How To Draw Mohr Coulomb Failure Envelope

Effigy 1: View of Mohr–Coulomb failure surface in 3D space of principal stresses for

The Mohr–Coulomb ^[v] failure benchmark represents the linear envelope that is obtained from a plot of the shear strength of a material versus the applied normal stress. This relation is expressed every bit

where τ is the shear force, σ is the normal stress, c is the intercept of the failure envelope with the τ axis, and φ is the slope of the failure envelope. The quantity c is often called the cohesion and the angle φ is called the angle of internal friction . Compression is assumed to be positive in the post-obit discussion. If pinch is assumed to be negative so σ should be replaced with − σ.

If φ = 0, the Mohr–Coulomb criterion reduces to the Tresca criterion. On the other hand, if the Mohr–Coulomb model is equivalent to the Rankine model. Higher values of φ are non allowed.

From Mohr's circle we have

where

and σ₁ is the maximum principal stress and σ₃ is the minimum principal stress.

Therefore the Mohr–Coulomb criterion may also be expressed as

This form of the Mohr–Coulomb benchmark is applicable to failure on a plane that is parallel to the σ₂ management.

Mohr–Coulomb failure criterion in three dimensions

The Mohr–Coulomb criterion in three dimensions is oftentimes expressed as

The Mohr–Coulomb failure surface is a cone with a hexagonal cross section in deviatoric stress space.

The expressions for τ and σ tin exist generalized to three dimensions by developing expressions for the normal stress and the resolved shear stress on a aeroplane of arbitrary orientation with respect to the coordinate axes (footing vectors). If the unit of measurement normal to the plane of interest is

where are 3 orthonormal unit basis vectors, and if the principal stresses σ₁,σ₂,σ₃ are aligned with the basis vectors , and so the expressions for σ,τ are

The Mohr–Coulomb failure criterion can then be evaluated using the usual expression

for the six planes of maximum shear stress.

[prove]Derivation of normal and shear stress on a plane

Mohr–Coulomb failure surface in Haigh–Westergaard space

The Mohr–Coulomb failure (yield) surface is often expressed in Haigh–Westergaad coordinates. For instance, the function

can be expressed equally

Alternatively, in terms of the invariants p,q,r we can write

where

[prove]Derivation of alternative forms of Mohr–Coulomb yield function

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