Post-Buckling of Web-Core Sandwich Plates Based on Classical Continuum Mechanics

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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

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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

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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 applications. 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 microstructure. 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´rmań 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 plate is composed of ratio, lmicro/lmacro, is not infinitesimal, the micro- global bending deflection of the mid-surface and local fluctuations of stress at unit cell level contribute deflection due to warping of face and web plates. Thus, significantly to the total normal stress response of the the total deflections along the three coordinate direc- face and web plates. tions of the faceplate can be represented as In order to show these aspects, we first formulate the geometrically non-linear differential equations of 123 Meccanica

Fig. 2 Web-core sandwich plate, notations and the kinematics of the First order Shear Deformation Theory (FSDT) in Equivalent Single Layer (ESL) formulation

of the mid-plane of the entire plate is denoted by / , uxðÞ; y; z; xl; yl; zl ¼u0ðÞþx; y z/g;xðÞx; y g the in-plane displacements by u and v and deflection þ zl/x;lðÞx; y; xl; yl ð1Þ 0 0 by wg. The global and local rotations are vxðÞ; y; z; xl; yl; zl ¼v0ðÞþx; y z/g;yðÞx; y dw du dw dv / ¼ c À g ¼ ; / ¼ c À g ¼ ð4Þ þ zl/y;lðÞx; y; xl; yl ð2Þ x;g xz dx dz y;g yz dy dz wxðÞ; y; z; xl; yl; zl ¼wgðÞþx; y wlðÞx; y; xl; yl : ð3Þ dwl dwl /x;l ¼À ; /y;l ¼À ð5Þ where subscript 0 denotes the in-plane membrane dx dy displacements at the geometrical mid-plane, as shown which means the sandwich plate behaves according to in Fig. 2. The subscripts g and l denote bending the FSDT. The relative strains, expressed in column actions at the geometrical mid-plane of the entire vector, are sandwich plate and face plate respectively. The global 8 9 8 9 8 9 8 9 deflection of webs is defined completely by the face < exx = < exx;0 = < exx;g = < exx;l = fge ¼ eyy ¼ eyy;0 þ eyy;g þ eyy;l plate deflection at their intersection, wweb = wg.In : ; : ; : ; : ; addition, due to perfect connection with the faces, the cxy cxy;0 cxy;g cxy;l webs bend in the y-direction due to the local deflection ð6Þ of the faces, w . This deflection of the webs is defined l with strains defined as completely by the face plate rotations at the web-to- face-interface. Thus, additional kinematical variables are not needed for the web deformations. The rotation

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8 9 8 9 2 < = < = du0 1 dwg d/x;g Nxx h=2 rxx exx;0 ¼ þ ; exx;g ¼ zg ; exx;l dx 2 dx dx fgN ¼ : Nyy ; ¼ r : ryy ;dz; fgM h= 2 Nxy À 2 rxy d wl 8 9 ¼Àzl ð7Þ < r = dx2 h=2 xx ¼ r : ryy ;zdz; ð13Þ h= 2 À 2 rxy dv0 1 dwg d/y;g eyy;0 ¼ þ ; eyy;g ¼ zg ; eyy;l dy 2 dy dy Due to the periodic microstructure, smearing the shear 2 d wl modulus of the core by an integrating over the ¼Àzl ð8Þ dy2 thickness of the sandwich does not result in correct stiffness values. In the x-direction this is due to the du0 dv0 dwg dwg shear flow of the thin-walled section. In the y-direction cxy;0 ¼ þ þ ; cxy;g dydx dx dy this is due to the higher order warping deformations of 2 d/x;g d/y;g d wl the unit cells. Instead, we incorporate the shear ¼ zg þ ; c ¼Àzl ð9Þ dy dx xy;l dxdy stiffness that accounts directly these two effects and write: where the underlined terms denote the von Ka´rmań ÈÉ strains, which are accounted at the macroscale. The Qxx DQ1 0 cxz fgQ ¼ ¼ ½DQ cz ¼ plates are assumed to be made of isotropic material Qyy 0 DQ2 cyz and follow Hooke’s law. Thus, the elasticity matrices ð14Þ of the different layers (subscripts: t = top face, b = bottom face, w = web) of the sandwich plate are The equilibrium of a plate element is governed by 2 3 the following equations: Q11 Q12 0 4 5 dN dN ½¼Qt ½¼Qb Q12 Q22 0 xx þ xy ¼ 0 ð15Þ Q dx dy 2 00366 t;b E tE 6 0 7 dNyy dNxy 6 1 À t2 1 À t2 7 þ ¼ 0 ð16Þ ¼ 6 tE E 7 ð10Þ dy dx 4 0 5 1 À t2 1 À t2 dQ dQ 00G t;b xx yy Ã þ þ N wg þ q ¼ 0 ð17Þ 2 3 dx dy t 6 Ew 007 dM dM dM dM 4 s 5 xx xy xx;l xy;l ½¼Qw 000 ð11Þ þ À Qxx þ þ ¼ 0 ð18Þ dx dy |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}dxl dyl 000 local where the web properties are smeared equally over the dMyy dMxy dMyy;l dMxy;l unit cell width by the rule-of- mixtures. The stresses þ À Qyy þ þ ¼ 0 ð19Þ per layer are: dy dx |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}dyl dxl local fgri ¼ ½Ei fgei ; i ¼ t; w; b ð12Þ d dw dw The stress resultants are obtained by through- NÃw ¼ N g þ N g g dx xx dx xy dy thickness integration and are d dw dw þ N g þ N g ð20Þ dy xy dx yy dx where the underlined terms are associated with local shear-induced warping response. It should be also noticed that due to the von Ka´rmań strains, the in-

