STIFFNESS EVALUATION OF THE COMPOSITE LAMINATES WITH WAVY PLIES AND THEIR STABILITY ANALYSIS

Transcription

1 THE 9 TH ITERATIOA COFERECE O COMPOITE MATERIA TIFFE EVAUATIO OF THE COMPOITE AMIATE WITH WAVY PIE AD THEIR TABIITY AAYI H. Dalir *, J. E. Brunel, F. Dervault, A. andry Department of Composites Development, Bombardier Aerospace, Montreal, Canada * Corresponding author (hamid.dalir@aero.bombardier.com) Keyords: ply aviness, avy laminate, elastic properties, compression and shear buckling Introduction Creation of the ply aviness during different composite fabrication processes is a critical issue for the analysis of the composite structures. It is a source of geometrical non-uniformity hich makes the composite analysis and modeling complicated and computation-intensive. For simplicity, hen analyzing laminated composites, idealized lamina ith perfectly straight fibers is often assumed. Hoever, the aviness usually produces different mechanical properties for such laminates [-8]. Ply aviness could be either the outcome of the local deformation of the fibers, or be caused by the residual stresses accumulated during the curing process. It usually happens through the thickness of the composite parts and can be often revealed by their destructive inspection. everal researchers have introduced different theories, micro-mechanical finite element methods (M-FEM) and eperimental techniques to investigate the elastic behavior of composite laminates having ply aviness. in et al. [9] developed a formulation to predict the compressive failure of the composite laminates ith avy fibers and a nonlinear matri. Wisnom et al. [0] discussed the effect of the misalignment angle beteen the fibers and the loading ais and shoed that small misalignment angles have a very strong effect on compressive strength. Camponeschi et al. [] designed an interface measurement technique using an optical microscope equipped ith an -z indeing table to measure aviness at the mid-plane of the laminate and then analyzed and established a theoretical relationship beteen fiber aviness and reduction of compressive strength. o et al. [] developed a model to predict the compressive strength of composite laminates using a combined analytical and semi-empirical approach. In the tension case, Rai et al. [] and Chan et al. [4] investigated the effect of fiber aviness on stiffness of composites by a numerical method. Bogetti et al. [5,6], developed an analytical model for evaluating the stiffness reduction of a laminate having aviness. Hoever, no eplicit analytical formulation as obtained in their model. Garnich et al. [7], developed an incremental scheme to predict the general stress-strain response for a curved fiber composite lamina. Chun et al. [8], evaluated the influence of in-plane aviness on the structural response and its sensitivity on a composite beam. Ray et al. [9] and Reddy et al. [0] developed finite element micromechanical models for stiffness of composite plates having aviness. The main objective of this paper is to present eplicit formulas to evaluate the influence of the out-of-plane fiber aviness on the stiffness of the composite panels. tability sensitivities due to the variation of fiber aviness ould also be assessed by determination of their effects on the critical buckling loads of the composite laminates under compression and shear. Our proposed approach to the problem of aviness is to calculate the equivalent homogenized properties of the composite laminate under consideration, and subsequently use these properties for structural analysis. Results of the developed method are compared ith those of M-FEM of Garnich et al. [7]. et, the effects of the ply aviness on the buckling behavior of the composite laminates under compression and pure shear are investigated ith to different Finite Element Model (FEM) approaches hich ould be presented in details in the folloing section. Finally, the results are compared ith those of analytical solutions. Effective Mechanical Properties. Analytical Formulation The cross section of a monolithic laminate ith local ply aviness has been shon in Fig.. To obtain the reduced stiffness values, the average strain values over the length of the laminate should be calculated.

