On the Transient Hygroscopic Stresses in Polymer Matrix Laminated Composites Plates with Cyclic and Unsymmetric Environmental Conditions

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1 On the Tansient Hygoscopic Stesses in Polyme Matix Laminated Composites Plates with Cyclic and Unsymmetic Envionmental Conditions Laboatoie des Matéiaux et Hydologie, Univesité de Sidi Bel Abbes, BP 89 Cité Ben M hidi Sidi Bel Abbes, Algéie Received: 1 Apil 2004 Accepted: 8 Novembe 2004 SUMMARY The use of aicaft stuctual pats made of polyme matix composites subjected to sevee envionment conditions calls fo bette knowledge of thei long tem behaviou, with an emphasis on thei ability to withstand impotant cyclic vaiations of moistue and tempeatue. The influence of tempeatue and moistue on such stuctues is eceiving special attention, because it induces tansient esidual stesses within the plies. Such stesses must be taken into account in the design of composite mateials, paticulaly aeospace stuctues, e.g. aicaft. In the pesent pape the tansient hygoscopic stesses induced by cyclic and unsymmetic envionmental loadings which simulate an aeonautical application ae investigated. It is shown that the heteogeneity and anisotopy of such plates geneally esult in tansient stess distibutions which ae vey diffeent fom the equilibium stess distibution. Some stacking sequences exhibit impotant stesses within the plies. These stesses have to be taken into account in the design of composite stuctues submitted to a moist envionment. 1. INTRODUCTION A knowledge of long tem behaviou unde cyclic envionmental conditions is one of the most impotant conditions fo an inceasing use of stuctues made of polyme matix composites. This poblem has been extensively investigated 1 4, especially fo aeonautical stuctues, to calculate the moistue content of composite mateials exposed to ai by vaying tempeatue and elative humidity simulating long tem sevice. In these latte studies, the moistue absoption within the composite plate is modelled by a Fick s law whose diffusion coefficient depends on the tempeatue though an Ahénius law. Howeve, the tempeatue inside the mateial eaches equilibium much faste than the moistue concentation. Theefoe, the tempeatue field within the thickness of the plate is given by the pemanent solution. On the othe hand, paticula attention must be paid to the calculation of the moistue concentation though laminated plates. tou_abdel@yahoo.com Let us conside a plate with thickness L, made of composite died in an oven afte manufactuing until weight stabilisation is obtained. The plate is exposed to moistue distibutions C L t () t vaying 2 () and C + L 2 with time t on its faces at positions z = L 2 and z =+ L 2 along the z axis though the thickness. The moistue concentation field within the thickness is the solution of the system (1) and (2) below: ( ) D() t 2 C z,t z 2 = C( z,t) t (1) C L 2,t = C L t () and C + L 2,t = C L () t (2) When the envionmental conditions ae cyclic, the diffusivity D(t) and moistue distibution on the faces C () + L t C L 2 and () t 2 ae cyclic functions of time, with peiod τ. 489

