Mohr diagram

Figure 2.5 shows the horizontal surface of a cohesionless soil mass with a rectangular section shown as a free body. On the lower surface at the depth z the normal pressure is N. Along the sides of the free body the normal pressure varies from zero at the surface to a maximum value of Nh at the depth z (the variation is linear, except when arching occurs). In Figure 2.6 the value of N = z (unit weight) is plotted on a Mohr diagram. 

See Figure 6.5, where diagram (b) is a Mohr diagram and diagram (a) will be called a sine-wave diagram. 

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). 

The shear stress (x) generated along a defined plane depends on the normal stress (a) exerted on this plane. If a material is subjected to a shearing action, a characteristic relation is obtained between normal and shear stresses for each material. This relationship is graphically shown in o-x coordinates (Mohr diagrams) and the straight line obtained finally is the yield locus for a bulk material [11]. 

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 most accurate calculations of the self-energy have been carried out in the point-nucleus Coulomb case by Mohr and collaborators [28]. Techniques that work for the general, non-Coulomb case have also been developed [29], and it is now possible to carry out a complete one-photon calculation for any potential with relative ease. We present the overall effect of the one-photon diagrams on the 2p3/2 — 2si/2 splitting in the second row of Table 1, and give a breakdown for the individual states in Tables 2 and 3. 

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. 

Figure 7.5 Diagrams of limiting stressed state of specimens made of steel St3, fixed with Sprut-5M (a) in air, (b) under water, (c) in oil. (1-7) Design diagrams on the criteria of Balandin (1), Miroliubov (2), Botkin (3), Yagn (4), Pisarenko-Lebedev (5), Skudra-Kirulis (6), and Mohr (7). Curve (8) is the experimental diagram.

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,... 

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 . 

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. 

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. 

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... 

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. 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... 

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... 

When stress conditions in real rocks in the Earth s crust are studied, the Mohr s diagram is modified. Rocks are typically porous and contain pressurized fluids which influence overall stress in the rock (Scholz 2002). For example, fluids present inside a fault reduce the normal stress along the fault. If the fluid pressure is sufficiently high, fluids can even cause opening of the fault (see Fig. 3a). For this reason, effective normal stress (7 is introduced as a difference between normal stress cr and pore-fluid pressure p... 

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... 

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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