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Postbuckling analysis of three-dimensional textile composite cylindrical shells under axial compression in thermal environments
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COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 872–879 www.elsevier.com/locate/compscitech
Postbuckling analysis of three-dimensional textile composite cylindrical shells under axial compression in thermal environments Zhi-Min Li a, Hui-Shen Shen b
a,b,*
a School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China
Received 1 October 2006; received in revised form 10 August 2007; accepted 15 August 2007 Available online 24 August 2007
Abstract A postbuckling analysis is presented for a three-dimensional textile composite cylindrical shell of finite length subjected to axial compression in thermal environments. Based on a micro–macro-mechanical model, a three-dimensional textile composite may be as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the cylindrical shell. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on a higher order shear deformation shell theory with a von Ka´rma´n–Donnell-type of kinematic nonlinearity and including thermal effects. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, braided composite cylindrical shells with different values of geometric parameter and of fiber volume fraction in different cases of thermal environmental conditions. The results show that the shell has lower buckling load and postbuckling path when the temperature-dependent properties are taken into account. The results reveal that the temperature changes, the fiber volume fraction, and the shell geometric parameter have a significant effect on the buckling load and postbuckling behavior of textile composite cylindrical shells. 2007 Elsevier Ltd. All rights reserved. Keywords: A. Textile composites; Thermal effect; C. Buckling; C. Shell theory
1. Introduction Recently, textile composites which are manufactured by fabrication methods derived from the textile industry have drawn considerable attention. Unlike laminated composites, in which cracking and debonding may occur at high temperature due to the material property mismatch at the interface of two discrete materials, the textile composites are able to eliminate the delamination due to the interlacing of the tows in the through-thickness direction. Numerous investigations for determining their physical, mechanical and thermal properties are available in the literature, see, for example, [1–4]. Textile composites are now developed for general use as structural compo*
Corresponding author. Address: School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China. E-mail address: [email protected] (H.-S. Shen). 0266-3538/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.08.009
nents in aerospace, marine and offshore industries. One of the problems deserving special attention is the study of the postbuckling of cylindrical shells subjected to mechanical edge loads and in thermal environments. Many experimental studies have observed buckling and many attempts have been made to predict such phenomenon for textile composite cylindrical shells [5–14]. Among those, Jensen and Pai [5,6] discussed the influence of fiber undulations on buckling of thin filament-wound cylinders in axial compression and concluded that the stiffness coupling effects due to fiber undulation need to be properly accounted for in the analytical model. Chyssanthopoulos et al. [7–10] reported their experimental results and examined the validation of finite element models for the buckling of woven graphite fiber reinforced plastic (GFRP) shells and concluded that they have less sensitive to the effects of initial geometric imperfections than their isotropic counterparts. Harte and Fleck [11] examined the
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
deformation and failure behavior of glass fiber epoxy braided composite tubes under axial compression, torsion and their combination and found two types of diamond shaped buckling of the tube and fiber microbuckling in axial compression. Shu et al. [12,13] examined the planar biaxial compression/tension response of carbon twodimensional triaxial braided composites. In their analysis the three-dimensional finite element based on micromechanics study was carried out and the buckling mode was determined for the finite element representative unit cell under uniaxial and biaxial loading cases. Kuo et al. [14] examined the compression-induced damage in a threedimensional woven composite and concluded that the molding temperature is a critical parameter affecting the compressive strength. In the aforementioned investigations [5–14], however, material properties were considered to be independent of temperature. Moreover, Zeng and Wu [15] presented a postbuckling analysis for a stiffened textile composite cylindrical thin shell subjected to combined loading of external pressure and axial compression. In their analysis, a three-cell model was used and a boundary layer theory for the shell buckling suggested by Shen [16] was adopted. In the above studies, because the shells were considered as being relatively thin and therefore the transverse shear deformation was not accounted for. In fact, the loaded braiding cells may play a great role in moderately thick shells. However, studies on postbuckling of shear deformable textile composite cylindrical shells subjected to axial compression in thermal environments and with temperature-dependent material properties have not been seen in the literature. In the present study, we focus on the three-dimensional textile composite cylindrical shells and develop a micro– macro-mechanical model, from which four interior cells and two surface cells are performed. The geometry of each cell deeply relied on its position in the cross-section of the textile composite cylindrical shell. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on Reddy’s higher order shear deformation shell theory with a von Ka´rma´n–Donnell-type of kinematic nonlinearity and including thermal effects. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. The numerical illustrations show the full nonlinear postbuckling response of textile composite cylindrical shells under axial compression in thermal environments. 2. Theoretical development Consider a circular cylindrical shell made of threedimensional textile composites. The length, mean radius and total thickness of the shell are L, R and t. The shell is assumed to be relatively thick, geometrically imperfect, and is subjected to axial compression P0. The shell is
873