2 Geneally speaking, the solution C(x,t) is composed of a tansient pat, a fluctuating pat and a pemanent pat 1 4. It tends to a time peiodic solution. With the objective of completing studies on the behaviou of composite mateials unde cyclic envionmental loading, we analyse hee the tansient hygoscopic stesses induced by envionmental conditions such as tempeatue diffusion and moistue soption. In the liteatue, a few studies ae devoted to this type of eseach, but most of them suppose constant envionmental conditions on both faces of plate. Fo example, Hahn and Kim 5 have discussed the distibution of such stesses in thee paticula cases of stacking sequences. Benkeddad et al 6 have poposed a method to compute these stesses whee they assume a linea distibution of moistue concentation within the sub-ply. Tounsi et al 7 have developed a staightfowad and quick analytical method to detemine the tansient hygoscopic stesses in a polyme matix composite plate exposed to symmetic envionmental conditions. Using the same methodology, the authos 8 have extended the latte method to compute such stesses duing moistue desoption. The effect of anisotopy on the distibution of the tansient hygoscopic stesses within laminates was studied by Tounsi et al 9 10 in ode to find laminates with moe esistant stacking sequences to the envionmental condition. Recently, Tounsi et al have studied the distibution of the tansient hygoscopic stesses within the thickness of the laminated composite plates unde given cyclic and symmetic envionmental conditions. We have also noted in the liteatue that seveal authos have studied the poblem of the calculation of the hygothemal stesses, and paticulaly themal stesses. Chamis et al. 13 poposed an integated theoy fo pedicting the hygothemomechanical esponse of the composites. Using classical laminated plate theoy, Hape and Weitsman 14 studied the effect of envionmental conditioning on the esidual stesses in composite laminates. Vel and Bata 15 have pesented an analytical solution fo thee dimensional themomechanical defomations of a simply suppoted functionally gaded ectangula plate subjected to time dependent themal loads on its top and/o bottom sufaces. In the pesent pape, we epot an analysis of the tansient hygoscopic stesses within the plies, induced by cyclic and unsymmetic envionmental conditions. Though the theoetical study pesented hee, we hope to contibute to an undestanding of the hygothemal behaviou of composite laminated plates. 2. CALCULATION OF THE NON- MECHANICAL STRESSES DUE TO THE MOISTURE CONCENTRATION DISTRIBUTION The moistue concentation distibution inside the plate, which is induced by cyclic and unsymmetic envionmental conditions, gives ise to esidual stesses. Such stesses can be computed using the method descibed by Tounsi et al 10 fo cyclic and symmetic envionmental conditions. This latte is an extension of the method given by Benkeddad et al 6 fo constant envionmental conditions in the two faces of the plate. Time t being given, the moistue concentation distibution was obtained by using the compute code W8GAIN 4. The fist step was to compute the on-axis fee expansions e x and e y defined in Figue 1. These expansions wee computed at each point z k of the thickness whee the moistue concentation C was given by the compute code W8GAIN. The value of k vaied fom 0 to N, whee N was the numbe of plies and sub-plies (in counting, the sub-plies wee consideed as a ply). In all the laminates consideed, we took N to be 72. The fee expansions can be expessed as follows: e (K) x =α x T +β x c(z K,t) e (K) y =α y T +β y c(z K,t) e S (K) = 0 (3) The non-mechanical stesses wee then easily computed in the ply fame σ i n(k) = Q ij e j (K), i, j = x,y,s (4) In the laminate fame defined in Figue 2, these stesses became, σ α n(k) α = 1, 2, 6, and the nonmechanical in-plane stess components N α and moment components M α wee N α n = L/2 n σ α L/2 L/2 dz and M n α = zσ n α dz α=1,2,6 L/2 (5) 490

3 N n and α Mn wee computed assuming a linea α n(k) distibution of the non-mechanical stesses σ α between the points z k and z k+1, k = 0 N-1 In the ply fame, these non-mechanical stains became ε i n(k). The esidual stains wee then defined by the diffeence between the non-mechanical stains and the fee expansions: N 1 2 N n (k) (z α = a k+1 z 2 k ) (k) α + b α (z k+1 z k ) 2 k=0 N 1 M n (k) (z α = a k+1 z 3 k ) (k) α + b α 3 k=0 3 2 (z k+1 z 2 k ) 2 (6) ε i (k) =ε i n(k) e i (k) (11) Then we could obtain the esidual stesses σ i (k) in the ply fame as follows: With: a (k) α = σ n(k+1) n(k) α σ α ; b k α =σ n(k) α a (k) α z k z k+1 z k (7) σ i (k) = Q ij ε j (k) (12) z k < z < z k+1 σ α n (k) = a α (k) z + b α (k) (8) The non-mechanical stains wee deduced using the usual stain-stess elation in laminated plates 16. Figue 2. Laminate fame (1,2) and local fame (x,y) of one of the plies { ε 0n }= α { k n }= β { }+ β { }+ δ [ ] N n [ ] t N n { } { } [ ] M n [ ] M n (9) The non-mechanical stains ε α n(k) wee then computed at each point z k : ε α n(k) =ε α 0n + z k k α n (10) Figue 1. Definition of the diffeent stains involved in the calculation of the esidual stesses in the i (x o y) diection: (a) initial state, (b) fee expansions, (c) actual final state 491

