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where after substitution: Ex 笑 07 Ox Ey [T] E 1E 0[T] Yxy. 0 0 Gu- Txy new matrix of elastic coefficients in x,y axes When all calculations are performed,one obtains the following constitutive relation, written in the coordinates x,y that make an angle 0 with the axes ,t.The elastic moduli and Poisson coefficients appear in these relations.One can also see the existence of the coupling coefficients n and u,which demonstrates that a normal stress can produce a distortion. Ex a Ox Ey Cy 是 莞 with: 1 Ex(θ)= 1 E,(0)= ++-增 (11.5) G(0)= 1 4(++2) Gu 0=+s-2+总 器=-2a怎-+-管动 急(0=-2c管--e-s管) Recall that the matrix of elastic coefficients is symmetric,meaning in particular: and Ho/Gsy=My/Ey 7 See example described in Section 3.1. 2003 by CRC Press LLCwhere after substitution: When all calculations are performed, one obtains the following constitutive relation, written in the coordinates x,y that make an angle q with the axes ,t. The elastic moduli and Poisson coefficients appear in these relations. One can also see the existence of the coupling coefficients h and m, 6 which demonstrates that a normal stress can produce a distortion.7 (11.5) 6 Recall that the matrix of elastic coefficients is symmetric, meaning in particular: hxy /Gxy = hx/Ex and mxy /Gxy = my /Ey. 7 See example described in Section 3.1. ex e y Óg xy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ [ ] T ¢ 1 E ----- nt -Et -------- 0 nt E –------ 1 Et ---- 0 0 0 1 Gt ------- [ ] T sx sy Ótxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = new matrix of elastic coefficients in x,y axes Ï Ô Ô Ô Ô Ì Ô Ô Ô Ô Ó ex e y Óg xy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 E -- x nyx Ey –------- hxy Gxy -------- n xy Ex –------- 1 E -- y mxy Gxy -------- hx Ex ----- my Ey ----- 1 Gxy -------- sx sy Ótxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = with: Ex( ) q 1 c E -----4 s 4 Et ---- c 2 s 2 1 Gt ------- 2 nt Et – ------ Ë ¯ Ê ˆ + + = ---------------------------------------------------------- Ey( ) q 1 s E -----4 c 4 Et ---- c 2 s 2 1 Gt ------- 2 nt Et – ------ Ë ¯ Ê ˆ + + = ---------------------------------------------------------- Gxy( ) q 1 4c 2 s 2 1 E ----- 1 Et ---- 2 nt Et + + ------ Ë ¯ Ê ˆ c 2 s 2 ( ) – 2 Gt + ---------------------- = ----------------------------------------------------------------------- nyx Ey -------( ) q nt Et ------ c 4 s 4 ( ) + c 2 s 2 1 E ----- 1 Et ---- 1 Gt + – ----- Ë ¯ Ê ˆ = – hxy Gxy --------( ) q 2cs c 2 E ----- s 2 Et – ---- c 2 s 2 + ( ) – nt Et ------ 1 2Gt – ----------- Ë ¯ Ê ˆ Ó ˛ Ì ˝ Ï ¸ = – mxy Gxy --------( ) q 2cs s 2 E ----- c 2 Et ---- c 2 s 2 ( ) – nt Et ------ 1 2Gt – ----------- Ë ¯ Ê ˆ – – Ó ˛ Ì ˝ Ï ¸ = – TX846_Frame_C11 Page 227 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC