Mohr circle diagram

Analysis of yielding at a notch (a) shear band patterns seen in a thin section cut from a polycarbonate specimen (b) slip line field pattern for yielding (c) Mohr circle diagram for the states of stress at points A and B in (b) (d) stress components on the surface of the prism marked out by neighbouring a and j3 slip lines. 

For the analysis of combined stress in the two-dimensional situation the Mohr circle diagram (see Appendix A1.8) is of value. Normal stresses are represented along the x axis and shear stresses along the y axis, so the Mohr circle thus... 

Figure 11.12 Mohr circle diagram for two states of stress that produce yield in a material satisfying the Tresca yield criterion (a) and the Coulomb yield criterion (h)...

There are two other ways in which these results can be presented. First, recalling Section 11.2.6 and Figure 11.12, the Mohr circle diagram can be constructed from the data, as shown in Figure 11.19 where Bowden and Jukes s results appear as crossed points. This diagram leads naturally to a Coulomb yield criterion. 

For the analysis of combined stress in the two-dimensional situation the Mohr circle diagram (see standard texts [13]) is of value. Normal stresses are represented along the 1 axis and shear stresses along the 2 axis, so that the Mohr circle thus represents a state of stress, with each point representing the stresses on a particular plane. The direction of the plane normal is given relative to the directions of the principal stresses by the rule that a rotation in real space of 0 in a clockwise direction, corresponds to a rotation in Mohr circle space of 20... 

Summary. The variation of compressive stress with direction in a principal plane can be represented in three equivalent ways—by an ellipse, a sine-wave or a Mohr circle. The variation is fixed by just the two extreme values, tjy and (T3 or 02 and (T3 or and ffj- The different diagrams are simply visualizations of the original statement... 

The Mohr circle representation (Fig. 9.6c) is a graphical method of relating stress components in different sets of axes. When the axes in the material rotate by an angle B, the diameter of the circle rotates by an angle 2 B. If the material yields, the circle has radius k, the constant in the Tresca yield criterion. The axes of the Mohr diagram are the tensile and shear stress components. Thus, in the left-hand circle, representing the stresses at A in Fig. 9.6b, the ends of the horizontal diameter are the principal stresses. The principal axes are parallel and perpendicular to the notch-free surface. There is a tensile principal stress Ik parallel to the surface, and a zero stress perpendicular to the surface. The points at the ends of the vertical diameter represent the stress components in the a)3 axes, rotated by 45° from the principal axes. In the a/3 axes, the shear stresses have a maximum value k, and there are equal biaxial tensile stresses of magnitude = k (the coordinate of the centre of the circle). 

As in the two-dimensional case, the direct stresses are on the horizontal axis and the shear stresses are on the vertical axis. For the construction of the Mohr s circle, three circles are required. The stresses on any plane at any rotation, when plotted in the three-dimensional Mohr s circle diagram, are represented by a point located either on one of the three circles or within the area between the largest and the two smaller circles. The maximum shear stress is given by the radius of the largest circle. When constructing the Mohr s circle, the angle of rotation is double that of the real stress system. Shear stresses are positive if they cause clockwise rotation,... 

Obviously, concerning the formulation of failure conditions at the particle contacts we can follow the Molerus theory [8, 9], but here with a general supplement for the particle contact constitutive model Eq.( 2). It should be paid attention that the stressing pre-history of a cohesive powder flow is stationary (steady-state) and delivers significantly a cohesive stationary yield locus in radius-centre-stresses of a Mohr circle or in a t-a-diagram [28], see Fig. 2,... 

Fig. 6. Mohr s circle diagram for a granular material with passive failure at wall [4].

In this entry, the basic concept of stress is introduced and its relation to earthquake mechanisms is explained (for description of focal mechanisms, see entry Earthquake Mechanism Description and Inversion ). Mohr s circle diagram and simple failure criteria are described and used for defining the fault instability, principal faults and principal focal mechanisms. Methods of determining stress from observed earthquake mechanisms are reported and their robustness is... 

Earthquake Mechanisms and Stress Field, Fig. 2 Mohr s circle diagram. Quantities T and t are the normal and shear stresses along a fault, Ci, T2 and Cs are the principal stresses. All permissible values of a and T acting on a fault must lie in the shaded area of the diagram... 

Earthquake Mechanisms and Stress Field, Fig. 5 Griffith failure criterion, (a) Scheme of a cylindrical specimen with a fracture created by loading with stresses tri > (72 = left) and the corresponding Mohr s circle diagram (right), (b) Position of different... 

If the Mohr-Coulomb failure criterion is satisfied (red area in Fig. 6), the fault becomes unstable and an earthquake occurs along this fault. The higher the shear stress difference. At = t, the higher the instability of the fault and the higher the susceptibility of the fault to be activated. A fault most susceptible to failure is called the principal fault (Vavrycuk 2011a) being defined by the point in which the Mohr-Coulomb failure criterion touches the Mohr s circle diagram (blue point in Fig. 6)... 

Fig. 10 Principal focal mechanisms, (a) Full Mohr s circle diagram, (b) principal nodal lines and P/T axes, and (c, d) principal focal mechanisms. The blue dot in (a) marks the principal fault plane, the arrows in (c) and (d) denote the nodal lines corresponding to the principal fault plane. Point B in (c) and (d) denotes the neutral axis, N denotes north (After Vavrycuk (2011a))...

Although the condition of maximizing the SSSC value looks as a rather rough approximation, its use in practice is advocated by Angelier (2002) for the following reasons. Firstly, the activated fault planes are located mostly close to the top of the Mohr s circle diagram. For example, friction of... 

Fig. 12 Example of data used in numerical tests of stress inversions. The plots show 200 noise-free focal mechanisms selected to satisfy the Mohr-Coulomb failure criterion. Left/right plots - dataset with a full/ reduced variety of focal mechanisms, (a, b) Mohr s circle diagrams, (c, d) P/T axes and (e, f) corresponding nodal lines. The P axes are marked by the red circles and T axes by the blue crosses in (c) and (d). The tti,

FIGURE 7.10 Mohr s diagram t vs. cylindrical sample under the influence of horizontal stress [Pg.278]

This can be achieved by keeping at least one stress component in the compressive region, thus shifting Mohr s circle to the left of the diagram. This method is used in metal forming like forging or rolling. 

Fig. 11 shows a cr, t-diagram. The curve represents the maximum shear stress x the sample can support under a certain normal stress o it is called the yield locus. Parameter of a yield locus is the bulk density Ai,. With higher preconsolidation loads the bulk density Ai, increases and the yield loci move upwards. Each yield locus terminates at point E in direction of increasing normal stresses a. Point E characterizes the steady state flow which is the flow with no change in stresses and bulk density. Two Mohr stress circles are shown. The major principal stresses of the two Mohr stress circles are charcteristic of a yield locus, Oi is (he major principal stress at steady state flow, called major consolidation stress, and cTc is the... 

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). 

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