Effect of anisotropy on laminated composite plates containing circular holes

Transcription

1 Indan Journal of ngneerng & Materals Scences Vol. 1, June 005, pp ffect of ansotropy on lamnated composte plates contanng crcular holes H Murat Arslan Cukurova Unversty, Cvl ngneerng Department, Adana, Turkey Receved 15 Aprl 004; accepted 1 Aprl 005 In ths study, an analyss of fbre renforced, mult-layered composte plates, contanng crcular holes, has been carred out. Frst, stress concentratons around a crcular hole are studed. A number of dagrams are drawn to show stress concentratons around a hole, for each layer, usng dfferent materal pars wth dfferent 1 / ratos (rato of elastcty modulus of fbre drecton to that of transverse drecton). The graphs are gven, for varous 1 / values, for the crcumferental stress values around the hole versus angular locaton of ponts, for two dfferent fbre orentaton angles. Second, the falure of the lamnated composte plate s studed. For ths purpose Tsa-Hll falure crteron and Hoffman falure crteron are ntroduced. To determne the frst-ply falure of a lamnated plate, these two falure crtera are employed. Fnally, to show the effect of 1 / rato on the falure of plates a number of tables are gven for dfferent fbre orentaton angles. IPC Code: 04C/00 Composte materals consttute a group of materals formed by puttng together at least two dfferent materals. A renforced concrete beam and a car tyre are examples for materals of ths type. The am of ths three dmensonal composton s to obtan a property whch none of the consttuents possesses. In other words, the target s to produce a materal that possesses hgher performance propertes than ts consttuent parts for a partcular purpose. Some of these propertes are mechancal strength, corroson resstance, hgh temperature resstance, heat conductblty, stffness, lghtness and appearance. In accordance wth ths defnton, the followng condtons must be satsfed by the composte materal. It must be manmade and not natural. It must comprse of at least two dfferent materals wth dfferent chemcal components separated by dstnct nterfaces. Dfferent materals must be put together n a three dmensonal unty. It must possess propertes whch none of the consttuents possesses alone and that must be the am of ts producton. The materal must behave as a whole,.e. the fbre and the matrx must be perfectly bonded. Lamnaton s used to combne the best aspect of the consttuent layers and bondng materal n order to acheve a more useful materal. The propertes that can be emphaszed by lamnaton are strength, stffness, low weght, corroson resstance and thermal nsulaton. Lamnates, as many other structures, could -mal: hmarslan@cukurova.edu.tr have holes to serve varous purposes. An obvous purpose s to accommodate a bolt (Fg. 1). Stress concentraton for orthotropc plates contanng crcular holes was ntroduced by Green and Zerna 1. Later, Broockman and Serakowsk carred out an nvestgaton on boron-alumnum compostes wth crcular cut-outs subjected to loads parallel to the fbres of the compostes. The stress dstrbuton of glass-epoxy composte plates was gven by Hyer and Lu 3 usng photo-elastc methods. Kaltakc 4 ponted out the effect of fbre orentaton on the stress concentraton around the holes of ansotropc plates and ther falures. In ths paper, the effect of 1 / rato on the stress concentratons n lamnated composte plates wth crcular holes and ther falure crtera are studed. Theoretcal Analyss Stress concentratons around holes A crcular hole n a plate could be requred for varous purposes. If a small crcular hole s made n the mddle of the plate, the stress dstrbuton n the neghborhood of the hole wll be changed. The stress dstrbuton around a hole has been studed by several researchers. As descrbed earler 1, Green studed the problem usng, F = F0 + A0log r1+ B0log r Ancos nθ 1 Bncos nθ (1) + + n n n n n= 1 (1 + γ1) r1 (1 + γ) r