4 3. MATERIAL AND LAMINATED PLATES USED The mateial used was a gaphite/epoxy laminated plate fo which the popeties ae summaised in Tables 1 to 3 and the thickness of the laminated plate was 12.7 mm Table 1. Popeties of mateial (Spinge 17 ) Diffusivity, mm 2 /sec D = 0.57exp(-4993/T), T: tempeatue ( K) Moistue concentation at the suface as a function of elative humidity in atmosphee,% C 0 = 0.015RH, RH: elative humidity (%) Table 2. Mechanical popeties of mateial (Tsai 16 ) E x (GPa) E y (GPa) ν x G xy (GPa) α x (10-6 K -1 ) α y (10-6 K -1 ) β x β y 4. RESULTS AND DISCUSSIONS The esults below ae given unde the following assumptions. The initial moistue concentation within the plate was zeo. The vaiation in tempeatue and elative humidity etained ae shown in Figues 3, 4 and 5 as a function of the educed time, t = t (τ is the peiod). These τ envionmental conditions ae simila to those descibed by Spinge 3 to simulate the 20-yea sevice life of a militay aicaft. The tempeatue and humidity wee assumed to vay in a cyclical way. Fo the 6-day peiod (144 hous), nomal atmospheic conditions wee active fo 123 hous, followed by a flight fo 11 hous, and nomal atmospheic conditions once moe fo 10 hous. In Figue 3, the elative humidity of the envionment was assumed to be educed as the tempeatue inceased, except duing flight, when the elative humidity deceased to zeo. In Figue 4, the elative humidity of the envionment was assumed to be constant (82%), except duing flight when the elative humidity deceased to zeo. Table 3. Stength data of mateial (Tsai 16 ) Mateial X (MPa) X (MPa) Y (MPa) Y (MPa) S (MPa) Gaphite/Epoxy Finally, we note that in all the examples consideed, the tansient hygoscopic stesses wee computed at the end of each cycle. Figue 3. Ambient elative humidity befoe and afte the flight vesus the educed time: t = t/ (6 hous), t/ (6 days), t/ (6 weeks), adapted fom Spinge 4 492

5 Figue 4. Constant elative humidity befoe and afte the flight vesus the educed time: t = t/ (6 hous), t/ (6 days), t/ (6 weeks), adapted fom Spinge 4 Figue 5. Ambient tempeatue befoe and afte the flight vesus the educed time: t = t/ (6 hous), t/ (6 days), t/ (6 weeks), adapted fom Spinge Effect of the Peiod on the Tansient Hygoscopic Stesses Having maintained the same fom of bounday conditions (Figue 5 fo tempeatue and Figue 4 fo elative humidity (RH) at the left side of the plate and a constant moistue at the ight side (RH = 82%), only the tansvesal tansient stess σ y was analysed and this was because of its impotance 6 12 compaed with the othe stesses, i.e. longitudinal tansient stess σ x and esidual shea stess σ S. Figues 6 to 9 illustate the distibution of tansvese stess σ y though the thickness of the plate, computed at the same time but with a diffeent peiod. This investigation was intended to show the effect of the peiod on the e-patitioning of such stesses within the laminate. Though these esults, we established that the tansvese esidual stesses (σ y ) ae independent of the peiod. Consequently, we kept to a peiod of six days. 4.2 Evolution of Tansient Hygoscopic Stesses in a Unidiectional Laminate In the pesent investigation, the evolution of the tansvese hygoscopic stess σ at thee paticula y positions of the unidiectional laminate was studied as a function of the envionmental loading cycles. Two cases of the envionmental conditions wee envisaged: 493

