The Through-Thickness Compressive Strength of a Composite Sandwich Panel With a Hierarchical Square Honeycomb Sandwich Core

Transcription

1 F. Côté B. P. Russell V. S. Deshpnde N. A. Fleck Deprtment of Engineering, University of Cmridge, Cmridge CB2 1PZ, UK The Through-Thickness Compressive Strength of Composite Sndwich Pnel With Hierrchicl Squre Honeycom Sndwich Core Sndwich pnels with luminum lloy fce sheets nd hierrchicl composite squre honeycom core hve een mnufctured nd tested in out-of-plne compression. The prismtic direction of the squre honeycom is ligned with the norml of the overll sndwich pnel. The cell wlls of the honeycom comprise sndwich pltes mde from glss fier/epoxy composite fces nd polymethcrylimide fom core. Anlyticl models re presented for the compressive strength sed on three possile collpse mechnisms: elstic of the sndwich wlls of the honeycom, elstic, nd plstic of the fces of the honeycom. Finite element clcultions confirm the vlidity of the nlyticl expressions for the perfect structure, ut in order for the finite element simultions to chieve close greement with the mesured strengths it is necessry to include geometric imperfections in the simultions. Comprison of the compressive strength of the hierrchicl honeycoms with tht of monolithic composite cores shows sustntil increse in performnce y using the hierrchicl topology. DOI: 1.111/ Keywords: honeycoms, composite, strength, sndwich pnels 1 Introduction 1 The concept of thick nd light core sndwiched etween two strong thin fcings hs een used to produce stiff nd lightweight pnels since the 19th century 1. The underlying theory nd design methods for sndwich constructions re well known 2,3. For exmple, it is generlly recognized tht the stiffness nd strength of sndwich pnel under mcroscopic loding ending, end loding, nd torsion re sensitive to the thickness of the core nd to the in-plne properties of the fce sheets. Stochstic fom nd periodic lttice mterils re two rod clsses of core incresingly employed in sndwich construction 4. Closed cell foms ehve s continuous core nd provide continuous interfce for onding to the fces, ut they possess low specific stiffness nd strength. In contrst, lttice mterils hve high specific stiffness nd strength due to their high nodl connectivity. Three ctegories of periodic lttice mterils hve een developed: i prismtic cores, ii 3D trusses, nd iii honeycoms. In this study we shll explore the properties of hierrchicl sndwich core in the form of squre honeycom. Hexgonl honeycoms mde from Nomex or from luminum lloy re extensively employed in sndwich construction due to their high specific stiffness nd strength in out-of-plne compression nd in longitudinl sher. However, these hexgonl honeycoms hve low vlues of in-plne stiffness nd strength due to nodl connectivity of only 3 6. Squre honeycoms hve higher nodl connectivity of 4, nd therey hve enhnced inplne properties. Further, the out-of-plne properties of the squre 1 Trde nme of Dupont referring to n rmid pper impregnted with epoxy resin. Contriuted y the Applied Mechnics Division of ASME for puliction in the JOURNAL OF APPLIED MECHANICS. Mnuscript received August 9, 27; finl mnuscript received Jnury 11, 29; pulished online July 23, 29. Review conducted y Kenneth M. Liechti. honeycom re comprle to those of the hexgonl honeycoms, see, for exmple, recent mesurements of out-of-plne compressive strength 7 nd of longitudinl sher strength 8. It is of interest to explore sndwich pnel cores tht possess high stiffness nd strength yet hve low density. Honeycom cores mde from monolithic composite sheets hve higher specific stiffness nd strength thn mny metllic honeycoms. The out-of-plne compressive strength of sndwich pnels with composite or metllic honeycom core is limited y the elstic of the cell wlls when the core hs low reltive density.. Here, we shll explore strtegy for incresing the elstic strength of the cell wlls y employing sndwich construction for the cell wlls 9. The microstructure is hierrchicl in topology: The mcroscopic sndwich pnel comprises luminum lloy fce sheets nd squre honeycom core, with the cell wlls of the core mde from mesoscopic sndwich pnels see Fig. 1. These mesoscopic pnels comprise polymethcrylimide PMI fom sndwiched etween woven glss