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To derive the equation of the Mohr circle for the two-dimensional cases of plane stress and plane strain, first consider a two-dimensional infinitesimal material element around a material point (Figure 4), with a unit area in the direction parallel to the -plane, i.e., perpendicular to the page or screen.
Mohr–Coulomb theory is a mathematical model (see yield surface) describing the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials follow this rule in at least a portion of their shear failure envelope.
This equation defines the yield surface as a circular cylinder (See Figure) whose yield curve, or intersection with the deviatoric plane, is a circle with radius , or . This implies that the yield condition is independent of hydrostatic stresses.
A graphical representation of this transformation law is the Mohr's circle for stress. The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations: it is a central concept in the linear theory of elasticity.
The formula reduces to the Tresca criterion if =. Figure 5 shows Mohr–Coulomb yield surface in the three-dimensional space of principal stresses. It is a conical prism and determines the inclination angle of conical surface. Figure 6 shows Mohr–Coulomb yield surface in two-dimensional stress space.
Mohr's circle, Lame's stress ellipsoid (together with the stress director surface), and Cauchy's stress quadric are two-dimensional graphical representations of the state of stress at a point. They allow for the graphical determination of the magnitude of the stress tensor at a given point for all planes passing through that point.
Undrained strength is typically defined by Tresca theory, based on Mohr's circle as: σ 1 - σ 3 = 2 S u. Where: σ 1 is the major principal stress σ 3 is the minor principal stress is the shear strength (σ 1 - σ 3)/2. hence, = S u (or sometimes c u), the undrained strength.
In structural geology, differential stress is used to assess whether tensile or shear failure will occur when a Mohr circle (plotted using and ) touches the failure envelope of the rocks. If the differential stress is less than four times the tensile strength of the rock , then extensional failure will occur.