123 Meccanica plane membrane and out-of-plane shear forces are d2v dw d2w d2u dw d2w B 0 þ g g þ B 0 þ g g coupled, by Eq. (20). The differential equations in 22 dy2 dy dy2 12 dxdy dx dxdy terms of displacements are then: 2 2 d /y;g d /x;g þ D22 2 þ D12 d2u dw d2w d2v dw d2w dy dxdy 0 g g 0 g g 2 2 A11 þ þ A12 þ d / d / dx2 dx dx2 dxdy dy dxdy y;g x;g þ D66 2 þ d2/ d2/ d2/ d2/ dx dxdy x;g y;g x;g y;g 2 2 2 2 þ B11 2 þ B12 þ B66 2 þ d v0 d u0 d wg dwg dwg d wg dx dxdy dy dxdy þ B66 þ þ þ 2 2 2 2 dx2 dxdy dxdy dx dy dx2 d u0 d v0 d wg dwg dwg d wg þ A66 þ þ þ dw dM dM dy2 dxdy dxdy dy dx dy2 À D g þ / À y;l À xy;l Q2 dy y;g dy dx ¼ 0 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}l l 21 local ð Þ ¼ 0 2 2 2 2 d v0 dwg d wg d u0 dwg d wg ð25Þ A22 2 þ 2 þ A12 þ dy dy dy dxdydx dxdy The membrane, membrane-bending coupling and 2 2 2 2 d /y;g d /x;g d /y;g d /x;g bending stiffnesses are þ B22 þ B12 þ B66 þ dy2 dxdy dx2 dxdy 2 2 2 2 h=2 ÀÁ d v0 d u0 d wg dwg dwg d wg A; B; D r Q 1; z; z2 dz; i t; w; b 26 þ A66 þ þ þ ½¼½i ¼ ð Þ dx2 dxdy dxdy dx dy dx2 Àh=2 ¼ 0 The shear stiffness is obtained by unit cell analysis ð22Þ of the periodic structure. In x-direction (see Ref. [21]) this is: d2w d/ d2w d/ D g þ x;g þ D g þ y;g þ NÃw Qx dx2 dx Qy dy2 dy g 2 tw DQ1 ¼ k11 Gttt þ Gbtb þ Gwhc ð27Þ þ q s vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 u u 1 k ¼ u ; i ¼ t; w; b ð28Þ ð23Þ 11 t P 2 si A i r Q s tidsi 2 2 2 2 Q;x d u0 dwg d wg d v0 dwg d wg B11 2 þ 2 þ B12 þ dx dx dx dxdy dy dxdy where the shear correction factor is obtained by 2 2 d /x;g d /y;g integration of shear flow through top, web and bottom þ D11 þ D12 dx2 dxdy faces; for details see Ref. [21]. In the y-direction the 2 2 d /x;g d /y;g shear stiffness is computed by unit frame analysis (see þ D66 þ dy2 dxdy Ref. [21] and Fig. 3b) and is given by d2u d2v d2w dw dw d2w þ B 0 þ 0 þ g g þ g g 66 2 2 12Dw dy dxdy dxdy dy dx dy DQ2 ¼ ð29Þ 2 Dw d Dw d dwg dMx;l dMxy;l s kQ þ 6 þ 12 b À 2 Db s k s s À DQ1 þ /x;g À À h dx |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}dxl dyl local i Qys t b t b kh ¼ i ; k1 ¼ 1 À kQ; k1 ¼ kQ; k2 ¼ 2 À 3kQ; k2 ¼ 0 hc ð24Þ ¼ 3kQ À 1; ð30Þ Dt 1 1 Dt d 1 þ 12 s kt À b þ 6 D s h kh w kQ ¼ ð31Þ 1 þ 12 Dt d þ Dt Dw s Db

