Numerical Buckling Analysis of Hybrid Honeycomb Cores for Advanced Helmholtz Resonator Liners

Article Numerical Buckling Analysis of Hybrid Honeycomb Cores for Advanced Helmholtz Resonator Liners

Moritz Neubauer , Martin Dannemann * , Michael Kucher , Niklas Bleil, Tino Wollmann and Niels Modler

Institute of Lightweight Engineering and Polymer Technology (ILK), Technische Universität Dresden, Holbeinstraße 3, 01307 Dresden, Germany; [email protected] (M.N.); [email protected] (M.K.); [email protected] (N.B.); [email protected] (T.W.); [email protected] (N.M.) * Correspondence: [email protected]; Tel.: +49-351-463-38134

Abstract: In order to realize novel acoustic liners, honeycomb core structures specially adapted to these applications are required. For this purpose, various design concepts were developed to create a hybrid cell core by combining flexible wall areas based on thermoplastic elastomer films and rigid honeycomb areas made of fiber-reinforced thermoplastics. Within the scope of the presented study, a numerical approach was introduced to analyze the global compressive failure of the hybrid composite core structure, considering local buckling and composite failure according to Puck and Cuntze. Therefore, different geometrical configurations of fiber-reinforced tapes were compared with respect to their deformation as well as their resulting failure behavior by means of a finite element analysis. The resulting compression strength obtained by a linear buckling analysis agrees

largely with calculated strengths of the more elaborate application of the failure criteria according to Puck and Cuntze, which were implemented in the framework of a nonlinear buckling analysis.

Citation: Neubauer, M.; The findings of this study serve as a starting point for the realization of the manufacturing concept, Dannemann, M.; Kucher, M.; Bleil, N.; for the design of experimental tests of hybrid composite honeycomb core structures, and for further Wollmann, T.; Modler, N. Numerical numerical investigations considering manufacturing as well as material specific aspects. Buckling Analysis of Hybrid Honeycomb Cores for Advanced Keywords: buckling; composite; Cuntze criterion; failure criteria; finite element analysis (FEA); Helmholtz Resonator Liners. J. Helmholtz resonator liner; honeycomb core; Puck criterion Compos. Sci. 2021, 5, 116. https:// doi.org/10.3390/jcs5050116

Academic Editor: Salvatore Brischetto 1. Introduction The Honeycomb core structures are used for a variety of different applications, fo- Received: 7 April 2021 cusing lightweight sandwich structures with speciﬁc high ﬂexural properties and a high Accepted: 20 April 2021 Published: 23 April 2021 bending stiffness. In the sense of a function-integrative lightweight design, as described by Modler et al. [1], these structural properties can be combined with functional properties

Publisher’s Note: MDPI stays neutral such as acoustic damping. A very well-known example is the acoustic liner, which is used with regard to jurisdictional claims in to reduce engine-introduced noise in various engineering applications, such as in automo- published maps and institutional affil- biles and aircrafts [2–5]. The separated cavities of honeycomb cores are particularly suitable iations. for these applications in acoustic liners [6] and are established in the aerospace industry. Due to the acoustical characteristics of the conventional Helmholtz resonator (HR) liners, high transmission losses are only obtained for a narrow frequency band [2]. A concept for the improvement of broadband noise absorption for acoustical liners was carried out by Barnobi [7], Follet et al. [8], and recently by Dannemann et al. [9] and Knobloch et al. [10]. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. These preliminary investigations showed an improvement in the damping performance This article is an open access article of the acoustic liners due to their hybrid structure, where flexible walls of thin film are distributed under the terms and integrated into a rigid core geometry [11]. conditions of the Creative Commons Even though acoustic liners mainly focusing acoustical applications, they have to with- Attribution (CC BY) license (https:// stand mechanical loads, such as during maintenance work or in case of impact loads. The creativecommons.org/licenses/by/ combination of isotropic films as flexible wall sections and fiber-reinforced thermoplastic 4.0/). tapes serving as a support structure allows optimization of both the acoustic and structural

J. Compos. Sci. 2021, 5, 116. https://doi.org/10.3390/jcs5050116 https://www.mdpi.com/journal/jcs J. Compos. Sci. 2021, 5, 116 2 of 14

properties of the core. Due to the intended application of the acoustic liners as function- integrative lightweight structure with combined acoustical and structural properties, the Helmholtz resonance frequency as well as the deformation and failure behavior depend on the cell geometry, the material of the cell walls and its configuration. Therefore, a numerical approach for preliminary investigations on design, manufacturing, and mechanical testing of honeycomb core structures made of composite materials is required. In the last decades, numerous publications were carried out considering especially an improvement of the core designs. Some of these studies analyzed the mechanical response of honeycomb structures and similar thin-walled structures under various loads (such as [12–17]). With regard to the simulation of complex microstructures, Vondˇrejcand Geus [18] compared the FEA and numerical methods based on the fast Fourier transform with respect to their numerical efficiency and showed potentials and limitations of the designated methods. A numerical formulation was introduced by Liu and Jeffers [19] which is based on isogeometric analysis enabling the simulation of functionally graded materials and geometrically nonlinear effects. The response to compression of honeycomb structures by considering the cell size and cell materials was investigated by Wu et al. [20]. They investigated that cores with a smaller cell size and height had a better structural performance under quasi-static and impact-loading conditions. The prediction of the macroscopic deformation behavior using detailed mechanic models on meso-scale was analyzed by Seemann and Krause [21] as well as Hähnel et al. [22]. The deformation behavior of honeycomb cores under various loads was investigated for Kevlar staple fibers with random orientation which was impregnated with phenol resin using a linear-elastic material behavior [22] and for different phenolic resin-impregnated aramid papers using a stochastic approach for the modelling imperfections of the material behavior [23]. In addition, recent work on the mechanical behavior of composite truss core structures under various loads is relevant to the present work because of the similarities to the support structure of the targeted hybrid honeycomb core [24–26]. Compared to these previous studies, a numerical approach for the investigation of the failure behavior for honeycomb core structure with partial fiber-reinforcement is introduced for the first time. The aim of the presented work is to analyze the failure behavior of hybrid honeycomb core structures partially made of composite materials under compression load. Different design concepts of prismatic hybrid core structures are demonstrated and the compression behavior of a preferred variant is analyzed. The numerical model, the implementation of boundary conditions, as well as geometrical imperfection are described. In order to assess the effect of the geometrical design of the hybrid honeycomb core on its failure behavior, various configurations of the composite support structure are analyzed, and different failure criteria as well as the resulting fiber failure modes of the composite core structure are discussed.

2. Materials and Methods 2.1. Structural Design The basic cell shapes of honeycomb cores are the hexagon, the square, and the flex- core [27]. Thereby, hexagonal cells are by far the most common adhesively bonded honeycomb core structures while most resistance welded and brazed cores have square cells and obtain very narrow cell sizes. With regard to the manufacturing, different designs of square and hexagonal honeycombs made of flexible films and stiff fiber-reinforced materials can be realized (Table1). For square honeycomb, wall straps with cross-slots could be assembled (Table1, Config. 1) [ 9,28]. Changing the design of the connection areas leads to further configurations (Table1, Config. 2, Config. 3). These square cores require a high manual effort during the assembly process. For the integration of thermoplastic materials, the expansion or corrugation method [27] as well as the manufacturing of folded thermoplastic honeycomb cores [29,30] seem to be more applicable. With regard to advanced acoustic liners, configurations 4 and 5 have the great advantage of enabling J.J. Compos.J.Compos.J. Compos. J.Compos. Compos.J. Compos. Sci.Sci. Sci. Sci. 20212021 Sci. 2021 2021 Sci. ,,2021 55, , ,5 x2021 x5, ,FOR,,xFOR 5x FOR,, ,FORxx 5 FORPEER,FORPEER x PEER FORPEER PEERPEER REVIEWREVIEW PEERREVIEW REVIEW REVIEWREVIEW REVIEW 33 of3of3 of 14 14of3 14 of 143 14of 14