referred to a coordinate system (X, Y, Z), in which X and Y are in the axial and circumferential directions of the shell and Z is in the direction of the inward normal to the middle surface. The corresponding displacement are designated by U , V and W . Wx and Wy are the rotations of normal to the middle surface with respect to the Y- and X-axes, respectively. The origin of the coordinate system is located at the end of the shell on the middle plane. Denoting the initial deflection by W ðX ; Y Þ, let W ðX ; Y Þ be the additional deflection and F ðX ; Y Þ be the stress function for the stress resultants defined by N x ¼ F ;yy , N y ¼ F ;xx and N xy ¼ F ;xy , where a comma denotes partial differentiation with respect to the corresponding coordinates. Based on the micro–macro-mechanical model, a threedimensional textile composite may be as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the cylindrical shell. We assume that the yarn is transversely isotropic and the matrix is isotropic, from which the stiffness matrix can be expressed as ½C ¼ V f ½T ½C f ½T T þ V m ½T ½C m ½T T
ð1Þ
where Vf and Vm are the yarn and the matrix volume fractions and are related by Vf þVm ¼1
ð2Þ
and [Cf] and [Cm] are stiffness matrix respectively, and can be expressed as 2 cf11 cf12 cf13 0 0 6 cf22 cf23 0 0 6 6 6 cf33 0 0 ½C f ¼ 6 6 cf44 0 6 6 4 Sym cf55 2
for yarn and matrix, 3
0
7 7 7 7 7 0 7 7 7 0 5 0 0
cf66 cm11
6 6 6 6 ½C m ¼ 6 6 6 6 4
cm12
cm13
0
0
0
cm22
cm23 cm33
0 0
0 0
0 0
cm44
0
Sym
cm55
ð3aÞ
3
7 7 7 7 7 0 7 7 7 0 5
ð3bÞ
cm66 in which ð1 vf12 vf21 ÞEf1 ; D ð1 vf12 vf21 ÞEf2 ¼ D
cf11 ¼ cf22
cf13 ¼ cf12 ;
cf23 ¼ cf44 ¼
cf12 ¼
ð1 þ vf23 Þvf21 Ef1 ; D
ðvf23 þ vf12 vf21 ÞEf2 ; D
cf55 ¼ cf66 ¼ Gf12 D ¼ 1 2vf12 vf21 ð1 þ vf23 Þ v2f23 cm11 ¼ cm22 ¼ cm33 ¼
Em ; 1 v2m
cm12 ¼ cm13 ¼ cm23 ¼
vm E m 1 v2m
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Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
cm44 ¼ cm55 ¼ cm66 ¼ Gm ¼
Em 2ð1 þ mm Þ
ð3cÞ
where Ef1, Ef2, Gf12, mf12 and mf21 are the Young’s modulus, shear modulus and Poisson’s ratio, respectively, of the fiber, and Em, Gm and mm are corresponding properties for the matrix. In Eq. (1), [T] is the transform matrix and can be expressed as 3 2 2 l1 m21 n21 2m1 n1 2n1 l1 2l1 m1 6 l2 m22 n22 2m2 n2 2n2 l2 2l2 m2 7 7 6 1 7 6 2 2 2 7 6 l1 m n 2m n 2n l 2l m 3 3 3 3 3 3 3 3 7 ½T ¼ 6 6l l m m n n m n þ m n n l þ n l l m þ l m 7 6 2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 27 7 6 4 l3 l1 m3 m1 n3 n1 m3 n1 þ m1 n3 n3 l1 þ n1 l3 l3 m1 þ l1 m3 5 l1 l2 m1 m2 n1 n2 m1 n2 þ m2 n1 n1 l2 þ n2 l1 l1 m2 þ l2 m1 ð4Þ where lk, mk, nk (k = 1, 2, 3) are the direction cosines of a yarn with braiding angle c and inclination angles b, and can be defined as l1 ¼ cos c; l2 ¼ sin c cos b; l3 ¼ sin c sin b m1 ¼ 0; m2 ¼ sin b; m3 ¼ cos b n1 ¼ sin b;
n2 ¼ cos c;
and
ðAij ; Bij ; Dij ; Eij ; F ij ; H ij Þ ¼
n3 ¼ cos c sin b
ð5aÞ
c ¼ ðc1 þ c2 Þ=2 cj ¼ tan1 ð4Lj =hÞ ðj ¼ 1; 2Þ Lj ¼ 2riþj1 sinðp=2N Þ sinðuiþj1 þ p=2N Þ þ d=2 tan uiþj1
ðj ¼ 1; 2Þ
d ¼ h tan ci cos ui =2 b ¼ p=2 ui ui ¼ a cosf½d cosðp=N Þ=½4ri sinðp=2N Þg þ p=2N riþ1 ¼ ri þ d=ð2 sin ui Þ; r1 ¼ Din =2 þ d; r2m1 ¼ Dout =2 d
ði ¼ 1; 2; . . . ; 2M 2Þ
ð5bÞ
where ui is the solid cross-sectional orient angle of the yarn, M is referred to [M · N], being the number of radical columns of the braiding carriers, Din and Dout are the inner and outer diameters of the cylindrical shell and d is short axis of cross-sectional of the yarn, and the value of pitch length h can be measured from the preform exterior. Therefore, the transformed elastic constants Qij can be written as Z Z 1 c t=2 Qij ¼ ½Cdz dy ð6Þ ct 0 t=2 from which the ‘‘lamina’’ stiffnesses can be calculated by
þ2 Z c=2 Z zl M þ2 Z c=2 Z tlþ1 X 1 M 1 X ðQij Þ2 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy þ ðQij Þ1 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy 2c l¼4 0 2c tl zl 0 l¼4 M Z c=2 Z fl M Z c Z slþ1 X X 1 1 þ ðQij Þ3 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy þ ðQij Þ4 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy 2c l¼2 0 2c sl fl 0 l¼2 M Z c Z zl M Z c Z slþ1 1 X 1 X 2 3 4 6 þ ðQij Þ1 ð1; z; z ; z ; z ; z Þdz dy þ ðQij Þ2 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy 2c l¼2 c=2 sl 2c l¼2 c=2 zl M þ2 Z c Z fl M þ2 Z c Z tlþ1 1 X 1 X 2 3 4 6 ðQij Þ4 ð1; z; z ; z ; z ; z Þdz dy þ ðQij Þ3 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy þ 2c l¼4 c=2 tl 2c l¼4 c=2 fl Z Z 1 t=2þts 1 t=2 2 3 4 6 þ ððQij Þ5 þ ðQij Þ6 Þð1; z; z ; z ; z ; z Þdz dy þ ððQij Þ5 2 t=2 2 t=2ts
þ ðQij Þ6 Þð1; z; z2 ; z3 ; z4 ; z6 Þdz dy ði; j ¼ 1; 2; 6Þ X Z c=2 Z zl X Z c=2 Z tlþ1 1 Mþ2 1 Mþ2 ðAij ; Dij ; F ij Þ ¼ ðQij Þ2 ð1; z2 ; z4 Þdz dy þ ðQij Þ1 ð1; z2 ; z4 Þdz dy 2c l¼4 0 2c l¼4 0 tl zl M Z c=2 Z fl M Z c Z slþ1 1 X 1 X þ ðQij Þ3 ð1; z2 ; z4 Þdz dy þ ðQij Þ4 ð1; z2 ; z4 Þdz dy 2c l¼2 0 2c l¼2 0 fl sl M Z c Z zl M Z c Z slþ1 1 X 1 X þ ðQij Þ1 ð1; z2 ; z4 Þdz dy þ ðQij Þ2 ð1; z2 ; z4 Þdz dy 2c l¼2 c=2 sl 2c l¼2 c=2 zl M þ2 Z c Z fl þ2 Z c Z tlþ1 X 1 X 1 M ðQij Þ4 ð1; z2 ; z4 Þdz dy þ ðQij Þ3 ð1; z2 ; z4 Þdz dy þ 2c l¼4 c=2 tl 2c l¼4 c=2 fl Z Z 1 t=2þts 1 t=2 þ ððQij Þ5 þ ðQij Þ6 Þð1; z2 ; z4 Þdz dy þ ððQij Þ5 þ ðQij Þ6 Þð1; z2 ; z4 Þdz dy 2 t=2 2 t=2ts
ð7aÞ
ði; j ¼ 4; 5Þ
ð7bÞ
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
where ðQij Þ1 , ðQij Þ2 , ðQij Þ3 , ðQij Þ4 , ðQij Þ5 and ðQij Þ6 are referred to stiffness matrix of four interior cells and two surface cells in different positions. In Eqs. (6) and (7) (with l = 2, 3, . . ., M + 2) c ¼ h tan cl zl ¼ y= tan ul þ ð2l M 7Þd=2 sin ul fl ¼ y= tan ul þ ð2l M 3Þd=2 sin ul tl ¼ ð2l M 7Þd=2 sin ul
ð0 6 y 6 cÞ
ð8Þ
ð11Þ ~41 ðW Þ þ L ~42 ðWx Þ þ L ~43 ðWy Þ þ L ~44 ðF Þ L ~45 ðN Þ L ~46 ðS Þ ¼ 0 L T
T
ð12Þ ~ij ð Þ and the nonlinear operin which all linear operators L ~ ator Lð Þ are defined in [18]. Note that the geometric nonlinearity in the von Ka´rma´n sense is given in terms of ~ Þ in Eqs. (8) and (9). Lð The thermal forces, moments and higher order moments caused by temperature rise DT are defined by 2 T 3 2 3 N x M Tx P Tx Ax M Z tk X 6 NT MT PT 7 6 7 4 y 4 Ay 5 ð1; Z; Z 3 ÞDT dZ y y 5 ¼ k¼1
tk1
Axy
k
ð13aÞ and 2 T3 2 T3 2 T3 Sx Mx Px 4 6 T7 6 ST 7 6 M T 7 4 y 5 ¼ 4 y 5 2 4 Py 5 3t M Txy P Txy S Txy
l22 m22 2l2 m2
3 7 a11 5 a22
in which a11 and a22 are the thermal expansion coefficients measured in the fiber and transverse directions, respectively, and defined by [19]
~11 ðW Þ L ~12 ðWx Þ L ~13 ðWy Þ þ L ~14 ðF Þ L ~15 ðN T Þ L ~16 ðM T Þ L 1 ~ þ W ; F Þ ð9Þ F ;xx ¼ LðW R ~22 ðWx Þ þ L ~23 ðWy Þ L ~24 ðW Þ L ~25 ðN T Þ þ 1 W ;xx ~21 ðF Þ þ L L R 1~ ¼ LðW þ 2W ; W Þ ð10Þ 2 ~31 ðW Þ þ L ~32 ðWx Þ þ L ~33 ðWy Þ þ L ~34 ðF Þ L ~35 ðN T Þ L ~36 ðS T Þ ¼ 0 L
P Txy
Q26
32 2 Q16 l1 76 Q26 54 m21 Q66 2l1 m1
ð14Þ