6 Figue 6. Residual stesses σ y though the thickness of the unidiectional laminate at 6 days with diffeent peiod: peiod of six hous and six days. Tempeatue fom Figue 5 at both sides of the plate, moistue fom Figue 4 at the left side of the plate and constant moistue at the ight side (82%) Figue 7. Residual stesses σ y though the thickness of the unidiectional laminate at 42 days with diffeent peiod: peiod of six days and six weeks. Tempeatue fom Figue 5 at both sides of the plate, moistue fom Figue 4 at the left side of the plate and constant moistue at the ight side (82%) in case 1, tempeatue fom Figue 5 at both sides of the plate and moistue fom Figue 4 at the left side, constant moistue at the ight side (RH=82%). In case 2, tempeatue fom Figue 5 at both sides of the plate and moistue fom Figue 3 at the left side, fom Figue 4 at the ight side. Figues 10 and 11 show that duing the moistue absoption, the extenal faces of the plate wee in a compessive state, in contast to the mid plane, which is in tension. In addition, we have emaked that the tansvese esidual stess at both sides of the plate deceased with inceasing numbes of loading cycles and it took impotant values at the beginning of the absoption pocedue. In the mid plane of the laminate, the tansvese esidual stess σ inceased at the beginning of the y moistue absoption and eached its maximum value afte about 90 cycles, whee it began to educe. 494

7 Figue 8. Residual stesses σ y though the thickness of the unidiectional laminate at 126 days with diffeent peiod: peiod of six hous, six days and six weeks. Tempeatue fom Figue 5 at both sides of the plate, moistue fom Figue 4 at the left side of the plate and constant moistue at the ight side (82%) Figue 9. Residual stesses σ y though the thickness of the unidiectional laminate at days with diffeent peiod: peiod of six hous, six days and six weeks. Tempeatue fom Figue 5 at both sides of the plate, moistue fom Figue 4 at the left side of the plate and constant moistue at the ight side (82%) Finally, it is shown in studies cited in efeences 6,7,8 and 10, that at satuation the tansvese esidual stess σ within a unidiectional laminate becomes y almost zeo in the thee paticula positions of the unidiectional laminate: both side of the plate, and the mid plane. 495

8 Figue 10. Tansvese esidual stess σ vesus numbe of cycles in both sides and mid plane of plate fo envionmental y loading of type 1 (case 1) Figue 11. Tansvese esidual stess σ vesus numbe of cycles in both sides and mid plane of plate fo envionmental y loading of type 2 (case 2) 4.3 Tansient Hygoscopic Stesses in Angle ply [+θ, -θ] 2S Laminates The following angle-ply [+θ,-θ] 2S laminates wee studied: θ = 10, 20, 30, 45. In all cases, it has been shown in ecent studies 6,7,8,10,11,12 that the longitudinal stess σ x was negligible compaed to X and X (Table 3). Apat fom the case θ = 45, the shea stess σ S was not zeo. A typical cuve is plotted in Figue 12. The maximum value of the shea stess fo θ = 20 eached about 28.5% of the shea stength S given in Table 3. We also notice that these stesses wee linea in each ply fo any time, but they wee not constant as is emaked in 6,7,10. This is due to the absence of any symmety of the bounday conditions (envionmental loading), which is a condition fo these stesses to be constant in each ply, as is emaked in 6,7,8,10. The tansvese stess σ y is plotted in Figue 13 to 16 fo the diffeent values of θ. As fo the unidiectional laminate, the extenal plies wee subjected to a compessive tansvese stess fom the beginning to the end of the moistue pocedue. In contast, the intenal plies wee intension, except when appoaching satuation. 496

9 Figue 12. Residual shea stesses σ though the thickness of a gaphite/epoxy [+20,-20 ] (envionmental conditions S 2S of case 1) Figue 13. Residual tansvese stesses σ Y though the thickness of a gaphite/epoxy [+10,-10 ] 2S (envionmental conditions of case 1) In all cases, the maximum tensile tansvese stess could not be neglected compaed to the tansvese tensile stength Y. In fact, the maximum value of the tansvese stess σ eached about 51.86% of y We also noticed the appeaance of edge effects, which epesent the fluctuating pat of the tansient esidual stesses in the left skin of the plate. the tensile tansvese stength Y fo [+10,-10 ] 2S laminate, 50.02% fo [+20,-20 ] 2S laminate, 47.98% fo [+30,-30 ] 2S laminate and 55.23% fo [+45, -45 ] 2S laminate. 497

