GENERAL THEORY FOR THE MODELLING OF LAMINATED MULTI-LAYER PLATES AND ITS APPLICATION FOR THE ANALYSIS OF SANDWICH PANELS WITH PLY DROPS

Transcription

1 GENERAL THEORY FOR THE MODELLING OF LAMINATED MULTI-LAYER PLATES AND ITS APPLICATION FOR THE ANALYSIS OF SANDWICH PANELS WITH PLY DROPS Ole Thybo Thomsen Insttute of Mechancal Engneerng, Aalborg Unversty Pontoppdanstraede 101, DK-9220 Aalborg East, Denmark SUMMARY: Mult-layer sandwch type structures can be dentfed n many structural assembles nvolvng composte materals. These mult-layer composte assembles are charactersed by havng nterchangng layers of hgh densty and hgh stffness separated by low densty and complant nterface layers. A characterstc feature of such mult-layer assembles s that severe and hghly localsed stress concentratons appear under certan condtons. The quanttatve understandng of these phenomena nvolves a detaled descrpton of the complcated mechancal nteracton of the stff layers through the complant nterface layers. The paper presents a hgh-order plate theory formulaton, whch nherently ncludes a descrpton of the global response as well as the local responses of the ndvdual layers of an arbtrary multple layer structural plate assembly. The features of the hgh-order mult-layer plate theory are demonstrated through analyss of the elastc response of a CFRP/honeycomb cored sandwch panel wth symmetrc eteror ply drops. KEYWORDS: mult-layer plates, sandwch type structures, stff sold lamnates, complant nterface layers, hgh-order plate theory, dscontnutes, termnatng ples, ply drops. INTRODUCTION Mult-layer sandwch type structures can be dentfed n many structural assembles nvolvng composte materals. Such mult-layer composte assembles are charactersed by havng nterchangng hgh stffness layers separated by low stffness nterface layers. The smplest eample of such a layered composte assembly s the classcal sandwch confguraton, where two stff face sheets are separated by a low stffness core materal. Another eample s a composte lamnate consstng of an arbtrary number of ples, where dstnct nterface layers can be dentfed. A characterstc feature of such mult-layer assembles s that severe and strongly localsed stress concentratons appear under certan condtons, where the stff layers nteract through the complant nterface layers. A well known eample of ths s the free edge

2 problem known from composte lamnates (N stff layers and (N-1) complant nterface layers). The understandng of these phenomena nvolves a detaled descrpton of the complcated nteracton between the stff layers through the complant nterface layers. There are several suggestons for the development of theores of varyng order for the analyss of mult-layer plates. The smplest approach s to use the classcal lamnate theory (CST) based on the Love-Krchhoff assumptons, as descrbed n [1]. Ths theory does not account for nterlamnar and transverse shearng effects, but these can be ncluded by adoptng hghorder theores, as dscussed by among others Reddy [2] and Savoa et al. [3], n whch hghorder dsplacement felds are assumed across the thckness of the lamnated assembles. A specal hgh-order formulaton for the analyss of symmetrc sandwch plates (3 layer assembly) s suggested by Lbrescu [4], ncludng the presence of transverse normal stresses n the core materal. None of the quoted references nclude the transverse fleblty, however, thus gnorng the effects assocated wth changes of the lamnate thckness. The present paper presents a hgh-order mult-layer plate theory, whch nherently ncludes a descrpton of the global response as well as the local responses of the ndvdual layers of an arbtrary mult-layer structural assembly. The theory ncludes the transverse fleblty of the nterface layers n the modellng. FORMULATION OF HIGH-ORDER MULTI-LAYER PLATE THEORY The hgh-order theory concept adopted n ths paper was orgnally ntroduced by Frostg et al. [5-7], for 3-layer sandwch beams, and by Thomsen and Rts [8], Thomsen [9] for 3-layer sandwch plates. The deas behnd these hgh-order sandwch theores are etended to multlayer plate confguratons as defned by Fg. 1. Fg. 1: Geometrcal defnton of mult-layer plate element consstng of N hgh stffness sold lamnates and (N-1) complant nterface/core layers. Basc Assumptons Wth reference to Fg. 1 the hgh-order plate theory ncludes the followng features: The mult-layer plate conssts of N hgh stffness and hgh strength layers separated by (N- 1) nterface layers of low stffness.

