THE 9 TH INTERNATIONA CONFERENCE ON COMPOSITE MATERIAS UNEXPECTED TWISTING CURVATURE GENERATION OF BISTABE CFRP AMINATE DUE TO THE UNCERTAINTY OF AY-UP SEQUENCE AND NEGATIVE INITIA CURVATURE J. Ryu, J-G. ee, S-W. Kim, J-W. ee, K-J. Cho, M. Cho * Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea, * Corresponding author (mhcho@snu.ac.kr) Keywords: bistable CFRP laminate, Tilting behavior, Initial curvature effect Introduction Inducing curvature to the CFRP bi-stable structure for curvature tailoring has been studied []. Final curvature of the one of the stable state can be easily tailored by curing it on the cylinder shape tool plate. The schematic of the inducing curvature is illustrated in Fig. ; concave cylinder shape tool plate is defined as negative curvature. Remarkable advantage of the methodology is that the final curvature of mode, stable state whose non-zero curvature direction is parallel to tool plate curvature direction, can be controlled in linear manner. Typical results of the curvature for cross-ply CFRP laminate, whose layup sequence of [0 n /90 n ], are illustrated in Fig.. Final curvature of the mode is the sum of the tool plate curvature and the final curvature without curvature. Meanwhile, final curvature of the mode is not changed regardless of curvature value. A simple relationship between tool-plate curvature and final curvature of the bi-stable CFRP laminate can be easily applied to the applications of bi-stable structures, such as flytrap robot []. Fig. curvature effect of CFRP bi-stable laminate Fig. curvature for curvature tailoring However, certain layup sequence, such as [0/90] n with negative curvature, unexpected twisting curvature of CFRP bi-stable laminate cab be generated. It can show bi-stabilities and deformation behaviour of it is illustrated in Fig. 3, Principal
curvature directions of the laminates are ±45 and signs of the curvature are same. the deformation. The generation of the unexpected twisting curvature are properly simulated by including the global force/moment equilibriums as constraints to the strain energy function. In addition, we propose a simplified optimization method to find the final curvatures of CFRP bi-stable laminate by limit analysis, assuming the side length of laminates are infinite. Theoretical Backgrounds Our claims for the reason of the phenomenon, the generation of the unexpected twisting curvature, can be easily verified through FEM simulation with ABAQUS. ay-up sequence with imperfection, [0./90/0./90] T, shows twisting curvature while perfect lay-up sequence, [0/90/0/90], shows conventional principal curvature direction; negative radius of curvature, -50mm, is applied to both laminates. Fig. 3 CFRP bi-stable laminates with unexpected twisting curvature The reasons of this phenomenon can be divided into two parts. One is based on the imperfection of the lay-up angle, such as [0./90/0./90] T. The imperfection induces constant shear strain in the overall domain of the laminate. As a result, the constant shear strain causes global moment unbalance which results in the unexpected twisting curvature. The other is negative curvature. Negative curvature induces the decrease of the in-plane constant normal strain of mode and the stability of the bi-stable structure. These claims are discussed in more detail manner in chapter. In this study, we use and expand the classical methods for bi-stable laminates [3-9], which are based on CT (Classical aminate Theory) and Rayleigh-Ritz method to describe the two unexpected twisting curvatures equilibrium state. Classical strain energy function, however, cannot describe correctly this phenomenon because it does not consider the global force/moment balance after Fig. 4 Verification with ABAQUS
PAPER TITE Tilting phenomena occurs depending on the curvature values. It means that for parametric study of tilting behaviour requires repeated FE meshes for various curvature configurations. Therefore, it is desirable to develop analytical simple model which is pertinent to the deformation behaviour including tilting action. In this sense, the Rayleigh-Ritz method is an efficient method to investigate the threshold range of imperfection lay-up angles and negative curvature. First of all, the computing time to calculate the final shape of the CFRP laminate is much smaller than conventional FEM with non-linear shell element. Second, applying the curvature effect in the Rayleigh-Ritz approximation method is much simpler than doing it in FEM because the curvature effect can be applied by changing one parameter while overall model should be reconstructed in FEM. Strain field for Rayleigh-Ritz method to describe curvature effect is proposed in our previous study []. Strain field of previous studies [3-9] can be easily extended to handle curvature effect by introducing reference state, which is illustrated in Fig. 5. The strain field between reference state and state is defined as strain field and the strain fields between reference state and final state is defined as final strain field. Those strain fields can be directly derived from the previous studies [3-9] because the reference state is flat. The strain field with curvature, which is described in Eq. (), is the difference between final strain field and strain field because the