Chapter 1 Brittle material behaviour 1.1 Reading assignment Twiss and Moore’s chapter nine is essential reading for this stuff, and parts of chapter ten are deﬁnitely relevant. 1.2 Failure criteria 1.2.1 Tensile stress σ1 = σ2 >> σ3 And σ3 is negative. Open tension fractures containing σ1 and σ2 and perpendic ular to σ3 . Draw a Mohr circle for this situation: S1 shear stress S3 S2 extension fractures open parallel to sigma 3 !! normal stress Figure 1.1: Mohr construction for tensile stresses 1.2.2 Shear fractures in noncohesive material For instance, loose sand. Byerlee in the 50s conducted various experiments look ing at the strength of these sorts of materials, where strength is understood as the amount of shear stress necessary to initiate motion, given a certain amount of nor mal stress. If you plot shear stress necessary to initiate motion against normal stress, you get a line which makes an angle φ with the normal stress axis. The equation de scribing this relationship is Byerlee’s law, and written as τc = σN tan φ = µσN N where φ is called the angle on internal friction and µ is the coefﬁcient of fric tion. We can create a Mohr construction that illustrates how this equation provides a physical law for predicting when failure will occur in a material: Figure 1.2: Mohr construction for shear failure according to Byerlee’s law (no cohe sion). As an exercise, label the ﬁgure with τ, σN , 2α, α, σ1 , σ3 , and phi . Also, use this diagram to derive the relationship α = 45 + φ/2 1.3 Cohesive material Given the above failure criterion, in the absence of a conﬁning stress, the shear stress for failure will be zero. But materials have strength even in the absence of conﬁning stress – this is known as the cohesion. A modiﬁed failure criterion incorporating this is the MohrCoulomb failure criterion: τ = C + σN tan φ Draw the Mohr circle construction for this. shear stress e lur lo ve en pe fai fluid pressure normal stress Figure 1.3: 1.4 Effect of pore ﬂuid pressures The effect of a pressurized ﬂuid on a fault plane is to decrease the normal stress by that pressure. In this case, the MohrCoulomb failure criterion becomes τ = C + σN e f f tan φ = C + (σN − P f ) tan φ where P f is the ﬂuid pressure. On a Mohr diagram, the effect of ﬂuid pressure is to shift the Mohr circle to the left of the diagram. As ﬂuid pressures go up, the possibil ity that the Mohr circle intersects the failure envelope is increased. 1.5 Review questions Lab 1 on stress, particularly the orientation of principal stresses in relation to fault types and tectonic environments. Lab 5 on faults, particularly the problems on joints and Downie’s slide. What does a generalized failure envelope taking into account both tensile and shear failure look like? In general, what is the effect of higher mean stress (pressure)? What implications does this have for the brittle failure of materials at greater and greater depths?