ANALYSIS OF BI-STABILITY AND RESIDUAL STRESS RELAXATION IN HYBRID UNSYMMETRIC LAMINATES

Transcription

1 ANALYSIS OF BI-STABILITY AND RESIDUAL STRESS RELAXATION IN HYBRID UNSYMMETRIC LAMINATES Fuhong Dai*, Hao Li, Shani Du Center for Composite Material and Structure, Harbin Institute of Technolog, China,5 * Corresponding author (daifh@hit.edu.cn) Abstract: Unsmmetric bi-stable composite laminates are receiving interests in adaptive and deploable structures. The low load-bearing capacit and the residual stress relaation of such laminates are significant limitations for application. The hbridization of unsmmetric composite laminates has potential to solve the problem. In this paper, the hbrid bi-stable laminates are investigated b analtical, finite element and eperimental techniques. The curvature decrease of hbrid laminates along with the residual stress relaation in the polmer matri is investigated. The critical snap-through loads of hbrid laminates are numericall and eperimentall studied. The improved critical load of hbrid laminate and the limited over time reducing of critical load are discussed. Comparisons of cured shape and critical loads between hbrid and pure bi-stable laminates show that the hbridization enable to accomplish both the high load-bearing capacit, the large shape change and the limited response to the residual stress relaation in the polmer matri. Kewords: Unsmmetric laminate; Hbrid composites; Bi-stable. Introduction There have been a lot of works on the generation of bi-stable unsmmetric fiber reinforced composite laminates. When cooled down to room temperature, unsmmetric laminates which are sufficientl wide and thin, deform into two possible stable configurations with almost clindrical shape. However, it is needed to overcome several shortcomings in reling upon laminate asmmetr to produce the required curvatures []. Such as the restrictive geometries determined b the stacking sequences and the negative influences of moisture effect []. One interesting idea recentl to emerge is the hbridization of unsmmetric laminates that can give more design freedom of cured curvatures [3], in which an analtical stud on cured shape of hbrid bi-stable laminates were proposed []. It is possible to accomplish a series of bi-stable laminates which have continuous curvature change, b the design of metal laer. In this paper, both the analtical and finite element methods are developed to predict the cured shape of hbrid composite laminates. How the metal pl contributes to the bi-stabilit of hbrid laminates is analzed. The influence of moisture effect on hbrid laminate and pure fiber reinforced laminate is compared and discussed. The critical snap through load of hbrid laminate is predicted b FEA. The influence of moisture effect on hbrid laminate and pure fiber reinforced laminate is compared and discussed. In the end, the advantages of hbrid laminate comparing with pure fiber reinforced laminate are summarized.. Analtical model The Raleigh-Ritz method is widel used to predict bi-stable unsmmetric laminates cured shape. As with the application of minimum energ principles to an problem, stable equilibrium shapes are found b minimizing the total potential energ, Π, given b [5-8] dv () v C T ij i j i i () Where:ω represents the strain energ densit. C ij denote the material elastic constants. β i are the coefficients related to the elastic constants and thermal epansion coefficients. ε i are the strains. T is the difference in temperature. For unsmmetric laminates, the thermall induced directional deformation of the single laer results in

