Homogenization Model for the Folded Core Sandwich Plates under the Transverse Loading

Transcription

1 IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 50-30, ISSN (p): Vol. 07, Issue 08 (August. 07), V Homogeizatio Model for te Folded Core Sadwic lates uder te Trasverse Loadig Duog am Tuog Mi* ad Nguye Quag Hug Tai Nguye Uiversity of Tecology, Tai Nguye city, Vietam Correspodig autor: Duog am Tuog Mi Astract: - I te developmet of egieerig sciece, material egieerig plays a pioeerig role. I te use of material, te proper use of te structure also rigs to te great efficiecy. Composite materials, especially composite sadwic paels ave sow outstadig advatages i may fields i egieerig, terefore, te calculatio to determie teir mecaical eavior is very ecessary. Tere ave ee may autors suggestig metods for calculatig sadwices, ut wit large ad usymmetrical sadwic paels, tese metods are still very difficult. I tis paper, we preset a metod to uild a equivalet model (D) tat ca e used to determie mecaical eavior of sadwic paels ad replaces to origial model (3D). Tis work elps to sigificatly reduce te computatioal time as well as time to uild te geometry of model. To demostrate te performace, i tis work, we carry out te simulatio of folded core sadwic i case of trasverse loadig. Tis metod ca e used, of course, for ay type of folded core as well as may types of sadwic paels. Keywords: Folded core, Homogeizatio, Ortotropic plate, Sadwic paels, Trasverse sear Date of Sumissio: Date of acceptace: I. INTRODUCTION Nowadays, structural sadwic paels are widely used ad tey play a importat role i may idustrial areas ecause of teir good compromise etwee stiffess ad ligtess. Sadwic paels are ofte formed y aderig two ig-desity ti plates called face seets wit a low-desity core possessig less stregt ad stiffess. We ca otai various properties ad desired performaces, especially ig stregt to weigt ratio y varyig te core materials, core structures ad core tickess, or material of face-seets. May differet core sapes ave ee applied to te costructio of sadwic structures suc as folded, solid, foam, truss, ad we core. Folded core sadwic is oe of te most used materials to make partitios or roof ski costructio or automoile. Te folded core sadwic plate is produced y a maufacturig process, i wic tree or more layers are lamiated togeter. Te flat layers are called liers ad te folded cores are referred to as flutes (Fig. ). Te maufacturig process gives tree caracteristic directios: Te macie directio (MD), te cross directio (CD), ad te tickess directio (ZD). Fig.. Te model of sadwic plate wit folded core Several approaces to te modelig of tese sadwic paels are discussed, i wic fiite elemet metod is kow as a effective metod ad used popularly []. We we use te fiite elemet i te commercial software, a actual geometry of te model is represeted []; owever, tis metod ca e easily doe wit a tiy, ormal size ad symmetrical plates. I fact, te sizes of tese plates are usually large ad usymmetrical, tus uildig model will e difficult ad demad ig computer cost as well as log calculatio time. Several autors proposed to use a equivalet cotiuum istead of te core (oeycom core, for example), ad te comie it wit a 3D model [3] or a i-dimesioal plate model [4]. To determie its mecaical eavior, we ca always completely discretize te core ad facigs y sell elemets (3D sell modellig), ut te modellig will e extremely tedious ad time-cosumig. It is muc more advatageous to omogeize te sadwices i order to otai a equivalet ortotropic plate (D plate or sell modellig) [5]. 45 a g e