J. Compos. Sci. 2021, 5, x FOR PEER REVIEWthethethethe expansion expansionthe expansion expansionthe expansion expansion oror or orcorrugationcorrugation orcorrugation corrugation orcorrugation corrugation methodmethod method method method method [27][27] [27] [27] [27] asas as [27] aswellwell aswell well aswell asas wellas asthethe asthe the manufacturingasmanufacturingthe manufacturing manufacturingthe manufacturing manufacturing ofof of foldedoffolded offolded folded offolded folded thermo-thermo-3 thermo- ofthermo- thermo-14 thermo- plasticplasticplasticplasticplasticplastic honeycombhoneycomb honeycomb honeycomb honeycomb honeycomb corescores cores cores cores [29,30][29,30]cores [29,30] [29,30] [29,30] [29,30] seemseem seem seem seem totoseem to tobebe tobe be more moreto bemore more bemore applicable.applicable.more applicable. applicable. applicable. applicable. WithWith With With With regardregardWith regard regard regard regard toto to toadvancedadvanced toadvanced advanced toadvanced advanced acousticacousticacousticacousticacousticacoustic liners,liners, liners, liners, liners, liners, configurationsconfigurations configurations configurations configurations configurations 44 and4 and4 and 4and and 545 5haveand have5 have 5have have 5 thethe have the the greatthegreat great greatthe great advantage advantagegreat advantage advantage advantage advantage ofof of enablingofenabling ofenabling enabling ofenabling enabling aa acontinuous continuousa continuous acontinuous continuous a continuous manufacturingmanufacturingmanufacturingmanufacturingmanufacturingmanufacturing process,process, process, process, process, process, allowingallowing allowing allowing allowing allowing complexcomplex complex complex complex complex cellcell cell cell cellshapesshapes shapes cellshapes shapes shapes andand and and and obtainingobtaining obtaining andobtaining obtaining obtaining largelarge large large large flexibleflexiblelarge flexible flexible flexible flexible wallwall wall wall wall wall J. Compos. Sci. 2021, 5, 116 the expansion or corrugation method [27] as well as the manufacturing of folded thermo-3 of 14 areas.areas.areas.areas.areas. areas. plastic honeycomb cores [29,30] seem to be more applicable. With regard to advanced InInIn Inthethe Inthe the Incurrentthecurrent current currentthe current current study,study, study, study, study, study, configurationconfiguration configuration configuration configuration configuration 44 4was was4 was 4was was chosen4chosen chosenwas chosen chosen chosen becausebecause because because because because itit it offersoffersit offers itoffers offersit highoffershigh high high high potentialpotential highpotential potential potential potential forfor for for for for acoustic liners, configurations 4 and 5 have the great advantage of enabling a continuous processingprocessingprocessingprocessingprocessingprocessing thermoplasticthermoplastic thermoplastic thermoplastic thermoplastic thermoplastic film-basedfilm-based film-based film-based film-based film-based materials,materials, materials, materials, materials, materials, enablesenables enables enables enables enables aa acontinuous continuousa continuous acontinuous continuousa continuous manufacturingmanufacturing manufacturing manufacturing manufacturing manufacturing pro-pro- pro- pro- pro- pro- manufacturing process, allowing complex cell shapes and obtaining large flexible wall acess, continuouscess,cess,cess,cess, andand cess,and and and achievesachieves achieves and manufacturingachieves achieves achieves thethe the the largestlargestthe largest largestthe largest process, largest flexibleflexible flexible flexible flexible flexible allowing wallwall wall wall wall areaarea wallarea area complexarea (Table(Table area(Table (Table (Table (Table 1,1, cell 1, Config.1,Config. Config.1, Config.shapes Config.1, Config. 4).4). 4). 4). and 4). 4). obtaining large areas.ﬂexible wall areas. In the current study, configuration 4 was chosen because it offers high potential for TableTableTableTableTable 1.1.Table 1. DesignDesign1. Design1. Design Design1. Design variantsvariants variants variants variants variants ofof of newofnew newof new newof acousticacoustic acousticnew acoustic acoustic acoustic liners.liners. liners. liners. liners. liners. processingTable thermoplastic 1. Design variants film-based of new materials, acoustic liners. enables a continuous manufacturing pro-

SquareSquarecess,Square and achievesSquareSquare theSquare largest flexible Square Square wall Square area (TableHexagonalHexagonal 1,Hexagonal Config. 4). HexagonalHexagonalHexagonal SquareSquareSquareSquare Square SquareSquareSquareSquare Square Square Square Square HexagonalHexagonalHexagonal Hexagonal HexagonalHexagonalHexagonal Hexagonal Table 1. Design variants of new acoustic liners. CellCellCellCell Cell Cell Cell structure structurestructurestructurestructurestructurestructure Square Square Square Hexagonal Hexagonal

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[31] indicatesindicates indicates [31]indicates indicates indicates anan an an altealte an alte alte an rnativernativealternative rnativealternativernative methodmethod method method method method forfor for for realizing realizingfor realizing realizingfor realizing realizing thethe the the supportthesupport support supportthe support support struc-struc- struc- struc- struc- struc- ofcarbon a carbon fiber-reinforced fiber-reinforced polyamide polyamide 6 tape. 6 tape. For For the the manufacturing, manufacturing, the the concept concept of of folding fold- turetureturetureture ofof ofture configurationofconfiguration ofconfiguration configuration ofconfiguration configuration 4.4. 4. 4. 4. 4. ingthermoplastic thermoplastic honeycomb honeycomb cores cores structures structures introduced introduced by Kucherby Kucher et al.et al. [30 [30]] is targeted.is targeted. 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FigureFigureFigureFigureFigure Figure1.1. 1. ((1.aa )(1. ) a( ConventionalaConventional) ( )1.Conventionala Conventional) ( Conventionala) Conventional HelmholtzHelmholtz Helmholtz Helmholtz Helmholtz Helmholtz resonatorresonator resonator resonator resonator resonator lineline line liner rline withrwith rlinewith withr with perforatedrperforated withperforated perforated perforated perforated upperupper upper upper upper layerupperlayer layer layer layer (1),(1), layer(1), (1), cece(1), cell ll ce(1), llcorecorecell core llcorece core (2),ll(2), (2), core(2), andand(2), and and(2), bottomandbottom bottom andbottom bottom bottom layerlayer layer layer layer (3),(3), layer(3), (3), (3), (3), ((bb())b( newbnew)( )bnew new)( bnew conceptconcept) conceptnew concept concept concept ofof of aofa hybrid ahybridof a hybrid hybridaof hybrid a hybrid acousticacoustic acoustic acoustic acoustic acoustic linerliner liner liner liner withwith withliner with with fiber-refiber-re fiber-rewith fiber-re fiber-re fiber-reinforcedinforcedinforcedinforcedinforcedinforced tapetape tape tape tape (4)(4) (4)tape (4) andand (4) and and (4) flexibleandflexible flexible andflexible flexible flexible cellcell cell cell wall wallcell wall wallcell wall (5)(5) (5)wall (5) andand (5) and and (5) (and(cc )() cand( detaildetailc) ()detailc detail) (detailc ) ofof detail of theofthe theof the hybrid hybridtheof hybrid hybrid the hybrid hybrid cellcellcellcell corecorecell core corecell core structure.structure. structure. corestructure. structure. structure.

Figure 1. (a) Conventional HelmholtzHelmholtz resonatorresonator liner liner with with perforated perforated upper upper layer layer (1), (1), cell ce corell core (2), (2), and and bottom bottom layer layer (3), (3), (b) (b) new concept of a hybrid acoustic liner with fiber-reinforced tape (4) and flexible cell wall (5) and (c) detail of the hybrid new concept of a hybrid acoustic liner with ﬁber-reinforced tape (4) and ﬂexible cell wall (5) and (c) detail of the hybrid cell cell core structure. core structure.

In order to investigate the effects of the geometry of the ﬁber composite structures on the structural mechanical behavior of the core, the width wt, and thickness tt of the tape J. J.J.Compos. Compos.Compos. Sci. Sci.Sci. 2021 20212021, 5,, ,55 x,, xxFOR FORFOR PEER PEERPEER REVIEW REVIEWREVIEW 4 44of ofof 14 1414 J. J. Compos. Compos. Sci. Sci. 2021 2021, 5,, ,5 x,, x xFOR FORFOR PEER PEERPEER REVIEW REVIEWREVIEW 4 4of of 14 14 J. J. Compos. Compos. Sci. Sci. 2021 2021, 5, 5, x, x FOR FOR PEER PEER REVIEW REVIEW 44 of of 14 14

InInIn order orderorder to toto investigate investigateinvestigate the thethe effects effectseffects of ofof the thethe ge gegeometryometry of of the the fiber fiber composite composite structures structures on on InIn order order to to investigate investigate the the effects effects of of the the ge geometryometry of of the the fiber fiber composite composite structures structures on on thethethe structural structuralstructural mechanical mechanicalmechanical beha behabehaviorvior of of the the core, core, the the width width 𝑤 𝑤t,t tand,, andand thickness thicknessthickness 𝑡 𝑡𝑡 of of of the thethe tape tapetape thethe structural structural mechanical mechanical beha behaviorvior of of the the core, core, the the width width 𝑤𝑤t,t ,and and thickness thickness 𝑡𝑡 of of the the tape tape thethe structural structural mechanical mechanical beha behaviorvior of of the the core, core, the the width width 𝑤𝑤,t ,and and thickness thickness 𝑡𝑡 of of the the tape tape werewere varied, varied, while while the the film film thickness thickness 𝑡 f𝑡𝑡 ff and and and the thethe height heightheight of ofof the thethet t core corecore ℎ ℎc cc were were were kept keptkept constant. constant.constant. werewere varied, varied, while while the the film film thickness thickness 𝑡f𝑡 ff and and the the height height of of the the core core ℎℎc c were were kept kept constant. constant. J. Compos. Sci. 2021, 5, 116 ThewereThewere chosen chosenvaried, varied, base basewhile while material material the the film film was was thickness thickness a aunidirectional unidirectional 𝑡f𝑡 f and and the the carbon carbon height height fiber-reinforced fiber-reinforced of of the the core core ℎℎc c wereplastic wereplastic kept kept (CFRP) (CFRP) constant. constant.4 of tape 14 tape TheThe chosen chosen base base material material was was a a unidirectional unidirectional carbon carbon fiber-reinforced fiber-reinforced plastic plastic (CFRP) (CFRP) tape tape withThewithThe chosen achosen athickness thickness base base materialof materialof 0.13 0.13 mm. mm.was was To a Toa unidirectional unidirectionalcreate create differ different entcarbon carbon thicknesses thicknesses fiber-reinforced fiber-reinforced of of the the reinforcing reinforcing plastic plastic (CFRP) (CFRP) structure, structure, tape tape withwith a a thickness thickness of of 0.13 0.13 mm. mm. To To create create differ differentent thicknesses thicknesses of of the the reinforcing reinforcing structure, structure, thewiththewiththe tape tape tapea a thickness thicknesswas waswas stacked stackedstacked of of 0.13 0.13on onon top mm. toptopmm. of ofof To eachTo eacheach create create other otherother differ differresulting resultingresultingentent thicknesses thicknessesin inin total totaltotal thickness thicknessthickness of of the the reinforcingof reinforcing ofof 0.26 0.260.26 mm, mm,mm, structure, 0.52structure, 0.520.52 mm, mm,mm, thethe tape tape was was stacked stacked on on top top of of each each other other resulting resulting in in total total thickness thickness of of 0.26 0.26 mm, mm, 0.52 0.52 mm, mm, andand 0.78 0.78 mm. mm. The The width width of of the the tapes tapes varied varied up up to to a aaproportion proportionproportion of ofof 100 100100 %, %,%, 75 7575 %, %,%, or oror 50 5050 % %% of ofof wereandand varied, 0.78 0.78 mm. mm. while The The the width width film of thicknessof the the tapes tapestf varied andvaried the up up height to to a a proportion proportion of the core of hof c100 100were %, %, kept75 75 %, %, constant. or or 50 50 % % of of thethethe side sideside length lengthlength of ofof 𝑎 𝑎c,cc =,, ==12 1212 mm, mm,mm, respectively. respectively.respectively. This ThisThis resulted resultedresulted in inin nine ninenine different differentdifferent geometric geometricgeometric con- con-con- Thethethe chosen side side length length base material of of 𝑎𝑎c,c ,= = 12 was 12 mm, mm, a unidirectional respectively. respectively. carbonThis This resulted resulted fiber-reinforced in in nine nine different different plastic geometric (CFRP) geometric tape con- con- figurations,thefigurations,thefigurations, side side length length ranging rangingranging of of 𝑎𝑎c ,cfrom ,=fromfrom = 12 12 mm,a mm, aathickness thicknessthickness respectively. respectively. of ofof 0.26 0.260.26 This mmThis mmmm resulted andresulted andand a aawidth widthwidth in in nine nine of ofof 6different different 66mm mmmm (T0.26W6) (T0.26W6)(T0.26W6) geometric geometric for forfor con- thecon- thethe withfigurations,figurations, a thickness ranging ranging of 0.13 from from mm. a a Tothickness thickness create differentof of 0.26 0.26 mm mm thicknesses and and a a width width of the of of 6reinforcing 6 mm mm (T0.26W6) (T0.26W6) structure, for for the the thinnest/narrowestfigurations,thinnest/narrowestfigurations,thinnest/narrowest ranging ranging configuration from configurationconfigurationfrom a a thickness thickness to toto a of of aa thickness 0.26 thickness0.26thickness mm mm and of and ofof 0.78 a 0.780.78a width width mm mmmm of ofand and6and 6 mm mma aa width (T0.26W6) width(T0.26W6)width of ofof 12 for12 12for mm themmmm the thethinnest/narrowestthinnest/narrowest tape was stacked on configurationconfiguration top of each other toto aa resulting thicknessthickness in of totalof 0.780.78 thickness mmmm andand of 0.26 aa widthwidth mm, 0.52 ofof 12 mm,12 mmmm (T0.78W12)thinnest/narrowest(T0.78W12)thinnest/narrowest(T0.78W12) for forfor the thethe thickest/widest thickest/widestconfigurationthickest/widestconfiguration tape to tapetotape aa configuration thickness configurationconfigurationthickness ofof (Table0.78 0.78(Table(Table mmmm 2). 2).2). and and aa widthwidth ofof 1212 mmmm and(T0.78W12)(T0.78W12) 0.78 mm. for for The the the width thickest/widest thickest/widest of the tapes tape tape varied configuration configuration up to a proportion (Table (Table 2). 2). of 100%, 75%, or 50% (T0.78W12)(T0.78W12)InInIn order orderorder for forto toto therefer the referrefer thickest/widest thickest/widest to toto the thethe stress stressstress state state statetape tape of ofofconfiguration configurationth ththe eCFRP CFRP tape, tape, (Table (Table the the introduced 2).introduced 2). global global coordinate coordinate of the sideInIn order order length to to referof referac to= to 12the the mm, stress stress respectively. state state of of th thee CFRP ThisCFRP resultedtape, tape, the the in introduced introduced nine different global global geometric coordinate coordinate systemsystemsystemInIn (seeorder order(see(see Figure Figure Figureto to refer refer 1b) 1b)1b) to towith withthewith the stressaxes stress axesaxes 1, state1, 1, state2, 2,2, and andandof of th3 th 33ise eis is CFRPaligned CFRP alignedaligned tape, tape, according accordingaccording the the introduced introduced to toto the thethe orthotropic orthotropicorthotropic global global coordinate coordinate coordi- coordi-coordi- configurations,systemsystem (see (see Figure Figure ranging 1b) 1b) fromwith with aaxes axes thickness 1, 1, 2, 2, and and of 3 0.26 3 is is aligned aligned mm and according according a width ofto to 6the the mm orthotropic orthotropic (T0.26W6) coordi- coordi- for natesystemnatesystem system system (see (see Figureof Figureof a arepresentative representative 1b) 1b) with with axes axes volume volume1, 1, 2, 2, and and elem elem 3 3 is entis entaligned aligned of of a aunidirectional unidirectionalaccording according to to thelaminate, thelaminate, orthotropic orthotropic where where coordi- coordi-the the 1- 1- thenatenate thinnest/narrowest system system of of a a representative representative configuration volume volume to aelem elem thicknessentent of of a ofa unidirectional unidirectional 0.78 mm and laminate, laminate, a width where of where 12 mm the the 1- 1- directionnatedirectionnate system system corresponds corresponds of of a a representative representative to to the the ‖ ‖-direction volume-direction-direction volume elem elem(fiber (fiber(fiberentent direction) ofdirection)direction) of a a unidirectional unidirectional and andand the thethe 2- laminate,2-2- laminate,respectively respectivelyrespectively where where 3-direc- 3-direc-3-direc- the the 1- 1- (T0.78W12)directiondirection forcorresponds corresponds the thickest/widest to to the the ‖ ‖-direction-direction tape configuration (fiber (fiber direction) direction) (Table and 2and). the the 2- 2- respectively respectively 3-direc- 3-direc- directiondirection corresponds corresponds to ⊥to⊥ the the ‖ ‖-direction-direction (fiber (fiber direction) direction) and and the the 2- 2- respectively respectively 3-direc- 3-direc- tiontiontion corresponds correspondscorresponds to toto the thethe ⊥ ⊥-direction-direction-direction (transverse (transverse(transverse to toto fiber fiberfiber direction) direction)direction) [32]. [32].[32]. tiontion corresponds corresponds to to the the ⊥ ⊥-direction-direction (transverse (transverse to to fiber fiber direction) direction) [32]. [32]. Table 2. Analyzed tape configurations and the corresponding geometric dimensions. TableTable 2. 2. Analyzed Analyzed tape tape configurations configurations and and th the eecorresponding correspondingcorresponding geometric geometricgeometric dimensions. dimensions.dimensions. Table 2. Analyzed tape configurations and the corresponding geometric dimensions. TableTable 2. 2. Analyzed Analyzed tape tape configurations configurations and and th thee corresponding corresponding geometric geometric dimensions. dimensions. Tape Width wt [mm] TapeTape Width Width 𝒘 𝒘 [mm [mm[mm] ]]

TapeTape Width Width 𝒘 𝒘𝒕 𝒕𝒕 [mm[mm] ] 𝒕 Tape Width 𝒘𝒕 [mm] TapeTape Width Width 𝒘𝒘 𝒕 [mm[mm] ] 𝒕 12𝒕 9 6 1212 𝒕 9 9 6 6

1212 99 66

12 9 6 1212 99 66

0.780.780.78 0.780.78 [mm] [mm] [mm] 0.78 0.78 [mm] [mm]

[mm] 𝑡 𝑡 𝑡 [mm] [mm] [mm] [mm] 𝑡 𝑡 t 𝑡

t 𝑡 𝑡 𝑡

0.520.52 0.520.520.52 0.520.52

0.260.26 Tape thickness Tape thickness Tape thickness 0.260.26 Tape thickness Tape thickness 0.26 Tape thickness Tape thickness 0.260.260.26

Tape thickness Tape thickness Tape thickness 2.2.2.2. Material Material and and Failure Failure Behavior Behavior of of Polymer Polymer and and Composite Composite Structures Structures 2.2.2.2. Material Material and and Failure Failure Behavior Behavior of of Polymer Polymer and and Composite Composite Structures Structures InThe orderThe core core to referstructure structure to the consists consists stress state of of two two of thecomponents components CFRP tape, (TPU (TPU the introducedfilm, film, CFRP CFRP globaltape). tape). coordinateThe The films films inin- TheThe core core structure structure consists consists of of two two components components (TPU (TPU film, film, CFRP CFRP tape). tape). The The films films inin- systemfluencedfluencedfluencedThe (seeThe the Figurecore thecorethe acoustic acousticacousticstructure structure1b) with liners linersliners consists axes consists damping dampingdamping 1, 2, of andof two two performance 3 performanceperformance is components components aligned according as asas (TPUdescribed (TPU describeddescribed film, to film, the in CFRPin in orthotropicCFRP the thethe studies studiestape).studies tape).coordinate [9,10,33]The The[9,10,33][9,10,33] films films and andandin- in- fluencedfluenced the the acoustic acoustic liners liners damping damping performance performance as as described described in in the the studies studies [9,10,33] [9,10,33] and and systemwerefluencedwerefluenced made of made a the the representative of acousticof acousticTPU. TPU. Since Sinceliners liners volumethe thedamping damping film film plays element plays performance performance a asubordinate subordinate of a unidirectional as as described described role role with with in laminate,in regard theregard the studies studies to to wherethe the [9,10,33] [9,10,33] stability stability the 1-and and of of werewere made made of of TPU. TPU. Since Since the the film film plays plays a a subordinate subordinate role role with with regard regard to to the the stability stability of of directionthewerethewerethe structure structurestructure made made corresponds of ofdue duedueTPU. TPU. to toto totheirSince Sincetheirtheir the significantly k the significantlysignificantlythe-direction film film plays plays (fiber lowe lowelowe a a subordinate direction)rsubordinate rrYoung’s Young’sYoung’s andmodulus modulusmodulusrole role the with with 2- (compare respectively (compare(compareregard regard to toTable TableTablethe the 3-direction stability 3),stability 3),3), an anan iso- iso- iso-of of thethe structure structure due due to to their their significantly significantly lowe lower rYoung’s Young’s modulus modulus (compare (compare Table Table 3), 3), an an iso- iso- correspondstropic,thetropic,thetropic, structure structure linear linearlinear to thedueelastic dueelasticelastic⊥ to -directionto their material their materialmaterial significantly significantly (transverse behavior behaviorbehavior lowe lowewa towawas fiberr ss rassigned Young’s assignedYoung’sassigned direction) modulusto modulustoto reduce reducereduce [32]. (compare (comparethe thethe necessary necessarynecessary Table Table computa-3), computa-3),computa- an an iso- iso- tropic,tropic, linear linear elastic elastic material material behavior behavior wa wass assigned assigned to to reduce reduce the the necessary necessary computa- computa- tionaltropic,tionaltropic,tional resources resourceslinear resourceslinear elastic elastic for forfor solving solvingsolvingmaterial material the thethe behavior behaviormodel. model.model. wa wass assigned assigned to to reduce reduce the the necessary necessary computa- computa- tionaltional resources resources for for solving solving the the model. model. 2.2.tionaltional MaterialToTo resources resourcesincrease increase and Failure for the forthe solving stabilitysolving Behaviorstability the theand of and model. Polymermodel. thus thus the andthe critical critical Composite buckling buckling Structures load load of of the the core, core, CFRP CFRP tapes tapes ToTo increase increase the the stability stability and and thus thus the the critical critical buckling buckling load load of of the the core, core, CFRP CFRP tapes tapes areareThe integrated integratedToTo core increase increase structure in in eachthe theeach consistsstability stabilitycell cell as as a of and asupportand twosupport thus thus components structurethe structurethe critical critical (TPU(see (see buckling buckling Figure film,Figure CFRP 1).load load1). The The of tape).of tapesthe tapesthe Thecore, core, consist consist films CFRP CFRP of influ-of poly-tapes poly-tapes areare integrated integrated in in each each cell cell as as a a support support structure structure (see (see Figure Figure 1). 1). The The tapes tapes consist consist of of poly- poly- encedamideareamideare integrated the integrated 6 acoustic 6reinforced reinforced in in liners each each by by damping unidirectionalcell unidirectionalcell as as a a performancesupport support aligned aligned structure structure as ca ca describedrbonrbonrbon (see (see fibers. fibers.fibers.Figure Figure in theThis This This1). 1). studies materialThe Thematerialmaterial tapes tapes [9,10 has ,hashasconsist33 consist ]a aandatransversal transversaltransversal of wereof poly- poly- amideamide 6 6 reinforced reinforced by by unidirectional unidirectional aligned aligned ca carbonrbon fibers. fibers. This This material material has has a a transversal transversal madeisotropicamideisotropicamide of TPU. 6 6 reinforced materialreinforced material Sincethe behavior. behavior.by by film unidirectional unidirectional plays a subordinate aligned aligned roleca carbonrbon with fibers. fibers. regard This This to thematerial material stability has has of a a thetransversal transversal struc- isotropicisotropicisotropic material materialmaterial behavior. behavior.behavior. tureisotropicisotropic due to theirmaterial material significantly behavior. behavior. lower Young’s modulus (compare Table3), an isotropic, linear elasticTableTable material 3. 3. Material Material behavior properties properties was of assignedof investigated investigated to reduce components componentscomponents the necessary [34] [34][34] (p. (p.(p. 202), 202), computational202), [35]. [35].[35]. resources for TableTable 3. 3. Material Material properties properties of of investigated investigated components componentscomponents [34] [34][34] (p. (p.(p. 202), 202),202), [35]. [35].[35]. solvingTableTable the3. 3. Material Material model. properties properties of of investigated investigated components components [34] [34] (p. (p. 202), 202), [35]. [35]. PropertyProperty 1 11 SymbolSymbol UnitUnit QuantityQuantity To increase the stability1 1 and thus the critical buckling load of the core, CFRP tapes are PropertyProperty 1 1 SymbolSymbol UnitUnit QuantityQuantity integratedThermoplasticThermoplastic in each Polyurethane cell Polyurethane as a support structure (see Figure1). The tapes consist of polyamide ThermoplasticThermoplastic Polyurethane Polyurethane 6 reinforcedYoung’sYoung’s by unidirectional modulus modulus aligned carbon𝐸𝐸 fibers. This materialMPaMPa has a transversal16471647 isotropic Young’sYoung’s modulus modulus 𝐸𝐸 MPaMPa 16471647 materialYoung’s behavior.Young’sPoissonPoisson modulus modulus ratio ratio 𝜐𝜐 MPaMPa- -- 0.407816470.40780.40781647 PoissonPoisson ratio ratio 𝜐𝜐 - - 0.40780.4078 For thePoissonPoisson estimationDensityDensity ratio ratio of the failure behavior𝜌𝜐𝜌 of the compositeg/cm³g/cm³- - core structure,0.40781.1780.40781.178 the Puck DensityDensity 𝜌𝜌 g/cm³g/cm³ 1.1781.178 criterion [36]Density andDensity the Cuntze criterion [37𝜌]𝜌 were applied. Forg/cm³g/cm³ each criterion, the1.1781.178 relevant TensileTensile Strength Strength 𝑅𝑅 MPaMPa 48.8448.84 TensileTensile Strength Strength 𝑅𝑅 MPaMPa 48.8448.84 failureCarbonCarbon modeTensile Tensilefiber fiber was reinforced reinforcedStrength determinedStrength poly- poly- using the engineering stress andMPaMPa the relevant strengths48.8448.84 [32 ,38]. CarbonCarbon fiber fiber reinforced reinforced poly- poly- amideamide 6 6 amideamide 6 6 YoungsYoungs modulus modulus in in ∥ ∥ 𝐸𝐸∥ ∥∥ MPaMPa 103400103400 YoungsYoungs modulus modulus in in ∥∥ 𝐸𝐸∥ ∥ MPaMPa 103400103400 YoungsYoungs modulus modulus in in ∥∥ 𝐸𝐸∥ ∥ MPaMPa 103400103400

