Mechanics of Materials and Structures

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1 Journl of Mehnis of Merils nd Sruures A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES Renfu Li nd George Krdomes Volume 4 Nº 7-8 Sepember 29 mhemil sienes publishers

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3 JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vol. 4 No A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES RENFU LI AND GEORGE KARDOMATEAS This pper presens nonliner high-order heory for ylindril sndwih shells wih flexible ores exending previously presened high-order heory for sndwih ples. The ouer nd inner fes re ssumed o be relively hin ompred o he ore nd he effes from he ore ompressibiliy re ddressed in he soluion by inorporing he exended nonliner ore heory ino he onsiuive relions of he ylindril shells. The governing equions nd boundry ondiions for he ylindril shells re derived using vriionl priniple. Numeril resuls re presened for he ses where he wo fes nd he ore re mde of orhoropi merils. These resuls show h his model ould pure he nonlineriy in he rnsverse sress disribuion in he ore of he ylindril sndwih shell. Numeril resuls re presened on he deils of he sress nd displemen profiles for ylindril sndwih shell under lolied exernl pressure. This sudy ould hve signifine for he opiml design of dvned ylindril sndwih shells. 1. Inroduion Unique properies suh s high siffness/weigh nd srengh/weigh rios presen inresing promise for ppliions of ylindril sndwih shells in erospe nd mrine vehiles suh s irrf fuselge seions rokes nd submrine hulls. A ylindril sndwih shell onsiss of ouer nd inner siff hin fes mde eiher from homogeneous melli merils or omposie lmines sepred by hik ore of sof fom or honeyomb. In he nlysis of he sndwih onsruion i is rouinely ssumed h he fe shees rry he in-plne nd bending lodings nd he ore rnsmis he rnsverse norml nd sher lods [Plnem 1966; Vinson These lssil heories lso onsider he rnsverse displemen of he ore o be he sme s he displemens of he middle surfe of he wo fe shees. The vriion in hikness ompressibiliy of he ore is ofen negleed. However reen sudies show h he ore ould experiene signifin hnges in hikness [Ling e l. 27; Nem-Nsser e l. 27; Li e l. 28. As onsequene here is n inresing onern on he influene of ore ompressibiliy on he behvior of sndwih sruures. Effors o ddress his issue re demonsred hrough he formulion of vrious dvned high-order sndwih models in he lierure [Frosig e l. 1992; Pi nd Ploo 21; Hohe nd Libresu 23; Li nd Krdomes 28. Models onsidering he ore ompressibiliy my no only give more ure soluion o simpler problems bu my lso help o nlyilly ddress some oherwise diffiul problems suh s debond behvior [Li e l. 21 shok wve propgion nd energy bsorpion in sndwih sruures. In previous work we derived high-order sndwih ple heory [Li nd Krdomes 28 in whih he rnsverse displemen of he ore is no longer ssumed onsn bu i is fourh order funion Keywords: omposie sndwih shells ompressibiliy high-order heory exernl pressure. 1453

4 1454 RENFU LI AND GEORGE KARDOMATEAS of he rnsverse oordine. The in-plne displemens vry s fifh order funions of he rnsverse oordine. The urren pper presens n dpion of his nonliner high-order ore model o he onfigurion of ylindril sndwih shells. The derivion proedure of his heory is similr o he one in [Li nd Krdomes 28 bu ommoded o he speifi geomery of ylindril sndwih shells. In he developmen of he dvned ylindril sndwih shell model he following ssumpions hve been mde: 1 The fe shees sisfy he Kirhhoff Love ssumpions nd heir hiknesses re smll ompred wih he overll hikness of he sndwih seion. The rnsverse displemens in he fes do no vry hrough he hikness. In he urren pper he wo fe shees re onsidered o hve idenil hikness. 2 The ore is ompressible in he rnsverse direion h is is hikness my hnge. 3 The bonding beween he fe shees nd he ore is ssumed perfe. The pper is orgnied s follows: We firs exend he high-order sndwih ple ompressible ore heory o he ylindril sndwih shell. In he derivion he ylindril oordine sysem x s is inrodued nd loed he middle plne of he ore or he fe shees. The rnsverse displemen of he iniil mid-plne is onsidered s n unknown funion of he oordines x s. The xil irumferenil nd rnsverse displemens in he ore re hen expressed s funions in erms of he displemens of he wo fe shees nd he displemen of he ore iniil mid-plne. The displemen oninuiy ondiions long he inerfe beween he fe shee nd he ore re employed. We hen formule he governing equions boundry ondiions nd soluion proedure for ylindril sndwih shells. As represenive he equions for n orhoropi sndwih shell re sudied in deil. Nex he numeril resuls for ypil ylindril sndwih shell wih hree orhoropi phses wo fe shees nd ore re presened. Finlly we drw some onlusions nd suggesions on fuure work. 2. Exension of high-order sndwih ple heory o shells Le oordine sysem x s be loed he middle plne of he fe shees or he ore wih x in he xil direion s in he irumferenil direion nd in he ouwrd norml direion Figure 1 nd u v w be he orresponding displemens. R i nd R o re he rdii of he middle surfe of he inner nd ouer fe respeively; L is he shell lengh; he ouer nd inner fes re ssumed o hve n idenil hikness h f nd he ore hikness is h. Also se R = R o + R i /2. 2A. Displemen field represenion. In he lssil sndwih model he ompressibiliy of he ore in he hikness direion is ignored. This my give good pproximion in simple nd preliminry sudies. However in mny more demnding ses suh s sndwih sruure subje o bls/imp loding onsiderion of he rnsverse ompressibiliy of he ore my be needed. In he high-order ore heory proposed in [Li nd Krdomes 28 he rnsverse displemen in he ore -h /2 h /2 is in he form w x s = 1 22 h w 2 h 4 x s + h h 4 wx s + 43 h h 3 wx s 1