2 large out-of-plane deformation. To take into account the large deformation which are often man times over the laminate thickness, the linear strain displacement relations must be etended b non-linear terms. It is assumed that the laminate obes the Kirchhoff hpothesis, and that hpothesis is used in conjunction with the von Karman approimation to the geometrical nonlinear strain-displacement relations [8-] : u w w z v w w z (3) u v w w w z Where the inde refers to the laminate neutral plane, where u and v are the - and -reference-surface displacements, w is the z-displacement. ε, ε and ε are the total strains. Substituting equation (3) into equation () and (), total potential energ of the laminate is given b [9] Q ij i j i it dv V L / L / H / L / L / H / Q Q Q Q Q Q Q Q Q T Q Q Q T Q Q Q T dddz () Where, L and L are the side lengths of the laminate and H is the laminate thickness. In Eq. (), the Q terms represent the transformed reduced stiffness, α is the thermal epansion coefficient. A practical wa to solve this problem is the use of approimate epressions for both curvatures and strain-displacement relations, as functions of unknown coefficients. Geometrical assumptions for the out-of-plane displacements of hbrid unsmmetric laminates lead to the general second-order approach w, a b (5) Where, the coefficients a and b represent the curvatures in the - and -directions respectivel. Both curvatures are assumed constant throughout the plate. For the description of the in-plane deformations, several approimations can be found in the literature [9, -]. The robustness and reliabilit of the modeling framework depend on the scaling and conditioning number of the sstem of solving equations. A high order complete polnomial approimating to the displacement field has been proposed b Pirrera and Cerami [5, ]. The carried out calculations at orders 3, 5, 7, 9 and showed that increasing orders do not lead to major differences in the bifurcation diagram. Actuall, b respecting the essential boundar conditions and imposing smmetries, this number can be reduced. Here, the mid-plane displacements are approimated b using the following polnomials with unknown coefficients: u v 3 3 a a a a a a a b b b b b b b () substituting the displacement approimations (5) and () into the strain displacement relation (3) and substituting the resulting epressions into (),the total potential energ of the laminate becomes a function which is dependent on the coefficients a m, b m (m=,,,7). The principle of the minimum total potential energ requires the first variation ofπ to be zero: am bm (7) a b m To satisf this condition, ever summand in equation (7) must be zero, which results in a coupled non-linear algebraic equation sstem. The equation group is solved with speciall-written mathematical software. These solutions should be checked for their stabilit b means of, which has to be positive and definite for a stable deformation state. 3 Finite Element Method In addition to analtical method, Finite Element Method (FEM) is a robust wa to stud the bi-stable laminates. The commercial finite element code ABAQUS is widel emploed to predict the cued shape. A cool-down analsis is used to predict the equilibrium cured shapes of bi-stable laminates. The phsical process itself takes a considerable time to complete (a few hours depending on the curing ccle that the material requires) and therefore it can be considered as a quasi-static process. In FEA, the m

3 laminate is modeled using the four-node-square shell elements (SR), and the model is clamped at the center point. An initial temperature of the elevated curing temperature is applied on the model in the initial step, and in the *Static analsis step the temperature of the model decreases to a final room temperature [7]. At the end of the *Static analsis step, the stable states of the hbrid bi-stable laminate are obtained. After the cool-down simulation, the snap-through simulation of the bi-stable configurations can be conducted. The snap-through problem of unsmmetric laminates is a buckling problem in essence. This problem has to be solved either dnamicall or with the aid of artificial damping. The STABILIZE technique in Static analsis introduces artificial damping to deal with local instabilit and also perform the pseudo dnamic analsis for the snap through process.. Eperimental procedure Within the current research, several eperimental studies have been carried out on hbrid laminates made of CFRP and Steel. The materials and their properties are given in Table. Two CFRP-metal hbrid laminates were manufactured with the laup [] (.3 mm steel) and dimensions of 3 mm, whose bi-stable shapes are shown in Fig.. The other four laminates were manufactured with the laups [] (.37mm steel and mm), [] (.3mm steel and 8 8mm), [/9] (3 mm) and [//9/9] (3 mm), see Fig.. The laminates were autoclave manufactured at the cured temperature 7 C and cooled to the room temperature C. The curvatures of the two shapes were measured. In order to stud the influence of the moisture effect, the curvatures of different laminates are measured again at the 5th da after the manufacturing process. The snap through load is measured, the bi-stable laminate is placed on a smooth glass plate and load is applied at the central point of laminates, see Fig.3. Table Thermo-mechanical properties of CFRP and metal laer CFRP E =37.7 GPa,E =.7 GPa, G=7.7 GPa, ν =.3, α =.37 -/ C, α =.9 - / C, thickness=.5mm Steel E m =GPa, ν =.3, α m = - / C, thickness=.3mm and.37mm Fig. Two stable shapes of hbrid bistable laminates Fig. Manufactured laminates 3

