§15.6 Theories of Elastic Failure 405 Pure shear to yield D Pure compression Pure tensior to yield to yield Fig.15.1.Mohr theory on o-t axes. As a close approximation to this procedure Mohr suggests that only the pure tension and pure compression failure circles need be drawn with OA and OBequal to the yield or fracture strengths of the brittle material.Common tangents to these circles may then be used as the failure envelope as shown in Fig.15.2.Circles drawn tangent to this envelope then represent the condition of failure at the point of tangency. Fig.15.2.Simplified Mohr theory on o-t axes. In order to develop a theoretical expression for the failure criterion,consider a general stress circle with principal stresses of a and o2.It is then possible to develop an expression relatingo,2,the principal stresses,andthe yield strengths of the brittle material in tension and compression respectively. From the geometry of Fig.15.3, KL JL KMMH Now,in terms of the stresses, KL=(a1+02)-01+0%=(c%-01+02) KM=o%+和y.=(o%+c) JL=(a1+02)-0y=(o1+02-0 MH=0%.-o%=(og-0y)$15.6 Theories of Elastic Failure 405 Y I Fig. 15.1. Mohr theory on 0-T axes. As a close approximation to this procedure Mohr suggests that only the pure tension and pure compression failure circles need be drawn with OA and OB equal to the yield or fracture strengths of the brittle material. Common tangents to these circles may then be used as the failure envelope as shown in Fig. 15.2. Circles drawn tangent to this envelope then represent the condition of failure at the point of tangency. r Fig. 15.2. Simplified Mohr theory on g-7 axes. In order to develop a theoretical expression for the failure criterion, consider a general stress circle with principal stresses of o1 and 02. It is then possible to develop an expression relating ol, 02, the principal stresses, and o,,, o,,, the yield strengths of the brittle material in tension and compression respectively. From the geometry of Fig. 15.3, KL JL KM MH -=- Now, in terms of the stresses, KL =$(.I +o,)-oa, +$c~,=$(D~,-Q~ +a,) K M = $a,, + *oyc = f (oY, + on) JL = $(01+ 02) -$o,, = $(GI + 62 - oy,) MH='~ -Lo -o ) 2 Yc 2 Y, 2 Yc Y