10 Figue 14. Residual tansvese stesses σ Y though the thickness of a gaphite/epoxy [+20,-20 ] 2S (envionmental conditions of case 1) Figue 15. Residual tansvese stesses σ Y though the thickness of a gaphite/epoxy [+30,-30 ] 2S (envionmental conditions of case 1) In Figues 17, 18 and 19 we have give some examples of the esults obtained by the pesent method showing the influence of fibe oientation on the vaiation of the tansvese esidual stess computed at both sides of the plate and the mid plane. Figues 17 and 18 show that the extenal plies wee subjected to a compessive tansvese stess fo all consideed values of θ. In addition, we can emak that this stess deceased ove time. 498

11 Figue 16. Residual tansvese stesses σ Y though the thickness of a gaphite/epoxy [+45,-45 ] 2S (envionmental conditions of case 1) Figue 17. Vaiation of the esidual tansvese stess σ Y vesus the fibe oientation θ on the ight side of a gaphite/ epoxy [+θ, - θ] 2S (envionmental conditions of case 1) The vaiation of the tansvese stess σ with the y angle of the fibe oientation on the mid plane of the [+θ,-θ] 2S laminate is plotted in Figue 19. The mid plane was subjected to a compessive tansvese stess at the beginning of the moistue absoption, afte which it passed to a tensile state fo the emaining time. 499

12 Figue 18. Vaiation of the esidual tansvese stess σ vesus the fibe oientation θ on the left side of a gaphite/ Y epoxy [+θ, - θ] 2S (envionmental conditions of case 1) Figue 19. Vaiation of the esidual tansvese stess σ Y vesus the fibe oientation θ on the mid-plane of a gaphite/ epoxy [+θ, - θ] 2S (envionmental conditions of case 1) 4.4 Tansient Hygoscopic Stesses in Quasi isotopic Laminates Two quasi isotopic laminates wee studied hee: [0/+45/90/-45] s and [60/-60/0/60/0/60/0/60/- 60] 4. The esults obtained ae plotted in Figues 20 and 21. The pesent distibution of the tansvese stess σ within these two quasi isotopic laminates Y was simila to the distibution in the above [+45, -45 ] 2S laminate. The explanation of this coincidence has aleady been given in 8,9,10 in which we have used the notion of the degee of anisotopy to undestand this behaviou. 500

13 Figue 20. Residual tansvese stesses σ Y though the thickness of a gaphite/epoxy [0/+45/90/-45]ls (envionmental conditions of case 1) Figue 21. Residual tansvese stesses σ Y though the thickness of a gaphite/epoxy [60/-60/0/60/0/60/0/60/-60] 4 (envionmental conditions of case 1) 5. CONCLUSIONS In the pesent pape, the tansient hygoscopic stesses have been computed fo polyme matix laminated composite plates exposed to cyclic and unsymmetic hygothemal conditions. Fom the esults pesented hee, it can be concluded that thee is a need to analyse popely the tansient hygoscopic stesses in composite plates. Some stacking sequences, such as those of unidiectional, angle ply laminates and quasi isotopic laminates have been studied. No effect of the peiod on the vaiation of the tansvese esidual stesses was obseved. An extension of the appoach is in pogess to include impoved models fo the mechanical behaviou, which also depend on the moistue concentation. 501