4 Fig3. Illustration of loading at the central point of laminates 5 Results and Discussion The curvatures of five bi-stable laminates from eperiments, analtical epressions and finite element analsis are given in Fig.. The eperimental data were measured immediatel after the manufacture process. The comparisons in Fig. show a good agreement among the eperiments, analtical model and finite element simulation. The cured curvature of the pure T3/Epo laminate with the laup [/9] is.3 m -, while the cured curvature with the laup [//9/9] is. m -. The result demonstrates that the cured curvature decreases greatl as the laers increase. It is difficult for pure bi-stable composite laminates to have other curvatures because the thickness of the prepreg is often normalized. The hbrid laminate with laups [] deforms to be a clindrical shell with a curvature of. m -, which is almost the same as the curvature of the [//9/9] laminate. The thermal strain mismatch between composite plies is the main reason of bi-stabilit for pure composite laminate. For the hbrid composite laminate, the metal pl also has large thermal strain after cooling down. To identif the main role of giving rising to bi-stabilit, an assumption that the thermal coefficients of composite plies are zero is made. FEM is emplo to calculate the shapes of hbrid composite laminates in Fig. again based on the assumption. The results are presented in Table. It indicates that the curvatures of the hbrid laminates decrease little even the thermal strain of composite plies are ignored. So, it is concluded that the thermal strain of the metal pl plas the main role of giving rise to the bi-stabilit. Table FEM calculated curvatures of hbrid laminates with the assumption of zero thermal coefficients for composite plies and the curvature decrease comparing with normal hbrid laminates Hbrid laminate 3 mm.3mm st 8 8mm.3mm st mm.37mm st Curvature m Curvature Decrease 5.5% 5.5% 3.% Table 3 Eperimental measured curvatures of hbrid laminates and pure bi-stable laminates at the 5th da after the manufacturing process and the curvature decrease comparing with the original curvatures. Bi-stable laminate 3 mm.3mm st 8 8mm.3mm st mm.37mm st //9/9 3 mm /9 3 mm Curvature m Curvature Decrease.53%.% 3.7%.% 39.7%

5 Curvature m - 8 Eperiments Analtical FEM No. Table Measured critical loads for different bistable laminates Size (mm ) Laups Total thickness (mm) Critical Loads (N) 3 [ /M/9 ] [ /9 ] / *mm.3mm st 8*8mm.3mm st *mm.37mm st //9/9 3*mm /9 3*mm Fig. Cured curvatures of different laminates For a pure unsmmetric laminate, the residual thermal stress will continuousl decrease because of the moisture effect of polmer matri and the elongation of molecule chain of polmer matri. And then results in continuous decreasing of laminate curvature. However for a hbrid laminate, the thermal strain of the metal pl is not affected b the moisture, and has no molecule chain elongation problem as the high polmer material. As a result, the curvatures of hbrid laminates have no obvious change even a long time after the manufacture process. Table 3 presents the measured curvatures of hbrid and pure bi-stable laminates at the 5th da after the manufacturing process. It indicates that the curvature keeping abilit of hbrid laminates are much more improved than the pure unsmmetric laminates. The critical loads to snap the bi-stable laminates are given in Table. The critical loads to snap the pure laminate is ver small, which is the disadvantage of pure bi-stable laminates. The load-bearing capacit of pure composite laminates can be improved b increasing the thickness, but meanwhile the curvatures of stable shapes are decreased. In comparison, the critical load of hbrid composite laminate increases greatl (up to times) due to the metal laer which enhances the laminates stiffness. The hbridization over pure bi-stable composite laminates is able to accomplish both the high load-bearing capacit and the large shape change. Meanwhile, similar as their shape keeping abilit, the load-bearing capacit of the hbrid laminates will not decrease much because of the residual stress rela in the polmer matri. To validate that, the snap-through process of the hbrid laminate is simulated b FEM. The finite element models of the hbrid bi-stable laminates with the laup [] and dimension 3 mm are numericall generated during the cool-down process, see Fig.5. The load in vertical direction was applied to the central point of the laminate. In the simulation, an analtical plane is required to simulate the glass plate in the eperiments. The surface-to-surface contact between the laminate model and the analtical plane is established to simulate the interaction between the specimen and the glass plate. The boundar and loads are shown in Fig.5, in which a friction force is applied at the right edge along horizontal direction. Fig.5 The boundar and load for hbrid bi-stable laminate The FEA on the prediction of the load carring capacit is performed twice. The first time it is assumed that there is no stress relaation in the polmer matri of hbrid laminate. The second time it is assumed that the residual thermal stress in the polmer matri is completel relaed. The curves of load vs. curvature are potted in Fig.. The largest load at the curve peak is the critical load of the hbrid laminate. The FEM predicted critical loads of hbrid laminates with and without residual stress in polmer matri are. N and 3.5 N respectivel. Without considering the thermal stress in polmer matri, the critical load of hbrid laminate has a decrease of.%. However, it is impossible that the residual thermal stress of the polmer matri completel relaes. 5