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Table 3. Material properties of investigated components [34] (p. 202), [35].

Property 1 Symbol Unit Quantity Thermoplastic Polyurethane Young’s modulus E MPa 1647 Poisson ratio υ - 0.4078 Density ρ g/cm3 1.178 Tensile Strength R MPa 48.84 Carbon ﬁber reinforced polyamide 6

Youngs modulus in k Ek MPa 103,400 Youngs modulus ⊥ E⊥ MPa 3600 Poisson ratio k/⊥ υk⊥ - 0.29 Poisson ratio ⊥/⊥ υ⊥⊥ - 0.37 Shear modulus k/⊥ Gk⊥ MPa 1810 Shear modulus matrix G⊥⊥ MPa 1193 Tensile strength k (+) MPa 1637 Rk ⊥ (+) Tensile strength R⊥ MPa 18.4 Compression strength k (−) MPa 324 Rk ⊥ (−) Compression strength R⊥ MPa 91 Shear strength k/⊥ Rk⊥ MPa 36.6 Shear strength ⊥/⊥ R⊥/⊥ MPa 19.5 Density ρ g/cm3 1.450 1 In ﬁber direction (k) and perpendicular to the ﬁber direction (⊥).

2.3. Modelling and Numerical Implementation of Honeycomb Core Buckling A ﬁnite element analysis (FEA) of the composite core structure under quasi-static compression was carried out using the simulation environment (Ansys Workbench 2021 R.2., Ansys, Canonsburg, PA, USA). A linear static-structural-mechanical analysis, a linear buckling analysis, and a nonlinear static-structural-mechanical analysis considering large deformations were performed in order to obtain the deformation behavior of the core. Preliminary investigations regarding the geometry, element selection, and boundary conditions determined the setup of the core modeling. The simulation with solid elements revealed that the normal stresses in the direction of lateral compression (x1-direction) are signiﬁcantly higher compared to the normal stresses perpendicular to them (x2-, and x3-direction). Since the cell walls and tapes are very thin, only very low shear stresses are generated in the tapes and honeycombs. Consequently, the use of shell elements instead of solid elements was advisable, since the stresses in the thickness direction were low. Thus, the simulation was performed using the thin shell element SHELL181 with four nodes (QUAD4) and linear basis functions. The element size of 0.5 mm was determined in a convergence study representing a reasonable compromise between computational effort and numerical accuracy. Depending on the tape width, the model consisted of 6480 to 7200 elements.

2.3.1. Geometry and Boundary Conditions

A periodic unit cell of the core was deﬁned (Figure2). The displacements ui were oriented in the direction of the global coordinates xi, and the displacements ui were oriented in the directions of the local coordinates xi. The unit cell contains two tapes aligned parallel to the double honeycomb walls and four adjacent bisected single honeycomb walls. The compression of the shell model was modeled in accordance to the ASTM-C365 [39] by applying a constant displacement in the x1-direction at the upper edges of the core. A total displacement of 0.02 mm was applied in the linear static analysis and a displacement of 0.5 mm was considered in the nonlinear deformation analysis, which corresponds to approximately 1.67% of the core height. J. Compos. Sci. 2021, 5, 116 6 of 14 J. Compos. Sci. 2021, 5, x FOR PEER REVIEW 6 of 14

FigureFigure 2. (a) 2. Selection(a) Selection of unit of unit cell cell for for buckling buckling analysis, analysis, (b)) unitunit cell cell with with applied applied loads loads and and boundary boundary conditions. conditions.

The displacement in all other directions (global x - and x -directions) was fixed. A The displacement in all other directions (global2 𝑥- and3 𝑥-directions) was fixed. A fixationfixation in inall all directions directions was was realizedat at the the lower lower edges edges of theof the core. core. Depending Depending on the on the desired manufacturing process, the tapes are either welded or glued to the honeycombs. desired manufacturing process, the tapes are either welded or glued to the honeycombs. Therefore, the connection between the adjacent shells of the model was set as a compound. Therefore,In order tothe represent connection the periodicity between ofthe the adjacent structure, shells boundary of the conditions model was can set be as introduced a compound. In atorder the sectionto represent planes ofthe the periodicity honeycomb of edges the structure, (in the shell boundary geometry inconditions the form of can section be intro- ducededges). at the For section the further planes numerical of the honeycomb analyses, symmetry edges (in constraints the shell perpendiculargeometry in the to the form of sectionsection edges). planes For of thethe free-standingfurther numerical cell walls anal wereyses, applied symmetry by fixing constraints the displacement perpendicular in to thethe sectionx3-direction planes andof the setting free-standing the displacement cell wa inlls eachwere transverse applied by direction fixing the as free. displacement This approach has proven to be useful in previous investigations with respect to the realistic in the 𝑥-direction and setting the displacement in each transverse direction as free. This approachrepresentation has proven of the structuralto be useful failure in previous [17,40,41]. investigations with respect to the realistic representation2.3.2. Implementation of the structural of Geometrical failure Imperfection [17,40,41]. The structure is predisposed to stability failure in the form of buckling due to the 2.3.2.geometry Implementation of the walls. of In Geometrical the Ansys FEA Imperfection software, eigenvalue buckling analysis was used toThe determine structure the bucklingis predisposed strength to of thestability core structure. failure in When the form the critical of buckling buckling due load to the geometrywas reached, of the the walls. structure In the started Ansys to FEA deform software, according eigenvalue to the first buckling buckling analysis mode. Thiswas used to determineled to a drop the in buckling the reaction strength force andof the ultimately core structure. to a stability When failure the critical of the structure.buckling load wasThis reached, analysis the is alsostructure known started as linear to bucklingdeform according analysis. However, to the first since buckling imperfections mode. This such as nonlinear material behavior, manufacturing tolerances or, in the case of fiber led to a drop in the reaction force and ultimately to a stability failure of the structure. This composites, fiber waviness, resin concentration or thickness variations occur in reality, the analysistheoretical is also buckling known strengths as linear are buckling rarely achieved analysis. in However, practice. With since the imperfections help of a linear such as nonlinearbuckling material analysis, behavior, the realistic manufacturing buckling behavior tolerances cannot or, be describedin the case because of fiber the composites, force fiberapplication waviness, point resin shifts concentration during deformation or thickness which variations is not taken occur into in account reality, in the lineartheoretical bucklinganalysis. strengths Therefore, are for rarely large displacementsachieved in practi that occurce. With during the buckling help of ofa thinlinear structures, buckling it analysis,is necessarythe realistic to perform buckling a geometric behavior nonlinear cannot analysis. be described A static because structural the analysis force was application first pointperformed shifts during where the deformation geometry and which initial is conditions not taken of ainto unit account load case in are the defined. linear In analysis. the Therefore,following for coupled large lineardisplacements eigenvalue that buckling occur analysis, during the buckling deformations of thin related structures, to the firstit is nec- buckling shape are calculated, which occurs at the critical buckling load of the structure essary to perform a geometric nonlinear analysis. A static structural analysis was first per- for the defined load case. Finally, a nonlinear analysis was carried out to determine the formed where the geometry and initial conditions of a unit load case are defined. In the following coupled linear eigenvalue buckling analysis, the deformations related to the first buckling shape are calculated, which occurs at the critical buckling load of the structure for the defined load case. Finally, a nonlinear analysis was carried out to determine the failure of the composite support structure. Therefore, the calculated deformations of