2 08 INDIAN J. NG. MATR. SCI., JUN 005 Fg. 1 Two layered, cross-ply lamnate (0 and 90 ) Fg. 3 (a) Lamnated composte plate wth a crcular hole (b) an ansotropc layer wth a crcular hole where F s the Ary stress functon. For the orthotropc plate shown n Fg. 1, loaded n x-drecton (tenson or compresson) the crcumferental stress around the hole s gven as 1 (1 + γ ) (1 + γ ) (1 + γ + γ γ γ cos θ) θ1 = x (1 + γ1 γ1 cos θ) (1 + γ γ cos θ) () where x s the stress appled n x-drecton, θ, measured from the x-axs n counterclockwse drecton, s the angle showng the drecton n whch the stress value s calculated, and γ 1, γ are defned as γ = γ 1 Fg. An orthotropc plate loaded by unform shear stress = ν 1 + ν1 1 G1 G1 1 ν 1 + ν1 + 1 G1 G1 1 ν 1 ν1 1 G1 G1 1 ν 1 ν G1 G1 1 (3) where 1 s the elastcty modulus n the fbre drecton (x-drecton n ths case), s the elastcty modulus n the transverse drecton, ν 1 s the Posson s rato and G 1 s the shear modulus. When the plate s subjected to unform shear stress, S (Fg. ), the crcumferental stress around the hole s found by 1 4( γ γ 1)snθ (1 γ γ cos θ) (1 γ γ cos θ) S (4) 1 θ1 = Usng the crcumferental stress, the stresses n the layers are calculated by x = θ1 sn θ y = θ1 cos θ = snθ cosθ xy θ1 (5) The stress concentratons around the hole, n ansotropc plates (see Fg. 3), could be found by superposton. Usng qs (), (4) and the other form of q. (), whch s wrtten for transverse drecton, one can get the crcumferental stress equaton for a layer shown n Fg. 3b as 4 ( N + N + N ) (1 + γ γ cos ( θ α))(1 + γ γ cos ( θ α)) (6) 1 3 x θ1 = 1 1 where, N1 = (1 + γ1) (1 + γ) [ 1 + γ1+ γ γ1 γ cos ( θ α) ] N = 4 [ γ1+ γ (1 + γ1 γ) cos( θ α) ] sn α N3 = 4 ( γ1 γ 1)sn ( θ α) snα cosα (7) Falure of fbre renforced ansotropc layers Predcton of falure of structural components s

3 ARSLAN: FFCT OF ANISOTROPY ON LAMINATD COMPOSIT PLATS CONTAINING CIRCULAR HOLS 09 usually accomplshed by comparng functons of the stresses or strans to materal strength lmts. Gven a two-dmensonal stress state, the calculaton of the magntudes of prncpal stresses s suffcent to predct the falure of sotropc materals. Determnaton of the falure of a lamnated composte materal presents several dffcultes. Delamnaton, varaton of stresses n the thckness drecton and stress concentratons are some of those dffcultes. The falure of a lamnated composte s more complcated than that of a sngle plate, due to the stackng together of several layers of materal wth dfferent orentatons and propertes. Snce stresses and strengths are dfferent n dfferent layers, t s possble that, as the loadng s ncreased, the stresses n one or more of the layers of a lamnate would reach ther lmtng strength earler than others. It s, therefore, lkely that the lamnate would suffer damage n the form of local falures n those layers before t fals completely. For some applcatons the frst falure of any layer s not acceptable because t degrades the strength and stffness of the lamnate. Falure predcton based on the frst falure s commonly referred to as the frst-ply falure crteron. There are varous quadratc falure crtera for composte layers. The most frequently used ones are the Tsa-Hll, Hoffman and Tsa-Wu crtera. In ths study, Tsa-Hll and Hoffman crtera are used to predct the falure of lamnated composte plates. Hll proposed an extenson of the von Mses yeld crteron to ansotropc materals wth equal strengths n tenson and compresson. For three dmensonal stress state Hll s crteron s gven by f f f f f f x y + x z + z x f f f yz xz xy F( ) G( ) H( ) + L( ) + M ( ) + N ( ) < 1 Table 1 Materal propertes Materal propertes Graphte-epoxy Glass-epoxy 1 (kg/cm ) (kg/cm ) G 1 (kg/cm ) (X t ) 1 (kg/cm ) (X t ) (kg/cm ) 4 81 (X c ) 1 (kg/cm ) (X c ) (kg/cm ) G (kg/cm ) ν (8) Under plane stress assumpton, Tsa extended ths crteron as 1 f 1 f 1 f f 1 f ( x) + ( y) xy + ( xy) < 1 Xt Yt Xt G (9) where G s the shear strength, X t and Y t are the tensle strengths of the materal n the fbre and the transverse f drectons, respectvely. s the shear stress and f x f y xy,, are the stresses n fbre and transverse drectons, respectvely. Ths equaton s known as Tsa- Hll falure crteron. A generalzaton of the Tsa-Hll falure crteron s proposed by Hoffman, takng nto account dfferng tensle and compressve strengths of the materal. Hoffman offers 1 f 1 f 1 f f ( x) + ( y) xy + XX t c YY t c XX t c 1 1 f 1 1 f 1 f x + y + ( xy ) < 1 Xt Xc Yt Yc G (10) for the falure of lamnated compostes. Here, X c and Y c are compressve strengths of the materal n the x- and the transverse drectons, respectvely. Results and Dscusson Two plates, whch are loaded parallel to x- drecton, are studed as examples. In the frst example, a two layered graphte-epoxy composte plate s consdered. The orentaton of the fbres n the upper layer s 0 and that of the lower s 90 (0 /90 ) wth the x-axs. A four layered, glass-epoxy composte wth 0 /45 /-45 /90 fbre orentatons s used as the second example. The materal propertes of the graphte-epoxy and glass-epoxy, shown n Table 1. In ths table, subscrpts 1 and refer to fbre drecton and the transverse one, respectvely. For both plates, stress concentraton values around the hole are presented graphcally for each layer. The varaton of the stress concentraton values for, graphte-epoxy composte, dependng on the locaton (the angle θ vares from 0 to 180 ) are drawn, as shown n Fg. 4. It must be remembered that f the stress s appled to a layered composte plate, the dstrbuton of that stress to layers could not be equal. When Fg. 4 s examned, t s noted that the maxmum stress concentraton value s for the upper layer, whereas, that for the lower layer s.519