assumed strain field is not the strain with von-karman non-linearity but Green- agrangian strain. E y yy x 0 xy init init z T yy init 0 xy () The above strain field should be extended further to describe the twisting curvature because the equation does not include the twisting curvature as a variable. It should be noted that the final shape of the CFRP laminate is a cylinder if the side lengths of the laminates are sufficiently long because nondevelopable surface, such as saddle shape, requires too much shear strain and strain energy based on Gauss Theorema egregium. In general, however, principal curvature direction may not be aligned with the original coordinate system, such as angleply. Therefore, introducing principal curvature direction as a new variable is essential for the generalization of Eq. () to simulate the generation of the twisting curvature. This approach is based on the previous study of the Jun and Hong [6]. Detail formulation for the strain field is described in Eq. (). The variables m and n, in Eq. (), means cos θ and sin θ where θ indicates the angle difference between coordinate system and principal curvature direction of CFRP laminate after the curing. Fig. 5 Definition of reference state for strain field 3
m n mn y' E yy n m mn x' z T yy z xy mn mn m n x ' y ' 0 xy 0 () To calculate second P-K stress, linear elastic constitutive equation is used because the order of the Green-agrangian strain field is same with thermal strain; i.e. infinitesimal. C E T (3) ˆij ijkl kl kl Strain energy function with the generalized Green- agrangian strain field, Eq. (), is defined as follows. U Cijkl Eij ijtekl kltdv (4) V The above strain energy function can handle almost the whole general deformed behaviour of CFRP laminate. It can handle the curvature effect not only cross-ply but also angle-ply CFRP laminate. Verification with ABAQUS is illustrated in Fig. 6. Red meshes are the result of the ABAQUS and blue circle is the result of the minimization of the strain energy in Eq. (4) However, it cannot handle the generation of the unexpected twisting curvature because the strain energy function does not include the global force/moment balance. The unbalance of the global force/moment is closely related with the magnitude of the constant shear strain, zero value in the case of the perfect lay-up sequence. Fig. 7 illustrated the moment unbalance which is induced by the constant shear strain. Resultant moment by integrating shear stress along the curved edge is smaller than that of the straight edge because the stress distribution along curved edge is not aligned. Fig. 6 verification with ABAQUS Fig. 7 global moment unbalance due to constant shear strain by imperfection of lay-up sequence imperfection It should be noted that the global force/moment balance satisfies automatically in the case of the
PAPER TITE cross-ply laminate with perfect lay-up sequence; global moment balance satisfied for each edges. Fig. 8 illustrates the moment balance for each edge. As a result, minimization of the strain energy function without the global force/moment balance as constraints has showed a satisfactory result. global moment imbalance which is induced by imperfect laminate sequence becomes dominant. Fig. 8 global moment balance due to other shear strains The other reason for the unexpected twisting curvature is negative curvature. Initial curvature is related with the in-plane strain along x edge of mode, zero curvature direction []. In the case of negative curvature, strain energy of mode and intermediate saddle mode, local maximum point is decreased because of the change of the in-plane strain along x-direction. Moreover, negative curvature decreases the final curvature of mode while the positive curvature increases the final curvature of mode. As a result, conventional shape of mode, cylinder shape with positive normal curvature along x-direction, lost its stability and Fig. 9 the change of the strain energy level by curvature effect To simulate the twisting curvature generation by imperfect lay-up angles and negative curvature, global force/moment balance should be introduced to the strain energy function for Rayleigh-Ritz method as constraints. Considering the force/moment balance along the z-direction is enough because the strain field in Eq. () is x- symmetric and y-symmetric; force/moment balance along the x-direction and y-direction satisfies automatically. Global force/moment balance along the z-direction is described in Eq. (5). Introducing those balance equations in Eq. (4), are introduced to strain energy function as equality constraints because free boundary condition should be applied. As a result, the twisting curvature generation by the imperfect lay-up sequence and negative curvature can be simulated by minimizing the modified energy function, which is described in Eq. (6), 5