2 May omogeizatio models were otaied y aalytical, umerical ad experimetal metods [5-3]. Luo et al. [6] made a aalytic study o te edig stiffesses of corrugated oard. Nordstrad et al [7-9] preseted some omogeized properties of te corrugated oard y a aalytical metod. Tey ave studied te ucklig ad post-ucklig eaviors of a ortotropic plate icludig te trasverse sear effect. Aoura et al. [0] also developed a aalytical omogeizatio model ased o te teory of lamiated plate ad compared its results wit umerical ad experimetal results. Buaic et al. [] proposed a omogeizatio teory ased o te asymptotic expasio metod ad preseted te FE computatio of te effective eavior properties. Biacolii [] used a FE umerical approac for evaluatig te stiffess parameters. Amog te existig aalytical models, tere are still some questioale prolems suc as te eaviors uder te trasversal sear efforts ad te torsio momets. I tis paper, we preset a omogeizatio model to simulate te mecaical eaviors of folded core sadwic paels. Te omogeizatio is carried out y calculatig aalytically te gloal rigidities of te folded core sadwic ad te tis 3D structure is replaced y a equivalet omogeized D plate. Te simulatios i case of trasverse loadig of Aaqus-3D ad H-D model for te folded core sadwic will e studied i tis article. Tis D omogeizatio model is very fast ad as close results comparig to te 3D model usig te Aaqus sell elemets. Te compariso sow may outstadig advatages of proposed model suc as reduced time for modelig, time for calculatio We ca use tis model, of course, for oter core structures, types of load or may oter types of sadwic paels. II. MINDLIN S THEORY AND THEORY OF LAMINATED LATE Midli s teory ad teory of lamiated plates are ofte used i may prolems to calculate ad aalyze te eavior of composite structures. Te memrae forces, edig, torsioal momets, ad trasverse sear forces are otaied y itegratio of te costraits o te tickess. N x y x y N x x N ( x, y ) N d z y y () M x y x y M x x M ( x, y ) M z d z y y () T ( x, y ) y y z T x x z d z T (3) If we cosider a composite pael cosistig of several layers, te resultig forces defied aove may e comied i layers: N x x Q Q 0 k x x N d z Q Q 0 z d z y y y y k N k 0 0 Q x y x y 3 3 k x y x y m (4) M x x Q Q 0 k x x M z d z Q Q 0 z z d z y y y y k M k 0 0 Q x y x y 3 3 k x y x y m (5) k x x z x z T C 0 d z d z T 0 C (6) y y z k k k y z 46 a g e

3 After te itegratio alog te tickess, we otai te overall stiffess matrix tat liks te geeralized deformatios wit resultat forces: N x A A 0 B B x m N y A A 0 B B y m N xy 0 0 A 0 0 B x y m M x B B 0 D D x M y B B 0 D D y M xy 0 0 B 0 0 D x y T x F 0 x z T y F yz i wic k k k k k ij ij ij A Q Q t k k k k k k k k ij ij ij B Q Q t z k k k 3 3 t k k k k k k D ij Q Q t ij ij z 3 k k k k k k k ij ij ij F C C t k k Te law of eavior aove ca e writte i matrix form: N A B 0 m M B D 0 (9) T 0 0 F s were {N}, {T} ad {M} are te iteral forces ad momets; [A], [D], [B] ad [F] are te stiffess matrices related to te memrae forces, te edig-torsio momets, te edig-torsio-memrae couplig effects ad te trasverse sear forces respectively; {ε m } is te memrae strai vector, {κ} is te curvature vector ad {γ s } is te trasverse sear strai vector. III. HOMOGENIZATION MODEL FOR THE FOLDED CORE SANDWICH Te sadwices wit folded core are made of multilayer icludig face-seets ad core tat ave flutig cores ad te cavities etwee layers. To apply te calculatio metod usig te teory of plates, te matrix (9) otaied y te teory of lamiated plates sould e modified [0, ]. Cosiderig a folded core sadwic plate ad usig a,, ad c to represet te lower lier, te folded core ad te upper lier (Fig. ). To omogeize a folded core sadwic plate, we cosider a represetative volume elemet (RVE). We ote tat, tis RVE must e small eoug comparig to te size of te etire plate. Accordig to te structure of te plate, we take out a period legt of te core as a RVE. We calculate te average mecaical properties of RVE ad use tem to model tis 3D structure y a omogeeous D plate. However, te folded core as a vertical positio varies wit x directio so we cut it ito very little vertical slices (tickess dx) ad itegrate alog te tickess (or sum up te participats of tree layers) o eac slice. 3 (7) (8) 47 a g e

4 Fig.. Geometry of folded core plates Notig tat te mecaical properties of te core acieved y experimets are valid oly i its face, so we eed to calculate te local coordiate system. Oce te overall stiffess of eac slice is otaied y itegratig over te tickess of te plate, omogeizatio alog x will e performed to calculate te average stiffess of all slices i oe period [5]. A A x d x B B x d x ( ) ; ( ) 0 0 D D x d x F F x d x ( ) ; ( ) 0 0 (0) 3. Tractio ad edig stiffesses related to N x, M x, N y, M y Te vertical positio (z) of a groove portio (ds) is a fuctio of x ad a tickess over its vertical sectio is a fuctio of te agle of icliatio of te groove. Eq. (8) ca e write: a a t c c ij ij ij ij A Q t Q Q t c o s a a a t c c c ij ij ij ij B Q t z Q z Q t z c o s a a a a t t c c c c ij ij ij ij D Q t z t Q z Q t z t c o s c o s were a c t t a c a t c t a z ; z ; z x t x () 3. Trasverse sear stiffesses related to T y I lamiate teory, te sear stiffess relative to te sear force T y o a CD sectio is calculated y te sum of te layers. However, te CD sectio of te sadwic is ot a cotiuous medium ad te trasverse sear deformatio is ot costat or liear o te sectio, so te teory of lamiates is o loger valid. Te sear force T y o a CD sectio causes a couplig etwee edig ad trasverse sear. Tus, it is very difficult to directly determie te trasverse sear stiffess o a CD sectio relatig to T y. To avoid te couplig etwee edig ad trasverse sear, ad to otai "pure" sear, accordig to te reciprocity teorem, Nordstrad et al. [7] ave proposed a orizotal sear model i wic trasverse sear uder te force T y (force alog z ad per uit legt alog x) is replaced y sear over te tickess uder te force T (alog y) (Fig. 3). Te sear modulus tus otaied is equivalet to te trasverse sear modulus. Te deformatios due to te searig of te flat skis are muc less ta tose due to te searig of te groove, ad terefore egligile. We make equivalece etwee oe-alf period of te folded core sadwic (wit ot skis ot sow) (Fig. 3a) ad a solid wit a dimesio of / (Fig. 3). A pair of sear forces T exerted o te groove y te upper ad lower faces gives te slidig v. Te sear of te omogeeous solid ca e defied y: 48 a g e

