Mechanics Analysis 2010 Part 12 potx

Chapter 8. Optimal composite structures 37 1 As a rule, helical plies are combined with circumferential plies as in Fig. 7.43. For this case, k = 3, hl = hl = h4/2, = -& = 4, h3 = h90, c$3 = 90°, and Eq. (8.17) gives (8.18) Because the thickness cannot be negative, this equation is valid for 0 6 4 6 40. For Q 4 6 90", the helical layer should be combined with the axial one, i.e., we should put k = 3, hl = hZ = hd/2, 4, = = 4 and h3 = ho, 43 = 0". Then (8.19) Dependencies corresponding to Eqs. (8.18) and (8.19) are presented in Fig. 8.3. As an example, consider a filament wound pressure vessel whose parameters are listed in Table 6.1. Cylindricalpart of the vessel shown in Figs. 4.14 and 6.23 consists of a f36" angle-ply helical layer and a circumferentiallayer whose thickness hl = hb and hZ = h90 are presented in Table 6.1. The ratio hyo/h& for two experimental vessels is 0.97 and 1.01, while Eq. (8.18) gives for this case hyo/h& = 0.96 which shows that both vcsscls are close to optimal structures. Laminates reinforced with uniformly stressed fibers can exist under some restrictions imposed on the acting forces Ify, F,., and iVyy. Such restrictions follow from Eqs. (8.13) and (8.14) under the conditions that hi 2 0, 0 6 sin' 4i, cos' 4i 6 1 and have the form 2 1.6 1.2 0.8 0.4 0 Fig. 8.3. Optimal thickness ratios for a cylindrical pressure vessel consisting of i$ helical plies combined with circumferential (90") or axial (0") plies. 372 Mechanics and analysis of composite materials Particularly, Eqs. (8.13) and (8.14) do not describe the case of pure shear for which only shear stress resultant, N,,, is not zero. This is quite natural because strength condition cry) = 51 under which Eqs. (8.12)-(8.14) were derived is not valid for shear inducing tension and compression in angle-ply layers. To study in-plane shear of the laminate, we should use both solutions of Eq. (8.7) and assume that for some layers, e.g., with i = 1,2,3,. ,n - 1, of) = 81 while for the other layers (i = n,n + I,n -t 2,. . . ,k), or) = -81. Then, Eqs. (8.1) can be reduced to the following forms: N, + N" = - h-) (8.20) (8.21) (8.22) where n- I k hf = h- = Ch; i= I i=n are the total thicknesses of the plies with tensile and compressive stresses in the fibers, respectively. For the case of pure shear (N, = N, = 0), Eqs. (8.20) and (8.21) yield h+ = h- and #j = f45". Then, assuming that +i = +45" for the layers with hi = hi+, while q5i = -45" for the layers with hi = hi we get from Eq. (8.22) The optimal laminate, as follows from the foregoing derivation, corresponds to 4~45' angle-ply structure shown in Fig. 8.2b. 8.2. Composite laminates of uniform strength Consider again the panel in Fig. 8.1 and assume that unidirectional plies or fabric layers, that form the panel are orthotropic, i.e., in contrast to the previous section, we do not neglect now stresses 02 and ~12 in comparison with 01 (see Fig. 3.29). Then, constitutive equations for the panel in plane stress state are specified by the first three equations in Eqs. (5.35), i.e. Chapter 8. Optimal composite structures 313 (8.23) where in accordance with Eqs. (4.72) and (5.28), (5.42) and In the general case, the panel can consist of layers made of different composite materials. Using the optimality criterion developed in the previous section for the fibrous structures we assume that the fibers in each layer are directed along the lines of principal strains, or principal stresses because $2 = GI~$ for an orthotropic layer and condition $3 = 0 is equivalent to condition = 0 (see section 2.4). Using the third equation in Eqs. (4.69) we can write these conditions as 2(g. - E,) sin di cos di + y.rI- cos 24; = 0 . (8.25) This equation can be satisfied for all the layers if we take &.v=&Y-&, y,,=0 . (8.26) Then, Eqs. (8.23) yield Nv = (BII + BI~)E, Nv = (&I + B22)~, Nry = (841 + B42)~ 374 