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Fig. 3 a Web-core sandwich under shear deformation in weak shear induced bending moment between the unit cells (periodic direction. b Reduction of the deformation from 3rd order behavior is shown by dashed lines, and average behavior by polynomial to 1st order polynomial in y-direction. c The shear- points) induced bending stresses in the unit cell and d variation of the Q sd sy y2 y3 where kh is the rotation stiffness of the laser-stake weld Qy i l l l wi ¼ k1 À þ 3 À 3 ; i ¼ t; b ð32Þ and k-parameters model the relative stiffness of faces 12Di d d ds and webs. For symmetric sandwich panels the param- with the k-factors defined in Eq. (30). The curvature is eter kQ = 1/2 indicating that equal amount of shear is 2 2 2 carried out by top and bottom face plates. d wi d wi d wi QQys i yl 2 ¼ ¼ 0; 2 ¼ k1 1 À 2 ð33Þ dxl dxdy dyl 2Di s 2.3 Unit cell analysis for out-of-plane shear The local bending moments and shear forces are The local response, opposite to stiffener direction, is Q s y M ¼ tM ; M ¼À Qy ki 1 À 2 l ; M ¼ 0 assumed to have the same values as the global x;l y;l y;l 2 1 s xy;l response at the unit cell edges in terms of displace- ð34Þ ment, wl(x,y)=wg(x,y), wl(x,y ? s)=wg(x,y ? s), as i shown in Fig. 3a and b. However, along the unit cell Qx;l ¼ 0; Qy;l ¼ QQyk1 ð35Þ the local displacement field in the y-direction can differ from global due to shear-induced warping (see The volume averages are Fig. 3b). Due to the classical continuum assumptions M MÃ ¼ r s x;l dV ¼ 0; MÃ ¼ 0; MÃ ¼ 0; ð36Þ the average shear strain of horizontal and vertical x;l 0 V y;l xy;l sliding, cyz=(cyz,h? cyz,v)/2, is utilized. Ã Ã t b The local warping deflection due to shear force Qyy Qx;l ¼ 0; Qy;l ¼ Qyy ¼ QQyk1 þ QQyk1 ð37Þ is given as (see Ref. [21]):

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Thus, with these average values the underlined It should be noted that when lmicro = s=0, these terms in equilibrium Eqs. (18) and (19) and differen- bending moments and resulting shear induced stresses tial Eqs. (24) and (25) based on displacements become become zero and result is the same as produced by the zero. The shear force Qyy is produced by the unit cell homogenized solution without any additional post- warping, see Fig. 3b. This gives us the homogenized processing. differential equations that are derived based on the classical continuum mechanics assumption, that lmicro/ lmacro = 0. These equations can be found from numer- 3 Example ous text books of composite materials(see, e.g., Ref. [40]). In these books the corresponding analytical 3.1 General solutions and ﬁnite element formulations are also presented. The corresponding ﬁnite element imple- The example presented here is taken from Ref. [39]; it mentations can be found from most of the commercial is extended here in order to explain the main post- FE codes. buckling phenomena. A square plate with length and width of L = B=3.60 m is considered. Thickness of the