fier/epoxy composite fces. The outline of this pper is s follows. First, competing collpse mechnisms re considered for the out-of-plne compressive strength of sndwich pnel with squre honeycom core. Anlyticl expressions re tken from the literture for ech mechnism, nd the dominnt collpse mechnism is plotted on mp with geometry s xes. Second, n l investigtion is conducted to proe the ccurcy of the nlyticl predictions. Third, the mesurements nd nlyticl predictions re compred with three-dimensionl 3D finite element FE simultions, nd the role of geometric imperfection in dictting compressive strength is ssessed. Finlly, the mesured nd predicted compressive strengths re compred with tht of monolithic composite squre honeycoms. Journl of Applied Mechnics Copyright 29 y ASME NOVEMBER 29, Vol. 76 / Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see

2 () () l (c) plte A plte B x 3 h unit cell x 1 x 2 unit cell c Fig. 1 Sketch of squre honeycom core with the cell wlls of the core mde from mesoscopic sndwich pnels, schemtic of the unit cell nd geometricl prmeters employed in the nlyticl model nd finite element nlysis, nd c torsionl-xil mode of squre honeycom core t 2 Anlyticl Models for the Competing Collpse Modes 2.1 Geometry of Core nd the Imposed Loding. Consider squre honeycom core perfectly onded to rigid fce sheets. The core hs sndwich wlls with squre cells of side-length l nd height h, nd the prismtic direction of the honeycom core is ligned with the out-of-plne direction of the overll honeycom Fig. 1. The sndwich wlls comprise n isotropic fom of thickness c nd density c onded etween two orthotropic elstic fces, ech of thickness t nd density f. This representtion is pproprite for the mteril system studied herein: woven glss fier/epoxy fces nd PMI fom core. The mcroscopic sndwich pnel is sujected to n out-of-plne compressive stress ; consequently, ech cell wll is sujected to line lod P per unit length where P = l 1 2 while the effective density of the squre honeycom core is =2 l 2 t 2t + c f + c 2t + c c 2 l l 2 l Three collpse mechnisms re considered: i elstic of the squre honeycom, ii elstic fce of the mesoscopic sndwich pnels tht mke up the cell wlls of the squre honeycom, nd iii plstic of the composite fce sheets of the mesoscopic sndwich pnels. The out-of-plne compressive strength of the mcroscopic sndwich pnel is dictted y the collpse mechnism tht hs the lowest vlue of line lod P cr for ny given set of geometric nd mteril prmeters. The compressive strengths for mechnisms i nd ii ove re sensitive to the elstic moduli of the fom core in the mesoscopic sndwich pnels. We proceed y summrizing the elstic moduli for closed-cell fom. 2.2 Elstic Properties of Closed-Cell Foms. Gison nd Ashy 4 extensively reviewed the mechnicl properties of open- nd closed-cell foms. They showed tht Young s modulus E c nd the sher modulus G c of n isotropic, closed-cell fom re directly relted to the reltive density c / s, where s is the density of the prent, solid polymer, such tht nd 2 s E c E s 2 c + 1 c s 3 G c 3 8 E c 4 Here, E s is Young s modulus of the solid polymer nd is the volume frction of cell wll mteril contined within the cell edges. For the PMI fom investigted in Sec. 3 of the present study, we ssume tht E s =36 MP, s =12 kg m 3, nd = Elstic Buckling of the Squre Honeycom Core. A recent l study 7 on the compressive collpse of squre honeycoms mde from type 34 stinless steel suggests tht the elstic mode resemles torsionl-xil of squre tue i.e., locl of the wlls with uilt-in top nd ottom edges, s sketched in Fig. 1 c. Finite element simultions reported lter in the current study confirm this mode of elstic for the sndwich-wlled squre honeycom. This mode is modeled y the of single plte with fully clmped top nd ottom edges nd simply supported sides. Ericksen nd Mrch 1 nlyzed this reduced prolem for sndwich plte mde from orthotropic fces nd n orthotropic core, nd we mke direct use of their nlyticl results. The ifurction line lod reds P uck = K 2 D l 2 where K is coefficient s prescried y Ericksen nd Mrch 1 nd D is the ending stiffness of the sndwich plte. In our prolem, we consider sndwich plte mde from orthotropic composite fces of identicl thickness t. Upon neglecting the contriution from the fom core to the