3 The N hgh stffness layers, denoted as sold lamnates, are modelled as generally orthotropc lamnated plates adoptng the classcal Love-Krchhoff assumptons. The (N-1) low stffness or complant nterface layers are modelled as a specal type of orthotropc sold, only possessng stffness n the out-of-plane drecton. Accordngly, the n-plane stresses are assumed to be nl n the nterface layers, whle transverse shear and transverse normal stresses are accounted for. Modellng of Sold Lamnates: For the purposes of the present paper, the general plate theory out-lned above s shown for the specal case of cylndrcal bendng. Wth reference to Fg. 1, cylndrcal bendng s defned as a state of deformaton where the fundamental varables are functons of the longtudnal coordnate alone. Consequently, the dsplacement feld s unform n the wdth drecton (y), and can be descrbed as follows: u = u ( ), v = v ( ), w = w ( ) (1) where u 0, v 0 and w are the mddle plane dsplacements n the, y and z drectons, respectvely, and refers to the th sold lamnate of the mult-layer plate (=1,...,N). As a consequence of Eqn 1, the followng s deduced: u = v = w = w = (2) 0, y 0, y, y, yy 0 where dfferentaton wth respect to the spatal coordnates s represented by a subscrpt comma. It should be noted that the concept of cylndrcal bendng s not unque, and that other defntons than the one used n the present formulaton can be adopted, Ref. 1. Accordng to the Love-Krchhoff assumptons, and by adoptng Eqn 2, the knematc relatons for the th sold lamnate (=1,...,N) are epressed as u = u0 + zβ, β = w,, v = v0 + zβ, β = w, = 0 (3) Substtuton of Eqn 2 nto the general consttutve relatons for a lamnated generally orthotropc plate yelds the consttutve relatons for the th sold lamnate (=1,...,N) n a state of cylndrcal bendng: y y N = A u + A v B w, M = B u + B v D w 11 0, 16 0, 11, 11 0, 16 0, 11, N = A u + A v B w, M = B u + B v D w yy 12 0, 26 0, 12, yy 12 0, 26 0, 12, N = A u + A v B w, M = B u + B v D w y 16 0, 66 0, 16, y 16 0, 66 0, 16, (4) where A jk, B jk, D jk (j,k=1,2,6) are the etensonal, couplng and bendng stffnesses of the th sold lamnate. N, N yy, N y are the n-plane stress resultants, and M, M yy, M y are the moment resultants of the th sold lamnate. Wth reference to Fg. 1 and Fg. 2, the equlbrum equatons for the th sold lamnate for the case of cylndrcal bendng (.e. all dervatves wth respect to y vansh) are gven by:

4 top N =, N =, Q = σ σ M 1 1 ( ) 1( bottom), z z y, zy zy, z z f = Q + = f { 1 z z } My Qy { 1, zy zy },, (5) where (top) z, zy are the shear stress components n the th nterface layer, σ z s the transverse normal stress component along the boundary between the th sold lamnate and -1 the th nterface layer (.e. the top boundary of the th nterface layer), σ (bottom) z s the transverse normal stress component along the boundary between the th sold lamnate and the (-1) th nterface layer (.e. the bottom nterface of the (-1) th nterface layer), f s the thckness of the th sold lamnate, and Q, Q y are the transverse shear stress resultants n the th sold lamnate. Fg. 2: Equlbrum condtons for the th sold lamnate. Modellng of Interface Layers As mentoned, the (N-1) complant nterface layers are modelled as a specal type of orthotropc lnear elastc sold, only possessng stffness n the out-of-plane drecton (z). Consequently, the stffness propertes of the th nterface layer are charactersed by the transverse Youngs modulus E cz and the two transverse shear modul G cz and G cyz. As a consequence of the assumed zero n-plane stffness the n-plane stresses are nl, and the stress feld n the th nterface layer s descrbed solely by the transverse normal stress component σ z and the two transverse shear stress components z, yz. Combnaton of the nterface layer equlbrum equatons wth the nterface layer knematc and consttutve relatons yelds a set of equatons, whch descrbes the complete nterface layer stress and dsplacement felds n terms of the transverse coordnate z (measured from the mddle plane of each nterface layer, see Fg. 1), and n terms of the sold lamnate dsplacement components. The response of the th nterface layer s coupled wth the responses of the surroundng sold lamnates (.e. sold lamnates (+1) and, see Fg. 1) by requrng contnuty of the dsplacements across the sold lamnate/nterface layer boundares. Ths mples that the sold lamnates and the nterface layers nteract through the transfer of surface tractons and shear stresses. For the specal case of cylndrcal bendng the resultng nterface layer feld equatons read:

