2.2 MATLAB Functions Used 15 function y IsotropicStiffness(E,NU) %IsotropicStiffness This function returns the stiffness matrix for isotropic % materials.There are two % arguments representing the two independent material constants.The size of the 名 stiffness matrix is 6 x 6. x=[1/E-NU/E-w/E000;-NU/E1/E-NU/E000; -NU/E-NU/E1/E000;0002*(1+N0U)/E00; 00002*(1+NU)/E0;000002*(1+NU)/E]; y=inv(x); Example 2.1 For an orthotropic material,derive expressions for the elements of the stiffness matrix Cii directly in terms of the nine independent material constants. Solution Substitute the elements of [S]from (2.1)into (2.5)along with using (2.6). This is illustrated in detail for Cl below.First evaluate the expression of S from (2.5)as follows: S=S11S22S33-S11S23S23-S22S13S13-S33S12S12+2S12S23S13 1111 -23 -32 E E2 Ex E1 E2 E3 1 -V3 -31 /1 E2 E 、E3 +2(器) 1-V23V32-V13V31-M221-2y12V2331 E1E2E3 1-0 EE2E3 (2.9a) where vo is given by V0=2332+y331+1221+212V2331 (2.9b) Next,C1l is calculated as follows2.2 MATLAB Functions Used 15 function y = IsotropicStiffness(E,NU) %IsotropicStiffness This function returns the % stiffness matrix for isotropic % materials. There are two % arguments representing the % two independent material % constants. The size of the % stiffness matrix is 6 x 6. x = [1/E -NU/E -NU/E 0 0 0 ; -NU/E 1/E -NU/E000; -NU/E -NU/E 1/E 0 0 0;000 2*(1+NU)/E00; 0000 2*(1+NU)/E 0;00000 2*(1+NU)/E]; y = inv(x); Example 2.1 For an orthotropic material, derive expressions for the elements of the stiffness matrix Cij directly in terms of the nine independent material constants. Solution Substitute the elements of [S] from (2.1) into (2.5) along with using (2.6). This is illustrated in detail for C11 below. First evaluate the expression of S from (2.5) as follows: S = S11S22S33 − S11S23S23 − S22S13S13 − S33S12S12 + 2S12S23S13 = 1 E1 1 E2 1 E3 − 1 E1 −ν23 E2  −ν32 E3  − 1 E2 −ν13 E1  −ν31 E3  − 1 E3 −ν12 E1  −ν21 E2  +2 −ν12 E1  −ν23 E2  −ν31 E3  = 1 − ν23ν32 − ν13ν31 − ν12ν21 − 2ν12ν23ν31 E1E2E3 = 1 − ν0 E1E2E3 (2.9a) where ν0 is given by ν0 = ν23ν32 + ν13ν31 + ν12ν21 + 2ν12ν23ν31 (2.9b) Next, C11 is calculated as follows