Mohr envelope

Mohr envelope The envelope of a sequence of Mohr circles that represents conditions at failure for a given material. 

Figure 8.4. Mohr circle and Mohr-Coulomb failure envelope.

The Mohr-Coulomb failure criterion can be recognized as an upper bound for the stress combination on any plane in the material. Consider points A, B, and C in Fig. 8.4. Point A represents a state of stresses on a plane along which failure will not occur. On the other hand, failure will occur along a plane if the state of stresses on that plane plots a point on the failure envelope, like point B. The state of stresses represented by point C cannot exist since it lies above the failure envelope. Since the Mohr-Coulomb failure envelope characterizes the state of stresses under which the material starts to slide, it is usually referred to as the yield locus, YL. 

For each porosity, there is a particular yield locus, a family of three 3deld loci is shown in Figure 12.38. Many experiments [72] have established that the envelope of the Mohr circles through the points Ei that lead to steady state flow for different porosities is, to a veiy close... 

The lines OA and OB are tangent to all the stress circles for any value of Py provided the material is noncohesive. They form the Mohr rupture envelope. With cohesive solids or solid masses the tangents forming the envelope do not pass through the origin but intercept the vertical axis at points above and below the horizontal axis. ... 

The failure envelope /(CTi (J3) = 0 is defined from point to i by the Mohr-Coulomb failure criterion /, = 0 with... 

The soft clays were modelled using effective stress parameters, assuming linear elastie behaviour bounded by a Mohr-Coulomb envelope t/s = sin

shear stress/normal stress diagram for plane strain, in which the Mohr-Coulomb envelope is marked, together with a typical effective stress path for undrained behaviour of a normally consolidated clay. This path reaches failure at point F on the Mohr-Coulomb envelope with undrained strength c . 

Figure 19. The Mohr-Coulomb envelope as a plastic potential.

For a cohesive particulate material, each YL curve ends at a point where the normal stress equals the consolidation pressure. Mohr circles can now he drawn that are tangent to the end point of the various yield loci. The envelope of these circles is called the effective yield locus (EYL). This is generally a straight line passing through the origin see Fig. 6.5. 

Balmer G. G. Shearing strength of concrete under high tiiaxial stress—computation of Mohr s envelope as a curve. Struct. Research Lab. Report No. SP-23 USBR, 1949. 

They can be plotted as the so called Mohr failure envelopes in the cr-r diagram together with Mohr s circles and employed for analysis of stability of fractures or faults under given stress conditions. If the outer Mohr s circle touches the failure envelope, there is one fracture or fault which is unstable and can fail, its orientation being defined by inclination 6 of the fault fi om the maximum stress direction (see Fig. 5a). 

The repeated loading of the fault plane by successive pressure increase is illustrated in Fig. 5. Here the effective stress decreases due to fluid pressure increase until the shear stress exceeds the failure envelope, which results in shearing and decrease of the differential stress represented by the diameter of the Mohr circle. Further increase of pore pressure leads to repeated failure. 

Earthquake Swarms, Fig. 5 Mohr-Coulomb diagram illustrating the reactivation of earthquake slip due to increase of the pore pressure Ap. The failure envelope, which is indicated by the inclined line, shows the failure shear stress Xf = — p), where p is the Mctirai coeffi-... 

A second prerequisite for tensile failure is that the shear stresses be small, or equivalently that the principal stresses be nearly equal. The diameter of the Mohr s circle in Fig. 5 is equal to the maximum shear stress (difference between the extreme principal stresses). If this diameter is too large, the circle can touch the failure envelope only along its straight portion, which corresponds to shear failure. Only if the shear stress, and thus the diameter, is small will the circle first touch the failure envelope in the tensile field to the left of the t axis. 

FIGURE 7.11 Mohr s circles and associated failure envelope for intact samples of Muderong Shale (Josh et ah, 2012 Dewhurst and Hennig, 2003). Unconfined compressive strength can be calculated at 8 MPa. 

The equations represent in a r —ffn plot circle, called Mohr s circle (Fig. 7.10). A series of experiments result in a number of circles (see Fig. 7.11). The envelope... 

Figure 7.11 shows as an example Mohr s diagram for the Muderong shale, a relatively weak material with a cohesive strength of 2.75 MPa and a friction coefficient of 0.34. The envelope delivers the equation (Dewhurst and Hennig, 2003) ... 

---