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 threedimensional 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)
§15.7 Theories of Elastic Failure 407 where a is zero and may be tensile or compressive as appropriate,i.e.2 may have a value less than o3 for the purpose of this development. The maximum principal stress theory then states that failure will occur when a or 2=y or ay.Assuming oy=yoy,these conditions are represented graphically on 1,2 coordinates as shown in Fig.15.5.If the point with coordinates (1,2)representing any complex two-dimensional stress system falls outside the square,then failure will occur according to the theory. 02 -Oy Fig.15.5.Maximum principal stress failure envelope (locus). (b)Maximum shear stress theory For like stresses,i.e.and2,both tensile or both compressive(first and third quadrants), the maximum shear stress criterion is (o1-0)=o,or(o2-0)=o, i.e. 01=0y0r02=0y thus producing the same result as the previous theory in the first and third quadrants. For unlike stresses the criterion becomes (o1-02)=0y since consideration of the third stress as zero will not produce as large a shear as that when o2 is negative.Thus for the second and fourth quadrants, These are straight lines and produce the failure envelope of Fig.15.6.Again,any point outside the failure envelope represents a condition of potential failure. (c)Maximum principal strain theory For yielding in tension the theory states that 01-vo2=0y
$15.7 Theories of Elastic Failure 407 where a3 is zero and a2 may be tensile or compressive as appropriate, i.e. a2 may have a value less than a3 for the purpose of this development. The maximum principal stress theory then states that failure will occur when a1 or a2 = a,,, or a,,,. Assuming a,,, = a,,, = a,,, these conditions are represented graphically on aI, a2 coordinates as shown in Fig. 15.5. If the point with coordinates (al, a2) representing any complex two-dimensional stress system falls outside the square, then failure will occur according to the theory. 02 t Fig. 15.5. Maximum principal stress failure envelope (locus). (b) Maximum shear stress theory For like stresses, i.e. a1 and a2, both tensile or both compressive (first and third quadrants), the maximum shear stress criterion is +(al -0) = $0, or $(a2 -0) =+ay i.e. a1 = ay or a2 =ay thus producing the same result as the previous theory in the first and third quadrants. For unlike stresses the criterion becomes +(a1 - 62) = 3.y since consideration of the third stress as zero will not produce as large a shear as that when a2 is negative. Thus for the second and fourth quadrants, These are straight lines and produce the failure envelope of Fig. 15.6. Again, any point outside the failure envelope represents a condition of potential failure. (c) Maximum principal strain theory For yielding in tension the theory states that 61 --a2 = by
408 Mechanics of Materials $15.7 02 6 Sheor diagonal Fig.15.6.Maximum shear stress failure envelope. and for compressive yield,with a2 compressive, 02-v01=0y Since this theory does not find general acceptance in any engineering field it is sufficient to note here,without proof,that the above equations produce the rhomboid failure envelope shown in Fig.15.7. 2 品 Shear diagonal Fig.15.7.Maximum principal strain failure envelope. (d)Maximum strain energy per unit volume theory With o3=0 this failure criterion reduces to a1+o3-2v0102= i.e. a+-2x(e)-1
408 Mechanics of Materials 515.7 Fig. 15.6. Maximum shear stress failure envelope and for compressive yield, with o2 compressive, Since this theory does not find general acceptance in any engineering field it is sufficient to note here, without proof, that the above equations produce the rhomboid failure envelope shown in Fig. 15.7. 4 Fig. 15.7. Maximum principal strain failure envelope. (d) Maximum strain energy per unit oolume theory With c3 = 0 this failure criterion reduces to a:+a;-2vo,02 = 6; i.e
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
410 Mechanics of Materials S15.8 (f)Mohr's modified shear stress theory (ay>oy) For the original formulation of the theory based on the results of pure tension,pure compression and pure shear tests the Mohr failure envelope is as indicated in Fig.15.10. In its simplified form,however,based on just the pure tension and pure compression results,the failure envelope becomes that of Fig.15.11. 402 ay, 9 Shear (a) (b) Fig.15.10.(a)Mohr theory on o-axes.(b)Mohr theory failure envelope on a-02 axes. C2 (a) (b) Fig.15.11.(a)Simplified Mohr theory ontaxes.(b)Failure envelope for simplified Mohr theory. 15.8.Graphical solution of two-dimensional theory of failure problems The graphical representations of the failure theories,or yield loci,may be combined onto a single set of o and o2 coordinate axes as shown in Fig.15.12.Inside any particular locus or failure envelope elastic conditions prevail whilst points outside the loci suggest that yielding or fracture will occur.It will be noted that in most cases the maximum shear stress criterion is the most conservative of the theories.The combined diagram is particularly useful since it allows experimental points to be plotted to give an immediate assessment of failure
410 Mechanics of Materials 515.8 (f) Mohr’s modijied shear stress theory (cJ,,, > cy,) For the original formulation of the theory based on the results of pure tension, pure compression and pure shear tests the Mohr failure envelope is as indicated in Fig. 15.10. In its simplified form, however, based on just the pure tension and pure compression results, the failure envelope becomes that of Fig. 15.11. Fig. 15.10. (a) Mohr theory on u-T axes. (b) Mohr theory failure envelope on u,-u2 axes. Q2 Fig. 15.1 1. (a) Simplified Mohr theory on u-T axes. (b) Failure envelope for simplified Mohr theory. 15.8. Graphical solation of two-dimensional theory of failure problems The graphical representations of the failure theories, or yield loci, may be combined onto a single set of ol and o2 coordinate axes as shown in Fig. 15.12. Inside any particular locus or failure envelope elastic conditions prevail whilst points outside the loci suggest that yielding or fracture will occur. It will be noted that in most cases the maximum shear stress criterion is the most conservative of the theories. The combined diagram is particularly useful since it allows experimental points to be plotted to give an immediate assessment of failure