2 PREDICTIO OF THE EATIC PROPERTIE FOR COMPOITE TAPE AMIATE WITH WAVY PIE Fig.. aminate configuration ith ply aviness The stress-strain relation in XZ coordinate system is defined as [], and y y y y () are the strain and stress tensors, respectively, and is the compliance matri hich is defined as, here 6 6 () () Here, cos, sin and is the angle beteen the fiber and direction X throughout the laminate and 0 E E ij 0 (4) E E 0 0 G here E, E,, and G are the stiffness properties of the lamina, i, j,,6. o if e assume the fiber aviness as a sinusoidal function that its amplitude is reducing linearly from h ma to zero at the etreme plies, e can assume the out-of-plane fiber aviness as: and z hma sin t (5) l z h t l ma tan cos l (6) hma and l are the maimum aviness height and length over all the laminate plies. t is the laminate thickness. Therefore the equivalent values for the compliance matri could be ritten as: t eqij ij ddz (7) t t 0 and the equivalent major Young s modulus ould be obtained as: E here l is the aviness fraction and i i i t t i i i i t t i i i t i t i (8) (9) and i t are the total number of laminate plies and thickness of the i th ply respectively; and

3 z i h ma l t. All other effective laminate properties could be obtained in the similar manner.. Evaluation of the aminate tiffness To validate the formulations obtained in the previous section, the stiffness results are compared ith those of finite element micromechanics modeling done by Garnich et al. [7]. In their study, the aviness as assumed to be sinusoidal. Based on the volume percentage of each constituent, the avelength and the amplitude of the aved fibers, the unit cell geometry as defined (see Fig. ). This unit cell as divided into heahedral finite elements as shon in Fig.. In order to determine the stiffness parameters, stress analysis of the avy unit cell as carried out for several independent load cases. Here e consider the same fiber and matri properties of a graphite/epoy system as ere employed by Garnich et al. [7]. The fiber is assumed to have transversely isotropic elastic properties of E 07.5 GPa, E 5 GPa, E G G 95 GPa, G 9. GPa, 0.4 and The polymer matri is assumed to have isotropic elastic properties of E 4.5 GPa and 0.4. For 40, 50, 60, and 70% of fiber volume fractions ( V ), the predicted equivalent major V f Young s modulus is shon in Fig. 4. As it can be seen the modulus values obtained from Eq. (8) are in close agreement ith those of comple M-FEM results of Garnich et al. [7]. Fig.. A avy fiber unit cell volume [7] Fig.. M-FEM discretization of part of the unit cell [7] E (GPa) Fig. 4. Major modulus ( E ) as a function of aviness Buckling of Composite Panels ith Wavy Plies. Compression Case.. Analytical Formulation In this section, e consider a rectangular composite plate having a region ith avy plies (see Fig. 5). We are specifically interested to see if it is a valid assumption to consider the effects of the ply aviness on the buckling analysis by applying the equivalent homogenized properties of the composite laminate under consideration. The plate is assumed to be simply-supported on all of its four edges. The out-of-plane displacement, y of the plate is assumed to be in the form of V f / V M m ny, Amn sin sin (0) m n y a ma / l Micromechanical FEM Analytical solution here m and n are odd numbers, and A mn are

4 PREDICTIO OF THE EATIC PROPERTIE FOR COMPOITE TAPE AMIATE WITH WAVY PIE Carrying out the integrations, and setting the derivative of ith respect to A mn equal to zero, leads to the generalized eigenvalue problem: A B () here the eigenvalue is the buckling load hen lbf in contains the constants Amn is applied; the eigenvector of Eq. (0), and the matrices A and B contain stiffness and geometry information of the problem. Therefore, in case of a square plate, the main critical buckling load is given as crit D D D D (4) Fig. 5. aminate ith aviness under pure compression unknon coefficients to be determined. Assuming that the layup is symmetric and balanced ith D6 D6 0, the buckling load can be determined by minimizing the total potential energy of the plate: 0 0 D 4D d dy y y D D 0 0 d dy () here is the applied compressive load per unit idth, and i,, 6, are the bending D ij stiffnesses at any location in the plate. The coupling terms D6 and D6 could be incorporated ith appropriate changes in Eq. (0) at a slight increase in compleity by introducing an anisotropy factor []. The unknons Amn are determined by minimizing the total energy 0 A mn () To compensate for the effects of transverse shear hich could become important hen analyzing thick laminate plates, one can rite [] crit G t (5) crit here G y and t are the shear modulus and thickness of the laminate plate, respectively. Eq. (5) is only applicable to balanced and symmetric laminates. To account for the anisotropy of the laminate plate here the coupling terms D6 and D6 could not be neglected, an anisotropy factor ould be considered as [] * crit 0. D D 6 D y D.. Finite Element Modeling 6 crit (6) Based on the formulation introduced in section., a macro as prepared using Visual Basic for Applications (VBA) enabling the users to calculate the equivalent homogenized properties of the composite laminate having avy plies (see Fig. 6). The users ill have to insert the geometry of the aviness, stiffness properties of the idealized lamina ith perfectly straight fibers along ith the laminates layup as the input. The output of the program ould be the averaged equivalent homogenized mechanical properties of the avy