overll ending stiffness D we hve E 1f E 3f t c + t 2 D = f 31f 2 where E if nd ijf re Young s modulus nd Poisson s rtios of the orthotropic fces long the x i = x 1,x 3 directions, s defined in Fig. 1. For the composite fces considered elow Young s moduli re equl long directions x 1 nd x 3. The formul for the coefficient K is explicit ut lengthy nd is omitted here. They re functions of geometry c, t, l, nd h s defined in Fig. 1, of the enforced oundry conditions long the edges of the plte, of Young s modulus E c nd sher modulus G c for the isotropic core, s well s of the sher modulus, Young s moduli, nd Poisson rtios of the fces in the x 1 x 3 plne / Vol. 76, NOVEMBER 29 Trnsctions of the ASME Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see

3 Tle 1 Mesured properties of the woven glss fier/epoxy fces nd PMI fom core Mteril Density kg m 3 Compressive strength MP Young s modulus GP Sher modulus GP Poisson s rtio GFRP f =197 f =2 E 3f =32 G 31f =4 31f =.1 PMI fom c =7 1.4 E c =.89 G c =.38 c = Elstic Fce Wrinkling. The sndwich pltes in the wlls of the squre honeycom cn collpse y n lterntive mechnism to those considered y Ericksen nd Mrch 1 : elstic of the fces of the sndwich plte. This mode is locl instility ssocited with short wve of the fces of the mesoscopic sndwich plte nd occurs when compressive stress within the fces ttins criticl vlue, see, for exmple, Ref. 3. Assume tht the sndwich plte is loded long the x 3 -direction. Then, the criticl line lod P wrink in the sndwich plte depends on Young s modulus E 3f in the fce nd on the moduli E c,g c of the core such tht P wrink = 1.7t E 3f E c G c 1/3 7 This formul gives the ifurction stress for the perfect structure in the sence of geometric imperfection. It is pprecited tht the presence of imperfections knocks down this strength y fctor of out 2 for most prcticl sndwich structures Plstic Fce Micro. It is generlly recognized tht the composite fces of sndwich plte cn fil y plstic, prticulrly when the composite hs wvy fiers due to its woven construction, see Ref. 11 for detils on plstic. Here, we tret the strength f s n intrinsic mteril property. Consequently, the line lod in the x 3 -direction on the sndwich pltes of the squre honeycom reds P m =2t f 8 where we neglect ny contriution from the compression of the core. 2.6 Collpse Mechnism Mp. In the reminder of this study we shll limit our ttention to the prcticl cse of squre honeycoms with spect rtio h/ l=1, therey reducing the numer of independent geometric vriles. The regimes of dominnce of the collpse modes cn then e illustrted in collpse mechnism mp, with xes t/c nd l/c. In construction of the mp, it is ssumed tht the opertive collpse mode is the one ssocited with the lowest line lod for ny given geometry. A collpse mechnism mp hs een constructed for the mterils used in the l study, s listed in Tle 1. This mp is plotted in Fig. 2 ; it ssumes fom core of reltive density =6% s used in the s. In ddition, mp is shown in Fig. 2 for higher reltive density of fom core =2% for comprison purposes. Although such high density PMI foms exist, they re more expensive nd less ville thn the =6% nd so no s were conducted on the higher density fom. Contours of nondimensionl compressive strength P cr / P m hve een dded to the mps, upon treting the line lod s n intrinsic reference property of the sndwich pltes, independent of c,l,h. The collpse mechnism mp of Fig. 2 =6% displys two filure modes: Core domintes the mp with much smller regime of fce. The dominnce of core cn e trced to the fct tht this mechnism is sensitive to low vlue of sher modulus in the fom core. Now increse to 2%. The lod nd fce lods increse, nd replces fce s n ctive mechnism, see Fig. 2. The ccurcy of the filure mp given in Fig. 2 hs een prtilly ssessed y conducting series of s on the geometries shown in the mp. It ws not prcticl to cover much lrger regime of the mp due to mnufcturing nd testing constrints. These s re now reported. 3 Experimentl Investigtion A series of compressive tests ws performed on squre honeycoms with cell wlls mde from glss fier/epoxy composite fce sheets nd =6% fom core. 3.1 Mnufcturing Technique. The sndwich cores were mde from closed-cell PMI fom of reltive density 6% trde nme Rohcell 71 IG nd thickness c=.7 mm. This fom σ =.7 l / c 2 B 1 C σ =.3 σ =.4 σ =. A σ = t / c fce l / c 2 σ = 1 1 σ =.8 σ = t / c Fig. 2 Collpse mechnism mp nd contours of normlized strength ÆP cr /P m of the hierrchicl squre honeycom mde from elstic fce nd polymeric fom core of reltive density =.6 nd =.2. The six tested geometries re mrked on the mp. Journl of Applied Mechnics NOVEMBER 29, Vol. 76 / Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see

