Composite Materials Research Progress

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36 Jacquemin Frédéric and Fréour Sylvain m* 1122 F m F1122 1 = ≠ − (58) m F 2 1111 Moreover, according to (38-42) an uniaxial macroscopic tension or compression along the transverse (or normal) direction induces local mechanical states in the matrix generally exhibiting no zero strain and stress on-diagonal components (see for instance the cases of the macroscopic loads I σ a and I σ b on Table 7). As a consequence, only the strength coefficient m F 1212 can be determined independently from the three others, from the single macroscopic I m m m load σ d . Concerning the calculation of F 1111, F1122 , F11 , one has to solve numerically the system (60) (cf. Table 7). Finally, the uniaxial microscopic ultimate stresses of the epoxy matrix embedded in the composite structure can be deduced from the set of equations (55) expressed at microscopic scale (i.e. replacing the subscripts I by the subscript m ), provided that the coefficients of the local failure envelope are already known: ⎧ ⎪X ⎪ ⎪ ⎪ ⎨X ⎪ ⎪ ⎪S ⎪ ⎩ m / m m = Y = m = Y m / 1 2 F = Z = Z m 1212 m 1 = 2 F m / m 1111 1 = 2 F ⎛ ⎜ ⎝ m 1111 ⎛ ⎜ ⎝ m 2 11 + 4 F m 2 11 F m 1111 + 4 F The method, developed in the present paragraph, enables the determination of a) the coefficients of the microscopic failure envelope of the epoxy matrix in stress and/or strain space from the macroscopic failure envelope of the ply and scale transition relations linking macroscopic loads to the corresponding local microscopic mechanical states experienced by the matrix, only thereafter, b) the local maximum strength of the matrix embedding the carbon fibers which can be evaluated from the classical formalism relating the strength to the coefficients of the failure envelope. This inverse method provides an alternative to the classical direct approach leading to the determination of the failure envelope from the maximum strength measured on pure epoxies, in the cases that the required data is not available or when the behaviour of the matrix embedded in the composite structure is expected to be significantly different from the behaviour of the pure matrix, as shown for example, in references (Garett and Bailey, 1977; Christensen and Rinde, 1979). F − F m 1111 m 11 ⎞ ⎟ ⎠ + F m 11 ⎞ ⎟ ⎠ (62)
Applied macroscopic load Corresponding macroscopic strain Corresponding microscopic stress according to (15-17) Corresponding conditions for finding the microscopic strength coefficients in stress space from (10, 19) I σ a I ε i Table 7. One possible set of trials enabling the determination of the microscopic strength coefficients of the matrix expressed in stress space. ⎡0 ⎢ = ⎢ 0 ⎢ ⎣0 = ⎡ ⎢ ε ⎢ ⎢ ⎢ ⎢ ⎣ i = a, b 0 0 0 I Y I 11i 0 0⎤ ⎥ 0 , I ⎥ σ b 0⎥ ⎦ 0 I ε 22i 0 I ε 33i ⎡0 ⎢ = ⎢0 ⎢ ⎣ 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0 I / Y 0 0⎤ ⎥ 0⎥ 0⎥ ⎦ I I I ε11i = S12 σ22i , I I I ε22i = S22 σ22i , I I I ε33i = S23 σ22i ⎡0 0 0 ⎤ I ⎢ I ⎥ σ c = ⎢ 0 σ22c 0 ⎥ , ⎢ I ⎣0 0 σ ⎥ 33c⎦ ⎧ I ⎛ I2 I2 ⎞ ⎪F2222 ⎜ ⎜σ 22c + σ33c ⎟ + ⎪ ⎝ ⎠ ⎨ ⎪ I I I I ⎛ I2 I2 ⎞ ⎪ 2 F2233 σ22c σ33c + F22 ⎜ ⎜σ22c + σ33c ⎟ −1 = 0 ⎩ ⎝ ⎠ ⎡ I ε ⎢ 11c I ε = ⎢ c 0 ⎢ ⎢ 0 ⎢⎣ 0 I ε 22c 0 0 ⎤ ⎥ 0 ⎥ ⎥ I ε ⎥ 33c ⎥⎦ I I ( σ + σ ) I I ε11c = S12 22c 33c , I I I I I ε22c = S22 σ22c + S23 σ33c , I I I I I ε33c = S23 σ22c + S22 σ33c ⎡ m σ ⎢ 11i m σ ⎢ i = 0 ⎢ ⎢ 0 ⎢⎣ 0 m σ 22i 0 0 ⎤ ⎥ 0 ⎥ , i = a, b, c ⎥ m σ ⎥ 33i ⎥⎦ (59) ⎧ F m + m + m ⎪ 1111 Ai F 1122 Bi F 11 Ci −1 = 0 ⎪ 2 2 2 ⎪A = m + m + m ⎪ i σ 11i σ 22i σ 33i , i = a, b, c ⎨ ⎡ ⎛ ⎞ ⎤ ⎪B = 2 ⎢σ m ⎜σ m + σ m ⎟ + σ m σ m ⎥ ⎪ i ⎢ 11i ⎜ 22i 33i ⎟ 22i 33i ⎥ ⎪ ⎣ ⎝ ⎠ ⎦ ⎪ ⎩ C = σ m + σ m + σ m i 11i 22i 33i (60) I σ d ⎡ 0 ⎢ I = ⎢S ⎢ ⎢ 0 ⎣ S ⎡ 0 ⎢ I ⎢ I ε d = ε ⎢ 12d ⎢ 0 ⎣ I 12d ε = S m σ d I 66 ⎡ 0 ⎢ = ⎢S ⎢ ⎢ 0 ⎣ m S I 0 0 I 0⎤ ⎥ ⎥⎥⎥ 0 0 ⎦ I ε12d S m 0 0 0 0 0⎤ ⎥ 0⎥ 0 ⎥ ⎥⎦ 0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎦ 2 2 F m S m 1212 -1 = 0 (61)
Page 3: COMPOSITE MATERIALS RESEARCH PROGRE
Page 6 and 7: Copyright © 2008 by Nova Science P
Page 8 and 9: vi Contents Chapter 9 Recent Advanc
Page 10 and 11: viii Lucas P. Durand scale transiti
Page 12 and 13: x Lucas P. Durand machine. In paral
Page 15 and 16: In: Composite Materials Research Pr
Page 17 and 18: Multi-scale Analysis of Fiber-Reinf
Page 25 and 26: T300 fibers (Soden et al., 1998; Ag
Page 29 and 30: 1 I L = L : r v Multi-scale Analysi
Page 37 and 38: I E ⎡0 ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0
Page 49: Multi-scale Analysis of Fiber-Reinf
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Page 67 and 68: Optimization of Laminated Composite
Page 71 and 72: ⎧σ x ⎫ ⎡ mE ⎪ ⎪ ⎢ x
Page 73 and 74: and γ 2 γ 0 Optimization of Lamin
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4 x 10 5 4 3 2 1 Compliance (Nmm) 0
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Optimization of Laminated Composite
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3.5 3 2.5 2 1.5 1 Optimization of L
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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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In: Composite Materials Research Pr
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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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