In view of macro-mechanism, a shear deformable shell model is adopted. Reddy and Liu [17] developed a simple higher order shear deformation shell theory, in which the transverse shear strains are assumed to be parabolically distributed across the shell thickness and which contains the same dependent unknowns as in the first order shear deformation theory, and no shear correction factors are required. Based on Reddy’s higher order shear deformation theory, and using von Ka´rma´n–Donnell-type kinematic relations, the governing differential equations for three-dimensional textile composite cylindrical shells are derived and can be expressed in terms of a stress function F , two rotations Wx and Wy , and transverse displacement W , along with initial geometric imperfection W . They are
M Txy
Q12 Q22
ð0 6 y 6 cÞ
sl ¼ ð2l M 3Þd=2 sin ul ts ¼ ðt ðM 1Þd= sin ul Þ=2:
N Txy
where 2 3 2 Ax Q11 6 7 6 4 Ay 5 ¼ 4 Q12 Axy Q16
875
ð13bÞ
V f Ef1 af1 þ V m Em am V f Ef1 þ V m Em a22 ¼ V m ð1 þ vm Þam þ V f ð1 þ vf12 Þaf2 v12 a11
a11 ¼
ð15aÞ ð15bÞ
where af1, af2 and am are thermal expansion coefficients of the fiber and matrix, respectively. The two end edges of the shell is assumed to be simply supported or clamped, so that the boundary conditions are X = 0, L: W ¼ Wy ¼ 0;
Mx ¼ Px ¼ 0
ðsimply supportedÞ
W ¼ Wx ¼ Wy ¼ 0 ðclampedÞ Z 2pR N x dY þ P 0 ¼ 0
ð16aÞ ð16bÞ ð16cÞ
0
where M x is the bending moment and P x is higher order moment as defined in [17]. Also we have the closed (or periodicity) condition Z 2pR oV dY ¼ 0 ð17aÞ oY 0 or Z 2pR
2 o2 F 4 oWx o F þ A þ B E 12 21 3t2 21 oX oX 2 oY 2 0 2 4 oWy 4 o2 W o W 2 E21 þ E þ B22 2 E22 22 3t 3t oY oX 2 oY 2 # 2 W 1 oW oW oW T T þ A12 N x þ A22 N y dY ¼ 0 R 2 oY oY oY A22
ð17bÞ Because of Eq. (17), the in-plane boundary condition V ¼ 0 (at X = 0, L) is not needed in Eq. (16). The average end-shortening relationship is Z 2pR Z L Dx 1 oU ¼ dX dY L 2pRL 0 0 oX Z 2pR Z L 1 o2 F o2 F 4 oWx A11 2 þA12 2 þ B11 2 E11 ¼ oX 2pRL 0 3t oY oX 0 2 2 2 4 oWy 4 oW oW 1 oW E þE12 2 þ B12 2 E12 oY 3t2 11 oX 2 3t 2 oX oY oW oW ð18Þ A11 N Tx þA12 N Ty dX dY : oX oX
876
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
It is assumed that Em is a function of temperature, then all braiding stiffnesses Aij, Bij, Dij, etc., are also functions of temperature. Furthermore, in Eqs. (17b) and (18) the reduced stiffness matrices ½Aij ; ½Bij ; ½Dij ; ½Eij ; ½F ij and ½H ij (i, j = 1, 2, 6) are functions of temperature, determined through relationships [18] A ¼ A1 ;
B ¼ A1 B;
E ¼ A1 E;
D ¼ D BA1 B;
F ¼ F EA1 B;
H ¼ H EA1 E ð19Þ
90 γ = 40
60
o *
1:
Present (w /t = 0.0)
2: 3:
Present (w /t = 0.26) Harte and Fleck [11]
*
σx (MPa)
where Aij, Bij, etc., are the shell stiffnesses as given in detail in Eq. (7). It is worthy to note that the governing differential Eqs. (9)–(18) for a three-dimensional textile composite shell are identical in form to those of unsymmetric cross-ply laminated shells. Hence we may employ a singular perturbation technique to determine the buckling loads and postbuckling equilibrium paths. The solutions are obtained in the same form as previously reported in [18], and can be solved numerically. Note that the modal geometric imperfection is taken into account in the present analysis. The major difference herein is that the shell stiffnesses are determined based on a micro-mechanical model.
for braided composite tubes are compared in Fig. 1 with experimental results of Harte and Fleck [11]. The material vf = 0.23, properties adopted are: Ef = 63 GPa, Em = 2.8 GPa, and vm = 0.34. The results show that when an initial geometric imperfection was present, i.e. W =t ¼ 0:26 for the tube with braiding angle c = 40 and W =t ¼ 0:06 for the tube with c = 55, the present results are in reasonable agreement with the experimental results. A parametric study has been carried out and typical results are shown in Tables 2 and 3 and Figs. 2–4. In all of the figures, W =t denotes the dimensionless maximum initial geometric imperfection of the shell. For these examples R/t = 40 and the total thickness of the shell is t = 4 mm, pitch length h = 2.8 mm and M · N = 9 · 400. The carbon fiber tows are used as braiding material and the material properties of carbon fiber and epoxy adopted,
3. Numerical results and discussions
1
Table 1 Comparison of buckling loads Pcr (N) for (±30/90)T filament-wound shells subjected to axial compression (L = 228.6 mm, R = 76.2 mm, t = 0.51 mm) Cylinder
Pcr (N) Theory [6]
Test [6]
Present
N4O2 N6O2 N8O2 N10O2
26,100 27,530 27,710 28,860
21,240 21,610 22,360 21,190
23,654.27 24,886.45 24,976.37 25,864.73
a b
(22.9%) (27.4%) (23.9%) (36.2%)
(11, 11)a (11.4%)b (11, 11) (15.2%) (11, 11) (11.7%) (11, 11) (22.1%)
The number in brackets indicate the buckling mode (m, n). Relative errors = (theory test)/test · 100%.
2 3
30
0 0.0
0.1
0.2
0.3
0.4
Δx (mm)
(a) tube with γ =40o 60 γ = 55
50
o *
1:
Present (w /t = 0.0)
2: 3:
Present (w /t = 0.06) Harte and Fleck [11]
*
40
σx (MPa)
Numerical results are presented in this section for perfect and imperfect, three-dimensional braided composite cylindrical shells subjected to axial compression. We first examine the buckling loads for filament-wound cylinders under axial compression, excluding temperature effects. The results are listed in Table 1 and compared with theoretical and experimental results of Pai and Jensen [6]. The material properties adopted are: Ef1 = 230 GPa, Ef2 = 14.9 GPa, Gf12 = 29.4 GPa, Gf23 = 14.3 GPa, vf12 = 0.24, Em = 3.5 GPa, vm = 0.35, and the fiber volume fraction Vf = 0.63. The relative errors between theoretical results and experiments are also listed in Table 1. The comparisons show that the present results are lower than theoretical ones of Pai and Jensen [6], and agree reasonably well with experimental results. The discrepancies between these theoretical solutions are in part due to applying different shell theories. In addition, the nominal stress–strain curves
30 1 2 3 20
10
0 0.0
0.1
0.2
0.3
0.4
Δx (mm)
(b) tube with γ =55o Fig. 1. Comparisons of nominal stress–strain curves for a braided composite tube under axial compression.
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
877
Table 2 Comparison of buckling loads Pcr (kN) for perfect braided composite cylindrical shells with different values of shell geometric parameter and fiber volume fraction subjected to axial compression (R/t = 40, DT = 100 C) Z ¼ 100
Vf
Z ¼ 200
T-ID a
0.4 0.5 0.6
T-ID
T-D
T-ID
T-D
901.47 (3, 5) 1108.0 (3, 5) 1314.4 (3, 5)
953.43 (4, 5) 1151.4 (4, 5) 1349.0 (4, 5)
900.61 (4, 5) 1107.0 (4, 5) 1313.3 (4, 5)
950.38 (6, 5) 1151.0 (6, 5) 1348.1 (6, 5)
897.90 (6, 5) 1106.8 (6, 5) 1312.4 (6, 5)
The number in brackets indicate the buckling mode (m, n).