4 () l () (c) L x 1 x 3 x 2 Fig. 3 Photogrphs specimen with l=7 mm nd t=.6 mm demonstrting the mnufcturing technique: wterjet mchined pnel, the ssemling technique, nd c the finl product ws selected for its low density c =7 kg m 3 nd its cpcity to sustin pressure up to.3 MP t 12 C, llowing utoclve curing. The fces were mde from one to three stcked plies of eight hrness stin weve 7781 E-glss pre-impregnted 42 wt % of resin efore curing in toughened epoxy mtrix resin trde nme ACG MTM28/GF1. This prepreg ws selected for its low curing temperture of 12 C, high dmge tolernce, nd ility to ond directly to the PMI fom without dditionl dhesive. After curing, ech ply hs thickness of.2 mm, volume frction of % glss fiers, nd density of f =197 kg m 3. The mnufcturing sequence of the mcroscopic squre honeycom is s follows. i ii The PMI fom ws dried for 2hinfurnce t 1 C in order to reduce the moisture content. Lyers of prepreg were plced symmetriclly on ech side of the fom, nd the 9 deg orienttions of the lminte were ligned with the x 1 nd x 3 directions see Fig. 1. Air pockets etween ech composite lyer were removed y deulking the pnel t room temperture under vcuum for 3 min. The sndwich pnel ws then cured in n utoclve for 1 h t 12 C nd guge pressure of.2 MP, with the usul vcuum gging technique. iii The sndwich pnel ws cropped y wterjet cutting mchine into long strips of height h=l nd length L=4l +2 L, where L is smll overhng length set t mm, see Fig. 3. Cross-slots were wterjet cut into the strips. These slots were of spcing l nd of width c+2t+e, where e is clernce of.1 mm to provide sufficiently tight fit during honeycom ssemly. iv Structurl dhesive trde nme 3M Scotch-Weld DP49 ws pplied over the slot res, nd the honeycom ws ssemled y slotting together the strips Fig. 3. Finlly, the honeycom ws onded to two 214-T3 luminum lloy fce sheets of thickness 3 mm using the sme structurl dhesive s for the core Fig. 3 c. A series of compression tests ws performed on the mcroscopic sndwich pnels. Ech specimen comprised 4 4 cells with cell size l vrying from 3 mm to 7 mm nd fce thickness t etween.2 mm nd.6 mm. Consequently, the core density vried from 38 kg m 3 to 167 kg m 3. The cell sizes nd densities of the tested specimens re listed in Tle Mteril Properties of the Composite-Fom Sndwich Pltes of the Squre Honeycom. The in-plne strength f, Young s modulus E 1f =E 3f, nd Poisson rtio 13 of Tle 2 Geometry, predicted compressive strength, nd mesured compressive strength of the mcroscopic sndwich pnels with squre honeycom core t mm l mm kg m 3 Predicted strength MP Wrinkling Micro Buckling First eigenvlue Mesured strength MP sher stress, MP sher compression engineering strin engineering strin Fig. 4 The mesured compressive nd sher responses of the eight hrness stin weve 7781 E-glss/epoxy composite nd the mesured unixil compressive response of the PMI fom Rohcell 71 IG used for the mesoscopic sndwich pnel / Vol. 76, NOVEMBER 29 Trnsctions of the ASME Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see

5 the composite fce sheet were mesured y compressing the 9 deg composite long the x 3 -direction within Celnese test rig 12 t nominl strin rte of 1 4 s 1. The Celnese tests were performed on specimens of width 1 mm, thickness 2 mm, nd guge length 8 mm. Strin guge rosettes were dhered to oth fces of the specimen nd confirmed tht the degree of mcroscopic ending in the tests ws negligile. The mesured compressive stress versus strin response is shown in Fig. 4 ; the mesured xil Young s modulus is E 3f =32 GP, the Poisson s rtio is 31f =.1, nd the strength is f =2 MP. These vlues re of the order expected from the specifiction given y the mnufcturer. The sher elstic properties of the composite fces were mesured vi unixil tensile test on 4 deg mteril 13. We denote the pplied tensile stress s x, nd the corresponding