5 A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES 1455 s v w x u x h r L hf Figure 1. A ylindril sndwih shell. nd he in-plne displemens in he ore re in he form u x s = ux s h /2ūx s + h f wx x s h v x s = vx s h /2 vx s + h f wy x s h In hese equions w x s is he rnsverse displemen of he middle surfe of he ore; wx s is he verge of he displemens of op fe shee w x s nd boom fe shee w b x s; nd wx s is hlf of he differene of hese displemens. Similr definiions hold for he orresponding in-plne displemens. This high-order ore heory ould be exended o oher geomeri onfigurions suh s shpes wih urvure provided he hikness of he fe shees is smll ompred o he ol hikness of he sndwih sruure. In his work i will be exended o ylindril sndwih shells wih orhoropi phses. The hin fe shees of he shell sisfy he Kirhhoff Love ssumpions. Therefore seing h = h + h f /2 one hs for he displemens in he ouer fe h /2 + h f h /2 he expressions u x s = u x s + hwț x x s v x s = v x s + hwț s x s w x s = w x s nd for he nd displemens in he inner fe h /2 h /2 + h f 2 3 u b x s = u b x s hwb xx s v b x s = v b x s hwb sx s w b x s = w b x s. 4

6 1456 RENFU LI AND GEORGE KARDOMATEAS In order o ke he ore ompressibiliy ino oun nonliner models n be proposed. The one proposed here sisfies ll he displemen oninuiy ondiions long he inerfe beween he ore nd he fe shees s shown in [Li nd Krdomes 28. 2B. Srin-displemen relion. For hin fe shees one n obin he srin ensor poin in he ouer fe shee of he ylindri sndwih shell s [ɛ = ɛx ɛs = ɛx ɛs u x + + h[κ = vs + w /R o + + h[κ. 5 γ xs γ xs u s + v x A similr expression holds for he srin ensor in he inner fe [ɛ b = ɛx b ɛs b = ɛx b ɛs b u b x + h[κ b = vs b + wb /R i + h[κ b. 6 γ b xs γ b xs u b s + vb x In hese equions [κ b = κx b κs b κxs b = w b xx w b ss 2w b xs. 7 The ore is onsidered undergoing lrge roion wih smll displemens nd is in-plne srins ould be negleed. Therefore one n derive he srin-displemen relions of he ore from equions 1 nd 2 s follows: ɛ = 1 2h + 2 h 2 62 h h 4 w xs h h 4 w xs+ 2h + 2 h h h 4 w b xs γ = 2 ūx s + η 1 w ț x h x s + η 2wx x s + η 3wx b x s 8 γs = 2 vx s + η 1 w ț s h x s + η 2ws x s + η 3w b v sx s r in whih R h /2 r R + h /2 nd 1 η 1 = η 2 = 1 + h f h 1 η 3 = 2 + h f h 2 + h f + h h h f h 2 h h f h 1 + 3h f h f 4 h h f h h h 2 2 h 2 h h f h 3 h 4 3 h 3 h h f 4 h h h f 4 h h 4. 9