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𝑓 thefailure first buckling of the composite mode are support considered structure. as Therefore, initial imperfection, the calculated a deformationsscaling factor of thes is applied; a scaling factor of 𝑓 = 1 represents the deformation magnitudes at the critical buck- first buckling mode are considereds as initial imperfection, a scaling factor fs is applied; a lingscaling load of factor the offirstfs =mode. 1 represents To show the deformationthe effect of magnitudes 𝑓s on the stability at the critical behavior buckling of the load structure,of the firstnonlinear mode. analysis To show was theeffect performed of fs on fo ther different stability behaviorscaling factors of the structure,between 0.05 the and 2 fornonlinear the first analysis buckling was mode performed (Figure for 3). different Therefore, scaling a tape factors thickness between of 0.05 𝑡t and = 0.26 2 for mm the and a t widthfirst of buckling 𝑤t = 12 mode mm (Figure and the3). Therefore,option of afree tape boundaries thickness of weret = 0.26 chosen. mm and Additionally, a width of the wt = 12 mm and the option of free boundaries were chosen. Additionally, the stress-strain stress-strain curve of a geometric linear analysis without imperfection (𝑓s = 0) was deter- f mined.curve of a geometric linear analysis without imperfection ( s = 0) was determined.

FigureFigure 3. Effect 3. Effect of geometric of geometric imperfection imperfection on on stress/strain behaviorbehavior of of the the structure structure using using different different scaling scaling factors ffactorss for the 𝑓s for the firstﬁrst buckling buckling mode. mode.

It can be observed that the stress increase was initially lower with a larger f , since It can be observed that the stress increase was initially lower with a largers 𝑓s, since thethe buckling buckling and and thus thus the the instability instability ofof thethe structure structure was was already already further further progressed. progressed. Thus, Thus, less force had to be applied to compress the structure. The two curves correspond to less force had to be applied to compress the structure. The two curves correspond to the the lowest factor f = 0.1 and 0.05 ran up to the linear-elastic buckling point close to the 𝑓 s lowestgeometric-linear factor s = solution.0.1 and 0.05 An initiallyran up analogousto the linear-elastic behavior also buckling occurs in point the study close of to Jang the geometric-linearet al. [40], wheresolution. the bucklingAn initially behavior analogous of over behavior expanded also metallic occurs honeycomb in the study cores of Jang is et al. [40],investigated. where the Despite buckling the lower behavior degree of of over geometric expanded imperfection, metallic thehoneycomb two curves cores with theis investigated.lowest Despitefs reached the comparativelylower degree lowerof geometri stress maxima.c imperfection, For the presentthe two study, curves the with scaling the lowest factor𝑓s reached was set comparatively to fs = 0.1 according lower to stress the similarity maxima to. For the the force-displacement present study, curvethe scaling from fac- the study by Jang et al. [40]. tor was set to 𝑓s = 0.1 according to the similarity to the force-displacement curve from the study2.3.3. by Determination Jang et al. [40]. of the Fiber Reinforced Plastic Failure Behavior In order to evaluate and assess the failure behavior of the CFRP tapes, the failure 2.3.3. Determination of the Fiber Reinforced Plastic Failure Behavior criteria according to Puck and Cuntze were integrated into the Ansys postprocessor in theIn form order of anto APDLevaluate script and as assess described the by failur Lucase [behavior42] and adapted of the toCFRP the current tapes, model.the failure criteriaThe failure according criteria to accordingPuck and toCuntze Puck and were Cuntze integrated allow theinto differentiation the Ansys postprocessor of Fiber failure in the form(FF) of and an APDL inter-ﬁber script failure as described (IFF), the by characterization Lucas [42] and of theadapted failure to mode the current and a realistic model. The failureassessment criteriaof according the stress to state. Puck In and order Cuntze to identify allow the the highest differentiation stress state of occurringFiber failure in (FF) andthe inter-fiber structure, failure both the (IFF), top and the thecharacterization bottom surface of of the shellfailure are mode considered and a and realistic the one assess- with the higher stress level was evaluated. In addition, the failure criteria according to ment of the stress state. In order to identify the highest stress state occurring in the struc- Puck allow the predetermination of the fracture angle. The failure criterion according to ture, both the top and the bottom surface of the shell are considered and the one with the higher stress level was evaluated. In addition, the failure criteria according to Puck allow the predetermination of the fracture angle. The failure criterion according to Cuntze also offers the possibility to calculate a failure of the structure due to inter-fiber shear failure (IFF2).

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Cuntze also offers the possibility to calculate a failure of the structure due to inter-fiber shear failure (IFF2). TheThe APDL APDL script script acquires acquires the the material material properties properties of of the the tapes, tapes, the the occurring occurring global global shearshear and and normal normal stresses, stresses, as as well well as as the the global global strains strains from from the the input input and and output output values values of theof the FEA. FEA. In caseIn case of theof the Puck Puck failure failure criteria, criteria the, the script script transforms transforms the the global global stresses stresses into into thethe stresses stresses in in the the active active plane plane on on an an element-by-element element-by-element basis. basis. The The relevant relevant failure failure criteria crite- forria eachfor each element element are thenare then calculated calculated using using the inclination the inclination parameters parameters of CFRP of CFRP [36] and[36] theand stressesthe stresses and failureand failure resistances resistances in the in active the acti plane.ve plane. For the For Cuntze the Cuntze criterion, criterion, the invariants the invar- ofiants the of occurring the occurring stresses stresses are calculated are calculated first. Then,first. Then, using using the curve the curve fitting fitting parameters parameters for CFRPfor CFRP [37], [37], the effectivethe effective efforts efforts for the for relevant the relevant failure failure modes modes are calculated. are calculated.

3.3. ResultsResults andand DiscussionDiscussion 3.1. Resulting Deformation Behavior and Composite Failure 3.1. Resulting Deformation Behavior and Composite Failure In order to assess the effect of geometrical variations of the tape structure, the nine In order to assess the effect of geometrical variations of the tape structure, the nine configurations (see Table2) were analyzed. The stress-compression responses of the unit configurations (see Table 2) were analyzed. The stress-compression responses of the unit cell for the different tape configurations were determined considering large deflections and cell for the different tape configurations were determined considering large deflections geometric imperfections (see Figure4). In the process, the structure is compressed, causing and geometric imperfections (see Figure 4). In the process, the structure is compressed, both the rigid CFRP tape and the flexible TPU walls to buckle. causing both the rigid CFRP tape and the flexible TPU walls to buckle.

FigureFigure 4. 4.Stress-strain Stress-strain curves curves of of the the tape tape conﬁgurations configurations with with theoretical theoretical structural structural response. response.

TheThe failure failure of of the the structure, structure, represented represented by by black black markers, markers, occurs occurs at at a a displacement displacement rangerange of of 0.13% 0.13% to to 0.22% 0.22% of of the the core core height. height. This This corresponds corresponds to ato deformation a deformation in x in1-direction 𝑥-direc- oftion 0.39 ofmm 0.39 andmm 0.66and mm,0.66 mm, respectively. respectively. Compressive Compressive strengths strengths of the of corethe core of 0.36 of 0.36 MPa MPa to to 4.6 MPa are achieved. The red marks represent the point where the deviation from the

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4.6 MPa are achieved. The red marks represent the point where the deviation from the initialinitial linear linear increase of the stress/compressionstress/compression ratio ratio reaches reaches the the threshold threshold of 5%. At this point, the limit of linear elastic increase of the deformation is approximately reached.reached. InIn contrast contrast to to the the initial initial course, course, where where the the critical critical buckling buckling stresses stresses reach reach much higher values (see Figure 33),), thethe modeledmodeled configurationsconfigurations failfail atat anan earlierearlier stage.stage. Consequently,Consequently, thethe failure failure of of the structure is is a material fa failureilure occurring at at the the micro micro and meso levels, levels, ratherrather than than a a stability stability failure failure of of the the entire entire structure. structure. However, However, the formation formation of instabilities instabilities inin the the form form of of buckling buckling shapes shapes have have a significa a significantnt impact impact on the on thefailure failure behavior behavior of the of core the geometry.core geometry. The numerical analysis shows, that for a tape thickness of 0.26 mm, all configurations configurations already exhibit the second buckling shape at th thee point of failure (Figure5 5).). InIn contrast,contrast, thethe thickerthicker configurations configurations exhibit a deformation behavior similar to configurationconfiguration T0.78W12 (see Figure 55b),b), correspondingcorresponding toto theirtheir firstfirst bucklingbucklingshape. shape.