5 E cz + 1 z (, y, z) = z ( ), yz (, y, z) = yz ( ), σ z (, y, z) = σ z (, z) = { w w } z, z c c = c = + w (, y, z) w (, z) w + 1 { w w } c c z ( c) z, z 2E cz w, z 3c w, z c z c uc (, y, z) = uc (, z) = u0 + f z + + z + + z 2 c 4 2 c 4 Gcz z ( c) z ( c) + + z, 2E cz yz c vc (, y, z) = vc (, z) = v0 + z G 2 cyz 2 (6) where uc, vc, wc are the n-plane and out-of-plane dsplacements, and c s the thckness of the th nterface layer. From Eqn 6 t s observed that z and yz are constant, that σ z vares lnearly, that w c vares quadratcally, that u c vares cubcally and that v c vares lnearly across the thckness of the th nterface layer. Complete Set of Governng Equatons Combnaton of the sold lamnate equatons, Eqn 1-5, wth the nterface layer equatons, Eqn 6, yelds the complete set of governng dfferental equatons for the th sold lamnate and the th nterface layer. For the case of cylndrcal bendng, where all the fundamental varables are functons of alone, the governng equatons can be epressed n the form: th sold lamnate: u0, = α1n 2Ny 3M, v0, = α 4N 5Ny 6M w, = β, β, = α 7N 8Ny 9M f N, = N, = M, = Q + 2 Q w E 1 c q w E 1 1 cz 1 1 cz E cz cz, = + q c w E c 1 2 c 1 c 2 c th nterface layer: q q 1 1 { z z} y { zy zy} { 1,, z z} = q, yz = qy 12E cz ( f + c) + 1 ( f+ 1 + c) + 1 c = u + β + u + β ( c ) 2 2 Gcz z,,, y, { Gcyz = kq 7 + k8 z + k9 k8 z k9 z c k + k + k k k Q yz 3 3 yz 3 yz 7 } 1 z (7) (8) where α m (m=1-7) and k m (m=3 and 7-9) are coeffcents determned from the sold lamnate etensonal, couplng and bendng stffnesses Amn, Bmn, Dmn (m,n=1,2,6). For the th sold lamnate nterface layer the soluton vectors contanng the fundamental varables defned by:

6 { β y } { z yz y} u, v, w,, N, N, M, Q =,..., N 0 0 1,, q, q = 1,..., N 1 (9) are determned from the soluton of the feld equatons, Eqn 7, 8. When the soluton gven by Eqn 9 has been obtaned, the followng sold lamnate quanttes can be determned drectly from the consttutve relatons, Eqn 4, and the equlbrum equatons, Eqn 5: In addton the nterface layer quanttes { yy yy y y} N, M, M, Q = 1,..., N (10) { c c c z} u, v, w, σ = 1,..., N 1 (11) can be determned from Eqn 6. The quanttes gven by Eqn 10 can be physcally nterpreted as the stress and moment resultants, whch have to be appled along the edges y=constant n order to mantan a state of cylndrcal bendng n the mult-layer plate. Mult-segment Method of Integraton For an arbtrary mult-layer plate, composed of N sold lamnates, and (N-1) complant nterface layers, the order of the set of governng equatons s N 8+(N-1) 4, snce the necessary number of boundary condtons to be specfed along an edge =constant eactly equals half ths number. The governng equatons, Eqn 7, 8, together wth the statement of the boundary condtons, consttute a multple pont boundary value problem, whch n the present nvestgaton has been solved numercally usng the mult-segment method of ntegraton. Ths method s based on a transformaton of the orgnal multple pont boundary value problem nto a seres of nterconnected ntal value problems. The consdered mult-layer plate confguraton s dvded nto a fnte number of segments, and the soluton wthn each segment s derved by drect ntegraton. Contnuty of the soluton vectors across the separaton ponts between the segments, as well as fulflment of the boundary condtons s ensured by formulatng and solvng a set of lnear algebrac equatons. EXAMPLE: CFRP/HONEYCOMB CORED SANDWICH PANEL WITH EXTERIOR PLY DROPS The developed hgh-order mult-layer plate theory enables the predcton of the mechancal response characterstcs dsplayed by a number of complcated structural problems ncludng: ply drop effects and free edge effects n composte lamnates, adhesve bonded jonts, multlayer sandwch plates and many other problems nvolvng layered structural assembles. In ths paper the capabltes of the theory are demonstrated through analyss of a CFRP/honeycomb cored sandwch panel wth eteror ply drops placed symmetrcally on each sde, as shown schematcally n Fg. 3. Fg. 3 llustrates a smplfed verson of an mportant practcal problem, snce the thckness of composte sandwch panel face sheets s often ncreased locally n order to provde for the load transfer around hghly loaded locatons such as jonts, rvets, bolts or nserts. Usually ples are dropped nternally, but eteror ply drops are also used qute frequently. At the statons where ply drops are made, the local bendng stffness of