int x' y' y' x' Fz x' x' sin x' y' sin dx dz y' y' sin x' y' sin dy dz x y int x' y' y' x' Mz x' x' sin x' y' sin dx zdz y' y' sin x' y' sin dy zdz (5) x y ' ' ' ' y x x y x' yy ' ' cos xy ' ' cos dy dz y ' ' ' cos xy ' ' cos dx dz y x where, x and y represent principal curvature direction of laminate. U,,,, U F M (6) * int int ij ij ij z z It should be noted that we assume the constant curvature in overall domain of laminate. Moreover, we neglect the deformation of the edges by in-plane strain because those strains are all infinitesimal. In the experimental points of view, directions of the principal curvature are ±45º if the twisting curvature is generated by the imperfect laminate sequence and negative curvature. This is a quite intuitive result because only two set of principal curvature, 0º ~ 90º and -45º ~ 45º, satisfies the x-symmetric and y-symmetric is the side lengths of the laminates are equal. In these cases, only one of the force balance and moment balance satisfies automatically. Proposing the simpler formulation to expect final curvatures of the CFRP laminates is possible based on this phenomenon. In the case of principal curvature direction is ±45º, only global force balance should be satisfied; global moment balance satisfies automatically although there is non-zero constant shear strain. Two constraints are derived from the global force balance. Those equilibriums are illustrated in Fig. 0. First, x-direction and y-direction normal stresses should be same to satisfy horizontal force equilibrium. Second those normal stresses should be same magnitude with shear stress. Those constraints are described in Eq. (7) yy xy (7) Fig. 0 force equilibrium for deformed state when unexpected twisting curvature is generated The constraints can be transformed to simpler form through the coordinate transform to the principal coordinate system. x' x' xy yy ' ' 0 (8) 0 xy ' '
PAPER TITE init init yy yy T init init z E' yy z 0 yy init init yy 0 yy On the other hand, strain field in Eq. () can be simplified by assuming the side lengths of the laminate is infinite and neglect the ξ xy because it is much smaller than ξ or ξ yy ; only 3 unknowns, ξ, ξ yy and κ, are remained. The simplified strain fields are described in Eq. (9) Applying zero equality constraints in Eq. (8), two global linear force equilibrium equations in classical plate theory can be identified. yy T A'' A'6' yy A'' A '6' Thermal A'6' A 6'6' N yy '' 0 Thermal B'' B '6' N'' 0 (0) B'' B '6' B '6' B 3' 6' As a result, strain energy function in Eq. (4) become a function with only one variable; simple line search can find the curvature value that minimizes the strain energy function. The comparison with ABAQUS is illustrated in Fig.. Table shows the detail material properties of the CFRP laminates which are used in the simulations. Table material properties of CFRP laminate Quantity Unit Value E GPa 60 E GPa G GPa 8 thickness mm 0.085 α C 0.9 0 6 α C 0.3 0 4 ΔT C 45 (9) Table shows the numerical comparison between the results of the ABAQUS, full strain energy minimization of Eq. (6) and simplified strain energy minimization; lay-up sequence of the simulation is [90/0./90/0.] and negative curvature is 50 (/mm). Table final curvature comparison between methodologies Quantity Value ABAQUS -5.56 Full energy minimization -5.57 Simplified energy minimization -5.548 Fig. the comparison of the result between FEM (ABAQUS) and proposed method Fig. and Table show that the proposed method shows accurate results in engineering sense. Error is smaller than 5% and computing cost is much smaller than that of non-linear FEM simulation. Moreover, parametric study of the present simplified approach is quite easy because only change of curvature is enough to generate final curvature. 7
3 Conclusions In this study, the reasons of the tilting behavior of bistable CFRP cross-ply laminate are presented. We modified the strain energy function for Rayleigh- Ritz method by introducing the additional constraint, global force and moment balance equation. The result by minimizing the modified strain energy is compared with that of FEM and it shows that the modified energy equation can simulate the tilting behaviour of bistable CFRP cross-ply laminate. [0] Dano M, Hyer MW. Thermally-induced deformation behavior of unsymmetric laminates. Int J Solids Struct. 998;35(7):0-0. Acknowledgement This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(mest)(no. 0K00368) References [] Ryu J, Kong JP, Kim SW, Koh JS, Cho KJ, Cho M, Curvature tailoring of unsymmetric laminates with an curvature. J Compos Mater, accepted [] Seung-Won Kim, Je-Sung Koh, Maenghyo Cho and Kyu-Jin Cho, Design & analysis a flytrap robot using bi-stable composite, IEEE International Conference on Robotics and Automation, pp. 5-0, 0. [3] Hyer MW. Calculations of the Room-Temperature Shapes of Unsymmetric aminates. J Compos Mater. 98;5(Jul):96-30. [4] Hyer MW. The room-temperature shapes of fourlayer unsymmetric cross-ply laminates. J Compos Mater. 98;6(4):38. [5] Jun WJ, Hong CS. Effect of Residual Shear Strain on the Cured Shape of Unsymmetric Cross-Ply Thin aminates. Compos Sci Technol. 990;38():55-67. [6] Jun WJ, Hong CS. Cured Shape of Unsymmetric aminates with Arbitrary ay-up Angles. J Reinf Plast Comp. 99;():35-366. [7] Cho M, Kim MH, Choi HS, Chung CH, Ahn KJ, Eom YS. A study on the room-temperature curvature shapes of unsymmetric laminates including slippage effects. J Compos Mater. 998;3(5):460-48. [8] Cho M, Roh HY. Non-linear analysis of the curved shapes of unsymmetric laminates accounting for slippage effects. Compos Sci Technol. 003;63(5):65-75. [9] Ren, Parvizi-Majidi A, i Z. Cured shape of crossply composite thin shells. J Compos Mater. 003;37(0):80.