xil strin nd trnsverse strin s x nd y, respectively. Then, the sher stress versus sher strin reltion is otined vi the connections = x /2 nd = x y. The sher modulus G 31f follows immeditely s the initil slope of the versus response. These unixil tensile tests were conducted on specimens of width 2 mm, thickness.4 mm, nd guge length mm t nominl strin rte of 1 4 s 1. Strin guge rosettes were used on oth sides of the coupons, nd the mesured sher modulus ws G 31f =4. GP. The derived sher stress nd strin response is included in Fig. 4. The unixil compressive response of the Rohcell 71 IG fom core ws mesured y compressing cylindricl specimens of dimeter mm nd height 6 mm etween luricted flt pltens t n pplied nominl strin rte of 1 3 s 1. The nominl stress ws estimted from the lod cell of the test mchine while the pplied strin ws deduced from the reltive displcement of the top nd ottom pltens using lser extensometer. The mesured response is plotted in Fig. 4 nd displys n initil elstic response nd plteu strength of pproximtely 1. MP nd densifiction of the fom commences t nominl of.4. The elstic modulus of the fom ws determined y conducting n unloding/reloding cycle, s shown in Fig. 4. However, difficulties in dhering strin guges to the fom specimens men tht elstic Poisson s rtio ws unle to e extrcted from these compression tests. We mesured the elstic properties of the Rohcell 71 IG fom core using viro-coustic technique sed on Chldni s lw 14. For n isotropic mteril, the method consists of identifying the x-mode resonnce frequency f x nd ring mode resonnce frequency f o of squre pltes with free edges. This technique is simple nd requires sic equipment, s discussed in Ref. 14. The squre plte is simply supported y soft locks of fom nd excited y loudspeker mounted eneth it. The loudspeker is driven y sinusoidl wve genertor, nd the frequency is djusted to resonnce. Chldni s ptterns re reveled y sprinkling light powder on the surfce. Once the f x nd f o frequencies hve een mesured, oth Young s modulus E c nd Poisson s rtio c re determined vi the prescriptions c 1.48 f 2 2 o f x f 2 o + f x 2 E c.46 1 c 2 f o 2 + f x 2 c 4 /c where is the squre plte length, c is the plte density, nd c is the plte thickness. In these mesurements, Rohcell 71 IG fom squre plte of dimension =31 mm, c =7 kg m 3, nd c=.7 mm ws employed. The resonnce frequencies f x nd f o were found to e 61.9 Hz nd 68.6 Hz, respectively, leding to c =.16 nd E c =89 MP. These vlues re within few percent given y Eq. 3 nd the textook vlues E s =36 MP, s =12 kg m 3, nd = Mesured Compressive Response of the Mcroscopic Sndwich Pnel With Squre Honeycom Core. Compression tests were performed on the mcroscopic sndwich pnels using 1 kn servohydrulic frme test mchine nd n pplied mcroscopic nominl strin rte of 1 4 s 1. The mcroscopic compressive stress on the core ws inferred from the lod cell of the test mchine while the verge through-thickness compressive strin ws deduced from the reltive displcement of the top nd ottom fces of the sndwich pnel, upon mking used of two lser extensometers. The mesured out-of-plne compressive response of ech sndwich pnel is plotted in Fig.. All disply n initil elstic response, with pek lod t strin level.2 followed y ctstrophic drop in lod. Ech test ws terminted fter the drop in lod, nd the specimens were then visully exmined to determine the collpse mechnism. It ws not possile to use visul evidence to distinguish etween the three nticipted collpse modes of elstic, fce, nd plstic : In ll cses the composite fces of the squre honeycom cresed in ending, with no evidence of the initil cuse of the filure. The nlyticl predictions of compressive strength Eqs. 1,, 7, nd 8 for the three competing collpse modes hve een dded to Fig.. In most cses ll three nlyticl predictions re sustntilly higher thn the mesured strength. The tests reported in Figs. c nd d re the exceptions to this oservtion: In these cses the elstic strength is comprle to the mesured strength. Note tht these two geometries lie remote from the elstic /fce oundry, s plotted in Fig. 2. In order to improve the ccurcy of prediction, finite element simultions re performed, nd the imperfection sensitivity is ssessed. 