7 A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES C. Consiuive relion. The fe shees of he shell re mde of orhoropi lmined omposies nd he ore is lso orhoropi. The sress-srin relionship for ny lyer of he fes reds s σ x Q 11 Q 12 Q 16 ɛ x σ s = Q 12 Q 22 Q 26 ɛ s or [σ = [Q[ɛ 1 Q 16 Q 26 Q 66 γ xs τ xs where he Q i j for i j = re he redued siffness oeffiiens. The sress-srin relions for he orhoropi ore re wrien s σ = E ɛ τ = G γ τ s = G s γ s. 11 Here we define he resulns for he ouer fe shee of he sndwih shell by h [N /2 h = [σ /2 d = [M = Nx Ns Nxs Mx Ms Mxs = = h /2+h f h /2 h /2+h f h /2+h f [σ d = [B[ɛ + [D[κ [Q [ɛ d = [A[ɛ + [B[κ 12 in whih he siffness oeffiiens re defined s [Â i j ˆB i j ˆD i j = h /2 h /2+h f Q i j [1 + h + h2 d. 13 Applying similr proedure one n obin he expressions for he resulns in he inner fe shee. 3. Equilibrium equions nd boundry ondiions The ylindril sndwih shell is ssumed subje o exernl nd inernl pressure q b x s. Le U denoe he srin energy nd W he work of exernl fores. The vriionl priniple equivlen o virul displemen pproh ses h δu W = 14 in whih δu = L L δw = h /2 + + h /2 h f σ x δɛ x + σ s δɛ s + τ xs δγ xs R o + d h /2 h /2 h /2+h f h /2 σ δɛ + τ δγ + τ s δγ R + d q b x sδw b dsdx + s σx b δɛb x + σ s b δɛb s + τ xs b δγ xs b R i + d dθ dx L N x x sδu ds dx. 15

8 1458 RENFU LI AND GEORGE KARDOMATEAS We now inrodue he noion α = h f h nd β = h R. For he hin fe shees of R i + = R i R o + = R o nd β 1 one n obin he equilibrium equions nd boundry ondiions by subsiuing he sress srin relions 1 11 srin-displemen relions 5 9 nd he displemen represenion equions 1 4 ino 15 hen ino 14 nd employing inegrion by prs. For he ouer fe shee his resuls in he governing equions δu : N xx 1 R o N xθθ + G 4 β u ub ζ 1 β R wț x R w ox ζ 1 β R wb x = δv : N xθx 1 R o N θθ + G s ζ6 v ζ 7v b + ζ 8w ț θ + ζ 9w θ + ζ 1w b θ = δw : Mxxx + 2 Mxθxθ R + 1 o Ro 2 Mθθθ β Nx + E w + R o 21β For he ompressive ore: β δw : E 15β For he inner fe shee: β w 15β β wb + ζ 1 RG u x ub x + G s ζ 11 v θ ζ 11 vb θ ζ 2 R 2 G wț xx ζ 12 G s wț θθ ζ 3 R 2 G w xx ζ 13 G s w θθ ζ 4 R 2 G wb xx ζ 14 G s wb θθ Q ox θ =. w β w β + w b 15β G u x ub x + G s ζ 11 v θ ζ 11 vb θ ζ 3 R 2 G wț xx ζ 12 G s wț θθ δu b : N b xx 1 R i N b θθ G ζ 5 R 2 G w xx ζ 13 G s w θθ ζ 3 R 2 G wb xx ζ 14 G s wb θθ =. 4 β u ub ζ 1 β R wț x R w x ζ 1 β R wb x = δv b : N b xθx 1 R i N b θθ G s ζ7 v ζ 6 v b ζ 8 w ț θ ζ 9 w θ ζ 1 w b θ = δw b : M b xxx + 2 R i M b xθxθ + 1 R 2 i M b θθθ 1 R i N b x 53 + E 15β w β w 15β 61 23β + w b 21β + ζ 1 RG u x ub x + G s ζ b 11 v θ ζ b 11 vb θ ζ 4 R 2 G wț xx ζ b 12 G s wț θθ ζ 3 R 2 G w xx ζ b 13 G s w θθ ζ 2 R 2 G wb xx ζ b 14 G s wb θθ Q ix θ =. The onsns ζ i nd ζ b i for i = in hese equions re funions of β nd α nd re lised in he Appendix. The orresponding boundry ondiions x = L re