Table 3 Imperfection sensitivity k* for imperfect braided composite cylindrical shells (R/t = 40, Z ¼ 500) Temperature changes
Vf
W =t 0.0
0.05
0.1
0.15
DT = 0 C
0.4 0.5 0.6
1.0 1.0 1.0
0.9587 0.9587 0.9585
0.8542 0.8541 0.8540
0.7804 0.7803 0.7799
DT = 100 C
0.4 0.5 0.6
1.0 1.0 1.0
0.9708 0.9679 0.9660
0.8656 0.8629 0.8610
0.7913 0.7885 0.7867
0.4 0.5 0.6
1.0 1.0 1.0
0.9849 0.9776 0.9729
0.8789 0.8718 0.8673
0.8035 0.7968 0.7924
DT = 200 C
1500
3
21
1000
500 *
W /t = 0.0 *
W /t = 0.1 0 -2
0
2
4
6
8
10
ΔX (mm)
(a) load-shortening 1500
1 2
Z = 500, R/t = 40 (m, n)=(6, 5), Vf = 0.5 1: ΔT = 0 K 2: ΔT = 100 K 3: ΔT = 200 K
3
1000
P (kN)
as given in [19–21], are: Ef1 = 220 GPa, Ef2 = 14 GPa, Gf12 = 34 GPa, Gf23 = 5.5 GPa, m12 = 0.20 GPa, m23 = 0.25, af1 = 0.36 · 106/C, af2 = 18.0 · 106/C, Em = (4.3 0.003DT) GPa, mm = 0.34 and am = 40.0 · 106/C. Table 2 presents buckling loads Pcr (kN) for perfect braided composite cylindrical shells with different values of shell geometric parameter Z (=100, 200, 500) and fiber volume fraction Vf (= 0.4, 0.5 and 0.6) subjected to axial compression under thermal environmental condition DT = 100 C. Here, T-D represents Em is temperaturedependent, whereas T-ID represents Em is temperatureindependent, i.e.Em = 4.3 GPa. It can be seen that the buckling stress is increased with increasing in fiber volume fraction. The percentage increase is about +41% under T-ID case, and is about +46% under T-D case for all these three shells from Vf = 0.4–0.6. It can also be seen that the temperature reduces the buckling load when the temperature dependency is put into consideration. Fig. 2 shows the effect of temperature rise on the postbuckling load–shortening and load–deflection curves for a braided composite cylindrical shell with Z ¼ 500 and Vf = 0.5 under different environmental conditions DT = 0, 100 and 200 C. It is found that an initial extension occurs as the temperature increases and the buckling loads are reduced with increases in temperature. Fig. 3 shows the effect of fiber volume fraction Vf (= 0.4, 0.5 and 0.6) on the postbuckling load–shortening and load–deflection curves for a braided composite cylindrical shell with Z ¼ 500 under DT = 100 C. It can be seen that the buckling loads are reduced with decreasing in fiber vol-
Z = 500, R/t = 40 (m, n)=(6, 5), Vf = 0.5 1: ΔT = 0 K 2: ΔT = 100 K 3: ΔT = 200 K
P (kN)
a
954.46 (3, 5) 1152.5 (3, 5) 1350.2 (3, 5)
Z ¼ 500
T-D
500 *
W /t = 0.0 *
W /t = 0.1 0 0.0
0.5
1.0
1.5
2.0
W/t
(b) load-deflection Fig. 2. Effect of temperature rise on the postbuckling behavior of a braided composite cylindrical shell under axial compression.
ume fraction, and the postbuckling path becomes lower when Vf is decreased. Fig. 4 shows the effect of shell geometric parameter Z (=100, 200, 500) on the postbuckling load–shortening and load–deflection curves of braided composite cylindrical shell with Vf = 0.5 under DT = 100 C. The results show
878
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879 1500
1500
R/t = 40, ΔT = 100 K Vf = 0.5 1: Z = 100, (m, n) = (3, 5) 2: Z = 200, (m, n) = (4, 5) 3: Z = 500, (m, n) = (6, 5) 1
Z = 500, R/t = 40 (m, n)=(6, 5), ΔT = 100 K 1: Vf = 0.6 2: Vf = 0.5 3: Vf = 0.4 1000
P (kN)
P (kN)
1000
1 2 3
500
2 3 500
*
*
W /t = 0.0
W /t = 0.0
*
*
W /t = 0.1 0 -2
0
2
4 6 Δx (mm)
W /t = 0.1
8
0 -2
10
0
(a) load-shortening
2
8
10
(a) load-shortening 1500
1500 Z = 500, R/t = 40 (m, n)=(6, 5), ΔT = 100 K 1: Vf = 0.6 2: Vf = 0.5 3: Vf = 0.4
R/t = 40, ΔT = 100 K Vf = 0.5 1: Z = 100, (m, n) = (3, 5) 2: Z = 200, (m, n) = (4, 5) 3: Z = 500, (m, n) = (6, 5)
1000
1
P (kN)
1000
P (kN)
4 6 Δx (mm)
1
2 3
2 500
3
500
*
W /t = 0.0
*
W /t = 0.0
*
W /t = 0.1
*
W /t = 0.1 0 0.0
0.5
1.0
1.5
2.0
0 0.0
0.5
1.0
1.5
2.0
W/t
W/t
(b) load-deflection
(b) load-deflection Fig. 3. Effect of fiber volume fraction on the postbuckling behavior of braided composite cylindrical shells under axial compression.
Fig. 4. Effect of shell geometric parameter on the postbuckling behavior of braided composite cylindrical shells under axial compression.
that the slope of the postbuckling load–shortening curve for the shell with Z ¼ 100 are larger than others, and the shell has considerable postbuckling strength. From Figs. 2–4, the well-known ‘‘snap-through’’ phenomenon could be found. The elastic limit loads for imperfect shells can be achieved and imperfection sensitivity can be predicted. The postbuckling load–shortening and load– deflection curves for imperfect braided composite cylindrical shells have been plotted, along with the perfect shell results, in Figs. 2–4. Table 3 gives imperfection sensitivity k* for the imperfect braided composite cylindrical shells with Z ¼ 500 and different values of fiber volume fraction Vf (=0.4, 0.5 and 0.6) under thermal environmental conditions DT = 0, 100 and 200 C. Here, k* is the maximum value of Px for the imperfect shell, made dimensionless by dividing by the critical value of Px for the perfect shell as shown in Table 2. The results show that the imperfection
sensitivity becomes weak when temperature is increased and fiber volume fraction only has small effect on the imperfection sensitivity. 4. Concluding remarks In order to investigate the postbuckling behavior of three-dimensional textile composite cylindrical shells subjected to axial compression in thermal environments, a fully nonlinear postbuckling analysis is presented based on a micro–macro-mechanical model. Numerical calculations have been made for perfect and imperfect, braided composite cylindrical shells with different values of geometric parameter and of fiber volume fraction in different cases of thermal environmental conditions. The results show that the shell has lower buckling loads and postbuckling paths when the temperature-dependent proper-
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
ties are taken into account. The results reveal that the temperature changes, the fiber volume fraction, and the shell geometric parameter have a significant effect on the buckling load and postbuckling behavior of braided composite cylindrical shells. The results also confirm that the postbuckling equilibrium path is unstable and the shell structure is imperfection-sensitive in the loading case of axial compression. Acknowledgements This work is supported in part by the National Natural Science Foundation of China under Grant 50375091. The authors are grateful for this financial support. References [1] Ishikawa T, Chou TW. Elastic behavior of woven hybrid composites. J Compos Mater 1982;16:2–19. [2] Naik RA. Failure analysis of woven and braided fabric reinforced composites. J Compos Mater 1995;29:2334–63. [3] Wang YQ, Wang ASD. Spatial distribution of yarns and mechanical properties in 3D braided tubular composites. Appl Compos Mater 1997;4:121–32. [4] Kwon YW, Cho WM. Multilevel, micromechanical model for thermal analysis of woven-fabric composite materials. J Therm Stresses 2004;27:59–73. [5] Jensen DW, Pai SP. Influence of local fiber undulation on the global buckling of filament-wound cylinders. J Reinforced Plast Compos 1993;12:865–75. [6] Pai SP, Jensen DW. Influence of fiber undulations on buckling of thin filament-wound cylinders in axial compression. J Aerospace Eng 2001;14:12–20. [7] Elghazouli AY, Chyssanthopoulos MK, Esong IE. Buckling of woven GFRP cylinders under concentric and eccentric compression. Compos Struct 1999;45:13–27.
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[8] Chyssanthopoulos MK, Elghazouli AY, Esong IE. Validation of FE models for buckling analysis of woven GFRP shells. Compos Struct 2000;49:355–67. [9] Elghazouli AY, Chryssanthopoulos MK, Spagnoli A. Experimental response of glass-reinforced plastic cylinders under axial compression. Mar Struct 1998;11:347–71. [10] Spagnoli A, Elghazouli AY, Chryssanthopoulos MK. Numerical simulation of glass-reinforced plastic cylinders under axial compression. Mar Struct 2001;14:353–74. [11] Harte AM, Fleck NA. Deformation and failure mechanics of braided composite tubes in compression and torsion. Acta Mater 2000;48: 1259–71. [12] Shu CQ, Anthony MW, Khaled WS, Venkatesh A. Compressive response and failure of braided textile composites: Part 1—Experiments. Int J Non-Linear Mech 2004;39:635–48. [13] Shu CQ, Anthony MW, Khaled WS, Venkatesh A. Compressive response and failure of braided textile composites: Part 1—Computations. Int J Non-Linear Mech 2004;39:649–63. [14] Kuo W-S, Ko T-H, Shiah Y-C. Compressive damage in three-axis woven thermoplastic composites. J Thermoplast Compos Mater 2006;19:357–73. [15] Zeng T, Wu L-Z. Post-buckling analysis of stiffened braided cylindrical shells under combined external pressure and axial compression. Compos Struct 2003;60:455–66. [16] Shen H-S. Post-buckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and axial compression. Comp Struct 1997;63:335–48. [17] Reddy JN, Liu CF. A higher-order shear deformation theory of laminated elastic shells. Int J Eng Sci 1985;23:319–30. [18] Shen HS, Li QS. Thermomechanical postbuckling of shear deformable laminated cylindrical shells with local geometric imperfections. Int J Solids Struct 2002;39:4525–42. [19] Shapery RA. Thermal expansion coefficient of composite materials based on energy principles. J Compos Mater 1968;2:380–404. [20] Adams DF, Crane DA. Combined loading micromechanical analysis of a unidirectional composite. Composites 1984;15: 181–92. [21] Lee MS. International encyclopedia of composites, vol. 6. New York, USA: VHC Publishers; 1991.