5 T v G () zy zy 0.5 Fig. 3. Te equivalet model to determie sear stiffess o CD sectio Te sear stress i te 3D folded core (Fig. 3a) is equivalet to te sear i te flatteed groove (Fig. 3c). Tis gives us: T v 0.5 T l G G v = (3) t 0.5 l G t Sustitutig Eq. () i (3), we otai te sear modulus of te solid: * 4 t G G (4) zy l Tus, we ave te sear stiffess o CD sectio: * G 4 t F G. (5) zy l 3.3 Trasverse sear stiffesses related to T x I lamiate teory, te trasverse sear stiffess relative to te sear force T x o MD sectio is also calculated y te sum of te layers. However, it is also difficult to determie tis stiffess ecause of te couplig etwee te edig ad trasverse sear. Nordstrad et al. [7] proposed to replace te trasverse sear uder te force T x (o MD sectio ad alog z) y searig o te tickess uder te force T = T x alog x. I fact, tis prolem is ot really a prolem of sear of te tree layers; it is domiated y te tesio ad compressio of te folded core. Te deformatios due to te searig of te flat skis are also muc less ta tose due to te searig of te groove, ad terefore egligile. I tis equivalet model, te folded core sadwic is cosidered to e a omogeeous material sujected to a orizotal sear etwee te two faces, te orizotal sear modulus (equal to te trasverse sear modulus) is defied as follows (Fig. 4): G * zx zx F / L F (6) u / u L zx Fig. 4. Te equivalet model to determie sear stiffess o MD sectio 49 a g e

6 Te prolem returs to determie te F/u ratio aalytically or umerically. Tus, te stiffess of te trasverse sear MD depeds o te Youg's modulus of te groove istead of teir sear modulus, sice te core layer eaves like ti walls i tesio ad compressio, ulike te eavior of layers i lamiate teory. To determie te F/u ratio aalytically, we cosider a period legt of te folded core sadwic as sow i Fig. 5. We fix te lower face ad apply te force F o te upper face to give te slidig x. We divide te displacemet x ito two compoets u ad v. Tese displacemets ca e determie as follows: N l u c o s v s i E A N l u s i v c o s E A wit F N N N ; l ; si ; co s = ; A t. co s 4 l l were N is te axial iteral force i te core, l is te legt of oe-alf period of core, is te eigt of core, ad is te period of core, E is te Youg's modulus of te folded core. (7) Fig. 5. Te displacemet model i a period of te core It sould e ote tat te v compoet is muc less ta u compoet, ad terefore egligile. So, te u displacemet ca e determie y: F l u (8) E A Fially, te sear stiffess of te folded core sadwic o MD sectio is: * E t. F G. zx l IV. VALIDATION OF HOMOGENIZATION MODEL To validate our omogeizatio model (H-Model), a folded core sadwic pael wit te dimesio L = 60 mm ad B = 50 mm is used. We first discretize te tree layers of folded core plate y sell elemets S4R of Aaqus to otai te model Aaqus-3D. Te, we discretize te middle surface of folded core plate y sell elemets S4R of Aaqus comied wit our H-Model (usig "user's suroutie UGENS") to otai H-D model. Te compariso of te results allows us to evaluate te efficiecy ad accuracy of our omogeizatio model. Te calculatios ad comparisos are made o a folded core sadwic pael avig CD sectio illustrated i Fig., te geometrical parameters are: t a = 0. mm, t = 0.5 mm, t c = 0. mm, = 4 mm ad = 8 mm. Te properties of material are give i te Tale. Te rigidities of D equivalet plate are calculated as sow i Tale. (9) 50 a g e