7 the face lamnates changes dscontnuously, thus nducng severe local bendng effects. These local bendng effects nduce severe stress concentratons, whch may ntate delamnaton, core crushng or sold lamnate bendng falure [10-12]. Fg. 3: CFRP/honeycomb cored sandwch panel wth eteror ply drops. Wth reference to Fg. 1 and Fg. 3, the sandwch plate confguraton s composed of a 3-layer structural assembly for 0 L 1 and a 7-layer assembly for L 1 L 1 +L 2 : Sold lamnates: 5245C T-800 from Narmco/BASF; toughened carbon (T-800 fbres)/bsmalemde prepreg system: E 11 =165 GPa, E 22 =9.7 GPa, ν 12 =0.31, G 12 =4.8 GPa, cured ply thckness=0.152 mm [12]. Interface layers 1 & 3: Prepreg bsmalemde resn: E z =3.3 GPa, G z =G yz =1.2 GPa [12]. Interface layer 2: Alumnum honeycomb; Hecel 3/ : E z =517 MPa, G z =310 MPa, G yz =152 MPa [12]. Stackng sequences: Lamnate 2 & 3: [0 /90 /0 ]. Lamnate 1: [90 /-45 /45 ]. Lamnate 4: [-45 /45 /90 ]. The quoted stackng sequences are counted from the top and downwards. Geometry: Boundary condtons: As shown n Fg. 3. Eternal load: =0: N 2 =N 3 =P/2, P=100 N/mm. L 1 =L 2 =25 mm, f =0.456 mm (=1-4), c 1 =c 3 =0.1 mm, c 2 =10 mm. The analysed confguraton s geometrcally symmetrc about the mddle plane of nterface layer 2 (the sandwch core), but sold lamnates 1 and 4 (dropped sub lamnates) are stacked asymmetrcally wth respect to ther own mddle planes as well as wth respect to the mddle plane of nterface layer 2. Consequently, the mechancal response wll dsplay couplng effects ncludng non-zero wdth drecton dsplacements (v 0, v c ), non-zero wdth drecton nterface layer shear stresses ( yz ), and non-zero n-plane shear stress and twst moment resultants (N y, M y ) n the sold lamnates. Fg. 4 shows the predcted dsplacements, and t s observed from the dstrbuton of w that severe local bendng effects are nduced near the ply drop poston at =L 1 =25 mm. In partcular t s notced that the dropped lamnates (=1 and 4) tends to peel of sold lamnates 2 and 3, thus ndcatng the presence of tensle transverse normal stresses (peelng stresses) n nterface layers 1 and 3. Non-zero wdth drecton dsplacements v 0 are also seen, as epected due to the asymmetrc stackng of the dropped lamnates 1 and 4. Fg. 5 and Fg. 6 dsplay the predcted nterface layer stresses, whch have been normalsed wth respect to the nomnal stress n sold lamnates 2 and 3 at =0 (pont of load ntroducton):

8 N N P = = = = f f 2 f 2 3 σ nom MPa (12) Fg. 4: Sold lamnate dsplacements u 0 (), v 0 (), w (); =1-4. Fg. 5: Normalsed nterface layer transverse normal stresses along the boundares to the sold lamnates. Fg. 5 shows the transverse normal stresses along the top (σ z (top) ) and bottom (σ z (bottom) ) boundares between the nterface layers and the sold lamnates. Accordng to Eqn 6 the varaton of the transverse normal stresses across the nterface layers s lnear, and t s seen that ths varaton s very large close to the free edge at =L 1 =25 mm of nterface layers 1 and 3. Thus, very large tensle nterface layer transverse normal stresses are nduced along the most remote boundary relatve to the mddle plane of the complete sandwch panel, whereas large compressve transverse normal stresses are nduced along the boundary closest to the sandwch panel mddle plane. It s further seen that the presence of transverse normal stresses n nterface layers 1 and 3 s very localsed, and a complete decay s observed a few nterface layer thcknesses from the ply drop poston. The transverse normal stresses n the core materal (nterface layer 2) are very small compared wth the stresses n nterface layers 1 and