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 872–879 www.elsevier.com/locate/compscitech
Postbuckling analysis of three-dimensional textile composite cylindrical shells under axial compression in thermal environments Zhi-Min Li a, Hui-Shen Shen b
a,b,*
a School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China
Received 1 October 2006; received in revised form 10 August 2007; accepted 15 August 2007 Available online 24 August 2007
Abstract A postbuckling analysis is presented for a three-dimensional textile composite cylindrical shell of finite length subjected to axial compression in thermal environments. Based on a micro–macro-mechanical model, a three-dimensional textile composite may be as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the cylindrical shell. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on a higher order shear deformation shell theory with a von Ka´rma´n–Donnell-type of kinematic nonlinearity and including thermal effects. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, braided composite cylindrical shells with different values of geometric parameter and of fiber volume fraction in different cases of thermal environmental conditions. The results show that the shell has lower buckling load and postbuckling path when the temperature-dependent properties are taken into account. The results reveal that the temperature changes, the fiber volume fraction, and the shell geometric parameter have a significant effect on the buckling load and postbuckling behavior of textile composite cylindrical shells. 2007 Elsevier Ltd. All rights reserved. Keywords: A. Textile composites; Thermal effect; C. Buckling; C. Shell theory
1. Introduction Recently, textile composites which are manufactured by fabrication methods derived from the textile industry have drawn considerable attention. Unlike laminated composites, in which cracking and debonding may occur at high temperature due to the material property mismatch at the interface of two discrete materials, the textile composites are able to eliminate the delamination due to the interlacing of the tows in the through-thickness direction. Numerous investigations for determining their physical, mechanical and thermal properties are available in the literature, see, for example, [1–4]. Textile composites are now developed for general use as structural compo*
Corresponding author. Address: School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China. E-mail address: [email protected] (H.-S. Shen). 0266-3538/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.08.009
nents in aerospace, marine and offshore industries. One of the problems deserving special attention is the study of the postbuckling of cylindrical shells subjected to mechanical edge loads and in thermal environments. Many experimental studies have observed buckling and many attempts have been made to predict such phenomenon for textile composite cylindrical shells [5–14]. Among those, Jensen and Pai [5,6] discussed the influence of fiber undulations on buckling of thin filament-wound cylinders in axial compression and concluded that the stiffness coupling effects due to fiber undulation need to be properly accounted for in the analytical model. Chyssanthopoulos et al. [7–10] reported their experimental results and examined the validation of finite element models for the buckling of woven graphite fiber reinforced plastic (GFRP) shells and concluded that they have less sensitive to the effects of initial geometric imperfections than their isotropic counterparts. Harte and Fleck [11] examined the
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
deformation and failure behavior of glass fiber epoxy braided composite tubes under axial compression, torsion and their combination and found two types of diamond shaped buckling of the tube and fiber microbuckling in axial compression. Shu et al. [12,13] examined the planar biaxial compression/tension response of carbon twodimensional triaxial braided composites. In their analysis the three-dimensional finite element based on micromechanics study was carried out and the buckling mode was determined for the finite element representative unit cell under uniaxial and biaxial loading cases. Kuo et al. [14] examined the compression-induced damage in a threedimensional woven composite and concluded that the molding temperature is a critical parameter affecting the compressive strength. In the aforementioned investigations [5–14], however, material properties were considered to be independent of temperature. Moreover, Zeng and Wu [15] presented a postbuckling analysis for a stiffened textile composite cylindrical thin shell subjected to combined loading of external pressure and axial compression. In their analysis, a three-cell model was used and a boundary layer theory for the shell buckling suggested by Shen [16] was adopted. In the above studies, because the shells were considered as being relatively thin and therefore the transverse shear deformation was not accounted for. In fact, the loaded braiding cells may play a great role in moderately thick shells. However, studies on postbuckling of shear deformable textile composite cylindrical shells subjected to axial compression in thermal environments and with temperature-dependent material properties have not been seen in the literature. In the present study, we focus on the three-dimensional textile composite cylindrical shells and develop a micro– macro-mechanical model, from which four interior cells and two surface cells are performed. The geometry of each cell deeply relied on its position in the cross-section of the textile composite cylindrical shell. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on Reddy’s higher order shear deformation shell theory with a von Ka´rma´n–Donnell-type of kinematic nonlinearity and including thermal effects. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. The numerical illustrations show the full nonlinear postbuckling response of textile composite cylindrical shells under axial compression in thermal environments. 2. Theoretical development Consider a circular cylindrical shell made of threedimensional textile composites. The length, mean radius and total thickness of the shell are L, R and t. The shell is assumed to be relatively thick, geometrically imperfect, and is subjected to axial compression P0. The shell is
873