4 10 INDIAN J. NG. MATR. SCI., JUN 005 Fg. 4 Stress concentratons ( θ1 x ) versus θ for two layered graphte-epoxy composte Fg. 6 Stress concentratons ( θ1 x ) versus θ for dfferent 1 / ratos Fg. 5 Stress concentraton ( θ1 x ) versus θ for four layered glass-epoxy composte both beng at the same locaton for whch θ=90. In the upper layer, the tensle stresses dctate the crtcal pont. In the lower layer the stuaton s qute dfferent. In ths layer, the compressve stresses must be consdered to fnd the crtcal pont. For θ=0 and θ=180, the stress concentraton value for the lower layer s 6.39, whereas, ths value s nearly zero for the upper layer. As a second example, a four layered glass-epoxy composte s chosen to study the effect of the orentatons of fbres on the stress concentraton factors. Startng from the top, the glass fbres make wth the x-axs, the followng angles, respectvely, 0, 45, 45, 90. If Fg. 5 s examned, one can agan note that n the lamnated composte plate, the stress concentraton value reaches a maxmum at θ=90 for layer 1 (fbre orentaton angle s 0 ). Ths value s a mnmum n the bottom layer. In lamnated composte plates, each layer can have dfferent concentraton values dependng on the fbre orentaton angle, as shown n Fg. 5. The maxmum concentraton value and the locaton, where t occurs, dffer as the fbre orentaton changes. Furthermore, the stress concentraton values change also, dependng on the rato of the elastcty modulus of the fbre drecton to the trans- Fg. 7 Stress concentratons ( 1 x θ ) versus θ for dfferent 1 / ratos verse one ( 1 / ). In the frst and second example plates, t s observed that the maxmum stress concentraton occurs when the fbres are orented parallel to the x-axs. The effect of the 1 / rato on the stress concentratons wll be studed consderng ths layer as the crtcal one. For the same fbre orentatons, dfferent concentraton factors are obtaned for two dfferent compostes wth dfferent 1 / ratos. To see the effect of ths rato on the stress concentraton value, a layer s consdered whch has 0 fbre orentaton angle. The stress concentraton values are obtaned for dfferent 1 / values usng q. (6). The same calculatons are performed for 90 fbre orentaton angle, also. In Fgs 6 and 7, the effect of 1 / rato s shown for 0 and 90 fbre orentaton angles. When the fbres are parallel to x-axs,.e., α=0, as the 1 / rato ncreases, the stress concentratons ncrease, also. When θ=90 and 1 / =100, t reaches But when values θ=0 and θ=180, the stress concentratons were not affected too much by the change n 1 / rato. The stuaton s vce-versa for

5 ARSLAN: FFCT OF ANISOTROPY ON LAMINATD COMPOSIT PLATS CONTAINING CIRCULAR HOLS 11 Table Comparson of falure crtera for graphte-epoxy Fbre orentaton angle (α ) Hoffman falure crteron Tsa-Hll falure crteron Under tensle or compressve stress Under tensle stress Under compressve stress Falure load Falure angle Falure load Falure angle Falure load Falure angle the second case (see Fg. 7). The postve stress concentraton values show no sgnfcant changes wth 1 / rato whereas, the negatve values change from 1 to nearly 10. In the present work, Tsa-Hll and Hoffman crtera are employed for the falure of lamnated composte plates. In lamnated compostes, the layers can have dfferent fbre orentaton angles. Hence, n Table are compared the falure loads for the two crtera for a sngle layer wth fbre orentaton angles 0-90 wth 5 ncrements under tensle and compressve loads. In Table, a graphte-epoxy layer s examned. Usng both crtera, falure loads and falure angles are determned for tensle and compressve loadng cases and for dfferent fbre orentatons. Usng ths table, one can see whch layer s the crtcal one and whch layer fals frst n a lamnated composte plate made wth dfferent fbre orentatons. Tsa-Hll crteron gves dfferent falure loads for tensle or compressve cases. However, Hoffman s crteron gves the same falure load value for both loadng cases. Graphte-epoxy s much stronger n tenson than n compresson. So, Tsa-Hll crteron gves hgher falure load values than Hoffman s, because the former consders only tensle strength of the materal, whereas, the latter consders both the tensle and the compressve strengths. Keepng n mnd ths pont, one can understand that the falure loads for compresson case n Tsa-Hll crteron s lower than that of Hoffman s. The same calculatons are performed for a glassepoxy composte and the results are presented n Table 3. It s observed from Tables and 3 that, wth an ncrease n fbre orentaton angle, there s a decrease n the falure load. Conclusons In ths paper, the stress concentratons around a crcular hole and the falure load of lamnated, ansotropc composte plates accordng to two crtera are studed. In the frst part, two lamnated composte plates wth dfferent materals are studed. For both plates, the stress concentraton factors n each layer, around the hole are presented graphcally n Fgs 4 and 5. To see effect of the materal propertes, 1 /