4 Finite Element Modeling 4.1 Model Detils. Finite element clcultions of the through-thickness compressive response of the sndwich pnels with squre honeycom core were performed using the finite element pckge ABAQUS stndrd Hiitt, Krlsson & Sorensen, Inc. HKS. All simultions reported here re performed on the unit cell shown in Fig. 1 in the form of cruciform section, including the nonliner effects of lrge displcements. The unit cell of the squre honeycom ws modeled using liner 3D rick elements, i.e., C3D8R following ABAQUS nottion. Typiclly, the model comprised four rick elements through the thickness of ech composite fce sheet nd five to ten rick elements through the depth of the fom core. In order to void excessive distortion, the element spect rtio ws kept elow for ll simultions. One exception is tht the specimen with t =.2 mm nd l=6 mm ws modeled with n spect rtio of 1 in order to reduce the model size. This meshing rule ssures convergence ut leds to lrge models, for exmple, the model with t=.6 mm nd l=7 mm contins over 1 6 degrees of freedom. In order to cpture the torsionl-xil mode Fig. 1 c nd the fce mode, ntiperiodic oundry conditions were imposed on the outer plte edges of the cruciform section. All nodes on the top nd ottom of the unit cell were fully clmped. Loding ws specified y prescriing n incresing reltive verticl displcement etween the top nd ottom fces. The PMI fom ws modeled s n isotropic elstic-idelly plstic solid, using the properties vlues presented in Sec. 3. The post-yield ehvior ws modeled with the crushle fom formultion ville in ABAQUS using n ssocited flow rule. The rtio of the unixil compressive strength to the hydrosttic strength ws set to 3, nd the tensile nd compressive hydrosttic strengths were ssumed to e equl. With these ssumptions the crushle fom model in ABAQUS reduces to the Deshpnde nd Fleck 1 fom model with plstic Poisson s rtio of. Mild Journl of Applied Mechnics NOVEMBER 29, Vol. 76 / 614- Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see

6 c e FE, ζ =.3% FE, ζ =.3% nd nd FE, ζ =.3% d f FE, ζ =.3% FE, ζ =.3% FE, ζ =.3% nd nd Fig. The mesured compressive response of squre honeycom specimens with l=6 mm nd t=.2 mm, l=3 mm nd t=.2 mm, c l=7 mm nd t=.6 mm, d l=6 mm nd t=.6 mm, e l=4 mm nd t=.6 mm, nd f l=3 mm nd t=.6 mm. Also dded in the figure is the compressive strength predicted y nlyticl models, FE nlysis, nd FE sttic nlysis with n imperfection of ςæw/ l=.3%. volumetric hrdening of the plstic response of the fom ws included in order to ssure numericl stility. The woven glss fier/epoxy composite ws treted s n orthotropic solid with the in-plne mesured properties stted in Sec. 3. Consider the representtive sndwich plte A, s shown in Fig. 1. The properties in the direction norml to tht of the composite pltes were tken s E 2 =7. GP, G 12 =G 32 =4. GP, nd 12 = 32 =.1. In order to mimic plstic, it ws ssumed tht the composite ws modeled s n elstic-idelly plstic solid. J2 flow-theory ws dopted with yield strength equls to the strength f =2 MP nd mild isotropic hrdening to ssure numericl stility. Although this description is only pproximte it suffices to cpture the onset of for the in-plne unixil stress stte imposed on the composite lyers. 4.2 Elstic Buckling Anlysis. The FE method hs een used to determine the lowest lod the first eigenvlue. The compressive strength is compred with the nlyticl models nd the mesured response in Fig. nd in Tle 2. In generl, the predicted first eigenvlues re in good greement with the lowest collpse lod predicted nlyticlly y the three competing mechnisms. There is one exception: The nlyticl model underestimtes the compressive strength of the specimen with t =.6 mm nd l=3 mm. It is rgued tht this is due to the fct tht the nlyticl model is inccurte for stuy pltes, l/c 6. The predicted eigenmode resemles the ssumed mode of elstic ucking model nd resemles the fce mode, depending on the geometry considered. Consider, for exmple, the geometry s specified y t/c=.3 nd l/c=.2 nd lel this specimen A in Fig. 2. This specimen undergoes elstic fce ccording to oth the nlyticl model nd the FE simultion, see Fig. 6 for the oserved eigenmode. Second, consider specimens with t/ c=.1, nd l/ c=1. nd l/ c=.2, nd lel them B nd C in Fig. 2. The lowest FE eigenmode for these two specimens is given in Figs. 6 nd 6 c, respectively. Plinly the modes re elstic of the sndwich pltes, nd these modes re in greement with the modes predicted y the nlyticl model of Ericksen nd Mrch Imperfection Sensitivity. The discrepncy etween mesured compressive strength nd the extrcted eigenvlues indictes possile sensitivity to geometric imperfection. This sensitivity ws scoped y FE clcultions of the collpse response for sndwich pltes with initil imperfections in the shpe of the first elstic eigenmode. The mgnitude of the initil imperfection is / Vol. 76, NOVEMBER 29 Trnsctions of the ASME Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see