Mechanics and analysis of composite materials These equations allow us to find strain, i.e. (8.27) and to write two relationships specifying the optimal structural parameters of the laminate Substitution of Bmn from Eqs. (8.24) results in the following explicit form of these conditions: (8.28) To determine the stresses that act in the optimal laminate, we use Eqs. (4.69) and (8.26) that specify the strains in the principal material coordinates of the layers as = ~2 = E, yI2 = 0. Applying constitutive equations, Eqs. (4.56), substituting E from Eq. (8.27) and writing the result in the explicit form with the aid of Eqs. (8.24) we arrive at: where (8.29) is the laminate stiffness coefficient. are simplified as If all the layers are made from one and the same material, Eqs. (8.28) and (8.29) Chapter 8. Optimal composite structures k 375 (8.30) (8.31) where Laminates of uniform strength exist under the following restrictions: For monotropic model of the unidirectional ply considered in the previous section, n = 0, m = I, and Eqs. (8.30) reduce to Eqs. (8.9) and (8.10). To determine the thickness of the optimal laminate, we should use Eqs. (8.31) in conjunction with one of the strength criteria discussed in Chapter 6. For the simplest case, using the maximum stress criterion in Eqs. (6.2), the thickness of the laminate can be found from the following conditions CTI = or 02 = a,, so that (8.32) Obviously, for the optimal structure, we would like to have hl = h2. However, this can happen only if material characteristics meet the following condition: (8.33) The results of calculation for typical materials whose properties are listed in Tables 3.5 and 4.4 are presented in Table 8.2. As can be seen, Eq. (8.33) is approximately valid for fabric composites whose stiffness and strength in the warp and fill directions (see section 4.6) are controlled by the fibers of one and the same nature. However for unidirectional polymeric and metal matrix composites, whose longitudinal stiffness and strength are governed by the fibers and transverse characteristics are determined by the matrix properties, a?/al << n. In accordance 376 Mechanics and analysis of composite materials Table 8.2 Parameters of typical advanced composites. Parameter Fabric-epoxy composites Unidirectional-epoxy composites Boron-AI Glass Carbon Aramid Glass Carbon Aramid Boron 0.83 0.022 0.025 0.012 0.054 0.108 - 0~/8l 0.99 0.99 n 0.85 1.0 1 .O 0.28 0. I 0.072 0.11 0.7 with Eqs. (8.32), this means that hl -K h2, and the ratio h2/h1 varies from 12.7 for glass-epoxy to 2.04 for boron+poxy composites. Now, return to the discussion presented in section 4.4.2 from which it follows that in laminated composites transverse stresses 02 reaching their ultimate value, &, cause cracks in the matrix which do not result in the failure of the laminate whose strength is controlled by fibers. To describe the laminate with cracks in the matrix (naturally, if the cracks are admitted for the structure under design), we can use the monotropic model of the ply and, hence, results of optimization presented in Section 8.1. Consider again the optimality condition Eq. (8.25). As can be seen, this equation can be satisfied not only by strains in Eqs. (8.26), but also if we take Y.VJ tan24i = - . Ex - E,. (8.34) Because the left-hand side of this equation is a periodic function with period 7c, Eq. (8.34) determines two angles, Le. (8.35) Thus, the optimal laminate consists of two layers, and the fibers in both layers are directed along the lines of principal stresses. Assume that the layers are made of the same composite material and have the same thickness, i.e. hl = h2 = h/2, where h is the thickness of the laminate. Then, using Eqs. (8.24) and (8.35) we can show that BII = B22 and B24 = -BI~ for this laminate. After some transformation involving elimination of y.!,,. from the first two equations of Eqs. (8.23) with the aid of Eq. (8.34) and similar transformation of the third equation from which and E: are eliminated using again Eq. (8.34) we get Nx = (BI I + 814 tan 24)~: + (B12 - 814 tan %)E!, N,, = (~12 - B14 tan 2414 + (BI 1 + B14 tan 24)~:~ Nx?; = (B44 f B14 cot 24)$,. . Upon substitution of coefficients B,,, from Eqs. (8.24) we arrive at Chapter 8. Optimal composite structures 377 Introducing average stresses a, = N,/h, 0,. = N,/h, and T.