84 6 Elastic Constants Based on Global Coordinate System function y =Etayxy(Sbar) %Etayxy This function returns the coefficient of % mutual influence of the first kind % ETAy,xy in the global coordinate system. % It has one argument -the reduced % transformed compliance matrix Sbar. Etayxy is returned as a scalar y Sbar(2,3)/Sbar(3,3); Example 6.1 Derive the expression for Ez given in (6.1). Solution From an elementary course on mechanics of materials,we have the following relation (assuming uniaxial tension with o0 and all other stresses zeros): =号 (6.14) However,from (5.10),we also have the following relation: Ex=S110 (6.15) Comparing (6.14)and (6.15),we conclude the following: 官=81 1 (6.16) Substituting for S1 from (5.16a)and taking the inverse of(6.16),we obtain the desired result as follows: 1 E B=m+(盒-2na)2m2+0m (6.17) In the above equation,we have substituted for the elements of the reduced compliance matrix with the appropriate elastic constants. MATLAB Example 6.2 Consider a graphite-reinforced polymer composite lamina with the elastic constants as given in Example 2.2.Use MATLAB to plot the values of the five elastic constants Er,vry,Ey,vur,and Gry as a function of the orientation angle 0 in the range -π/2≤0≤π/2.84 6 Elastic Constants Based on Global Coordinate System function y = Etayxy(Sbar) %Etayxy This function returns the coefficient of % mutual influence of the first kind % ETAy,xy in the global coordinate system. % It has one argument - the reduced % transformed compliance matrix Sbar. % Etayxy is returned as a scalar y = Sbar(2,3)/Sbar(3,3); Example 6.1 Derive the expression for Ex given in (6.1). Solution From an elementary course on mechanics of materials, we have the following relation (assuming uniaxial tension with σx = 0 and all other stresses zeros): εx = σx Ex (6.14) However, from (5.10), we also have the following relation: εx = S¯11σx (6.15) Comparing (6.14) and (6.15), we conclude the following: 1 Ex = S¯11 (6.16) Substituting for S¯11 from (5.16a) and taking the inverse of (6.16), we obtain the desired result as follows: Ex = 1 S¯11 = E1 m4 +  E1 G12 − 2ν12 n2m2 + E1 E2 n4 (6.17) In the above equation, we have substituted for the elements of the reduced compliance matrix with the appropriate elastic constants. MATLAB Example 6.2 Consider a graphite-reinforced polymer composite lamina with the elastic constants as given in Example 2.2. Use MATLAB to plot the values of the five elastic constants Ex, νxy, Ey, νyx, and Gxy as a function of the orientation angle θ in the range −π/2 ≤ θ ≤ π/2.