s15.7 Theories of Elastic Failure 409 This is the equation of an ellipse with major and minor semi-axes 1-可 and √/(1+) respectively,each at 45 to the coordinate axes as shown in Fig.15.8. 02 2-v 20+)】 ◆可 Shear diagonai 9 Fig.15.8.Failure envelope for maximum strain energy per unit volume theory. (e)Maximum shear strain energy per unit volume theory With o3=0 the criteria of failure for this theory reduces to [(o1-02)2+3+]= 1+01-0102=03 8+((侣g)-1 again an ellipse with semi-axes(2)o,and()o,at 45 to the coordinate axes as shown in Fig.15.9.The ellipse will circumscribe the maximum shear stress hexagon. 459 -Oy -O o, 0.5770y Shear diagonal Fig.15.9.Failure envelope for maximum shear strain energy per unit volume theory.01 5.7 Theories of Elastic Failure 409 This is the equation of an ellipse with major and minor semi-axes *Y *' and J(1 - 4 J(1 + 4 respectively, each at 45" to the coordinate axes as shown in Fig. 15.8. Fig. 15.8. Failure envelope for maximum strain energy per unit volume theory. (e) Maximum shear strain energy per unit volume theory With o3 = 0 the criteria of failure for this theory reduces to $[ (01 - a2)2 + *: + 41 = 0; a:+a;-rJa,a2 = 0; py+(;y-(:)(;)= 1 again an ellipse with semi-axes J(2)ay and ,/(*)cy at 45" to the coordinate axes as shown in Fig. 15.9. The ellipse will circumscribe the maximum shear stress hexagon. \Sheor diagonal I Fig. 15.9. Failure envelope for maximum shear strain energy per unit volume theory