~.~. = N,,/h and solving these equations for strains we have I E E.: = - (O.r - VSV), whcrc (8.36) (8.37) Changing strains for stresses in Eqs. (8.35) we can write the expression for the optimal orientation angle as (8.38) As follows from Eqs. (8.36), the laminate consisting of two layers reinforced along the directions of principal stresses behaves like an isotropic layer, and Eqs. (8.37) specify elastic constants of the corresponding isotropic material. For typical advanced composites, these constants are listed in Table 8.3 (the properties of unidirectional plies are taken from Table 3.5). Comparing elastic moduli of the optimal laminates with those for quasi-isotropic materials (see Table 5.1) we can see that for polymeric composites the characteristics of the first group of materials are about 40% higher than those for the second group. However, it should be emphasized that while the properties of quasi-isotropic laminates are the universal Table 8.3 Effective elastic constants of an optimal laminate. Property Glass Carbon- Aramid- Boron- Boron- Carbon- A1203- epoxy epoxy epoxy epoxy AI carbon AI Elastic modulus, E (GPa) 36.9 75.9 50.3 114.8 201.1 95.2 205.4 Poisson’s ratio, Y 0.053 0.039 0.035 0.035 0.21 0.06 0. I76 378 Mechanics and analysis of composite materials material constants, the optimal laminates demonstrate characteristics shown in Table 8.3 only if the orientation angles of the fibers are found from Eqs. (8.35) or (8.38) and correspond to a particular distribution of stresses ox, cy, and zxy. As follows from Table 8.3, the modulus of a carbon-epoxy laminate is close to the modulus of aluminum, while the density of the composite material is less by the factor of 1.7. This is the theoretical weight-saving factor that can be expected if we change aluminum for carbon+poxy composite in a thin-walled structure. Because the stiffness of both materials is approximately the same, to find the optimal orientation angles of the structure elements, we can substitute in Eq. (8.38) the stresses acting in the aluminum prototype structure. Thus designed composite structure will have approximately the same stiffness as the prototype structure and, as a rule, higher strength because carbon composites are stronger than aluminum alloys. To evaluate the strength of the optimal laminate, we should substitute strains from Eqs. (8.36) into Eqs. (4.69) and thus found strains in the principal material coordinates of the layers - into constitutive equations, Eqs. (4.56), that specify stresses ol and o~ (z12 = 0)acting in the layers. Applying the proper failure criterion (see Chapter 6) we can evaluate the laminate strength. Comparing Tables 1.1 and 8.3 we can see that boron-epoxy optimal laminates have approximately the same stiffness that titanium (but is lighter by the factor of about 2) and boron-aluminum can be used to substitute steel with a weight-saving factor of about 3. For preliminary evaluation, we can use a monotropic model of unidirectional plies neglecting stiffness and load-carrying capacity of the matrix. Then, Eqs. (8.37) acquire the following simple forms: (8.39) As an example, consider an aluminum shear web with thickness h = 2 mm, elastic constants E, = 72 GPa, v, = 0.3 and density pa = 2.7 g/cm3. The panel is loaded with shear stress z. Its shear stiffness is Bg, = 57.6 GPa mm and the mass of a unit surface is ma = 5.4 kg/m2. For the composite panel, taking ox = = 0 in Eq. (8.38) we get 4 = 45". Thus, the composite