tension, and foundation deformation. The challenges are further amplified in recently developed nanoindentation techniques for small-scale systems, such as 2D materials and cell membranes, where indenter size, shape, and foundation nonlinearity have been found to influence the results significantly. Here, we address these challenges by investigating a generalized indentation problem involving a pre-tensioned elastic sheet on a mattress foundation, considering both punch and spherical indenters. By linearizing the Föppl–von Kármán equations and the elastic foundation under small indentation depth, we obtain a set of asymptotic solutions that quantify the effects of pre-tension and indenter geometry on indentation stiffness. These solutions show excellent agreement with numerical solutions in various parameter regimes that we classify. We also discuss sources of nonlinearities arising from the kinematics in sheet stretching and the evolving contact radius in spherical indentation. The results should be of direct use for the nanometrology of layered materials where indentation remains one of the most accessible techniques for characterizing mechanical properties at small scales.
期刊介绍:
The International Journal of Solids and Structures has as its objective the publication and dissemination of original research in Mechanics of Solids and Structures as a field of Applied Science and Engineering. It fosters thus the exchange of ideas among workers in different parts of the world and also among workers who emphasize different aspects of the foundations and applications of the field.
Standing as it does at the cross-roads of Materials Science, Life Sciences, Mathematics, Physics and Engineering Design, the Mechanics of Solids and Structures is experiencing considerable growth as a result of recent technological advances. The Journal, by providing an international medium of communication, is encouraging this growth and is encompassing all aspects of the field from the more classical problems of structural analysis to mechanics of solids continually interacting with other media and including fracture, flow, wave propagation, heat transfer, thermal effects in solids, optimum design methods, model analysis, structural topology and numerical techniques. Interest extends to both inorganic and organic solids and structures.