6 1 INDIAN J. NG. MATR. SCI., JUN 005 Table 3 Comparson of falure crtera for glass-epoxy Fbre Hoffman falure crteron Tsa-Hll falure crteron orentaton Under tensle or compressve stress Under tensle stress Under compressve stress angle (α ) Falure load Falure angle Falure load Falure angle Falure load Falure angle rato, on the stress concentratons, a layer wth a crcular hole under the effect of a tensle stress s consdered (Fg. 6). It can be seen from the curves, n ths fgure that, for 1 / =100, the stress concentraton factor reaches 7.73 at θ=90. When θ=90, the crcumferental stress s parallel to the fbre drecton, causng more of the force transfer to take place through the fbres. But these values don t change very much near the ponts where θ equals 0 and 180. In those regons, the crcumferental stress s perpendcular to the fbre drecton and force transfer takes place through the matrx materal whch s weak and has low elastcty modulus. The stuaton s vceversa for the second case, n ths case, there s very lttle change around ponts when the loadng s n the transverse drecton to the fbres (see Fg. 7) where θ equals 90, whereas, a consderable change takes place at ponts where θ equals 0 and 180. Two pont to be noted also. Frst, as the 1 / rato ncreases, the stress concentraton values also ncrease. However, the rate of ths ncrease s not constant. Second, ths ncrease does not occur evenly all around the crcular hole. In Tables and 3, the falure loads obtaned from Hoffman and Tsa-Hll crtera are gven for two composte materals wth dfferent fbre orentaton angles. Tsa-Hll crteron consders compressve and tensle cases separately. Hence, the falure loads for the two cases are dfferent. However, Hoffman s crteron gves only one result. Dependng on the materal strength both crtera gve smlar results. If Tables and 3 are examned, one can see that, as the fbre orentaton ncreases, the falure load for both cases decreases. Keepng n mnd that, the loadng s n the x-drecton, as the fbre orentaton ncreases from 0, the load resstng capacty of the layer decreses and t takes the lowest value when the loadng s perpendcular to the fbres, n whch case the loads are carred manly by the matrx materal. Nomenclature =crcumferental stress θ1 x =prncpal stress n the x-drecton y =prncpal stress n the y-drecton xy =prncpal shear stress n the layer

7 ARSLAN: FFCT OF ANISOTROPY ON LAMINATD COMPOSIT PLATS CONTAINING CIRCULAR HOLS 13 x =appled external layer stress S =appled external layer shear stress θ =crcumferental angle α =fbre orentaton angle 1 =modulus of elastcty n fbre drecton =modulus of elastcty transverse to fbre drecton G 1 =shearng modulus ν 1 =Posson s rato X t =tensle materal strength n fbre drecton Y t =tensle materal strength transverse to fbre drecton X c =compressve materal strength n fbre drecton Y c =compressve materal strength transverse to fbre drecton G =materal strength n shear References 1 Green A & Zerna W, Theoretcal elastcty (Clarendon Press, London,U.K.), 1954 Broockman C & Serakowsk R L, Fbre Sc Technol 1 (1979) 1 3 Hyer M W & Lu D, J Compos Mater, 19 (1985) Kaltakc M Y, Comput Struct, 61 (1996) 67 5 Hyer M W, Stress analyss of fbre-renforced composte materals (McGraw-Hll Book Company, Sngapore), Ugural A C, Stresses n plates and shells (McGraw-Hll Book Company, U.S.A.), Reddy J N & Mravete A, Practcal analyss of composte lamnates (CRC Press, U.S.A.), Jones R M, Mechancs of composte materals (Taylor & Francs, U.S.A.), Gurdal Z, Haftka R T & Hajela P, Desgn and optmzaton of lamnated composte materals (John Wley & Sons Inc., U.S.A.), Hearmon R F S, Appled ansotropc elastcty (Oxford Unversty, London U.K.), Chen X & Lu Y, Compu Mater Sc, 1 (001) 86 1 Chen T & Zheng Q S, Int J Solds Struct, 37 (000) 591