9 A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES 1459 u = ũ or N x = Ñ x w = w or N x wț x +M xx +N xθ wț y +2M xθx +G ζ 1 Ru b u +ζ 2 R 2 w ț x +ζ 2 R 2 w x +ζ 4 R 2 w b x = Q x w ț x = wț x or M x = M x w = w or u b = ũb or N b x = Ñ b x w b = w b or Rub u + ζ 3 R 2 w ț x + ζ 5 R 2 w x + ζ 3 R 2 w b x = Q N b x wb x +Mb xx +Nb xy wb y +2Mb xθx +G ζ 1 Ru b u +ζ 4 R 2 w ț x + ζ 3 R 2 w x + ζ 2 R 2 w b x = Q x w b x = wb x or Mb x = M b x where he supersrip denoes he known exernl boundry vlues. A θ = nd 2π oninuiy ondiions hold. For he sndwih shell mde ou of orhoropi merils he governing equions for he ouer fe shee n be rewrien s A 11 2 x 2 + A 66 R 2 o A 21 + A 66 2 u R o x θ + D θ 2 4G β A 66 x 4 +2 D D 66 R 2 o + A 11 R o u + A 12 R 2 o 2 u + A 12 + A 66 2 v R o x θ + ζ1 β RG + A 12 w ț x R o x 2 + A 22 R 2 o RG w x + 4G β ub + ζ 1 β G Rwb x = 16 w ț θ 2 A θ 2 ζ 6G s v + 22 Ro 2 ζ 8 G s ζ 9 G s w θ + ζ 7G s vb ζ 1G s wb θ = 17 4 x 2 θ 2 + D 22 4 Ro 4 θ βE ζ 2 R 2 G 21β x 2 ζ 12 G s θ 2 + A 12 Ro 2 x + ζ 1 RG βE x u ub + ζ 3 R 2 G 2 15β x 2 ζ 13 2 G s θ 2 w v θ + 53E G s θ ζ 11 v ζ 11 vb + 15β ζ 4 R 2 G 2 x 2 ζ 14 2 G s θ 2 Similrly he equion for he ore n be res s 2 2 w w b = Q o x θ βE 15 G u x + G s ζ 11 v θ + ζ 3 R 2 G 2 15β x 2 ζ 12 2 G s θ 2 w 716E + 15β ζ 5 R 2 G 2 x 2 ζ 13 2 G s θ 2 w G ub x ζ 11 G s vb θ βE + ζ 3 R 2 G 2 15β x 2 ζ 14 2 G s θ 2 w b =. 19

10 146 RENFU LI AND GEORGE KARDOMATEAS Finlly for he inner fe shee: A b 11 2 x 2 + Âb 66 R 2 i A b 21 + Ab 66 2 u b R i x θ + [ D b 11 2 θ 2 4G u b β + Ab 12 + Ab 66 2 v b ζ R i x θ + 1 β RG + Ab 12 wx b R i A b 66 2 x 2 + Ab 22 R 2 i 2 θ 2 ζ 6 G s RG w x + 4G β v b β 1 G s Ab 22 Ri 2 w b θ u + ζ 1 G β Rwț x = 2 ζ 9 G s w θ + ζ 7G s v ζ 8 G s wț θ = 21 4 x 4 +2 Db Db 66 4 Ri 2 x 2 θ 2 + Db 22 4 Ri 4 θ βE ζ 2 R 2 G 2 21β x 2 +ζ 14 b 2 G s θ 2 + ζ 1 RG x u ub + Ab 11 u b βE R o x + β 3 R 2 G 2 15β x 2 ζ 13 b 2 G s θ 2 + Ab 12 v b Ri 2 θ + 53E G s θ ζ 11 b v ζ 11 b vb + 15β ζ 4 R 2 G 2 x 2 ζ 12 b 2 G s θ 2 + Ab 12 w R 2 i w b w = Q i x θ. 22 I should be noed h sine his new ore heory is hree-dimensionl pproximion model for he ore bu more effiien hn omplee hree-dimensionl elsiiy pproh none of he exising shell heories ould produe idenil governing equions. 4. A ylindril sndwih shell under exernl pressure In his seion he soluion proedure for he response of sndwih shells will be demonsred hrough he sudy of simply suppored ylindril shell under exernl pressure. The boundry ondiions re w = w = w b = ; M x = Mb x = for x = L. nd v w w v b wb Myy nd Mb yy se in he form re oninuous θ = 2π. As suh he displemens n be MN u = m= n= MN v = w = m= n= MN m= n= U mn V mn W mn MN mπ x os L os nθ ub = m= n= MN mπ x sin L sin nθ vb = m= n= U b mn V b mn os mπ x L sin mπ x L os nθ sin nθ MN mπ x sin L os nθ wb = W b mπ x mn sin L os nθ w = m= n= MN m= n= W mn sin mπ x L 23 os nθ