6 Therefore, the actual critical load reduction of hbrid laminate is much less than.%. It is much more improved comparing with the critical load reduction of a pure unsmmetric laminate, which is often more than % due to the moisture effect [8]. Curvature / m without thermal stress in polmer matri with full level of thermal stress in polmer matri Load / N Fig. Loads vs. curvatures curve during snap-through process of 3 mm, [] hbrid laminate. Conclusions In conclusion, the hbrid bi-stable composite laminate are investigated comparing with the pure composite bi-stable laminates. The cured shapes were studied using the analtical model, FEM and eperiments, and good agreements among them are observed. The FEA results indicate that the thermal strain of the metal pl contributes more on the bi-stabilit of the hbrid laminate. The hbridization provides more design freedoms to achieve required bi-stable laminates. Comparisons of cured shapes and critical loads between hbrid bi-stable laminates and pure composite laminates show that the hbridization enable to accomplish both the high load-bearing capacit and the large shape change. Further, the hbrid laminates have much more improved curvature keeping abilit and critical load keeping abilit than pure unsmmetric composite laminates in moisture circumstances. Acknowledgements We thank supports from National Natural Science Foundation of China (Grant No.5 and 8758) and the Major State Basic Research Development Program of China (973 Program) under grant No. CB3. References []. Manzo Justin,Gareia Ephrahim, Wickenheiser Adam,etc. Adaptive structural sstems and compliant skin technolog of morphing aircraft structures. Proc. of SPIE, Vol. 539, Bellingham, WA, []. B.D.Harper. The Effects of Moisture Induced Swelling Upon the Shapes of Anti-Smmetric Cross-Pl Laminates. Journal of Composite Materials 987; :3-8. [3]. Dai FH, Zhang BM, Du SY. A novel bi-stable hbrid composite laminate. In: 7th international conference on composite materials (ICCM-7), Edinburgh, UK, 9. []. Danes Stephen,Weaver Paul. Analsis of unsmmetric CFRP metal hbrid laminates for use in adaptive structures. Composites: Part A ;:7 78 [5]. Maengho Cho, Hee Yuel Roh. Non-linear analsis of the curved shapes of unsmmetric laminates accounting for slippage effects. Composites Science and Technolog 3; 3: 5 75 []. Gigliotti M, Wisnom MR, Potter KD. Development of curvature during the cure of AS/855 [/9] unsmmetric composite plates. Composite Science and Technolog. 3;3:87-97 [7]. Gigliotti Marco,Frederic Jacquemin,Alain Vautrin. On the maimum curvatures of /9 plates under thermal stress. Composite Structures 5;8: 77 8 [8]. Hufenbach W,Gude M,Czulak A. Actor-initiated snap-through of unsmmetric composites with multiple deformation states, Journal of Materials Processing Technolog ;75: 5 3 [9]. Hufenbach W, Gude M. Analsis and optimisation of multistable composites under residual stresses. Composite structure ;55:39-37 []. Pirrera A, Avitabile D, Weaver PM. Bi-stable plates for morphing structures: A refined analtical approach with high-order polnomials. International Journal of Solids and Structures ;7: 3 35 []. Gigliotti Marco,Wisnom Michael R, Potter Kevin D. Loss of bifurcation and multiple shapes of thin [/9] unsmmetric composite plates subject to thermal stress. Composites Science and Technolog ;:9 8

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