This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Romanoff, Jani; Jelovica, Jasmin; Reddy, J. N.; Remes, Heikki Post-buckling of web-core sandwich plates based on classical continuum mechanics Published in: MECCANICA DOI: 10.1007/s11012-020-01174-6 Published: 01/06/2021 Document Version Publisher's PDF, also known as Version of record Published under the following license: CC BY Please cite the original version: Romanoff, J., Jelovica, J., Reddy, J. N., & Remes, H. (2021). Post-buckling of web-core sandwich plates based on classical continuum mechanics: success and needs for non-classical formulations. MECCANICA, 56(6), 1287-1302. https://doi.org/10.1007/s11012-020-01174-6 This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Powered by TCPDF (www.tcpdf.org) Meccanica https://doi.org/10.1007/s11012-020-01174-6 (0123456789().,-volV)( 0123456789().,-volV) RECENT ADVANCES IN COMPUTATIONAL MECHANICS AND INNOVATIVE MATERIALS Post-buckling of web-core sandwich plates based on classical continuum mechanics: success and needs for non-classical formulations Jani Romanoff . Jasmin Jelovica . J. N. Reddy . Heikki Remes Received: 21 January 2020 / Accepted: 10 May 2020 Ó The Author(s) 2020 Abstract The paper investigates the post-buckling element models of the actual 3D-geometry. Finally, response of web-core sandwich plates through classi- the physical limits of the classical continuum mechan- cal continuum mechanics assumptions. The compres- ics are discussed in the present context. sive loading is assumed to be in the direction of the web plates. Equivalent Single Layer (ESL) plate Keywords Plate theory Á Post-buckling Á Structural formulation is used with the kinematics of the First design Á Sandwich Structures order Shear Deformation Theory (FSDT). During the initial, membrane-dominated loading stages, it is observed that the effect of finite size of the periodic microstructure is barely influences the plate responses. 1 Introduction At the higher loads, when bending is activated, the finite size of the microstructure activates secondary There is constant need to study new materials and shear-induced bending moments at the unit cells of the structural configurations for thin-walled structures. plate. A method to capture the envelope of the Weight reduction and strength increase enable lighter maximum values of these bending moments is and often more sustainable structural solutions that presented. The findings are validated with the shell can be used in civil, naval and aeronautical applica- tions. Structural efficiency in terms of strength-to- weight and stiffness-to-weight is obtained by posi- J. Romanoff (&) Á H. Remes tioning of the material according to the load-carrying Marine Technology, Department of Mechanical mechanism of the structure. In bending dominated Engineering, School of Engineering, Aalto University, applications, this often results in plates with periodic Otakaari 4, 00076 Aalto, Finland e-mail: jani.romanoff@aalto.fi microstructure, for example, single-sided stiffened plates and sandwich panels with visibly discrete core; J. Jelovica see Fig. 1. The periodic structure allows integration of Department of Mechanical Engineering, University of functions to the panel(e.g. air-conditioning and cable British Columbia, Applied Science Lane, Vancouver Campus, 2054-6250, Vancouver, BC V6T 1Z4, Canada tracks). Thus, unidirectional stiffening system with coarse spacing becomes attractive structural alterna- J. N. Reddy tive; Refs. [1–13]. These unidirectional plates, espe- Department of Mechanical Engineering, Center of cially in the form of steel sandwich plates, have been Innovation in Mechanics for Design and Manufacturing, Texas A&M University, J. Mike Walker’66, 180 Spence found to have excellent mechanical properties for Street, College Station, TX 77843, USA 123 Meccanica Fig. 1 Examples of periodic sandwich panels structural failures under different extreme loads. (Figures from: explosions Valdevit et al. [18]; full-scale ship collisions Ehlers et al. [37] and quasi-static plate punch experiments Ko¨rgesaar et al. [38]) buckling, bending and impact loads [14–21]. These the fact that often the failure of these panels occurs structures are, however, challenging in terms of first locally in the microstructural level in the face and structural analysis. The direct inclusion of the web-plates or at the welds by yielding or buckling, see microstructure to computational models of the entire Fig. 1. This means that certain level of accuracy is structure leads to expensive