11 A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES 1461 where Umn V mn W mn W mn U mn b V mn b nd W mn b re onsns o be deermined. The pplied exernl nd inernl loding Q o x θ nd Q i x θ n be respeively expressed in he form Q o x θ = MN m= n= ˆQ o mn MN mπ x sin L os nθ Q ix θ = m= n= ˆQ i mn sin mπ x L where θ 2π nd he oeffiiens re defined for m =... M nd n =... N s ˆQ o mn = 2 π L Q o x θ dx dθ ˆQ i mn = 2 π L os nθ 24 Q i x θ dx dθ. 25 Subsiuing equions ino he governing equions one n obin se of equions in mrix form: [K M N U mn = F mn 26 where he displemen veor U mn is defined s U mn = [U mn V mn W mn W mn U b mn V b mn W b mn T nd he loding veor F mn s [.. ˆQ o mn... ˆQ i mn T. The [K M N is 7 7 mrix whose enries re given on he nex pge. One he pplied loding is given he displemens n be found by solving 26 for eh pir m n unil he soluions in form of 23 onverge s m nd n inrese. Resuls for ylindril sndwih shell under lolied exernl pressure. Assume h onsn pressure loding is pplied on porion of he ouer fe shee: px θ = p x π 4 θ π 4. From equions 24 nd 25 one n obin he following loding in he rnsformed spe for m = : Q m = 2 mπ p sin 2 mπ 2 Q mn = 8 mnπ 2 p sin 2 mπ 2 sin nπ 4 n = The relionship for he Poisson s rio ν i j = ν ji E i /E j will be pplied sine he sndwih sruure onsiss of orhoropi phses. In he following sudy we se he rdius of he ore middle plne R =.8 m. Is ore hikness is h = β R wih β = 1/1. The hikness of wo fe shees is he sme h f = αh wih α = 1/2. The lengh of he sndwih shell is se s L = 1.5 m. The fe shees of his ylindril sndwih shell hve he following elsi onsns in GP: E f 1 = 4. E f 2 = 1. E f 3 = 1. G f 12 = 4.5 G f 23 = 3.5 G f f 31 = 4.5; Poisson s rios: ν12 =.65 ν f 31 =.26 ν f 23 =.4. The ore is mde of orhoropi honeyomb meril wih elsi onsns reding s in GP: E1 = E 2 =.32 E3 = E =.3 G 12 =.13 G 31 =.48 G 23 =.48; Poisson s rios: ν 12 = ν 31 = ν 32 =.25. In he ompuion of resuls M = 16 nd N= 1 in equions 23 is required for he numeril onvergene. The displemens re normlied by p h o /E f where h o is he ol hikness of he shell; he sress normlied by p in he following sudy. Figure 2 plos he normlied mid-plne displemens in he ouer fe shee ore nd inner fe shee s funion of x θ =. One n redily see h he displemens in he hree phses of he ylindril shell re no idenil implying h he urren heory n pure he ompressibiliy of he ore in he ylindril sndwih shells. I n lso be seen h he displemen differene in mgniude beween he ouer fe nd he ore mid-plne is

