406 Mechanics of Materials §15.7 Fig.15.3. Substituting, 0%-01+0201十02-0% 0%+c% 0y.-0% Cross-multiplying and simplifying this reduces to 01+2=1 (15.5) Oye Oye which is then the Mohr's modified shear stress criterion for brittle materials. 15.7.Graphical representation of failure theories for two-dimensional stress systems (one principal stress zero) Having obtained the equations for the elastic failure criteria above in the general three- dimensional stress state it is relatively simple to obtain the corresponding equations when one of the principal stresses is zero. Each theory may be represented graphically as described below,the diagrams often being termed yield loci. (a)Maximum principal stress theory For simplicity of treatment,ignore for the moment the normal convention for the principal stresses,i.e.>2>3 and consider the two-dimensional stress state shown in Fig.15.4 Fig.15.4.Two-dimensional stress state (3=0).406 Mechanics of Materials $15.7 T t Fig. 15.3. Substituting, ayI-ao,+a2 al+a2-ayl - CY1 + OYc CY, - QYI Cross-multiplying and simplifying this reduces to (15.5) 01 02 -+-= 1 by, CY, which is then the Mohr's modified shear stress criterion for brittle materials. 15.7. Graphical representation of failure theories for two-dimensional stress systems (one principal stress zero) Having obtained the equations for the elastic failure criteria above in the general three￾dimensional stress state it is relatively simple to obtain the corresponding equations when one of the principal stresses is zero. Each theory may be represented graphically as described below, the diagrams often being termed yield loci. (a) Maximum principal stress theory For simplicity of treatment, ignore for the moment the normal convention for the principal stresses, i.e. a1 > a2 > a3 and consider the two-dimensional stress state shown in Fig. 15.4 i-' Fig. 15.4. Two-dimensional stress state (as = 0)