7 () () (c) Fig. 6 Finite element predictions of the first eigenmode extrcted for the specimen with l=3 mm nd t=.2 mm, x 1 l=7 mm nd t=.6 mm, nd c l=3 mm nd t=.6 mm t l c ζ=.3% ζ=.3% ζ=.8% ζ=.8% ζ=.8% ζ=1.7% ζ=1.7% ζ=.3% ζ=1.7% Fig. 7 FE predictions of the compressive response of specimens with l=3 mm nd t=.2 mm, l=7 mm nd t =.6 mm, nd c l=3 mm nd t=.6 mm. Imperfections in the first eigenmode with mximum mplitude ςæw/ l re specified. x 3 x 2 Fig. 8 Sketch of conventionl monolithic squre honeycom showing the geometricl prmeters long with the mnufcturing technique of the monolithic squre honeycom defined s follows. The mximum trnsverse deflection of the wlls of the honeycom is set to w= l, where is in the rnge of 1.7%. The FE predictions of the compressive stress versus strin responses of the geometries leled A, B, nd C in Fig. 2 re plotted in Fig. 7. These predictions correspond to the mesured responses plotted in Figs., d, nd f, respectively. The FE clcultions clerly show tht the pek stress is sensitive to the mgnitude of initil imperfection, nd n initil imperfection on the order of =.3% is needed to reproduce the qulittive shpe of the mesured collpse response. The finite element predictions of the pek compressive strength of the squre honeycoms with =.3% re included in Fig.. Mixed greement is otined with the mesured vlues. The discrepncy etween the mesurements nd FE predictions is lrgest for the specimens with l 6 mm. For these specimens it would e necessry to consider smller imperfection in order to otin the mesured vlue of pek lod. A quntifiction of the ctul imperfections within the mcroscopic sndwich pnel is difficult to chieve in view of the 3D nture of the structure. This is suggested s topic of future study using, for exmple, computed xil tomogrphy CAT scn mchine ut is not pursued further here. Comprison of the Mesured Compressive Strength for Squre Honeycoms With Sndwich Wlls nd With Monolithic Wlls It is instructive to explore the degree to which hierrchicl sndwich construction is dvntgeous over more conventionl honeycom. Here, we compre the through-thickness compressive strength of mcroscopic sndwich pnel with squre honeycom core mde from monolithic pltes with the compressive strength of mcroscopic sndwich pnel with squre honeycom core mde from sndwich pltes. An dditionl series of compressive tests hs een performed for squre honeycoms mde from monolithic, woven glss fier/epoxy composite. The rchitecture is identicl to tht reported ove for the fom-cored sndwich pltes, ut now the squre honeycom is constructed with the fom core sent. It is uncler priori whether the presence of the fom core elevtes or depresses the compressive strength: The presence of fom core rises the ending stiffness D in Eq. 6 ut it lowers the sher stiffness of the section. Monolithic squre honeycoms with 6 6 cells were mnufctured from eight hrness stin weve E-glss lminte of thickness t=.4 mm using micromilling mchining nd epoxy onding, see Fig. 8 for detils of the geometric prmeters nd sketch of the mnufcturing route. The cell spect rtio of the specimens ws mintined t h/l=1. The cell size l ws vried from 1.4 mm to Journl of Applied Mechnics NOVEMBER 29, Vol. 76 / h Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see