panel consists of +45" and -45" unidirectional layers of the same thickness. The total thickness of the laminate is h = 2 mm, i.e., the same as for an aluminum panel. Substituting El = 140 GPa and taking into account that p = 1.55 g/cm3 for a carbon-epoxy composite that is chosen to substitute aluminum we get B& = 70 GPa mm and m, = 3.1 kg/m3. Stresses acting in the fiber directions of the composite plies are o; = f2z. Thus, the composite panel has 21.5% higher stiffness and its mass makes only 57.4% of the mass of a metal panel. Composite panel has also higher strength because the longitudinal strength of unidirectional carbon-poxy composite under tension and compression is more than twice higher than the shear strength of aluminum. Possibilities of the composite structure under discussion can be enhanced if we use different materials in the layers with angles @, and @? specified by Eqs. (8.35). Chapter 8. Optimal composite structures 379 According to the derivation of Obraztsov and Vasiliev (1 989), the ratio of the layers’ thicknesses is and elastic constants in Eqs. (8.37) are generalized as Superscripts 1 and 2 correspond to layers with orientation angles 4I and 4:, respectively. 8.3. Application to pressure vessels As an example of application of the foregoing results, consider filament wound membrane shells of revolution, that are widely used as pressure vessels, solid propellant rocket motor cases, tanks for gases and liquids, etc. (see Figs. 4.14 and 7.43). The shell is loaded with uniform internal pressure p and axial forces T uniformly distributed along the contour of the shell cross-section Y = ro as in Fig. 8.4. Meridional, Nz, and circumferential, Np, stress resultants acting in the shell Fig. 8.4. Axisymmetrically loaded membrane shell of revolution. 380 Mechanics and analysis of composite materials follow from the corresponding free body diagrams of the shell element and can be written as (see, e.g., Vasiliev, 1993) where z(r)specifiesthe form of the shell meridian, z’ = dz/dr, and Q = TPO +E (7’ - 6) (8.41) 2 Let the shell be made by winding an orthotropic tape a1 angles +4 and -4 with respect to the shell meridian as in Fig. 8.4. Then, N, and Np can be expressed in terms of stresses 61, 62 and 212, referred to the principal material coordinates of the tape with the aid of Eqs. (4.68), i.e. N, = h(ol cos2 4 + 62 sin2 4 - 212 sin 2&), NF = h(o1 sin2 4 + a2 cos2 4 + q2 sin 24) , (8.42) where h is the shell thickness. Stresses q,~, and 212 are linked with the corresponding strains by Hooke’s law, Eqs. (429, as while strains E~,EZ,and yI2 can be expressed in terms of the meridional, E,, and circumferential, ~p,strains of the shell using Eqs. (4.69), i.e. EI = E, cos2 4 + ~p sin2 4, ~2 = E, sin2 4 + cos2 4, y12 = (~p - E,)sin 24 . (8.44) Because the right-hand side parts of these three equations include only two strains, E, and ES,there exists a compatibility equation linking E!, ~2 and y12. This equation is (CI - ~2) sin24 + yI2cos2r$ = 0 . Writing this equation in terms of stresses with the aid of Eqs. (8.43) we get In conjunction with Eqs. (8.42), this equation allows us to determine stresses as [...]... 310-311 362 Fukuda, H 15 27 77 120 Fukui, S 205 222 Gere, J.M 108 120 Gilman, J.J 59 120 Gol'denblat, 1.1 271, 277, 286, 291 300 Gong, X.J 151, 156 223 Goodey, W.J 17 28 66 120 Grakova, T.S 318 362 Green, A E 124 222 Griffith, A.A 61-62 120 Gudmundson, P 180 223 Gunyaev, G.M 116 120 319 Gurdal, Z 41 53 Gurvich, M.R 95, 117 120 21 1 223 272 300 320, 332 363 Gutans, Yu.A 63 120 Ha, S.K 315 362 Hahn, H.T... potential 4, 44, 46, 61, 125 , 128 , 145 elastic potential energy 334 elastic solid 41, 126 elastic strains 128 , 145, 307 elastic volume deformation 129 elastic waves 344 elastic-plastic behavior of metal layer 128 elastic-plastic material model 5, 162 elastic-plastic material 133 elastic-plastic Poisson’s ratio 129 elastic-viscoelastic analogy 327, 329 elasticity theory 145 elastomers 124 elementary cross-ply... V.I 337 362 Kim, H.G 87 120 Kim, R.Y 112- 113 119 Kincis, T.Ya 96, 113 120 Kingston-Lee, D.M 308 363 KO, F.K 15 27 Kobayashi, R 205 222 Koltunov, M.A 118-1 19 120 Kondo, K 87 120 Kopnov, V.A 271, 277, 286, 291 300 Krock, R.H 271 300 329 363 Kruklinsh, A.A 95, 1 17 120 21 1 223 272 300 320, 332 363 Kruzhkova, E.Yu 318 363 Kurshin, L.M 239 269 Kutinov, V.F 336, 355 Lagace, P.A 96 120 186, 196 223 Lapotkin,... 