Figure 5. TotalTotal deformation deformation of of the the unit unit cell cell of ofthe the configurations conﬁgurations (a) (T0.26W12a) T0.26W12 and and (b) (T0.78W12b) T0.78W12 and and the stress the stress state state (𝑥- direction) of the tape at the point of failure of the configurations (c) T0.26W12 and (d) T0.78W12 at a deformation-scaling (x1-direction) of the tape at the point of failure of the conﬁgurations (c) T0.26W12 and (d) T0.78W12 at a deformation-scaling factor of 6.0.

The differences inin thethe deformation deformation behavior behavior and and buckling buckling shapes shapes of theof the configurations configura- tionsresult result in an alteredin an altered stress state.stress Instate. Figure In 5Fic,d,gure the 5c,d, fiber-parallel the fiber-parallel stresses instresses the x1 in-direction the 𝑥- directionof the tape of configurationsthe tape configurations T0.26W12 T0.26W12 and T0.78W12 and T0.78W12 at the point at ofthe failure point areof failure displayed. are displayed.The occurring The bucklingoccurring shapes buckling lead shapes to the lead induction to the ofinduction tensile and of tensile compressive and compressive stresses in stressesthe x2- and in thex1 -directions,𝑥- and 𝑥-directions, which reach which their maximumreach their in maximum the buckling in the center. buckling In addition, center. Inthe addition, buckling the induces buckling the shearinduces stresses the shearτ23, stressesτ31, and τ21.,τ Since, and these τ stresses. Since these are very stresses low, arethey very have low, only they a small have effect only a on small the failure effect on of thethe structure.failure of the structure. According to the applied Puck and Cuntze Cr Criteria,iteria, the different tape configurations configurations either fail by fiber fiber compression failure or by inter-fiberinter-fiber tensiontension failurefailure (see(see TableTable4 4).).

Table 4. Mode effort of the relevant failure criteria according to Puck and Cuntze as well as the corresponding structural stress and structural strain in the 𝑥-direction for the different tape configurations at the point of failure. Tape thickness [mm] 0.78 0.52 0.26 Tape width [mm] 12 9 6 12 9 6 12 9 6 Puck Fiber compression failure [-] 0.99 1.00 1.00 0.73 0.75 0.98 0.95 1.00 1.00 Inter-fiber tensile failure [-] 1.00 0.94 0.50 1.00 1.00 1.00 1.00 0.96 0.95 Inter-fiber compression failure [-] 0.26 0.22 0.14 0.25 0.21 0.19 0.38 0.30 0.24 Cuntze Inter-fiber tensile failure- IFF1 [-] 1.00 0.94 0.50 1.00 1.00 1.00 1.00 0.96 0.95 Inter-fiber shear failure- IFF2 [-] 0.02 0.01 0.02 0.02 0.01 0.04 0.07 0.08 0.18

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Table 4. Mode effort of the relevant failure criteria according to Puck and Cuntze as well as the corresponding structural

stress and structural strain in the x1-direction for the different tape conﬁgurations at the point of failure.

Tape thickness [mm] 0.78 0.52 0.26 Tape width [mm] 12 9 6 12 9 6 12 9 6 Puck Fiber compression failure [-] 0.99 1.00 1.00 0.73 0.75 0.98 0.95 1.00 1.00 Inter-fiber tensile failure [-] 1.00 0.94 0.50 1.00 1.00 1.00 1.00 0.96 0.95 Inter-fiber compression failure [-] 0.26 0.22 0.14 0.25 0.21 0.19 0.38 0.30 0.24 Cuntze J. Compos. Sci. 2021, 5, x FOR PEER REVIEW 10 of 14 Inter-fiber tensile failure- IFF1 [-] 1.00 0.94 0.50 1.00 1.00 1.00 1.00 0.96 0.95 Inter-fiber shear failure- IFF2 [-] 0.02 0.01 0.02 0.02 0.01 0.04 0.07 0.08 0.18 Inter-fiber compression failure- IFF3 [-] 0.19 0.17 0.09 0.17 0.16 0.07 0.11 0.11 0.11 Inter-fiber2 compression failure- IFF3 [-] 0.19 0.17 0.09 0.17 0.16 0.07 0.11 0.11 0.11 σ1 [N/mm ] 4.64 2.96 1.55 1.98 1.32 0.79 1.07 0.6 0.36 2 𝜎e1 [%][N/mm ] 0.19 4.64 0.182.96 0.161.55 0.131.98 0.131.32 0.79 0.17 1.07 0.21 0.6 0.20 0.36 0.22 ϵ[%] 0.19 0.18 0.16 0.13 0.13 0.17 0.21 0.20 0.22 3.2. Influence of Tape Configuration on Failure Behavior 3.2. Influence of Tape Configuration on Failure Behavior Fiber compressive failure occurs when the compressive stress in the fiber direction Fiber compressive failure occurs when the compressive(−) stress in the fiber direction exceeds the fiber-parallel compressive strength of R () = 324 MPa. In case of the config- exceeds the fiber-parallel compressive strength of 𝑅k = 324 MPa. In case of the configurations, the maximum compressive stress is reached∥ without exception on the surfaces ofurations, the inner the sides maximum of the bucklingcompressive centers. stress The is reached reason for without the matrix exception failure on is the due surfaces to the buckleof the inner induced sides tensile of the stresses buckling at the centers. surfaces The of reason the buckling for the centersmatrix (seefailure Figure is due6a). to The the appliedbuckle induced Cuntze criteriontensile stresses for inter-fiber at the surfaces tensile failure of the indicatesbuckling thecenter occurrences (see Figure of failure 6a). The at theapplied buckling Cuntze center, criterion which for is dueinter-fiber to the low tensile tensile failure strength indicates transverse the occurrence to the fiber of directionfailure at the(+) buckling center, which is due to the low tensile strength transverse to the fiber direc- Rk =() 18.4 MPa of the tapes (Figure6b). tion 𝑅∥ = 18.4 MPa of the tapes (Figure 6b).

Figure 6. (a) Plot of the distribution of 𝜎 stresses of the tape configuration T0.56W12 and (b) the Figure 6. (a) Plot of the distribution of σ stresses of the tape conﬁguration T0.56W12 and (b) the inter-fiber tensile failure mode (IFF1) for2 all elements of the tape at the point of failure. inter-ﬁber tensile failure mode (IFF1) for all elements of the tape at the point of failure.

TheThe tapetape configurations configurations that that have have a lower a lower width width to height to height ratio ratio usually usually fail in fail the in form the ofform fiber of compressionfiber compression failure failure instead instead of inter-fiber of inter-fiber tensile tensile failure failure due due to the to generationthe generation of of lower 𝜎2 normal stresses across the width. To display this effect, the 𝜎2 -normal lower σ2 normal stresses across the width. To display this effect, the σ2-normal stresses forstresses the three for the configurations three configurations T0.26W12, T0.26W12, T0.26W9, T0.26W9, and T0.26W6 and T0.26W6 at the point at the of failurepoint of were fail- comparedure were compared (Figure7). (Figure 7).

Figure 7. Comparison of the normal stress 𝜎 distribution of the three tape widths (a) 12 mm, (b) 9 mm and (c) 6 mm with a tape thickness of 0.26 mm at the point of failure.

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Inter-fiber compression failure- IFF3 [-] 0.19 0.17 0.09 0.17 0.16 0.07 0.11 0.11 0.11 2 𝜎 [N/mm ] 4.64 2.96 1.55 1.98 1.32 0.79 1.07 0.6 0.36 ϵ[%] 0.19 0.18 0.16 0.13 0.13 0.17 0.21 0.20 0.22

3.2. Influence of Tape Configuration on Failure Behavior Fiber compressive failure occurs when the compressive stress in the fiber direction () exceeds the fiber-parallel compressive strength of 𝑅∥ = 324 MPa. In case of the configurations, the maximum compressive stress is reached without exception on the surfaces of the inner sides of the buckling centers. The reason for the matrix failure is due to the buckle induced tensile stresses at the surfaces of the buckling centers (see Figure 6a). The applied Cuntze criterion for inter-fiber tensile failure indicates the occurrence of failure at the buckling center, which is due to the low tensile strength transverse to the fiber direc- () tion 𝑅∥ = 18.4 MPa of the tapes (Figure 6b).