7 Tale. Te material properties of tree layers formed folded core plate Layers E (Ma) E (Ma) G (Ma) a c Tale. Rigidities of te D equivalet plate A Rigidities A A D D D F F (N/mm) (N/mm) (N/mm) (N.mm) (N.mm) (N.mm) (N/mm) (N/mm) Value Fig.6 Simulatio of Aaqus-3D ad H-D Model i sear force o MD sectio Fig. 7 Simulatio of Aaqus-3D ad H-D Model i sear force o CD sectio Tale 3. Compariso etwee Aaqus-3D ad H-D model i edig Vertical force F 3 = 0N Aaqus-3D H-D Model Error (%) MD CD Displacemet U 3 (mm) CU time (s) times Displacemet U 3 (mm) CU time (s).. times We fix te plate at te left ed ad apply te vertical force (F = 0 N) at te rigt ed. I ot types of simulatios (Aaqus-3D model ad H-D model), a rigid plate is oded to te MD or CD sectio at te rigt ed of te folded core sadwic pael to etter apply forces. Te deformed sapes togeter wit te iso-values of displacemet of te pael uder trasverse loadig otaied y 3D sell Aaqus ad our H-D model simulatios are sow i Fig. 6. ad Fig. 7. Te comparisos of results otaied y te two models ad te percetages of error i H-D model compared to Aaqus-3D results for te case of trasverse loadig are preseted i Tale 3. We oserve tat te calculatios y our H-D model are very fast wile calculatios y Aaqus-3D are muc loger ( times) ad we ote tat te umerical results give y te two models are very close. 5 a g e

8 V. CONCLUSION I tis paper, we ave proposed a aalytic omogeizatio model for te edig ad trasverse sear prolems of a sadwic pael wit a folded core. Te compariso of te results otaied y te Aaqus 3D ad te Aaqus Uges H-D simulatios ave proved te validatio of te preset omogeizatio model for edig ad trasverse sear prolems. Te preset H-D model allows us to largely reduce ot oly te time for te geometry creatio ad FEM calculatio, ut also te computatioal ardware requiremets for te large sadwic paels. Tis omogeizatio model ca e used ot oly for corrugated cardoard plates, ut also for aval ad aeroautic composite structures. ACKNOWLEDGEMENTS Te autors are grateful to Tai Nguye Uiversity of Tecology TNUT, mecaical laoratory, for te elpful supports of laoratory equipmet as well as fruitful discussio we ad o tis suject. REFERENCES [] Noor AK, Burto WS, Bert CW, Computatioal models for sadwic paels ad sells, Applied Mecaics Reviews 55 (3), 995, [] Buaic N, Cartraud, Quesel T, Homogeizatio of corrugated core sadwic paels, Composite Structure, 59, 003, [3] Triplett MH, Scoerg W, Static ad dyamic fiite elemet aalysis of oeycom core sadwic structures, Structural Egieerig ad Mecaics, 6 (), 998, [4] Burto WS, Noor AK, Assessmet of cotiuum models for sadwic pael oeycom cores, Computer Metods i Applied Mecaics ad Egieerig, 45(3 4), 997, [5] Tali N, Batti A, Ayad R, Guo YQ, A aalytical omogeizatio model for fiite elemet modellig of corrugated cardoard, Composite Structure, 69, 005, [6] Luo S. Te edig stiffesses of corrugated oard. AMD-Vol. 45/MD-Vol. 36, Mecaics of cellulosic materials. ASME, 99, 5 6. [7] Nordstrad T, Carlsso LA, Alle HG, Trasverse sear stiffess of structural core sadwic, Composite Structure, 7, 994, [8] Carlsso LA, Nordstrad T, Westerlid B, O te elastic stiffess of corrugated core sadwic plate, Joural of Sadwic Structures ad Materials, 3, 00, [9] Nordstrad T, Aalysis ad testig of corrugated oard paels ito te post-ucklig regime, Composite Structure, 63, 004, [0] Aoura Z, Tali N, Allaoui S, Bezeggag ML, Elastic eaviour of corrugated cardoard: experimets ad modellig, Composite Structure, 63, 004, [] Buaic N, Cartraud, Quesel T, Homogeizatio of corrugated core sadwic paels. Composite Structure, 59, 003, [] Biacolii, Evaluatio of equivalet stiffess properties of corrugated oard, Composite Structure, 69, 005, IOSR Joural of Egieerig (IOSRJEN) is UGC approved Joural wit Sl. No. 484, Joural o Duog am Tuog Mi. Homogeizatio Model for te Folded Core Sadwic lates uder te Trasverse Loadig. IOSR Joural of Egieerig (IOSRJEN), vol. 7, o. 8, 07, pp a g e