12 1462 RENFU LI AND GEORGE KARDOMATEAS 11 Â mπ 2 11 Â n 2 66 G R /α 12 Â 12 + Â 66 mπ o mπ 15 G 15 G /α β 1G R mπ 22 Â mπ 2 66 Â 22 n R o n R o 2 + β2 G s 23 Â 22 /R2 o + 2+α 2 α β 4G s n β 3 G s 27 β 4 G s n 31 Rβ 6G 33 ˆD 111 mπ 4 ˆD ˆD 166 mπ 2n 2 Ro 2 + ˆD 122 Ro 4 34 ˆD 211 mπ 4 ˆD ˆD 266 mπ 2n 2 Ro 2 + ˆD 222 Ro 4 35 Rβ 6 G mπ 37 ˆD 311 mπ 4 ˆD ˆD 366 mπ Ro 2 mπ n αE 21α n αE 15α 13 Â 12 /R o + β RG mπ 21 Â 21 + Â 66 mπ n R o αβ 5G s n 32 β 4 2+α 2 α G s n + β 7 G 2+α 36 β 4 2 α G s n 2n 2 + ˆD 322 Ro 4 n E 15 β 11G + β 9 G mπ mπ 2 + β8 G s n2 2 + β1 G s n2 mπ 2 β12 G s n mπ αE 15 G 42 G s 2 + αβ 5n 43 β 15α 9 G mπ 2 β1 G s n E 15α + β 13G mπ 2 β14 G s n mπ 15 G 46 2 αg s β 5n αE β 15α 15 G mπ 2 β16 G s n2 51 G /α β G R mπ mπ 15 G 55 Â b mπ 2 11 Â b n 2 66 G R /α 56 Â b 12 + Âb 56 mπ n i R i 57 β 1 RG mπ Âb 12 /R o β 3 G s n 63 β 4G s n 64 2 αβ 5G s n 65 Â 21 + Â 66 mπ n 66 Â b mπ 2 R 66 + Â b n 2 Â i 22 + β2 G b 22 s 67 R i Ri 2 2 α 2+α β 4G s n 71 Rβ 1 G mπ 72 β 4 G s n 73 ˆD 111 b mπ 4 ˆD b + 2 ˆD b 166 mπ 2n 2 + ˆD 122 b n E 15 β 11G mπ 2 β12 G s n2 R 2 i R 2 i 74 ˆD 211 b mπ 4 ˆD b + 2 ˆD b 266 mπ 77 ˆD 311 b mπ 2n 2 + ˆD b 222 R 4 i 75 Rβ 1 G mπ 4 ˆD b ˆD 366 b mπ 2n 2 + ˆD 322 b R 2 i R 4 i R 4 i n αE 15α 76 β 4 2 α 2+α G s n n αE 21α + β 15 G + β 17 G mπ 2 + β16 G s n2 mπ 2 + β18 G s n2 Tble 1. Mrix [K M N in 26. The number 12 inrodues he enry M = 1 N = 2 e.

13 A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES 1463 Normlied Trnsverse Displemens W E f1 /p h o Ouer fe Core middle plne Green Color: Ouer fe Red Color: Core middle plne Blue Color: Inner fe Inner fe x /L Figure 2. Mid-plne rnsverse displemen in he ouer fe shee ore nd inner fe shee s funion of x θ =. lrger hn h beween he ore mid-plne nd he inner fe shee. This observion demonsres h he rdil displemen in he ore is nonliner funion wih respe o he rdil oordine. Figure 3 presens he ross-seionl shpes of he ouer fe shee mid-plne u hrough x = L/6 L/4 nd L/2. The undeformed shpe is lso ploed s referene. I n be seen h he ross-seion deforms he mos from is originl shpe he middle of he ylindril shell in he xil direion x = L/2 in priulr wihin he region π/4 θ π/4 of eh ross seion where he loding is pplied..5 X = L/2 X = L/4. Originl shpe -.5 X = L/ Figure 3. The deformed ross-seionl shpe of he mid-plne in he ouer fe shee x = L/6 L/4 nd L/2 long wih he undeformed ross-seionl shpe.

14 1464 RENFU LI AND GEORGE KARDOMATEAS Figure 4. Vriion of rnsverse sress hrough he ore of he shell for vrious θ. We lso invesiged he rnsverse rdil sress disribuion in he ore of he sndwih shell one of he mos ineresing issues in sndwih sruurl sudies. The resuls re ploed in Figures 4 nd 5 where + denoes expnsion pressure nd ompressive pressure. Figure 4 shows he rnsverse sress for he ross-seion x = L/2 differen θ. We see h he sress vries wih θ from ompleely ompressive θ = o ompleely expnsive pressure θ = π. The mximum sress in mgniude hppens long he inerfe beween he ore nd he ouer fe shee on whih he loding is pplied. This mximum sress is ompressive. The mximum expnsion sress hppens θ = π/2 lso he inerfe beween he ore he ouer fe shee. This suggess h hese ould be he possible posiions for dmge iniiion useful knowledge for he opiml design of ylindril sndwih shells. The vriion of he rnsverse sresses θ = for differen ross-seions is presened in Figure 5. The resuls show h he mximum ompressive sress for eh ross-seion ours long he inerfe.5 = Figure 5. Cross-seionl shpe of he mid-plne of he ouer fe shee for vrious x.