5 THE 9 TH ITERATIOA COFERECE O COMPOITE MATERIA Fig. 6. VBA program developed to obtain the equivalent homogenized properties of the laminates having avy plies lamina or laminates hich could be used for structural analysis of the composite structures having avy plies. Here e consider a 0 in 0 in square panel comprised of 88 plies [(±45 /0 ) /(±45 4 /0 ) ] of Carbon/Epoy tape laminate ith the lamina properties of E 9.0 msi, E.40 msi, E.9 msi, G msi, G 0.4 msi, G 0.7 msi, 0.7, 0.57 and Eight different FEM cases ere run, each ith different aviness lengths ( l ) of 0 (laminate having plies ith straight fibers), in, in, 4 in, 8 in, in, 6 in and 0 in. The maimum height of the aviness as considered to be 0.0 in. For the FEM, the astran softare (O 05) by MC as used. The model consisted of 600 fournoded quadrilateral elements ith 68 nodes. The geometry, loading, and typical results obtained from FEM are shon in Figs. 5 and 7. Fig. 7. Typical buckling results under compression To different approaches ere considered ith FEM. In the first called to-phase FEM approach, the panel as considered to be made of to different regions, regions at etremities having the properties of laminas ith straight fibers and the avy region

6 THE 9 TH ITERATIOA COFERECE O COMPOITE MATERIA Table. Comparison of different FEM and analytical approaches for the case of the pure compression l (in) : To-phase FEM (lbf/in) : One-phase FEM (lbf/in) : One-phase Analytical (lbf/in) Difference beteen & (%) Difference beteen & (%) / o One-phase Analytical One-phase FEM To-phase FEM l (in) Fig. 8. Compression buckling load reduction by aviness / o in the middle of the panel ith averaged reduced properties obtained from the VBA program by considering l (see Fig. 6). In the second approach called one-phase FEM, the equivalent homogenized properties of the composite laminate ith avy plies ere applied all over the panel. In this approach, in the VBA program as considered to be 0 in, and the average homogenized properties ere obtained for laminas ith different aviness length. The buckling loads predicted by the to FEM approaches have been compared in Table and Fig. 8 ith those obtained from Eq. 6. In all cases and ith the both FEM approaches used here, the FE-predicted buckling modes ere symmetric. The highest difference beteen the to FEM approaches as.5%. The FEM results ere successfully correlated to those obtained from Eq. (6) ith the maimum of 6.7% deviation (see Table ). Fig. 9 shos the reduction in the buckling load of the panel in compression as a function of aviness parameters ( l and h ma ). For eample, for l in and h ma 0.4 in, one can epect ~% reduction in the critical buckling load in compression Fig. 9. Effect of the aviness parameters on the critical compression buckling load of our 88-ply composite panel. hear Case.. Analytical Formulation For the case of shear, e considered the same rectangular composite plate as in section. having the same aviness characteristics (see Fig. 0). The plate as loaded by a shear load. Therefore the total potential energy of the plate (Eq. ) for the case of pure shear can be reritten as l (in) 0 0 D 0 0 D 4D y y h ma (0-4 in) D d dy y y d dy y (7)