Figure 6. (a) Plot of the distribution of 𝜎 stresses of the tape configuration T0.56W12 and (b) the inter-fiber tensile failure mode (IFF1) for all elements of the tape at the point of failure.

The tape configurations that have a lower width to height ratio usually fail in the form of fiber compression failure instead of inter-fiber tensile failure due to the generation J. Compos. Sci. 2021, 5, 116 of lower 𝜎2 normal stresses across the width. To display this effect, the 𝜎2 -normal11 of 14 stresses for the three configurations T0.26W12, T0.26W9, and T0.26W6 at the point of failure were compared (Figure 7).

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When looking at the failure modes (see Table 4), it is noticeable that the configuration with the thickness of 0.52 mm fails for all three widths due to inter-fiber tensile failure. In contrast, the thicker and thinner configurations fail only in the widest configuration due

to inter-fiber tensile failure caused by different buckling shapes. To illustrate the correla- Figure 7. Comparison of the normal stress 𝜎 distribution of the three tape widths (a) 12 mm, (b) 9 mm and (c) 6 mm with a Figure 7. Comparison of the normaltion, Figure stress σ82 showsdistribution the variation of the three of tape the widths stresses (a) 12𝜎 mm, versus (b) 9 the mm applied and (c) 6 normalized mm with a dis- tape thickness of 0.26 mm at the point of failure. tape thickness of 0.26 mm atplacement the point of failure.in the 𝑥-direction, for three configurations of the same tape width. It is clear that as the tape width decreases, the increase in stress becomes more progressive. WhenHowever, looking in atconfiguration the failure modes T0.26W9, (see Tablethe formation4), it is noticeable of the second that the buckling configuration shape withleads the to thicknessa flattened of linear 0.52 mm increase fails for in allstress. three Thus, widths the due T0.52W9 to inter-fiber configuration tensile failure.fails at de- In contrast, the thicker and thinner configurations fail only in the widest configuration due to formation of 0.13 % and utilizes only a compression strength of 75 % in the fiber direction, inter-fiber tensile failure caused by different buckling shapes. To illustrate the correlation, while the T0.26W9 configuration does not fail at a lower deformation but achieves a de- Figure8 shows the variation of the stresses σ versus the applied normalized displacement formation of 0.2 % exploiting the full strength2 in the fiber direction. Therefore, without in the x -direction, for three configurations of the same tape width. It is clear that as the the occurrence1 of the second buckling shape, the thin tape configurations would already tape width decreases, the increase in stress becomes more progressive. fail at a lower stress due to inter-fiber tensile failure.

Figure 8. Stresses transverse to fiber direction vs. normalized displacement in 𝑥-direction of the Figure 8. Stresses transverse to ﬁber direction vs. normalized displacement in x1-direction of the unitunit cell. cell.

3.3. However,Resulting Properties in configuration of the Composite T0.26W9, Core the formationStructure of the second buckling shape leads to a flattenedDepending linear on increase the chosen in stress. tape thickness Thus, the and T0.52W9 the tape configuration width the failsmaximum at deformation compres- ofsive 0.13% strengths and utilizes and area only densities a compression of the comp strengthosite honeycomb of 75% in the core fiber structure direction, resulted while (Fig- the ure 9). It can be observed that the thickest and widest configuration shows a 900% increase in the compressive strength of the structure with a 153% increase in weight compared to the thinnest and narrowest configuration. In most cases, the increase in compressive strength relative to the weight increase was nearly linear and two exceptions were the configurations T0.52W9 and T0.52W12. In contrast to the other tape configurations, these two configurations fail comparatively early due to inter-fiber failure. The advantages of the CFRP in the form of high fiber-parallel strengths are consequently not exploited by the configurations.

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T0.26W9 configuration does not fail at a lower deformation but achieves a deformation of 0.2% exploiting the full strength in the fiber direction. Therefore, without the occurrence of the second buckling shape, the thin tape configurations would already fail at a lower stress due to inter-fiber tensile failure.

3.3. Resulting Properties of the Composite Core Structure Depending on the chosen tape thickness and the tape width the maximum compressive strengths and area densities of the composite honeycomb core structure resulted (Figure9). It can be observed that the thickest and widest configuration shows a 900% increase in the compressive strength of the structure with a 153% increase in weight compared to the thinnest and narrowest configuration. In most cases, the increase in compressive strength relative to the weight increase was nearly linear and two exceptions were the configurations T0.52W9 and T0.52W12. In contrast to the other tape configurations, these two configurations fail comparatively early due to inter-fiber failure. The advantages of J. Compos. Sci. 2021, 5, x FOR PEER REVIEWthe CFRP in the form of high fiber-parallel strengths are consequently not exploited12 by of the 14

conﬁgurations.

Figure 9. ComparisonComparison of of calculated calculated composite composite failure failure of of the the nonlin nonlinearear analysis analysis and and the the critical critical buckling load load of of the linear analysis over the normalized area density.

InIn general, general, configurations configurations where where the the fiber fiber st strengthsrengths are are not not utilized utilized to to a a high high degree degree andand premature premature matrix matrix fracture fracture occurs should not be considered. The development of the bucklingbuckling shapes shapes is is not not critical, critical, as as long long as as this this does does not not lead to an excessive increase in stressesstresses transverse transverse to to the the fiber fiber direction. Th Thee buckling shapes shapes (here (here the second buckling shape)shape) can can even even have have a a positive, positive, stiffening stiffening effect that prevents inter-fiber inter-fiber fracture. ToTo prevent an excessive increase increase in in stresses stresses transverse transverse to to the the fi fiber,ber, various measures cancan be be taken, taken, including including increasing increasing the the tape tape thickness thickness as as well well as as the the adjustment adjustment of of the the fiber fiber layoutlayout and and matrix matrix material. material. The The resulting resulting effects effects of of these these structural structural adjustments adjustments on on the acousticacoustic performance performance of of the the liner liner must must be be considered considered and investigated.

4.4. Conclusions Conclusions AA finite ﬁnite element element analysis analysis of of the deformati deformationon behavior of hybrid honeycomb core structurestructure under under axial axial compression compression in in the the core core thickness thickness direction direction primarily primarily showed showed a a fiber ﬁber compressioncompression failure failure according according to to the Puck and Cuntze criteria. The resulting strengths werewere similar similar to to the the results results of of the the linear linear buck bucklingling analysis analysis of the ofthe core core and and depend depend on the on chosen cross-sectional dimensions of the fiber-reinforced structure. The mechanical response to compression of the core structures showed a failure behavior attributed to material failure at the micro and meso levels rather than stability failure of the entire structure. The variation of the tape width and thickness affected the failure behavior and resulting compression strength of the support structure. In this context, some of these tape configurations lead to the unfavorable occurrence of inter-fiber failure without exploiting the mechanical properties in the fiber direction. An optimized tape configuration offers the potential for the consideration of function integrative acoustic liners in future design processes.

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the chosen cross-sectional dimensions of the fiber-reinforced structure. The mechanical response to compression of the core structures showed a failure behavior attributed to material failure at the micro and meso levels rather than stability failure of the entire structure. The variation of the tape width and thickness affected the failure behavior and resulting compression strength of the support structure. In this context, some of these tape configurations lead to the unfavorable occurrence of inter-fiber failure without exploiting the mechanical properties in the fiber direction. An optimized tape configuration offers the potential for the consideration of function integrative acoustic liners in future design processes.

Author Contributions: Conceptualization, N.M., M.D., M.K. and M.N.; investigation, M.N., N.B., M.K. and T.W.; writing—original draft preparation, M.N. and N.B.; writing—review and editing, M.K., M.D., T.W., N.B. and M.N.; visualization, M.N., N.B. and M.K.; supervision, N.M., M.D.; project administration, M.D., M.N.; funding acquisition, N.M. and M.D. All authors have read and agreed to the published version of the manuscript. Funding: This study was financially supported by the German Research Foundation (DFG) within the framework of Hybsch “Bauweisenentwicklung und Technologiesynthese zur Fertigung zellularer Kunststoffhybridstrukturen für den Einsatz in Schalldämpfern”, project under grant agreement no. MO2231/2-1. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in the current study are available on request from the corresponding author. Acknowledgments: We thank our colleagues of the Institute of Fluid Mechanics and Engineering Acoustics of the Technische Universität Berlin, and the German Aerospace Center (DLR), Institute of Propulsion Technology, Department of Turbulence Research in Berlin for the valuable discussions regarding the application and requirements of the honeycomb core structure for the realization of novel acoustic liners. Conflicts of Interest: The authors declare no conflict of interest.

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