2.4 Calculation of envelope of periodic stresses face and web plates are tt= 2.5 mm and tw= 4.0 mm, from homogenized response respectively. Core height is hc= 40 mm and the web plate spacing is s = 120 mm giving lmicro/lmacro= s/ The periodic shear-induced bending moments and L = 1/30. The interface between web and face plate is normal stresses (see Fig. 3c) are important as the with assumed to be rigid in order to simplify the analysis. It these included the total stress always exceeds the has been shown in Refs. [21, 36] that non-local plate averaged, homogenized solution values. The charac- formulations are needed in the cases were the laser- teristic length, lmicro, of these normal stresses is equal stake weld is assumed to be flexible or if lmicro/lmacro- to web plate spacing, s, lmicro = s. These shear-induced ratio would be significantly larger. In cases where the bending moments and normal stresses have zero mean lmicro/lmacro-ratio would be significantly smaller, the as shown in Fig. 3d and Eq. (36). Next we assume shear-induced stress fluctuations would be less impor- that, as the homogenized structure is in equilibrium, tant. Material is assumed to be linear-elastic with the periodic structure must be in equilibrium too. Youngs modulus 206 GPa and Poisson ratio 0.3. Then, the periodic strains and stresses are (see Ref. The FSDT-ESL problem is solved numerically by [21]): using Finite Element Method. The non-linear analysis À1 is carried out in two steps. The first eigenmode is first fge0 ½A ½B fgN ÈÉfg0 ¼ À computed. It is used as the shape of the initial out-of- M fgj ½B ½D fgM Q plane imperfection and is given the magnitude of ð38Þ 0.01% of the plate length, L. Then, the geometrically non-linear analysis is carried out to trace the post- fge ¼ fge0 À zfgj ð39Þ buckling path. Abaqus software, version 6.9, is used. ÈÉ A subspace iteration solver is used for the eigenvalue r E e r 40 fg¼ ½fgþ Q ð Þ analysis and the modified Riks procedure for the post- where Eqs. (34) and (38) together create periodic buckling path. In order to secure converged results in strain field from homogenized, smooth, strain field. FSDT-ESL, a mesh of 50 9 50 S4R shell elements are From strength viewpoint shape of this micro-fluctua- used. Simply supported boundary conditions are tion is not important, but the maximum value is. Thus, considered, with the loaded edges kept straight and for linear distribution within unit cell, (see Eq. (34) the unloaded edges free to pull in. and Fig. 3c and d), the maxima and minima are simply Q s Q s Mmax ¼Æm ki yy ; Mmax ¼Æki yy ; Mmax ¼ 0 Q;x;i i 1 2 Q;y;i 1 2 Q;xy;i ð41Þ

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3.2 3D Finite Shell Element Analyses Eq. (20), into the out-of-plane shear Qyy. Three load for Validation levels are selected from the curves to show the different stages of the structure undergoing non-linear In order to validate the FSDT-ESL approach, a 3D deformations. Point A corresponds to linear regime, model of the actual periodic plate is used; see Fig. 4 point B corresponds to the intermediate stage of for details. The 3D-plate is modelled using shell transition from linear to non-linear post-buckling elements (S4) that follow the Kirchhoff hypothesis regime, and point C is the point of local buckling of (assumption in FSDT-ESL model). Concentrated the face plates where the assumption of linear nodal forces act at web plates in the nodes in the microstructure becomes violated. Comparison of the geometrical mid-plane. Six shell elements per web predicted shear force Qyy-distribution from mid-span plate height and between webs are used as this has of the panel, at x = L/2, is presented in Fig. 6 for load been shown to produce converged results in buckling points A and C. Figure 7 present the corresponding and bending problems [20, 41]. Simply supported normal stresses with shear-induced stresses included boundary conditions are considered, with the loaded and excluded. edges kept straight by constraint equations and the From Fig. 5 it is clear that the non-linear response is unloaded edges free to move in-plane. The transverse predicted very accurately with the FSDT-ESL in deﬂection is zero only at the nodes at the geometric comparison to the 3D-model of the actual geometry mid-plane. This allows the rotation of the plate around (3D-FEA). Both load-end-shortening and load–deﬂec- the mid-plane edge. tion curves overlap until local buckling occurs at the unit cells at point C. It is also clear that this point is 3.3 Results well beyond the panel level buckling, around point B.