8 compressive strength, MP sndwich nlyticl monolithic nlyticl sndwich, expt. monolithic, expt compressive strength, MP density, kgm -3 density, kgm sndwich nlyticl l/c <4 monolithic nlyticl sndwich, expt. monolithic, expt. Fig. 9 Comprison etween the mesured nd predicted compressive strength of squre honeycoms mde from monolithic GFRP lmintes nd sndwich wlls with fce sheets of t =.6 mm nd t=.2 mm 31.3 mm in order to otin three core densities of kg m 3, 1 kg m 3, nd 1 kg m 3. Agin, luminum lloy fce sheets of thickness 3 mm were onded to the top nd ottom of the squre honeycom using the structurl dhesive 3M Scotch-Weld DP49. Three nominlly identicl specimens were tested for ech core density, nd the mesured strength vried y only 1% for given core density. A postmortem visul exmintion of the filed specimens suggested tht ech filed y elstic of the honeycom wlls. The verge compressive strengths re compred with those of the hierrchicl squre honeycoms of fce thickness t=.6 mm in Fig. 9. Likewise, the verge compressive strengths re compred with those of the hierrchicl squre honeycoms of fce thickness t=.2 mm in Fig. 9. The nlyticl predictions of elstic strength Eq. for the monolithic nd hierrchicl honeycoms hve een dded to Fig. 9. For the monolithic honeycom, the nlyticl model of Ericksen nd Mrch 1 is gin employed ut with E 1f E 3f t 3 D = f 31f 12 This prediction is in good greement with the mesured strengths. As lredy noted, ifurction clcultion of Ericksen nd Mrch 1 is nonconservtive for the hierrchicl honeycoms. We note from Fig. 9 tht there is smll dvntge in using the hierrchicl honeycom over the monolithic honeycom t sufficiently low core densities. But this comprison is for the cse of fom core of low modulus. As discussed in Sec. 2, dditionl structurl dvntges re expected y considering hierrchicl honeycom for the choice of stiffer fom core. 6 Concluding Remrks The hierrchicl sndwich pnel with squre honeycom core introduced in this study shows promise s it hs sustntilly higher through-thickness compressive strength thn n equivlent sndwich pnel with monolithic composite core. The compressive strength of the hierrchicl sndwich pnel is dictted y set of competing filure modes nd these re est illustrted in the form of collpse mp, with geometry s xes. Anlytic models re presented here for the competing collpse modes; these hve limited ccurcy due to the presence of geometric imperfections. Finite element simultions cn give improved ccurcy ut these require s input the level of imperfection. A chrcteriztion of the imperfections remins chllenging tsk. Acknowledgment The uthors re grteful to the Engineering nd Physicl Sciences Reserch Council, UK for finncil support. References 1 Firirn, W., 1849, An Account of the Construction of the Britnni nd Conwy Tuulr Bridges,Wele, London, UK. 2 Allen, H. G., 1969, Anlysis nd Design of Structurl Sndwich Pnels, Pergmon, Oxford, UK. 3 Zenkert, D., 199, An Introduction to Sndwich Construction, EMAS, London, UK. 4 Gison, L. J., nd Ashy, M. F., 1997, Cellulr Solids, Structure nd Properties, 2nd ed., Cmridge University Press, Cmridge, UK. Deshpnde, V. S., Fleck, N. A., nd Ashy, M. F., 21, Effective Properties of the Octet-Truss Lttice Mteril, J. Mech. Phys. Solids, 49, pp Zhng, J., nd Ashy, M. F., 1992, The Out-Of-Plne Properties of Honeycoms, Int. J. Mech. Sci., 34, pp Côté, F., Deshpnde, V. S., Fleck, N. A., nd Evns, A. G., 24, The Out- Of-Plne Compressive Behvior of Metllic Honeycoms, Mter. Sci. Eng., A, 38, pp Côté, F., Deshpnde, V. S., nd Fleck, N. A., 26, The Sher Response of Metllic Squre-Honeycoms, J. Mech. Mter. Struct., 1, pp Budinsky, B., 1999, On the Minimum Weights of Compression Structures, Int. J. Solids Struct., 36, pp Ericksen, W. S., nd Mrch, H. W., 198, Effects of Sher Deformtion in the Core of Flt Rectngulr Sndwich Pnel. Compressive Buckling of Sndwich Pnels Hving Dissimilr Fcings of Unequl Thickness, US Forest Product Lortory, Report No. 183-B. 11 Fleck, N. A., 1997, Compressive Filure of Fier Composites, Adv. Appl. Mech., 33, pp ASTM D341/D341M-3, Stndrd Test Method for Compressive Properties of Polymer Mtrix Composite Mterils With Unsupported Gge Section y Sher. 13 Khsh, U. A., 24, In-Plne Sher Properties of Cross-Ply Composite Lmintes With Different Off-Axis Angles, Compos. Struct., 6, pp McIntyre, M. E., nd Woodhouse, J., 1988, On Mesuring the Elstic nd Dmping Constnts of Orthotropic Sheet Mterils, Act Metll., 36, pp Deshpnde, V. S., nd Fleck, N. A. J., 2, Isotropic Constitutive Models for Metllic Foms, J. Mech. Phys. Solids, 48, pp / Vol. 76, NOVEMBER 29 Trnsctions of the ASME Downloded 27 Jul 29 to Redistriution suject to ASME license or copyright; see