15 A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES 1465 beween he ouer fe shee nd he ore. Anoher ineresing observion in his sudy is h he globl mximum ompressive sress of is found round x =.2L θ = no x =.5L θ = where he rnsverse ompressive sress is If one uses he vlue x =.5L θ = s he design rierion i ould yield 12% error. This pproximion my be epble in some preliminry designs. For n ure design one my hve o find ou he ex globl mximum ompressive nd expnsion sresses. Therefore he sudy in his work n provide useful guidelines for he design of dvned ylindril sndwih shells. 5. Conlusions We hve developed n nlyil soluion for ylindril sndwih shell wih flexible ore. A nonliner high order model for ylindril sndwih shells is formuled by exending our previous work on sndwih ples. The governing equions nd boundry ondiions hus derived hve he ompressibiliy of he ore inluded. The soluion proedure for n orhoropi sndwih ylindril shell is sudied in deil. Numeril resuls for exernl pressure loding exered on porion of he ouer fe shee re presened. The observions from he numeril resuls sugges he following onlusions: 1 The mid-plne displemens of he ouer fe shee he ore nd he inner fe shee re no idenil. 2 The rnsverse displemen disribuion in he ore hrough is hikness is nonliner funion of he rdil oordine. 3 The mximum sress in mgniude ours he inerfe beween he ore nd he fe shees on whih he loding is pplied. 4 The presen nonliner model is ble o pure he nonliner sress nd displemen profiles nd predi he globl mximum sress nd is loion. Therefore his sudy n hve signifine for he design of dvned ylindril sndwih shells. Aknowledgmens The finnil suppor of he Offie of Nvl Reserh Grn N nd he ineres nd enourgemen of he Grn Monior Dr. Y. D. S. Rjpkse re grefully knowledged. Appendix: Consns ppering in he governing equions pge 1458 When onsns re given ogeher sepred by omms he upper signs orrespond o he symbols before he omm nd he lower signs o he symbols fer he omm. ζ 1 ζ 1 = 8β + 3αβ ± 4β 2 ± 11αβ 2 /3 ζ 2 ζ 2 = 116β + 746αβ α 2 β ± 47β 2 ± 315αβ 2 ± 517α 2 β 2 /126 ζ 3 ζ 3 = 74β + 74αβ 766α 2 β ± 37β 2 286α 2 β 2 /126 ζ 4 = 22β 22αβ + 161α 2 β/126 ζ 6 ζ 6 = 1 4β 2 [ ± 16β β 2 log 2+β ζ 5 = 776β + 776αβ α 2 β/126 ζ 7 = 1 4β 2 [ 16 β 2 log 2+β

16 1466 RENFU LI AND GEORGE KARDOMATEAS ζ 8 ζ 11 = 1 62+ββ 5 [ 2β [ 224 6β +2β 2 5β 3 4β 4 +13β 5 +α912 18β +22β 2 +18β 3 216β β 5 15 [ 64 16β 4β 4 +β 5 +2α68 72β 36β 2 +18β 3 23β 4 +3β 5 log 2+β ζ 8 = ζ 11 = 1 62+ββ 5 [ 2β [ 224+6β +2β 2 +5β 3 4β 4 +3β 5 +α β +58β 2 +3β 3 96β 4 +47β β [ 32+8β 4β 2 +2β 3 +β 4 +2α 34+68β 28β 2 +11β 3 +3β 4 log 2+β ζ 9 ζ 11 = ± 1 3β 5 [ 8β [ 6 15β +2β 2 5β 3 11β 4 +α72 15β +18β 2 35β 3 11β 4 15 [ 32 8β +8β 2 2β 3 4β 4 +β 5 +2α192 4β +32β 2 6β 3 8β 4 +β 5 log 2+β ζ 9 = ζ 11 = 1 3β 5 [ 8β [ 6 15β 2β 2 5β 3 +11β 4 +α 72 15β 18β 2 35β 3 +11β [ 32+8β +8β 2 +2β 3 4β 4 β 5 +2α192+4β +32β 2 +6β 3 8β 4 β 5 log 2+β ζ 1 ζ b 11 = ± 1 6β 5 [ 2β [ β 2β 2 +5β 3 +4β 4 +3β 5 +α β 58β 2 +3β 3 +96β 4 +47β 5 ζ 1 = ζ b 11 = +15 [ 64 16β 4β 4 +β 5 +2α68 168β 12β 2 6β 3 17β 4 +3β 5 log 2+β 1 [ 6β 5 2β [ β 2β 2 5β 3 +4β 4 +13β 5 +α β 22β 2 +18β β β [ 32+24β +12β 2 +6β 3 +β 4 +2α34+188β +76β 2 +29β 3 +3β 4 log 2+β ζ 12 ζ b 14 = 1 422±β 2 β 8 [ 4β [ β β 4 +55β 6 +2β 8 ±23β 9 ζ b 12 ζ 14 = +7α β +16β β β 4 ±1576β 5 188β 6 ± 484β 7 97β 8 ± 91β 9 +α β 896β 2 112β β 4 ±98β 5 +64β 6 ±52β 7 3β 8 ±243β ±β 2[ 8 4β +2β 2 β α β +432β 2 248β 3 +78β 4 21β 5 +4β 6 +4α β +2872β β β 4 86β 5 +13β 6 log 2+β 1 [ 424 β 2 β 8 4β [ β 2 756β 4 55β 6 +26β 8 +7α β 2 222β 4 316β 6 +89β 8 + α β β 4 64β β 8 15β 2 4 [ β 2 44+β α β 2 +13β 4 +2β 6 +4α β β β 6 log 2+β