referred to a coordinate system (X, Y, Z), in which X and Y are in the axial and circumferential directions of the shell and Z is in the direction of the inward normal to the middle surface. The corresponding displacement are designated by U , V and W . Wx and Wy are the rotations of normal to the middle surface with respect to the Y- and X-axes, respectively. The origin of the coordinate system is located at the end of the shell on the middle plane. Denoting the initial deflection by W ðX ; Y Þ, let W ðX ; Y Þ be the additional deflection and F ðX ; Y Þ be the stress function for the stress resultants defined by N x ¼ F ;yy , N y ¼ F ;xx and N xy ¼ F ;xy , where a comma denotes partial differentiation with respect to the corresponding coordinates. Based on the micro–macro-mechanical model, a threedimensional textile composite may be as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the cylindrical shell. We assume that the yarn is transversely isotropic and the matrix is isotropic, from which the stiffness matrix can be expressed as ½C ¼ V f ½T ½C f ½T T þ V m ½T ½C m ½T T
ð1Þ
where Vf and Vm are the yarn and the matrix volume fractions and are related by Vf þVm ¼1
ð2Þ
and [Cf] and [Cm] are stiffness matrix respectively, and can be expressed as 2 cf11 cf12 cf13 0 0 6 cf22 cf23 0 0 6 6 6 cf33 0 0 ½C f ¼ 6 6 cf44 0 6 6 4 Sym cf55 2
for yarn and matrix, 3
0
7 7 7 7 7 0 7 7 7 0 5 0 0
cf66 cm11
6 6 6 6 ½C m ¼ 6 6 6 6 4
cm12
cm13
0
0
0
cm22
cm23 cm33
0 0
0 0
0 0
cm44
0
Sym
cm55
ð3aÞ
3
7 7 7 7 7 0 7 7 7 0 5
ð3bÞ
cm66 in which ð1 vf12 vf21 ÞEf1 ; D ð1 vf12 vf21 ÞEf2 ¼ D
cf11 ¼ cf22
cf13 ¼ cf12 ;
cf23 ¼ cf44 ¼
cf12 ¼
ð1 þ vf23 Þvf21 Ef1 ; D
ðvf23 þ vf12 vf21 ÞEf2 ; D
cf55 ¼ cf66 ¼ Gf12 D ¼ 1 2vf12 vf21 ð1 þ vf23 Þ v2f23 cm11 ¼ cm22 ¼ cm33 ¼
Em ; 1 v2m
cm12 ¼ cm13 ¼ cm23 ¼
vm E m 1 v2m
874
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
cm44 ¼ cm55 ¼ cm66 ¼ Gm ¼
Em 2ð1 þ mm Þ
ð3cÞ
where Ef1, Ef2, Gf12, mf12 and mf21 are the Young’s modulus, shear modulus and Poisson’s ratio, respectively, of the fiber, and Em, Gm and mm are corresponding properties for the matrix. In Eq. (1), [T] is the transform matrix and can be expressed as 3 2 2 l1 m21 n21 2m1 n1 2n1 l1 2l1 m1 6 l2 m22 n22 2m2 n2 2n2 l2 2l2 m2 7 7 6 1 7 6 2 2 2 7 6 l1 m n 2m n 2n l 2l m 3 3 3 3 3 3 3 3 7 ½T ¼ 6 6l l m m n n m n þ m n n l þ n l l m þ l m 7 6 2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 27 7 6 4 l3 l1 m3 m1 n3 n1 m3 n1 þ m1 n3 n3 l1 þ n1 l3 l3 m1 þ l1 m3 5 l1 l2 m1 m2 n1 n2 m1 n2 þ m2 n1 n1 l2 þ n2 l1 l1 m2 þ l2 m1 ð4Þ where lk, mk, nk (k = 1, 2, 3) are the direction cosines of a yarn with braiding angle c and inclination angles b, and can be defined as l1 ¼ cos c; l2 ¼ sin c cos b; l3 ¼ sin c sin b m1 ¼ 0; m2 ¼ sin b; m3 ¼ cos b n1 ¼ sin b;
n2 ¼ cos c;
and
ðAij ; Bij ; Dij ; Eij ; F ij ; H ij Þ ¼
n3 ¼ cos c sin b
ð5aÞ
c ¼ ðc1 þ c2 Þ=2 cj ¼ tan1 ð4Lj =hÞ ðj ¼ 1; 2Þ Lj ¼ 2riþj1 sinðp=2N Þ sinðuiþj1 þ p=2N Þ þ d=2 tan uiþj1
ðj ¼ 1; 2Þ
d ¼ h tan ci cos ui =2 b ¼ p=2 ui ui ¼ a cosf½d cosðp=N Þ=½4ri sinðp=2N Þg þ p=2N riþ1 ¼ ri þ d=ð2 sin ui Þ; r1 ¼ Din =2 þ d; r2m1 ¼ Dout =2 d
ði ¼ 1; 2; . . . ; 2M 2Þ
ð5bÞ
where ui is the solid cross-sectional orient angle of the yarn, M is referred to [M · N], being the number of radical columns of the braiding carriers, Din and Dout are the inner and outer diameters of the cylindrical shell and d is short axis of cross-sectional of the yarn, and the value of pitch length h can be measured from the preform exterior. Therefore, the transformed elastic constants Qij can be written as Z Z 1 c t=2 Qij ¼ ½Cdz dy ð6Þ ct 0 t=2 from which the ‘‘lamina’’ stiffnesses can be calculated by
þ2 Z c=2 Z zl M þ2 Z c=2 Z tlþ1 X 1 M 1 X ðQij Þ2 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy þ ðQij Þ1 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy 2c l¼4 0 2c tl zl 0 l¼4 M Z c=2 Z fl M Z c Z slþ1 X X 1 1 þ ðQij Þ3 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy þ ðQij Þ4 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy 2c l¼2 0 2c sl fl 0 l¼2 M Z c Z zl M Z c Z slþ1 1 X 1 X 2 3 4 6 þ ðQij Þ1 ð1; z; z ; z ; z ; z Þdz dy þ ðQij Þ2 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy 2c l¼2 c=2 sl 2c l¼2 c=2 zl M þ2 Z c Z fl M þ2 Z c Z tlþ1 1 X 1 X 2 3 4 6 ðQij Þ4 ð1; z; z ; z ; z ; z Þdz dy þ ðQij Þ3 ð1; z; z2 ; z3 ; z4 ; z6 Þdz dy þ 2c l¼4 c=2 tl 2c l¼4 c=2 fl Z Z 1 t=2þts 1 t=2 2 3 4 6 þ ððQij Þ5 þ ðQij Þ6 Þð1; z; z ; z ; z ; z Þdz dy þ ððQij Þ5 2 t=2 2 t=2ts
þ ðQij Þ6 Þð1; z; z2 ; z3 ; z4 ; z6 Þdz dy ði; j ¼ 1; 2; 6Þ X Z c=2 Z zl X Z c=2 Z tlþ1 1 Mþ2 1 Mþ2 ðAij ; Dij ; F ij Þ ¼ ðQij Þ2 ð1; z2 ; z4 Þdz dy þ ðQij Þ1 ð1; z2 ; z4 Þdz dy 2c l¼4 0 2c l¼4 0 tl zl M Z c=2 Z fl M Z c Z slþ1 1 X 1 X þ ðQij Þ3 ð1; z2 ; z4 Þdz dy þ ðQij Þ4 ð1; z2 ; z4 Þdz dy 2c l¼2 0 2c l¼2 0 fl sl M Z c Z zl M Z c Z slþ1 1 X 1 X þ ðQij Þ1 ð1; z2 ; z4 Þdz dy þ ðQij Þ2 ð1; z2 ; z4 Þdz dy 2c l¼2 c=2 sl 2c l¼2 c=2 zl M þ2 Z c Z fl þ2 Z c Z tlþ1 X 1 X 1 M ðQij Þ4 ð1; z2 ; z4 Þdz dy þ ðQij Þ3 ð1; z2 ; z4 Þdz dy þ 2c l¼4 c=2 tl 2c l¼4 c=2 fl Z Z 1 t=2þts 1 t=2 þ ððQij Þ5 þ ðQij Þ6 Þð1; z2 ; z4 Þdz dy þ ððQij Þ5 þ ðQij Þ6 Þð1; z2 ; z4 Þdz dy 2 t=2 2 t=2ts
ð7aÞ
ði; j ¼ 4; 5Þ
ð7bÞ
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
where ðQij Þ1 , ðQij Þ2 , ðQij Þ3 , ðQij Þ4 , ðQij Þ5 and ðQij Þ6 are referred to stiffness matrix of four interior cells and two surface cells in different positions. In Eqs. (6) and (7) (with l = 2, 3, . . ., M + 2) c ¼ h tan cl zl ¼ y= tan ul þ ð2l M 7Þd=2 sin ul fl ¼ y= tan ul þ ð2l M 3Þd=2 sin ul tl ¼ ð2l M 7Þd=2 sin ul
ð0 6 y 6 cÞ
ð8Þ
ð11Þ ~41 ðW Þ þ L ~42 ðWx Þ þ L ~43 ðWy Þ þ L ~44 ðF Þ L ~45 ðN Þ L ~46 ðS Þ ¼ 0 L T
T
ð12Þ ~ij ð Þ and the nonlinear operin which all linear operators L ~ ator Lð Þ are defined in [18]. Note that the geometric nonlinearity in the von Ka´rma´n sense is given in terms of ~ Þ in Eqs. (8) and (9). Lð The thermal forces, moments and higher order moments caused by temperature rise DT are defined by 2 T 3 2 3 N x M Tx P Tx Ax M Z tk X 6 NT MT PT 7 6 7 4 y 4 Ay 5 ð1; Z; Z 3 ÞDT dZ y y 5 ¼ k¼1
tk1
Axy
k
ð13aÞ and 2 T3 2 T3 2 T3 Sx Mx Px 4 6 T7 6 ST 7 6 M T 7 4 y 5 ¼ 4 y 5 2 4 Py 5 3t M Txy P Txy S Txy
l22 m22 2l2 m2
3 7 a11 5 a22
in which a11 and a22 are the thermal expansion coefficients measured in the fiber and transverse directions, respectively, and defined by [19]
~11 ðW Þ L ~12 ðWx Þ L ~13 ðWy Þ þ L ~14 ðF Þ L ~15 ðN T Þ L ~16 ðM T Þ L 1 ~ þ W ; F Þ ð9Þ F ;xx ¼ LðW R ~22 ðWx Þ þ L ~23 ðWy Þ L ~24 ðW Þ L ~25 ðN T Þ þ 1 W ;xx ~21 ðF Þ þ L L R 1~ ¼ LðW þ 2W ; W Þ ð10Þ 2 ~31 ðW Þ þ L ~32 ðWx Þ þ L ~33 ðWy Þ þ L ~34 ðF Þ L ~35 ðN T Þ L ~36 ðS T Þ ¼ 0 L
P Txy
Q26
32 2 Q16 l1 76 Q26 54 m21 Q66 2l1 m1
ð14Þ