255 complementary elastic potential 125 -126 , 128 , 143 compliance 1 compliance coefficients 86, 152, 156, 230 compliance matrix 45-46 composite beam theory 157 composite body of a space telescope 305 composite bundles 66 composite fibers 182 composite laminate 121 , 124 composite laminates of uniform strength 372 composite lattice shear web structure 214 composite layer 121 , 140 composite materials 9, 21-22,... R.I 112 120 Hashimoto, S 205 222 Hashin, Z 94 120 180 222 Herakovich, C.T v, 142, 146, 162 222 271 300 359 362 379, 391 392 Hodgkinson, J.M 338 363 394 Hondo, A 205 222 Huber, M.T 293 Hyer, M.W 359 362 Ilyushin, A.A 133, 139 223 Ishida, T 340 363 Ivanovskii, V.S 352 363 Jackson, D 310-311 362 Jeong, T.H 87 120 Jones, R.M 91, 96, 107 120 143 223 278 300 Kanagawa, Y 340 363 Kanovich, M.Z 118-1 I9 120 ... Lee, D.J 87 120 Li, L 341 363 Limonov, V.A 338 362 Lungren, J.-E 180 223 Margolin, G.G 276 300 Matthews, F.L 338 363 Mikelsons, M.Ya 63, 337-338 362 Mileiko, S.T 79 120 Milyutin, G.I 318 362 Mises, R 293 Author index Miyazawa, T 77 120 Morozov, E.V 222 223 277 300 360-362 363 Morton, J 338 363 Murakami, S 340 363 Nanyaro, A.P 284 300 Natrusov, V.I 118-1 19 120 Ni, R.G 335 363 Obraztsov, I.F 79 120 200... Tatarnikov, O.V 277 300 Tennyson, R.C 284 300 Tikhomirov, P.V 78 120 Timoshenko, S.P 108 120 Toland, R.H 271 300 Tomatsu, H 77 120 Tsai, S.W 145, 150, 180 222-223 239 269 271,278 300 Tsushima, E 340 363 Turkmen, D 316 363 Van Fo Fy (Vanin), G.A 88,91 120 Varshavskii, V.Ya 115 120 395 Vasiliev, V.V 20 28 41 53 64 120 157, 169, 174, 180-181, 183, 215, 217 222-223 254, 261 269 285, 294 300 344 363 379-380,... 15 27 91 119 Broutman, L.J 271 300 329 363 Brukker, L.E 239 269 Bulavs, F.Ya 95, 117 120 21 1 223 272 300 320, 332 363 Bulmanis, V.N 318 362 Buskel, N.C.R 338 363 Chamis, C.C 156 222 Chen, H.-J 239 269 Cherevatsky, A.S 196 222 Chiao, T.T 77 119 181 222 Chou, T.W 15 27 338 363 Crasto, AS 112- 1 13 119 Curtis, A.R 112 120 Dow, N.F 107 Doxsee, L 341 363 Dudchenko, A.A 169 223 Dvorak, G.J 340 363 Elpatievskii,... 24 compression strength of composites 112 compressive cycle 338 compressive direct pulse 343 compressive stress 197 concept of the accumulation of material damage 332 constitutive equation 5, 324 constitutive equation of the deformation theory 139 constitutive equation of the hereditary theory 321 constitutive equations 3, 41, 45, 48, 53, 70, 81, 85, 92, 125 , 127 -128 , 130, 136, 140-141, 143-144, 146,... Vorobey, V.V 277 300 Wada, A 15 27 Wharram, G.E 284 300 Whitford, L.E 221 223 Whitney, J.M 253 269 Woolstencroft, D.H I12 120 Wostenholm, G 310-31 1 362 Wu, E.M 271 300 Yakushiji, M 15 27 Yates, B 310-311 362 Yatsenko, V.F 276 300 Yunshu, W 341 363 Yushanov S.P 78 120 Zabolotskii, A.A 115 120 Zakrzhevskii, A.M 3 18 362 Zhenlong, G 341 363 Zhigun, I.G 15 28 215-216 223 Zinoviev, P.A 200 222 334-335 363 . with circumferential (90") or axial (0") plies. 372 Mechanics and analysis of composite materials Particularly, Eqs. (8.13) and (8.14) do not describe the case of pure. has already been noted, upon 388 Mechanics and analysis of composite materials winding, an opening of radius ro is formed at the shell apex. However, the analysis of Eq. (8.65) for rl. characteristics are determined by the matrix properties, a?/al << n. In accordance 376 Mechanics and analysis of composite materials Table 8.2 Parameters of typical advanced composites.