GEOMETRICAL STABILITY OF CFRP LAMINATE CONSIDERING PLY ANGLE - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS GEOMETRICAL STABILITY OF CFRP LAMINATE CONSIDERING PLY ANGLE MISALIGNMENT Y. Arao 1* , J. Koyanagi 2, S. Takeda 2 , S. Utsunomiya 2 , H. Kawada 1 1 Department of Mechanical and Systems Engineering, Doshisha University,1-3 Miyakodani Tatara, Kyoutanabe, Kyoto, Japan 2 Japan Aerospace Exploration Agency, Institute of Space and Astronautical Science, 3-1-1, Yoshinodai, Sagamihara, Kanagawa, Japan 3 Department of Applied Mechanical and Aerospace Engineering, 3-4-1 Okubo, Shinjuku, Tokyo, Japan, Corresponding author Y.Arao (yoshihiko.arao@gmail.com) Keywords : CFRP, Laminates, Stacking Sequence, Thermal Deformation, Geometrical Stability Abstract the out-of-plane deformation, symmetric stacking Accurate geometrical stability is required for the sequence is usually adopted. The problem is that precise structures like telescopes. It was reported there is no ideal symmetric laminate. This is because that symmetrical CFRP (Carbon Fiber Reinforced ply angle misalignment is inevitable when we Plastics) laminates show unpredictable deformation stacked ply. The symmetrical laminates are due to the ply angle misalignment and temperature specifically asymmetric. Laminates show not only change. This ply angle misalignment is unavoidable. the in-plain deformation but also show out-of-plane One of the answer to mitigate the deformation due to deformation with the temperature change. Author et the ply angle misalignment is to chose effective al. found that a standard deviation of approximately 0.4 ° of ply angle misalignment exists when A person stacking sequence. We discussed here the effective stacking sequence to reduce the thermal deformation. who is an expert at fabricating laminates stacks ply by a hand tape placement method [8] . In order to 1 Introduction make good use of CFRP, it is important to discuss Since CFRP has excellent thermal stability in the proper stacking sequence of CFRP laminate that addition to high specific stiffness and strength, it can mitigate the effect of ply angle misalignment on the be used as the main material for precise structures deformation. like telescopes [1] . The main mirror can be made In this research, we discuss the effects of stacking larger than conventional mirrors by using CFRP, sequence on the thermal deformation of CFRP which can drastically improve the resolution of the laminate considering ply angle misalignment. The telescope. The main mirror requires accurate analysis including laminate theory, Monte Carlo dimensional stability for long-term. For instance, a method and Mohr’s curvature circle was performed mirror of 3.5m diameter must keep its surface- geometry deviation within 5 µ m RMS (root mean 2 Analytical procedure square) [2] . In general, the P-V (Peak to Valley) value 2.1 Laminate theory of reflecting mirror must be kept within λ /8. λ Thermal deformation of laminate can be means the wave length of interest. λ /8 is named calculated by the classical laminate theory. The main equation is as follows; Rayleigh limit. If the mirror deforms more than the   value of Reyleigh limits, the resolution of reflective ε T   − N   1 A A A B B B x   x   11 12 16 11 12 16 mirror decreases. Therefore, geometrical stability in ε   T N A A A B B B     y y   12 22 26 12 22 26 CFRP laminates is critical problem for using CFRP γ     T N A A A B B B   =  xy    16 26 66 16 26 66 xy (1) to precise structures. κ  B B B D D D  T    M  x 11 12 16 11 12 16 CFRP mirrors have been developed by lots of x κ     B B B D D D   T M researchers [3]-[7] . The smooth mirror surface can be y 12 22 26 12 22 26   y     κ   B B B D D D   T  M  16 26 66 16 26 66 xy created by transcription and polishing techniques. xt However, the unpredictable deformation with the A ij , B ij , and D ij are the laminate extensional stiffness, temperature change is unsolved problems. To reduce coupling stiffness, and laminate bending stiffness,

respectively. N T and M T mean hygrothermal force The relationship between curvature radius and due to the temperature change and the resultant height is given by moments per unit length. N T and M T are given by   2 1 c   ρ = + (7) h   { } [ ] [ ] { } n = ∑ ( )   − 2 4 h 1 α ∆ − T (2) N T Q T z z − k ij k k 1 Each letter definitions are shown in Fig. 3. c denotes = 1 k = ∑ [ ] [ ] { } { } ( ) length in horizontal direction and curvature can be n 1 − α ∆ − 1 T 2 2 (3) M T Q T z z obtained as a reciprocal number of curvature radius. − k ij k k 1 2 k = k 1 If the height in equation (7) is extremely small, the See reference [9] to understand detail descriptions second term of right side becomes negligible, and for laminate theory. In this analysis, matrix B ij is an curvature radius is in inverse proportion to the important parameter. It is obvious according to height. In other word, curvature is proportional with equation (1) that the curvature κ occurs with axial the height. In the case of asymmetric laminates and force N T if B ij matrix is not zero It means that the infinitesimal deformation, κ 1 and κ 2 show opposite out-of-plane deformation occurs by temperature sign each other. So h 1 and h 2 become opposite sign. change or moisture absorption ． In the case of P-V value h pv is written as ideally symmetric laminate, B ij matrix becomes zero. = − (8) h h h pv 1 2 However, the laminates have some ply angle Here κ pv is described as follow: misalignment, and B ij matrix is practically not zero ． κ = κ − κ = 2 (9) R The curvature κ occurred by temperature change can pv 1 2 We can evaluate the P-V value of the laminate with be calculated using equation (1). circle form by using equation (4)-(9). 2.2 Mohr’s circle of curvature From equation (1), curvatures κ x , κ y , κ xy can be κ tw ( κ x , κ xy /2) obtained. In general, dimensional stability of mirror is evaluated by P-V value. We introduce the procedure to determine P-V value using each curvatures. In order to determine the P-V value, κ 2 κ 1 main curvatures κ 1 and κ 2 should be calculated from C κ x , κ y and κ xy . Based on the main curvatures, we can κ b easily determine the P-V value. Hyer proposed the ( κ y ,- κ xy /2) procedure to calculate the main curvatures using Mohr’s circle of curvature [10] . Mohr’s circle of curvature is the same concept with the Mohr’s strain circle. Fig. 2 is the concept of mohr’s circle of Fig. 2 Mohr’s circle of curvature curvature. Horizontal axis is a bending curvature κ b and vertical axis is a twisted curvature κ tw . The center point of the circle is always on the horizontal axis, and the coordinate of the center circle is r described as follow: h c /2 c /2 h r c /2 c /2 h h ρ ρ ρ ρ κ + κ = x y (4) C 2 Radius of circle can be written as Fig. 3 Definition of each letters κ − κ 2 κ 2         = + x y xy (5) R      2   2  2.3 Monte Carlo Method Main curvature κ 1 and κ 2 are ， The Monte Carlo method is a statistical analysis technique for obtaining approximate values by κ = + R C (6) 1 κ = − iterating the calculation using random numbers. The R C 2 solution for an unsolvable problem can be