14 REFERENCES 1. Adda-Bedia E.A., Han W.S. and Vechey G., An asymptotic chaacteisation of the moistue diffusion in polyme matix composites with cyclic envionmental conditions, Intenational Jounal of Composite Stuctues, 49 (3), , Adda-Bedia E.A., Han W.S. and Vechey G., Simplified methods fo pediction of moistue diffusion in polyme matix composites with cyclic envionmental conditions, Intenational Jounal of Polyme & Polyme Composites, 6, 4, , Spinge G., Moistue content of composites unde tansient condition, Intenational Jounal of Composite Mateial, 11, , Spinge G., Numeical pocedues fo the solutions of one-dimensional Fickian diffusion poblems, Envionmental Effects on Composite Mateials, Technomic Publishing Co., Westpot, CT 06880, USA, , Hahn H.T. and Kim R.Y., Swelling of composite laminates, Advanced Composite Mateials- Envionmental Effects, ASTM STP 658, , Benkeddad A., Gediac M. and Vautin A., Computation of tansient hygoscopic stesses in laminated composite plates, Intenational Jounal of Composites Science and Technology, 56, , Tounsi A. and Adda-Bedia E.A., Simplified Method fo Pediction of Tansient Hygoscopic Stesses in Polyme Matix Composites with Symmetic Envionmental Conditions, Intenational Jounal of Applied Composite Mateials, 10 (1) edited by P.W.R Beaumont. Kluwe Academic Publishes, 1-18, Tounsi A, Adda-Bedia E.A. and Benachou A., New computational method fo pediction of tansient hygoscopic stesses duing moistue desoption in laminated composite plate with diffeent degee of anisotopy, Intenational Jounal of Themoplastic Composite Mateials, (in pess) Tounsi A. and Adda-Bedia E.A., Some obsevations on the evolution of tansvesal hygoscopic stesses in laminated composite plates: Effects of anisotopy, Intenational Jounal of Composite Stuctues, 59 (4) , Tounsi A., Adda-Bedia E.A., Vechey, G., Influence of anisotopy on the tansient hygoscopic stesses in polyme matix composites with cyclic envionmental conditions, Intenational Jounal of Composite Stuctues, 55 (4), , Tounsi A., Adda-Bedia E.A., Seei Z. and Vechey G., Analyse de l effet d anisotopie et des conditions d envionnement cycliques su les vaiations des containtes hygoscopiques tansitoies à l intéieu des plaques composites statifiées, Communication au 12 ème jounées nationales su les composites, JNC12, 15, 16 et 17 novembe 2000 (in Fench), Adda-Bedia E.A., Tounsi A. and Seei Z., A quantitative study on the influence of anisotopy on the hygothemal behaviou of the laminated composite plates, Eighth Intenational Confeence on Composite Engineeing ICCE 8, August 5-11, 2001, Teneife, Spain, Chamis C.C, Lak R.F. and Sinclai J.H., Integated theoy fo pedicting the hygothemomechanical esponse of advanced composite stuctual components, Advanced composite mateials Envionmental effects, ASTM STP 658, edited by J.R. Vinson, , Hape B.D. and Weitsman Y., On the effects of envionmental conditioning on esidual stesses in composite laminates, Intenational Jounal of Solids and Stuctues, 21 (8), , Vel S.S. and Bata R.C., Thee dimensional analysis of tansient themal stesses in functionally gaded plates, Intenational Jounal of Solids and Stuctues, 40, , Tsai S.W., Composite Design, (4 th Edition), edited by Tsai S.W, Think Composites, Dayton, Spinge G., Envionmental Effects, Section 16, Composites Design, edited by Tsai S.W, 4 th Edition, Think Composites, Dayton, OH 45419, USA,

15 APPENDIX List of pincipal symbols D C C 0 N N n α Diffusivity Moistue concentation Moistue concentation at the suface of laminate Numbe of the ply, o numbe of the subply. The in-plane stess esultant components. M n Moment components. α X Longitudinal tensile stength. X Longitudinal compessive stength. Y Tansvese tensile stength. Y Tansvese compessive stength. S Longitudinal shea stength. T Tempeatue ( K). L Thickness of plate. RH Relative humidity (%). z Axis tansvese to the plane of laminate; o thickness diection of a laminate. t Reduced time. {k n } Non-mechanical cuvatue. ΔT [Q] Room (o opeating) tempeatue cue tempeatue. Reduced stiffness matix fo plane stesses. t n Time. Designates non-mechanical, usually as a supescipt o intege. k e i ; (i = x,y,s) The fee expansions at the point z k. α i ; (i = x,y,s) Themal expansion coefficients. β i ; (i = x,y,s) Moistue expansion coefficients. σ n i σ n α θ [α] [β] [β] t [δ] ε i σ i (i = x,y,s) Non-mechanical stesses component in the ply axis. (α = 1,2,6) Non-mechanical stesses component in the laminates axis. Ply angle. The in-plane sub-matix of the compliance components of laminate. The in-plane/flexue coupling submatix of the compliance matix of laminate. Tansposed [β]. Flexual sub-matix of the compliance matix of laminate. Residual stain components in the ply axis. Residual stess components in the ply axis. {ε 0n } Non-mechanical in-plane stain components. 503