In view of macro-mechanism, a shear deformable shell model is adopted. Reddy and Liu [17] developed a simple higher order shear deformation shell theory, in which the transverse shear strains are assumed to be parabolically distributed across the shell thickness and which contains the same dependent unknowns as in the first order shear deformation theory, and no shear correction factors are required. Based on Reddy’s higher order shear deformation theory, and using von Ka´rma´n–Donnell-type kinematic relations, the governing differential equations for three-dimensional textile composite cylindrical shells are derived and can be expressed in terms of a stress function F , two rotations Wx and Wy , and transverse displacement W , along with initial geometric imperfection W . They are
M Txy
Q12 Q22
ð0 6 y 6 cÞ
sl ¼ ð2l M 3Þd=2 sin ul ts ¼ ðt ðM 1Þd= sin ul Þ=2:
N Txy
where 2 3 2 Ax Q11 6 7 6 4 Ay 5 ¼ 4 Q12 Axy Q16
875
ð13bÞ
V f Ef1 af1 þ V m Em am V f Ef1 þ V m Em a22 ¼ V m ð1 þ vm Þam þ V f ð1 þ vf12 Þaf2 v12 a11
a11 ¼
ð15aÞ ð15bÞ
where af1, af2 and am are thermal expansion coefficients of the fiber and matrix, respectively. The two end edges of the shell is assumed to be simply supported or clamped, so that the boundary conditions are X = 0, L: W ¼ Wy ¼ 0;
Mx ¼ Px ¼ 0
ðsimply supportedÞ
W ¼ Wx ¼ Wy ¼ 0 ðclampedÞ Z 2pR N x dY þ P 0 ¼ 0
ð16aÞ ð16bÞ ð16cÞ
0
where M x is the bending moment and P x is higher order moment as defined in [17]. Also we have the closed (or periodicity) condition Z 2pR oV dY ¼ 0 ð17aÞ oY 0 or Z 2pR
2 o2 F 4 oWx o F þ A þ B E 12 21 3t2 21 oX oX 2 oY 2 0 2 4 oWy 4 o2 W o W 2 E21 þ E þ B22 2 E22 22 3t 3t oY oX 2 oY 2 # 2 W 1 oW oW oW T T þ A12 N x þ A22 N y dY ¼ 0 R 2 oY oY oY A22
ð17bÞ Because of Eq. (17), the in-plane boundary condition V ¼ 0 (at X = 0, L) is not needed in Eq. (16). The average end-shortening relationship is Z 2pR Z L Dx 1 oU ¼ dX dY L 2pRL 0 0 oX Z 2pR Z L 1 o2 F o2 F 4 oWx A11 2 þA12 2 þ B11 2 E11 ¼ oX 2pRL 0 3t oY oX 0 2 2 2 4 oWy 4 oW oW 1 oW E þE12 2 þ B12 2 E12 oY 3t2 11 oX 2 3t 2 oX oY oW oW ð18Þ A11 N Tx þA12 N Ty dX dY : oX oX
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Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
It is assumed that Em is a function of temperature, then all braiding stiffnesses Aij, Bij, Dij, etc., are also functions of temperature. Furthermore, in Eqs. (17b) and (18) the reduced stiffness matrices ½Aij ; ½Bij ; ½Dij ; ½Eij ; ½F ij and ½H ij (i, j = 1, 2, 6) are functions of temperature, determined through relationships [18] A ¼ A1 ;
B ¼ A1 B;
E ¼ A1 E;
D ¼ D BA1 B;
F ¼ F EA1 B;
H ¼ H EA1 E ð19Þ
90 γ = 40
60
o *
1:
Present (w /t = 0.0)
2: 3:
Present (w /t = 0.26) Harte and Fleck [11]
*
σx (MPa)
where Aij, Bij, etc., are the shell stiffnesses as given in detail in Eq. (7). It is worthy to note that the governing differential Eqs. (9)–(18) for a three-dimensional textile composite shell are identical in form to those of unsymmetric cross-ply laminated shells. Hence we may employ a singular perturbation technique to determine the buckling loads and postbuckling equilibrium paths. The solutions are obtained in the same form as previously reported in [18], and can be solved numerically. Note that the modal geometric imperfection is taken into account in the present analysis. The major difference herein is that the shell stiffnesses are determined based on a micro-mechanical model.
for braided composite tubes are compared in Fig. 1 with experimental results of Harte and Fleck [11]. The material vf = 0.23, properties adopted are: Ef = 63 GPa, Em = 2.8 GPa, and vm = 0.34. The results show that when an initial geometric imperfection was present, i.e. W =t ¼ 0:26 for the tube with braiding angle c = 40 and W =t ¼ 0:06 for the tube with c = 55, the present results are in reasonable agreement with the experimental results. A parametric study has been carried out and typical results are shown in Tables 2 and 3 and Figs. 2–4. In all of the figures, W =t denotes the dimensionless maximum initial geometric imperfection of the shell. For these examples R/t = 40 and the total thickness of the shell is t = 4 mm, pitch length h = 2.8 mm and M · N = 9 · 400. The carbon fiber tows are used as braiding material and the material properties of carbon fiber and epoxy adopted,
3. Numerical results and discussions
1
Table 1 Comparison of buckling loads Pcr (N) for (±30/90)T filament-wound shells subjected to axial compression (L = 228.6 mm, R = 76.2 mm, t = 0.51 mm) Cylinder
Pcr (N) Theory [6]
Test [6]
Present
N4O2 N6O2 N8O2 N10O2
26,100 27,530 27,710 28,860
21,240 21,610 22,360 21,190
23,654.27 24,886.45 24,976.37 25,864.73
a b
(22.9%) (27.4%) (23.9%) (36.2%)
(11, 11)a (11.4%)b (11, 11) (15.2%) (11, 11) (11.7%) (11, 11) (22.1%)
The number in brackets indicate the buckling mode (m, n). Relative errors = (theory test)/test · 100%.
2 3
30
0 0.0
0.1
0.2
0.3
0.4
Δx (mm)
(a) tube with γ =40o 60 γ = 55
50
o *
1:
Present (w /t = 0.0)
2: 3:
Present (w /t = 0.06) Harte and Fleck [11]
*
40
σx (MPa)
Numerical results are presented in this section for perfect and imperfect, three-dimensional braided composite cylindrical shells subjected to axial compression. We first examine the buckling loads for filament-wound cylinders under axial compression, excluding temperature effects. The results are listed in Table 1 and compared with theoretical and experimental results of Pai and Jensen [6]. The material properties adopted are: Ef1 = 230 GPa, Ef2 = 14.9 GPa, Gf12 = 29.4 GPa, Gf23 = 14.3 GPa, vf12 = 0.24, Em = 3.5 GPa, vm = 0.35, and the fiber volume fraction Vf = 0.63. The relative errors between theoretical results and experiments are also listed in Table 1. The comparisons show that the present results are lower than theoretical ones of Pai and Jensen [6], and agree reasonably well with experimental results. The discrepancies between these theoretical solutions are in part due to applying different shell theories. In addition, the nominal stress–strain curves
30 1 2 3 20
10
0 0.0
0.1
0.2
0.3
0.4
Δx (mm)
(b) tube with γ =55o Fig. 1. Comparisons of nominal stress–strain curves for a braided composite tube under axial compression.
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
877
Table 2 Comparison of buckling loads Pcr (kN) for perfect braided composite cylindrical shells with different values of shell geometric parameter and fiber volume fraction subjected to axial compression (R/t = 40, DT = 100 C) Z ¼ 100
Vf
Z ¼ 200
T-ID a
0.4 0.5 0.6
T-ID
T-D
T-ID
T-D
901.47 (3, 5) 1108.0 (3, 5) 1314.4 (3, 5)
953.43 (4, 5) 1151.4 (4, 5) 1349.0 (4, 5)
900.61 (4, 5) 1107.0 (4, 5) 1313.3 (4, 5)
950.38 (6, 5) 1151.0 (6, 5) 1348.1 (6, 5)
897.90 (6, 5) 1106.8 (6, 5) 1312.4 (6, 5)
The number in brackets indicate the buckling mode (m, n).
