Until now,we have taken into account the symmetry with respect to plane 1,3.Consider now the coordinates 1,2,3 and I',II',III'(see Figure 13.1),which can be obtained from each other by a 180 rotation about the 2 axis (symmetry with respect to plane 1,2).One has -1 0 [cos"] 0 1 0 0 0-1 The same procedure as above will lead to Φ11=-p1112=p1112=0;Φ1m1rm=-p2212=p2212=0 Φmm1=-p3312=93312=0;①m1'm=-p2313=92313=0 Considering the symmetry of the coefficientsindicated in Relation9.1.2 we have written here the only nonzero terms.For the mechanical behavior,one obtains by simplification of Equation 9.2: e11 φ1111 φ1122 p1133 0 0 0 611 p2211 02222P2233 0 0 0 62 e38 03311 03322 03333 0 0 0 033 (13.2) Y23 0 0 0 492323 0 0 T23 Y13 0 0 0 0 401313 0 t 0 0 0 0 0 4φ1212 T12 There remain then only nine distinct elastic coefficients,which can be written in the form of Young's moduli and Poisson ratios as: e 1 岩 岩 00 0 011 e22 E 苦 00 0 022 e33 岩 E 言 000 033 (13.3) Y23 0 0 0 00 23 0 0 0 T13 0 0 0 0 T12 Recall the symmetry relations:===u 2003 by CRC Press LLCUntil now, we have taken into account the symmetry with respect to plane 1,3. Consider now the coordinates 1,2,3 and I’, II’, III’ (see Figure 13.1), which can be obtained from each other by a 180∞ rotation about the 2 axis (symmetry with respect to plane 1,2). One has The same procedure as above will lead to Considering the symmetry of the coefficients jmnpq indicated in Relation 9.1,2 we have written here the only nonzero terms. For the mechanical behavior, one obtains by simplification of Equation 9.2: (13.2) There remain then only nine distinct elastic coefficients, which can be written in the form of Young’s moduli and Poisson ratios as: (13.3) 2 Recall the symmetry relations: jijkl = jijlk; jijkl = jjikl; jijkl = jklij. cosI m [ ] –1 0 0 010 001– = FI¢I¢I¢II¢ = == = == –j1112 j1112 0; FII¢II¢I¢II¢ –j2212 j2212 0 FIII¢III¢I¢II¢ = == = == –j3312 j3312 0; FII¢III¢I¢III¢ –j2313 j2313 0 e 11 e 22 e 33 g 23 g 13 g 12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ j1111 j1122 j1133 000 j2211 j2222 j2233 000 j3311 j3322 j3333 000 0 0 04j2323 0 0 0 0 0 04j1313 0 0 0 0 0 04j1212 s11 s22 s33 t 23 t 13 t 12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ = e 11 e 22 e 33 g 23 g 13 g 12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ 1 E1 ---- –n21 E2 --------- –n31 E3 --------- 000 –n12 E1 --------- 1 E2 ---- –n32 E3 --------- 000 –n13 E1 --------- –n23 E2 --------- 1 E3 ---- 000 00 0 1 G23 ------- 0 0 00 00 1 G13 ------- 0 00 0 00 1 G12 ------- s11 s22 s33 t 23 t 13 t 12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ = TX846_Frame_C13 Page 261 Monday, November 18, 2002 12:29 PM © 2003 by CRC Press LLC