17 A HIGH-ORDER THEORY FOR CYLINDRICAL SANDWICH SHELLS WITH FLEXIBLE CORES 1467 ζ12 ζ } 13 1 [ ζ13 b ζ 14 = 212±ββ 8 2β [ β 2 136β 4 174β 6 +47β 8 +α 2 168±15456β +3188β 2 ±7336β β 4 ±3192β β 6 684β 7 +13β 8 +α24864±1344β +6272β 2 ±784β β 4 112β 5 235β 6 376β β 8 15 [ 4 β β 2 +β 4 +4α 2 8±736β +848β 2 ±288β 3 588β 4 18β 5 11β 6 7β 7 +4β 8 +α4736±256β +8β 2 ±128β 3 568β 4 16β 5 18β 6 8β 7 +9β 8 log 2+β ζ 13 = 1 15β 8 [ 4β [ β β β 6 +14α β 2 +28β 4 +81β 6 +α β 2 +54β β [ 8 2β 2 +β α β 2 146β 4 21β 6 +β 8 +4α β 2 48β 4 1β 6 +β 8 log 2+β Referenes [Frosig e l Y. Frosig M. Bruh O. Vilny nd I. Sheinmn High-order heory for sndwih-bem behvior wih rnsversely flexible ore J. Eng. Meh. ASCE 118: [Hohe nd Libresu 23 J. Hohe nd L. Libresu A nonliner heory for doubly urved nisoropi sndwih shells wih rnsversely ompressible ore In. J. Solids Sru. 4: [Li nd Krdomes 28 R. Li nd G. A. Krdomes Nonliner high-order ore heory for sndwih ples wih orhoropi phses AIAA J. 46: [Li e l. 21 R. Li Y. Frosig nd G. A. Krdomes Nonliner high-order response of imperfe sndwih bems wih delmined fes AIAA J. 39: [Li e l. 28 R. Li G. A. Krdomes nd G. J. Simises Nonliner response of shllow sndwih shell wih ompressible ore o bls loding J. Appl. Meh. ASME 75:6 28 #6123. [Ling e l. 27 Y. Ling A. V. Spusknyuk S. E. Flores D. R. Hyhurs J. W. Huhinson R. M. MMeeking nd A. G. Evns The response of melli sndwih pnels o wer bls J. Appl. Meh. ASME 74: [Nem-Nsser e l. 27 S. Nem-Nsser W. J. Kng J. D. MGee W.-G. Guo nd J. B. Iss Experimenl invesigion of energy-bsorpion hrerisis of omponens of sndwih sruures In. J. Imp Eng. 34: [Pi nd Ploo 21 P. F. Pi nd A. N. Ploo A higher-order sndwih ple heory ouning for 3-D sresses In. J. Solids Sru. 38: [Plnem 1966 F. J. Plnem Sndwih onsruion Wiley New York [Vinson 1999 J. R. Vinson The behvior of sndwih sruures of isoropi nd omposie merils Tehnomi Lneser PA Reeived 17 My 29. Revised 23 Jun 29. Aeped 9 Jul 29. RENFU LI: renfu.li@mil.hus.edu.n Deprmen of Aerospe Engineering Georgi Insiue of Tehnology Aln GA Unied Ses Curren ddress: Shool of Energy nd Power Engineering HuZhong Universiy of Siene nd Tehnology Wuhn 4374 Chin GEORGE KARDOMATEAS: george.krdomes@erospe.geh.edu Deprmen of Aerospe Engineering Georgi Insiue of Tehnology Aln GA Unied Ses

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