Table 3 Imperfection sensitivity k* for imperfect braided composite cylindrical shells (R/t = 40, Z ¼ 500) Temperature changes
Vf
W =t 0.0
0.05
0.1
0.15
DT = 0 C
0.4 0.5 0.6
1.0 1.0 1.0
0.9587 0.9587 0.9585
0.8542 0.8541 0.8540
0.7804 0.7803 0.7799
DT = 100 C
0.4 0.5 0.6
1.0 1.0 1.0
0.9708 0.9679 0.9660
0.8656 0.8629 0.8610
0.7913 0.7885 0.7867
0.4 0.5 0.6
1.0 1.0 1.0
0.9849 0.9776 0.9729
0.8789 0.8718 0.8673
0.8035 0.7968 0.7924
DT = 200 C
1500
3
21
1000
500 *
W /t = 0.0 *
W /t = 0.1 0 -2
0
2
4
6
8
10
ΔX (mm)
(a) load-shortening 1500
1 2
Z = 500, R/t = 40 (m, n)=(6, 5), Vf = 0.5 1: ΔT = 0 K 2: ΔT = 100 K 3: ΔT = 200 K
3
1000
P (kN)
as given in [19–21], are: Ef1 = 220 GPa, Ef2 = 14 GPa, Gf12 = 34 GPa, Gf23 = 5.5 GPa, m12 = 0.20 GPa, m23 = 0.25, af1 = 0.36 · 106/C, af2 = 18.0 · 106/C, Em = (4.3 0.003DT) GPa, mm = 0.34 and am = 40.0 · 106/C. Table 2 presents buckling loads Pcr (kN) for perfect braided composite cylindrical shells with different values of shell geometric parameter Z (=100, 200, 500) and fiber volume fraction Vf (= 0.4, 0.5 and 0.6) subjected to axial compression under thermal environmental condition DT = 100 C. Here, T-D represents Em is temperaturedependent, whereas T-ID represents Em is temperatureindependent, i.e.Em = 4.3 GPa. It can be seen that the buckling stress is increased with increasing in fiber volume fraction. The percentage increase is about +41% under T-ID case, and is about +46% under T-D case for all these three shells from Vf = 0.4–0.6. It can also be seen that the temperature reduces the buckling load when the temperature dependency is put into consideration. Fig. 2 shows the effect of temperature rise on the postbuckling load–shortening and load–deflection curves for a braided composite cylindrical shell with Z ¼ 500 and Vf = 0.5 under different environmental conditions DT = 0, 100 and 200 C. It is found that an initial extension occurs as the temperature increases and the buckling loads are reduced with increases in temperature. Fig. 3 shows the effect of fiber volume fraction Vf (= 0.4, 0.5 and 0.6) on the postbuckling load–shortening and load–deflection curves for a braided composite cylindrical shell with Z ¼ 500 under DT = 100 C. It can be seen that the buckling loads are reduced with decreasing in fiber vol-
Z = 500, R/t = 40 (m, n)=(6, 5), Vf = 0.5 1: ΔT = 0 K 2: ΔT = 100 K 3: ΔT = 200 K
P (kN)
a
954.46 (3, 5) 1152.5 (3, 5) 1350.2 (3, 5)
Z ¼ 500
T-D
500 *
W /t = 0.0 *
W /t = 0.1 0 0.0
0.5
1.0
1.5
2.0
W/t
(b) load-deflection Fig. 2. Effect of temperature rise on the postbuckling behavior of a braided composite cylindrical shell under axial compression.
ume fraction, and the postbuckling path becomes lower when Vf is decreased. Fig. 4 shows the effect of shell geometric parameter Z (=100, 200, 500) on the postbuckling load–shortening and load–deflection curves of braided composite cylindrical shell with Vf = 0.5 under DT = 100 C. The results show
878
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879 1500
1500
R/t = 40, ΔT = 100 K Vf = 0.5 1: Z = 100, (m, n) = (3, 5) 2: Z = 200, (m, n) = (4, 5) 3: Z = 500, (m, n) = (6, 5) 1
Z = 500, R/t = 40 (m, n)=(6, 5), ΔT = 100 K 1: Vf = 0.6 2: Vf = 0.5 3: Vf = 0.4 1000
P (kN)
P (kN)
1000
1 2 3
500
2 3 500
*
*
W /t = 0.0
W /t = 0.0
*
*
W /t = 0.1 0 -2
0
2
4 6 Δx (mm)
W /t = 0.1
8
0 -2
10
0
(a) load-shortening
2
8
10
(a) load-shortening 1500
1500 Z = 500, R/t = 40 (m, n)=(6, 5), ΔT = 100 K 1: Vf = 0.6 2: Vf = 0.5 3: Vf = 0.4
R/t = 40, ΔT = 100 K Vf = 0.5 1: Z = 100, (m, n) = (3, 5) 2: Z = 200, (m, n) = (4, 5) 3: Z = 500, (m, n) = (6, 5)
1000
1
P (kN)
1000
P (kN)
4 6 Δx (mm)
1
2 3
2 500
3
500
*
W /t = 0.0
*
W /t = 0.0
*
W /t = 0.1
*
W /t = 0.1 0 0.0
0.5
1.0
1.5
2.0
0 0.0
0.5
1.0
1.5
2.0
W/t
W/t
(b) load-deflection
(b) load-deflection Fig. 3. Effect of fiber volume fraction on the postbuckling behavior of braided composite cylindrical shells under axial compression.
Fig. 4. Effect of shell geometric parameter on the postbuckling behavior of braided composite cylindrical shells under axial compression.
that the slope of the postbuckling load–shortening curve for the shell with Z ¼ 100 are larger than others, and the shell has considerable postbuckling strength. From Figs. 2–4, the well-known ‘‘snap-through’’ phenomenon could be found. The elastic limit loads for imperfect shells can be achieved and imperfection sensitivity can be predicted. The postbuckling load–shortening and load– deflection curves for imperfect braided composite cylindrical shells have been plotted, along with the perfect shell results, in Figs. 2–4. Table 3 gives imperfection sensitivity k* for the imperfect braided composite cylindrical shells with Z ¼ 500 and different values of fiber volume fraction Vf (=0.4, 0.5 and 0.6) under thermal environmental conditions DT = 0, 100 and 200 C. Here, k* is the maximum value of Px for the imperfect shell, made dimensionless by dividing by the critical value of Px for the perfect shell as shown in Table 2. The results show that the imperfection
sensitivity becomes weak when temperature is increased and fiber volume fraction only has small effect on the imperfection sensitivity. 4. Concluding remarks In order to investigate the postbuckling behavior of three-dimensional textile composite cylindrical shells subjected to axial compression in thermal environments, a fully nonlinear postbuckling analysis is presented based on a micro–macro-mechanical model. Numerical calculations have been made for perfect and imperfect, braided composite cylindrical shells with different values of geometric parameter and of fiber volume fraction in different cases of thermal environmental conditions. The results show that the shell has lower buckling loads and postbuckling paths when the temperature-dependent proper-
Z.-M. Li, H.-S. Shen / Composites Science and Technology 68 (2008) 872–879
ties are taken into account. The results reveal that the temperature changes, the fiber volume fraction, and the shell geometric parameter have a significant effect on the buckling load and postbuckling behavior of braided composite cylindrical shells. The results also confirm that the postbuckling equilibrium path is unstable and the shell structure is imperfection-sensitive in the loading case of axial compression. Acknowledgements This work is supported in part by the National Natural Science Foundation of China under Grant 50375091. The authors are grateful for this financial support. References [1] Ishikawa T, Chou TW. Elastic behavior of woven hybrid composites. J Compos Mater 1982;16:2–19. [2] Naik RA. Failure analysis of woven and braided fabric reinforced composites. J Compos Mater 1995;29:2334–63. [3] Wang YQ, Wang ASD. Spatial distribution of yarns and mechanical properties in 3D braided tubular composites. Appl Compos Mater 1997;4:121–32. [4] Kwon YW, Cho WM. Multilevel, micromechanical model for thermal analysis of woven-fabric composite materials. J Therm Stresses 2004;27:59–73. [5] Jensen DW, Pai SP. Influence of local fiber undulation on the global buckling of filament-wound cylinders. J Reinforced Plast Compos 1993;12:865–75. [6] Pai SP, Jensen DW. Influence of fiber undulations on buckling of thin filament-wound cylinders in axial compression. J Aerospace Eng 2001;14:12–20. [7] Elghazouli AY, Chyssanthopoulos MK, Esong IE. Buckling of woven GFRP cylinders under concentric and eccentric compression. Compos Struct 1999;45:13–27.
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