pre-processing, analysis needed in the analysis of structural details. and post-processing times (i.e., computationally inten- Homogenization is the alternative method for a sive). The problem compounds if the geometry direct modeling of both micro- and macrostructure. In changes and analysis is needed to be repeated several terms of plates, it has been discussed by several times(e.g., during structural optimization or reliability authors(see, e.g., Refs. [1, 3, 4, 11–14, 20–36]). In analysis [18, 20]). Further complications are caused by plates, the main idea is to reduce the mathematical 123 Meccanica description to Equivalent Single Layer (ESL) repre- the problem at hand. We assume that the microstruc- sentation in which the deformations and stress resul- ture is linearly-elastic. Thus, at micro-scale, analyti- tants (forces and moments) are computed for known cal, stiffness and strength formulations are possible load and boundary conditions. In periodic sandwich and the prevailing physics can be explained in panels, the closed unit cells, including the interacting parametric form. Then we show how to eliminate the face and web plates, form a structure that warps in microstructural elements from the plate model. This bending due to the out-of-plane shear. With this elimination allows solution of the macro-scale prob- phenomenon included, Libove and Hubka [1] pre- lem with the classical ESL-FSDT formulations that sented an ESL-theory for corrugated-core sandwich can be found from numerous textbooks and the finite plates which follows First order Shear Deformation element implementations from several commercial Theory (FSDT). They derived the equivalent shear codes. Then, at the post-processing stage we focus on stiffness and provided series solutions to prevailing assessment of stress concentrations at faces, webs and differential equations for bending. After this work, welds and reconsider the finite size of the microstruc- several papers have been written for other microstruc- ture. This shows the limitations of this paper and also tures [26–30, 33, 37, 38] where the stiffness param- the commercial FSDT-ESL FE-codes on capturing the eters change, but the differential equations remain the prevailing post-buckling behavior. It also motivates same. The approach has been also validated to extensions to the non-classical continuum mechanics geometrically moderate non-linear global deforma- formulations. tions [26–28]. However, the problem arises in the assessment of stresses. As the local, microstructural length-scale can be close to that of macro-structural, 2 Theory the local oscillations of stresses within microstructure can have significant contribution to the overall stress 2.1 Notations state caused by macroscopic plate behavior. It has been experimentally shown that the out-of-plane shear The plate is assumed to consist of structural elements damages the periodic structures from plate edges or with small thickness representing the face and web close to point loads, thus the secondary stresses plate(s). This justifies the use of Kirchhoff hypothesis induced by shear need to be carefully assessed locally at the face and web-plates. The thicknesses of [20–22], see also Fig. 1. These findings are based on the top and bottom face plates are denoted by tt and tb, bending of plates. This situation can also occur during respectively, and these are positioned in the xy-plane. in-plane compression after buckling when the initially The web plates are in the xz-plane and have thickness membrane-dominated responses are complemented tw, spacing s, and height hc. The plate has two with out-of-plane responses [39]. coordinate systems, namely: global xyz and local xlylzl The objective of this study is to present the (see Fig. 2). The origin of the global coordinate phenomena related to post-buckling response of system is located at the geometrical mid-plane of the web-core sandwich plates and thereby extending the plate and the origin of the local coordinate is located at work presented in Ref. [39]. Primary focus is on the the geometrical mid-planes of face or the web plates bending responses due to the von Ka´rma´n strains. The under consideration. ESL formulation based on the FSDT is utilized in order to show the connection between in- and out-of- 2.2 Classical, homogenized FSDT for periodic plane deformations. In post-buckling regime, bending plates occurs, which activates secondary normal stresses of the microstructure. As the unit cell to plate length The deformation of the periodic