■In summary: stress reciprocity:Piee=Pytk strain definition:uee= (9.1) symmetry:ike =ety There remain 21 distinct coefficients The previous stress-strain relation can then be written as: > Sn 01111 0112201133 201123 201113 201112 6 62 p211 02222 02233 202225 202213 20212 62 6 03311 03322 φ3333 203325 203313 203312 033 E25 02311 023220233 202323 202313 2p2312 023 E13 01311 01322 01333 201323 201313 201312 013 E12 012110122201233 2p1223201213 201212 012 The matrix above does not have the general symmetry as in the general form (9x 9)presented previously.(Note the coefficients 2 in this matrix).One can get around this inconvenience by doubling the terms 823,E13,812,introducing the shear strains: Y23=2E23;Y13=2813;Y12=2812 from which the stress-strain behavior can then be written in a symmetric form as: 11 01111 01122 p1133 201123 201113 201112 011 62 0211 02222 02233 20223 202213 202212 02 633 03311 03322 03333 203323 203313 203512 63 (9.2) 2623 Y23 202311 202322 202333 402323 402313 402312 025 2813 Y13 201311 201322 201333 401323 401313 401312 013 2612 Y12 2p121 201222 201233 401223 401213 401212 012 9.2 ORTHOTROPIC MATERIALS Definition:An orthotropic material is a homogeneous linear elastic material having two planes of symmetry at every point in terms of mechanical properties,these two planes being perpendicular to each other. 2003 by CRC Press LLC
In summary: The previous stress–strain relation can then be written as: The matrix above does not have the general symmetry as in the general form (9 ¥ 9) presented previously. (Note the coefficients 2 in this matrix). One can get around this inconvenience by doubling the terms e23, e13, e12, introducing the shear strains: from which the stress–strain behavior can then be written in a symmetric form as: (9.2) 9.2 ORTHOTROPIC MATERIALS Definition: An orthotropic material is a homogeneous linear elastic material having two planes of symmetry at every point in terms of mechanical properties, these two planes being perpendicular to each other. stress reciprocity: strain definition: symmetry: There remain 21 distinct coefficients (9.1) jijk
= jij
k jijk
= jjik
jijk
= jk
ij jijk
e 11 e 22 e 33 e 23 e 13 e 12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ j1111 j1122 j1133 2j1123 2j1113 2j1112 j2211 j2222 j2233 2j2223 2j2213 2j2212 j3311 j3322 j3333 2j3323 2j3313 2j3312 j2311 j2322 j2333 2j2323 2j2313 2j2312 j1311 j1322 j1333 2j1323 2j1313 2j1312 j1211 j1222 j1233 2j1223 2j1213 2j1212 s11 s22 s33 s23 s13 s12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ = g 23 2e 23; g 13 2e 13 = = ; g 12 = 2e 12 e 11 e 22 e 33 2e 23 2e 13 2e 12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ g 23 g 13 g 12 j1111 j1122 j1133 2j1123 2j1113 2j1112 j2211 j2222 j2233 2j2223 2j2213 2j2212 j3311 j3322 j3333 2j3323 2j3313 2j3312 2j2311 2j2322 2j2333 4j2323 4j2313 4j2312 2j1311 2j1322 2j1333 4j1323 4j1313 4j1312 2j1211 2j1222 2j1233 4j1223 4j1213 4j1212 s11 s22 s33 s23 s13 s12 Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ = TX846_Frame_C09 Page 209 Monday, November 18, 2002 12:24 PM © 2003 by CRC Press LLC