9 3. The values obtaned for the top and bottom nterfaces of nterface layer 2 are dentcal, and follow a pattern, whch s dentcal to the out-of plane dsplacements of sold lamnates 2 and 3, see Fg. 4. Ths ndcates that the core materal acts as a smple elastc foundaton. Fg. 6: Normalsed nterface layer shear stresses (core stresses z 2 and yz 2 are not shown snce they are neglgbly small). Fg. 6 shows the dstrbuton of normalsed shear stresses n nterface layers 1 and 3, and the shear stresses are constant across the nterface layers accordng to Eqn 6. Non-zero shear stresses are also predcted for the core layer (nterface layer 2), but these are not shown, snce they are very small compared wth the other shear stress components. The longtudnal shear stresses n nterface layers 1 and 3 ( z 1, z 3 ) are of equal magntude but of opposte sgns, and the mposed free edge condtons of zero shear stresses at the ply drop poston, =L 1 =25 mm, are fulflled. The peak values of z 1, z 3 are obtaned a few nterface layer thcknesses from the ply drop poston. The nterface layer shear stresses n the wdth drecton,.e. yz 1, yz 3, are also shown n Fg. 6, and t s observed that they are small compared wth z 1, z 3. The elastc response of the CFRP/sandwch panel wth ply drops also ncludes severe stress concentratons n the sold lamnates. These stress concentratons are assocated wth the localsed bendng phenomena shown n Fg. 4, but no further results wll be presented heren. CONCLUSIONS A hgh-order mult-layer plate theory has been presented. The theory nherently ncludes a descrpton of the global response, as well as the local responses of the ndvdual layers, of an arbtrary mult-layer plate assembly, where N hgh stffness layers are separated by (N-1) complant nterface layers. The theory ncludes the transverse fleblty of the nterface layers, thus allowng the thckness of the complete mult-layer plate assembly to change durng deformaton. The theory provdes a complete soluton wth respect to the dsplacements, stress resultants and moment resultants of the sold lamnates, as well as the dsplacements, transverse normal stresses and shear stresses of the nterface layers. The hgh-order mult-layer plate theory can be used for the analyss of the mechancal response dsplayed by a number of complcated structural problems nvolvng mult-layer assembles. In the present paper the features of the theory has been demonstrated through

10 analyss of the elastc response of a CFRP/honeycomb cored sandwch panel wth eteror ply drops. For ths partcular eample, t has been demonstrated that the theory enables the predcton of very complcated load transfer mechansms, ncludng the nteracton of the sold lamnates (sandwch face sheets and dropped sub lamnates) through the complant nterface and core layers. REFERENCES 1. Whtney, J. M., Structural Analyss of Lamnated Ansotropc Plates, Technomc Publshng Company, Lancaster, Reddy, J. N., Energy and Varatonal Methods n Appled Mechancs, John Wley & Sons, New York, Savoa, M, Laudero, F. and Trall, A., A Two-dmensonal Theory for the Analyss of Lamnated Plates, Computatonal Mechancs, Vol. 14, 1994, pp Lbrescu, L., Elastostatcs and Knetcs of Ansotropc and Heterogeneous Shell-Type Structures, Noordhoff Internatonal Publshng, Leyden, Frostg, Y., Baruch, M., Vlna, O., Shenman, I., Hgh-order Theory for Sandwch Beam Bendng wth Transversely Fleble Core, Journal of ASCE, EM Dvson, Vol. 118, 1992, Frostg, Y., On Stress Concentraton n the Bendng of Sandwch Beams wth Transversely Fleble Core, Composte Structures, Vol. 24, 1993, Frostg, Y. and Shenhar, Y., Hgh-order Bendng of Sandwch Beams wth a Transversely Fleble Core and Unsymmetrcal Lamnated Composte Skns, Compostes Engneerng, Vol. 5, 1995, Thomsen, O. T. and Rts, W., Analyss and Desgn of Sandwch Plates wth Inserts: A Hgher Order Sandwch Plate Theory Approach, Compostes Part B, Vol. 29B, 1998, pp Thomsen, O. T., Sandwch Plates wth Through-the-thckness and Fully Potted Inserts: Evaluaton of Dfferences n Structural Behavour, Composte Structures, Vol. 40, 1998, pp Thomsen, O. T., Rts, W., Eaton, D. C. G. and Brown, S., Ply Drop-off Effects n CFRP/ honeycomb Sandwch Panels - Theory, Compostes Scence & Technology, Vol. 56, 1996, pp Thomsen, O. T., Rts, W., Eaton, D. C. G., Dupont, O. and Queekers, P., Ply Drop-off Effects n CFRP/honeycomb Sandwch Panels - Epermental Results, Compostes Scence & Technology, Vol. 56, 1996, pp Thomsen, O. T., Mortensen, F. and Frostg, Y., Interface Falure at Ply Drops n CFRP/Sandwch Panels, Journal of Composte Materals, n press.