Composite Materials Research Progress

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60 ⎧ N ⎫ ⎡A ⎨ ⎬ = ⎢ ⎩M⎭ ⎣B z k 3 n k 2 1 ⎧ N1 ⎫ ⎡ A11 ⎪ ⎪ ⎢ ⎪ N2 A ⎪ ⎢ 12 B⎤⎪⎧ 0 ε ⎪⎫ ⎪ N6 ⎪ ⎢A16 ⎥⎨ ⎬ ⇔ ⎨ ⎬ = ⎢ D⎦⎪⎩ κ ⎪⎭ ⎪M1 ⎪ ⎢ B11 ⎪M 2 ⎪ ⎢B12 ⎪ ⎪ ⎢ ⎩M 6 ⎭ ⎣B16 Michaël Bruyneel A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 (a) A laminate with its structural axes. h is the total thickness (b) Several unidirectional plies stacked on top of each other. Material axes related to the k th ply . t k B ⎤⎧ 0 16 ε ⎫ 1 ⎪ ⎪ B ⎥ 0 26 ⎥⎪ε 2 ⎪ B ⎥⎪ 0 ⎪ 66 ε6 ⎥⎨ ⎬ D16 ⎥⎪κ1 ⎪ D26⎥⎪κ ⎪ 2 ⎥⎪ ⎪ D66⎦⎪⎩ κ6 ⎪⎭ (c). Definition of the plies location through the laminate’s thickness. hk and hk-1 are used to locate the k th ply of the stacking sequence h k-1 h 2 h 1 h k h 0 (3.10) Figure 3.3. A laminate with n layers (a) Structural axes (b) Material axes of ply k (c) Position of each ply in the stacking sequence. 1
Optimization of Laminated <strong>Composite</strong> Structures… 61 3.2. The Possible Parameterizations of Laminates There exist several parameterizations for the laminates depending on the way the coefficients of the stiffness matrices in (3.10) are computed and depending on the definition of the design variables. The advantages and disadvantages of those different parameterizations are compared in the perspective of the optimal design of the laminated composite structures. 3.2.1. Parameterization with Respect to Thickness and Orientation When the ply thickness and the related fibers orientation are chosen to describe the laminate, the coefficients of the stiffness matrices can be written as follows: = ∑ = − − n Aij Qij k hk k 1 hk 1 ) )]( ( [ θ ⇔ = ∑ = n Aij [ Qij ( θ k )] tk k 1 1 n 2 2 Bij = ∑ [ Qij ( θ k )]( hk − hk−1 ) ⇔ 2 k= 1 = ∑ = n Bij [ Qij ( θ k )] tk zk k 1 (3.11) 1 n 3 3 Dij = ∑ [ Qij ( θk )]( hk − hk−1 ) ⇔ 3 k= 1 = ∑ + = n 3 2 tk Dij [ Qij ( θ k )]( tk zk ) , i , j = 1, 2, 6 k 1 12 where zk and hk define the position of the k th ply in the stacking sequence. tk and k θ are the ply thickness and the fibers orientation, respectively (Figure 3.3). With such a parameterization the local values (e.g. the stresses in each ply of the laminate) are available via the relations (3.1) and (3.4). On top of that the design problem is written in terms of the physical parameters used for the manufacturing of the laminated structures. Finally several different materials can be considered in the laminate when the parameterization (3.11) is used. However when fibers orientations are allowed to change during the structural design process the resulting mechanical properties are generally strongly non linear (see Figure 3.2) and non convex, and local minima appear in the optimization problem. This is also illustrated in Figure 3.4 that draws the variation of the strain energy density in a laminate over 2 fibers orientations. In Figure 3.5 it is shown that the structural responses entirely differ when either ply thickness or ply orientation is considered in the design, resulting in mixed monotonousnon monotonous structural behaviors. It turns that the optimal design task is more complicated since the optimization method should be able to efficiently take into account simultaneously both different behaviors. Additionally using such a parameterization increases the number of design variables that may appear in the optimal design problem since the thickness and fibers orientation of each ply are possible variables. Finally optimizing with respect to the fibers orientations is known to be very difficult and few publications are available on the subject. For a sake of completion, the sensitivity analysis of the structural responses of composites with respect to those variables can be found in Mateus et al. (1991), Geier and Zimmerman (1994), and Dems (1996).
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COMPOSITE MATERIALS RESEARCH PROGRE
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Copyright © 2008 by Nova Science P
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vi Contents Chapter 9 Recent Advanc
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viii Lucas P. Durand scale transiti
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x Lucas P. Durand machine. In paral
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In: Composite Materials Research Pr
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Multi-scale Analysis of Fiber-Reinf
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Page 23 and 24: Multi-scale Analysis of Fiber-Reinf
Page 25 and 26: T300 fibers (Soden et al., 1998; Ag
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Page 51 and 52: Applied macroscopic load Correspond
Page 65 and 66: In: Composite Materials Research Pr
Page 67 and 68: Optimization of Laminated Composite
Page 71 and 72: ⎧σ x ⎫ ⎡ mE ⎪ ⎪ ⎢ x
Page 73: and γ 2 γ 0 Optimization of Lamin
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Page 107 and 108: 3.5 3 2.5 2 1.5 1 Optimization of L
Page 121: Optimization of Laminated Composite
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110 W.H. Zhong, R.G. Maguire, S.S.
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112 Reinforcement: Advanced materia
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130 Yuanxin Zhou, Hassan Mahfuz, Vi
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140 Heat Flow (W/g) Heat Flow (W/g)
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144 Material Yuanxin Zhou, Hassan M
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146 SEM Analysis Yuanxin Zhou, Hass
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150 Governing Equation For the fibe
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152 ⎧ UU = ⎨ ⎩ ⎧ UD = ⎨
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156 Stress (MPa) 120 80 40 0 Yuanxi
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166 Giangiacomo Minak and Andrea Zu
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170 ⎟ ⎞ ⎠ s ⎜ ⎛ ⎝ = a E
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188 A B E (MPa) D 250000 200000 150
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190 D D 1.0 0.9 0.8 0.7 0.6 0.5 0.4
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204 Conclusion Giangiacomo Minak an
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Research Directions in the Fatigue
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Bending force [N] Research Directio
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Damage Variables in Impact Testing
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F peak [kN] 16 14 12 10 8 6 4 2 0 D
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Eletromechanical Field Concentratio
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MV/m. Eletromechanical Field Concen
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276 S.C. Tjong developed in the pas
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278 S.C. Tjong ultrasonic waves gen
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280 S.C. Tjong applications. Goujon
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282 S.C. Tjong nanoparticles were p
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284 S.C. Tjong where DL is lattice
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286 S.C. Tjong stress is temperatur
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288 S.C. Tjong threshold stress res
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290 S.C. Tjong NC Al, and the NC Al
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292 S.C. Tjong (a) (b) Figure 19. T
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294 S.C. Tjong Apart from forming u
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296 S.C. Tjong [29] Saravanan, M.;
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298 braids, 115 broadband, 211 bubb
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300 endothermic, 139 endurance, 32
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302 isostatic pressing, 279, 288 It
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304 piezoelectricity, 261 pitch, 12
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306 67, 69, 70, 71, 72, 73, 84, 94,
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Composite Materials Research Progress

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