After this point B, the out-of-plane shear, Qyy, The load-end-shortening, load–deflection and load- increases rapidly due to the von Ka´rmań non-linearity, out of plane shear behavior is presented in Fig. 5, see Eq. (20). Figure 6 shows that the spatial distribu- where also the comparison of the FSDT-ESL with tion of out-of-plane shear is accurately predicted with respect to full 3D-FE model is presented. The main the FSDT-ESL. Even at the point of local buckling, the idea is to show the influence of the von Ka´rmań term, FSDT-ESL and 3D-FEA results overlap each other in

Fig. 4 Description of the constrains implementation of 3D-FEA. Reproduced from Ref. [39] 123 Meccanica

Fig. 5 Comparison of FSDT-ESL (dashed line) and 3D-FEA (solid line) in load-end-shortening, load–deflection and load-shear responses. Shear force-load marked with crosses average sense. Figure 7 shows that the membrane linear regime this loading is carried out mostly by pure stresses of the face plates, predicted by both FSDT- membrane actions of face and web plates as Fig. 7 at ESL and 3D-FE methods, overlap at the load point A. point A indicates. As the post-buckling takes place, the In post-buckling, due to global bending, they start to membrane actions start to interact with the out-of- differ significantly from each other. It is also seen that plane deformation, due to the von Ka´rmań non- the membrane stress in the faces is significantly lower linearity. As the core is visibly discrete, lmicro/lmacro- than the total stress at the surface of the face plates = 1/30, and the unit cells warp in shear, the magnitude which is magnified by the secondary bending. It is seen of shear-induced secondary normal stresses at the that the stresses from 3D-FEA are within the maxi- faces and webs becomes significant. When these are mum stress envelope curves defined by Eq. (41). This added to the membrane stresses, the total stress can be highlights the importance of taking the finite size, significantly higher than the membrane stress only. lmicro/lmacro, of the microstructure into account when This effect becomes larger when the role of out-of- computing the stress response. The stress jumps at the plane shear cyz or Qyz increases and when the unit cell location of webs close to plate edges indicate signif- size increases. So, this phenomenon can be considered icant bending in the webs and welds due to shear- as type of size effect. Usually in homogenization induced warping of the unit cells. theories we assume that these are infinitely far apart,

i.e. lmicro/lmacro = 0. When departing from this assumption, very soon we end up dealing with another 4 Discussion type of size effect, the assumptions of non-classical continuum mechanics. The investigation presented above indicates that when This issue is seen through following example. Post- sandwich structures, with visible periodic core, are buckling analysis requires two stage analysis with homogenized, special attention must be paid on the initial analysis to deﬁne the initial deformation shape stress assessment, even in the case of buckling which is then followed by geometrically non-linear assessment where loading is of membrane type. In analysis. In present formulation, the bifurcation

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Fig. 6 Comparison of FSDT-ESL and 3D-FEA in terms of shear force Qyy at x = L/2 for Top: load-Level A, and Bottom: Load level C

buckling load for simply supported rectangular plate is where m and n are used to denote the number of half- given as [40]: waves in directions x and y respectively. In present c11 2 2 DQ2 case, the loading is assumed to be uniaxial, thus the c55 þ a þ b 1 DQy DQ1 load ratio factor k = 0, orthotropy ratio for shear, DQ1/ N0 ¼ 2 2 c11 1 c22 DQ2 c33 a þ kb 1 þ þ þ DQ2-ratio, is very high (e.g. 100-1000). Thus, the DQ2 DQ1 c11 DQ1 c11 mp np buckling load minimum is obtained when m = n =1. a ¼ ; b ¼ ; c ¼ c c À c2 ; a b 11 22 33 44 Problems with this equation occur when the load is 2 2 ð42Þ turned to be along y-axis or biaxial compression is c22 ¼ D11a þ D12b ; considered. In this case, the high orthotropy in shear 2 2 c33 ¼ D66a þ D22b c44 ¼ ðÞD12 þ D66 ab; causes a situation where minimum does not converge 4 2 2 4 c55 ¼ D11a þ 2ðÞD12 þ 2D66 a b þ D22b ; for m = n = 1, but decreases as function of n, while Nyy m = 1. In this case the buckling load in ﬁnite element k ¼ ; Nxx ¼ÀN0; Nyy ¼ÀkN0; Nxx solution becomes mesh size dependent; the smaller is

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Fig. 7 Comparison of FSDT-ESL (points, crosses) and 3D-FEA (solid line) at x = L/2. Top: x-normal stresses due to membrane action for load-levels A; Center at load-level C and Bottom: at load-level C with envelopes of the shear induced secondary normal stress