7 Table. Comparison of different FEM and analytical approaches for the case of the pure shear l (in) : To-phase FEM (lbf/in) : One-phase FEM (lbf/in) : One-phase Analytical (lbf/in) Difference beteen & (%) Difference beteen & (%) a polynomial-based displacement function hich consists of a boundary polynomial specifying the geometric and kinematic boundary conditions, multiplied by a complete to-dimensional simple polynomial. This displacement function has been used successfully for the modeling of bilateral plate buckling problems [4,5], and can be ritten as M n n, ak y (8) m0 n0 y y y Fig. 0. aminate ith aviness under pure shear here M is the degree of a to-dimensional polynomial, a k is the arbitrary Ritz coefficient and k is determined as k n (9) Fig. shos the typical shear mode shape of the panel under shear loading. The edges of the plate are all simply supported. Therefore, the shear buckling load can be epressed as [6] ycrit k s 0. D D 5 (0) here k s is the shear buckling factor and for our specific layup, it as determined to be equal to.8. Fig.. Typical buckling results under pure shear The Rayleigh-Ritz method as used to determine the buckling load. This method is set out in Timoshenko and Gere s tet []. The out-of-plane, y of the plate as considered as displacement.. Finite Element Modeling For the FEM, the same models as those introduced in section.. ere used by astran softare. The nodes located on the edge lines ere loaded under pure shear. The comparison of the analytical Rayleigh-Ritz method ith the results of the to 7

8 PREDICTIO OF THE EATIC PROPERTIE FOR COMPOITE TAPE AMIATE WITH WAVY PIE y / yo y / yo Fig.. hear buckling load reduction by aviness l (in) h ma (0-4 in) Fig.. Effect of the aviness parameters on the critical shear buckling load of our 88-ply composite panel different FEM approaches hich ere introduces in section.., has been shon in Table and Fig.. As it can be seen, there is an ecellent agreement beteen the to FEM approaches (orst deviation is -.%). Fig. shos the reduction in the buckling load of the panel under pure shear as a function of aviness parameters ( l h ). 4 ummary and Conclusions One-phase Analytical One-phase FEM To-phase FEM l (in) and ma The eplicit forms of the stiffness for a laminate containing out-of-plane fiber aviness ere derived. The results ere in ecellent agreement ith those obtained from comple M-FEM of Garnich et al. [7]. The reduction of the lamina stiffness E as found to be %, 0%, 40% and 70% for plies having %, %, 5% and 0% aviness, respectively. These equivalent homogenized properties ere then used for the compression and shear stability analysis of the composite panels having fiber aviness. For a 0 in 0 in square panel comprised of 88 plies [(±45 /0 ) /(±45 4 /0 ) ] of Carbon/Epoy tape laminate having aviness parameters of 0. l and a ma l 0.075, it as found that the reduction of the stiffness ill lead to 8% and 7% decrease in the buckling load for the case of pure compression and pure shear, respectively. To different FE modeling approaches ere selected and their results ere compared ith those of closed form analytical formulations. It as also shon that the results are in close agreement ith each other ith the maimum deviation of 6.7% and 0.% for the case of pure compression and pure shear, respectively. We also found that for thick laminates, the effects of the local changes of the fiber volume fraction hich is caused by the aviness (i.e., changes on the local ply stiffness and the aviness geometry), don t have considerable effect on the compression and shear buckling loads (results not shon). Therefore, to find the effective stiffness of a laminate having ply aviness, one need to first measure the aviness parameters ( l and a ma l ) and then use the formulations introduced in section. (VBA tool shon in Fig. 6). Once the equivalent homogenized properties of the laminate having aviness ere determined, they can be used subsequently for the stability analysis using the analytical formulations introduced in section. Although the aviness is usually local, hoever, it ould be accurate enough if the homogenized stiffness values be applied all over the panel instead of considering the panel ith several regions ith or ithout aviness. 5 Acknoledgements The ork described in this paper as sponsored by Bombardier Aerospace. The authors ould like to acknoledge the technical support of Dr. aid Jazouli and Mr. Christian Faure from Core Engineering of Bombardier Aerospace. pecial thank goes to Prof. Daniel Therriault from Ecole Polytechnique de Montreal for his valuable comments and several discussions. The vies and conclusions contained in this article should not be interpreted as representing the official policies, either epressed or implied, of the Bombardier Aerospace.

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