# lecture13 - Theory of Anisotropy l. The Hooke’s law for...

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Unformatted text preview: Theory of Anisotropy l. The Hooke’s law for three-dimensional anisotropic materials: a”. : C We“ where CW is called stiffness matrix and may be a 9X9 matrix 51.] = SWUk, where SW is called compliance matrix and may be a 9x9 matrix where i,j, k, 1:1, 2,3 Symmetry of shear stresses and shear strains: at]. = 0‘], and 8y. 2 8]., the stiffness matrix and the compliance matrix may be of a 6x6 matrix a = C..8. ’ U ./ 8i 2 5470.; where i,j= 1,2,3 Stiffness and compliance matrices are symmetric matrices: Cy- = Cit, Sy- = Sﬂ There will be 21 independent elastic constants in anisotropic materials. A material with 21 independent elastic constants is called an anisotropic material. If there is one plane with which a material is symmetric, the material is called a clinotropic material. There are 13 independent elastic constants in the clinotropic material. If there are two perpendicular planes with which a material is symmetric, the material will also be symmetric with respect to the third plane which is normal to the ﬁrst two planes. This material is called an orthotropic material. There are 9 independent elastic constants in the orthotropic material and they are three Young’s moduli, three Poison’s ratios and three shear moduli. In a two—dimensional domain, an anisotropic material has 6 independent elastic constants. In a two-dimensional domain, an orthotropic material has 4 independent elastic constants. The corresponding Hooke’s laws based on the material coordinate system are shown below. an “Q11 Q12 0 811 511 SH 512 0 0-11 (722 : Q12 Q22 0 822 822 : S12 322 0 022 T12 _ 0 O Q66 712 7/12 0 0 See T12 _1_ _ V21 0 E11 E22 " V12 _1m 0 E11 E22 0 0 _ G12 8. The Hooke’s laws for two-dimensional orthotropic materials based on the reference coordinate system are shown below. Q11 912 z 912 922 _Q16 Q26 g 9.26 Q66 8 XX 8y); 7X); xx yy yxy E11 S 12 S 16 a = \$2 3: 22 E26 0— 1: : _§16 —2() _6(2_ 7;; — 1 _ Vyx 773x Eyy ny _ — V»; I E m, E (W ny 77m 1 E w Go" ...
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## This note was uploaded on 07/25/2008 for the course ME 426 taught by Professor Liu during the Spring '06 term at Michigan State University.

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lecture13 - Theory of Anisotropy l. The Hooke’s law for...

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