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Fig. 8 Strength of welds by stress resultants. Observed bending of welds with impact of contact to the initial stiffness geometrical and material gradients in the welds. Failure mode and strength. See for details Refs. [43, 44] for stress resultants Nz with microrotation at HAZ and My in the the mesh size, the smaller is the buckling load. This is continuum formulations. This calls for micropolar another type of size effect which results from classical plate elements, and recently such study has been continuum mechanics assumptions. The decrease is reported by Nampally et al. [43]. unphysical and can be corrected by incorporation non- In order to assess the strength of real welded classical continuum mechanics assumptions into our structures following issue must be handled. The stress FSDT-ESL model. As shown by Romanoff and Varsta values seen in point C of the case study are very close [21], Jelovica and Romanoff [42] by thick-face plates to the material yield point. Typically, the steel used for sandwich theory and by Karttunen et al. [36]by these panels has a yield point at 355 MPa for the faces micropolar theory, the shear deformations of sandwich and 235 MPa or 355 MPa for the web plates. Thus, as panels can have only finite wave-lengths, that is, finite Fig. 7, shows it is crucial to recover the microstruc- n-values. When using thick-faces effect or micropolar tural stress if first fiber yield is to be assessed. The solution, physically correct behavior is obtained. This assessment of laser-stake welds is a bit more chal- is a result of the fact that we can split the out-of-plane lenging. Jutila [44] carried out experiments with shear strain to symmetric and antisymmetric parts Digital Image correlation system of pull-out strength [36]. Therefore, the present investigation should be of laser-stake welds. Strength values up to 1000 MPa extended in future to compression in transverse for tension for steel faces and webs of 355 MPa and direction and investigations based on non-classical 235 MPa, respectively are reported. The difference is

123 Meccanica also seen in the hardness of the welds, see Fig. 8. Due when bending is activated, the finite size of the unit to rapid changes in the hardness, the weld deforms cells activates secondary shear-induced bending only moderately in softer faces and webs and micro- moments. Due to this effect, the normal stress levels rotation is seen between these two, in the heat affected become significantly higher than the homogenized zone. This is indicating that non-classical continuum plate theory would predict. A method to capture the mechanics are needed in the weld modelling. How- envelope of the maximum values of these stresses is ever, there is another effect that requires careful presented and validated with 3D-Finite Element - investigation. This is the contact between the faces and models of the actual geometry. It is also discussed that webs at the laser-stake welds when being bended. Due classical continuum mechanics has its limits when to the contact, the stresses are redistributed and failure of welded web-core sandwich structures is significant variations are seen in the moment carrying concerned. The welds experience micro-rotation close capacity of the welds due to small variations in weld to failure point. Due to this also the antisymmetric out- geometrical properties. Measured strength values are of-plane shear strain activates at the plate level which

Mcr= 1400Nm/m (see Ref. [45] for details), which calls for non-classical continuum mechanics. These corresponds Qyz = 21.5kN/m in shear. Thus, the extensions are left for future work. strength of welds would not be reached yet in present context for individual stress components. Proper Acknowledgements Open access funding provided by Aalto modelling of these phenomenon in 3D FEA is University. The initial research presented in this paper was funded the Finland Distinguished Professor (FidiPro) to J.N. therefore a signiﬁcant challenge which we cannot Reddy through a project titled ‘‘Non-linear response of large, solve in the present context. Also, an experimental complex thin-walled structures’’ and supported by Tekes, Napa, study is needed to gain more understanding of SSAB, Deltamarin, Koneteknologiakeskus Turku and Meyer prevailing failure modes in real structures. Turku Shipyard. The paper was ﬁnalized in the Academy of Finland project (#310828) called ‘‘Ultra Lightweight and In the present case it is indicated that the Fracture Resistant Thin-Walled Structures through microstructure can buckle during the deformation, Optimization of Strain Paths’’. This support is gratefully which is known to reduce the stiffness globally. As the acknowledged. Thanks to IT Centre for Science for providing local and global deformations interact, there is a need the computational capabilities. for coupled models where the geometrical non- Open Access This article is licensed under a Creative Com- linearity in one length-scale can be mapped correctly mons Attribution 4.0 International License, which permits use, to the another one. Such example in context of sharing, adaptation, distribution and reproduction in any med- classical continuum mechanics has been presented in ium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Reinaldo Goncalves et al. [46] with one way coupling Commons licence, and indicate if changes were made. The and in for example Geers et al. [21] and Rabzcuk et al. images or other third party material in this article are included in [30] for two-way coupling. However, this work should the article’s Creative Commons licence, unless indicated be extended to the non-classical continuum mechanics otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your due to the reasons mentioned above. intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit 5 Conclusions http://creativecommons.org/licenses/by/4.0/.

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